## Big Ideas Math Answers Grade 5 Chapter 12 Patterns in the Coordinate Plane

Big Ideas Math Answers Grade 5 Chapter 12 Patterns in the Coordinate Plane will help the students to improve their math skills easily. Answers are provided for Big Ideas Math Answers Grade 5 Chapter 12 Patterns in the Coordinate Plane along with the explanation. All the solutions are explained clearly step-by-step in a simple manner. Every topic is explained by the math experts and given with real-time examples. Therefore, make your preparation simple by referring to our Big Ideas Math Answers Grade 5 Chapter 12 Patterns in the Coordinate Plane.

## Big Ideas Math Book 5th Grade Chapter 12 Patterns in the Coordinate Plane Answer Key

Students who are struggling to solve Big Ideas Math Book 5th Grade Answer Key Chapter 12 Patterns in the Coordinate Plane have reached the correct place. This BIM Book 5th Grade 12th Chapter Answer Key gives the most accurate answers to all the questions related to this chapter. Different methods of solving every question are given on this page. Verify the below links to know the different methods to solve problems. Be the first to learn the concepts and practice all the questions available here.

Lesson: 1 Plot Points in a Coordinate Plane

Lesson: 2 Relate Points in a Coordinate Plane

Lesson: 3 Draw polygons in a Coordinate Plane

Lesson: 4 Graph Data

Lesson: 5 Make and Interpret Line Graphs

Lesson: 6 Numerical Patterns

Lesson: 7 Graph and Analyze Relationships

Chapter: 12 – Patterns in the Coordinate Plane

### Lesson 12.1 Plot Points in a Coordinate Plane

Explore and Grow

Choose a location for your buried treasure on the grid. Choose a point where two grid lines My Treasure intersect. An example is shown.

Take turns with a partner guessing the location of each other’s buried treasure. Keep track of your guesses on the grid. After each guess, give a clue to help yourMy Guessespartner, such as “my treasure is northwest of your guess.”This is an image Continue to guess until a treasure is located.

Reasoning
The point where the horizontal number line and the vertical number line intersect is called the origin. Why do you think it is called that?

Think and Grow: The Coordinate Plane

Key Idea
A coordinate plane is formed by the intersection of a horizontal number line and a vertical number line. An ordered pair is a pair of numbers that is used to locate a point in a coordinate plane.

Example
Write the ordered pair that corresponds to point M.

The horizontal distance from the origin to point M is __ units. So, the x-coordinate is __.
The vertical distance from the origin to point M is ___ units. So, the y-coordinate is ___. The ordered pair is __.

The horizontal distance from the origin to point M is 5 units. So, the x-coordinate is 5.
The vertical distance from the origin to point M is 5units. So, the y-coordinate is 3. The ordered pair is (5, 3).

Show and Grow

Write the ordered pair corresponding to the point.

Question 1.
Point B

Explanation:
The horizontal distance from the origin to point B is _4_ units.
So, the x-coordinate is _4_.
The vertical distance from the origin to point B is __2_ units.
So, the y-coordinate is _2__. The ordered pair is _(4,2)_.

Question 2.
Point S

Explanation:
The horizontal distance from the origin to point S is 0  units.
So, the x-coordinate is _0_.
The vertical distance from the origin to point S is __4_ units.
So, the y-coordinate is _4__.
The ordered pair is _(0,4)_.

Plot and label the point in the coordinate plane.

Question 3.
F(5, 4)
Explanation:

Start the Origin. Move  Units right and units up. Then plot and label the point.
The point can be labeled as Y,(5,4)

Question 4.
P(3, 0)
Explanation:

Start the Origin. Move  Units right and units up. Then plot and label the point.
The point can be labeled as Y,(3,0)

Apply and Grow: Practice

Use the coordinate plane to write the ordered pair corresponding to the point.

Question 5.
Point M
Explanation:

The horizontal distance from the origin to point M is  units.
So, the x-coordinate is _8_.
The vertical distance from the origin to point M is __8 units.
So, the y-coordinate is 8_.
The ordered pair is _(8,8)_.

Question 6.
Point Q
Explanation:

The horizontal distance from the origin to point Q is 2  units.
So, the x-coordinate is _2_.
The vertical distance from the origin to point Q is __7_ units.
So, the y-coordinate is _7__.
The ordered pair is _(2,7)_.

Question 7.
Point N
Explanation:

The horizontal distance from the origin to point N is 7 units.
So, the x-coordinate is _7_.
The vertical distance from the origin to point N is _6_ units.
So, the y-coordinate is _6__.
The ordered pair is _(7,6)_.

Question 8.
Point R
Explanation:

The horizontal distance from the origin to point R is 2  units.
So, the x-coordinate is _2_.
The vertical distance from the origin to point R is __0_ units.
So, the y-coordinate is _0_.
The ordered pair is _(2,0)_.

Question 9.
Point P
Explanation:

The horizontal distance from the origin to point P is 6  units.
So, the x-coordinate is _6_.
The vertical distance from the origin to point P is __1_ units.
So, the y-coordinate is _4__.
The ordered pair is _(6,1)_.

Question 10.
Point T
Explanation:

The horizontal distance from the origin to point T is  units.
So, the x-coordinate is _0_.
The vertical distance from the origin to point P is __5_ units.
So, the y-coordinate is _5__.
The ordered pair is _(0,5)_.

Plot and label the point in the coordinate plane above.

Question 11.
S(0, 3)
Answer: The Point can be labeled as (0,3)
Explanation:

Start the Origin. Move Units right and  3 units up then plot and label the point.
The point can be labeled as Y,(0,3)

Question 12.
F(2, 5)
Explanation:

Start the Origin. Move  Units right and  5 units up then plot and label the point.
The point can be labeled as Y,(2,5)

Question 13.
W(0, 0)
Explanation:

Start the Origin. Move Units right and  0 units up then plot and label the point.
The point can be labeled as Y,(0,0)

Name the point for the ordered pair.

Question 14.
(5, 2)
Explanation:
Start the Origin. Move Units right and  2 units up then label the point.
The point can be Named as D(5,2)

Question 15.
(8, 4)
Explanation:

Start the Origin. Move Units right and  4 units up then label the point.
The point can be Named as E(8,4)

Question 16.
(0, 3)
Explanation:

Start the Origin. Move Units right and  3 units up then label the point.
The point can be Named as D(5,2)

Question 17.
Reasoning
How are the locations of the points A(0, 4) and B(4, 0) different in a coordinate plane?
Answer: A(0,4) is in Y axis as x is 0 and y is 4. B(4,0) is in X axis as x is 4 and y is 0.

Question 18.
DIG DEEPER!
Newton buries a bone in a park at the location shown. How can he use a coordinate plane to describe its location?

He can use the x-coordinate and y-coordinate to describe the location.

Think and Grow: Modeling Real Life

Example
In a video game, you move an aircraft carrier and a tugboat away from your base. Use the directions to plot and label the locations of the aircraft carrier and the tugboat.

• Aircraft carrier: Located 3 miles east and 4 miles north of your base.
• Tugboat: Located 8 miles east and twice as many miles north of your base as the aircraft carrier.
To find the location of the aircraft carrier, start at your base, which is at the origin.

Move __ units east, or right, and ___ units north, or up.
Plot and label the point as A(___, ___ ).
To find the location of the tugboat, start at your base, which is at the origin.
Move ___ units east, or right, and ___ × __ = ___ units north, or up.
Plot the label the point as T(__, ___).

Show and Grow

Question 19.
A guidebook describes how to get to various statues in Chicago, Illinois, from Willis Tower. Plot and label the location of each statue on the map.
Dubuffet’s Monument with Standing Beast: Walk 2 blocks east and 5 blocks north.
• Miró’s Sun, Moon, and One Star: Walk twice as many blocks east as you do to get to the Standing Beast, and 3 blocks north.

Question 20.
DIG DEEPER!
Which statue is closer to Moon, and One Star, Cloud Gate or Flamingo? Explain.

### Plot Points in a Coordinate Plane Homework & practice 12.1

Use the coordinate plane to write the ordered pair corresponding to the point.

Question 1.
Point E
Explanation:

The horizontal distance from the origin to point E is 5  units.
So, the x-coordinate is _5_.
The vertical distance from the origin to point E is __7_ units.
So, the y-coordinate is _7__.
The ordered pair is _(5,7)_.

Question 2.
Point H
Explanation:

The horizontal distance from the origin to point H is 0  units.
So, the x-coordinate is _0_.
The vertical distance from the origin to point H is __7_ units.
So, the y-coordinate is _7__.
The ordered pair is _(0,7)_.

Question 3.
Point F
Explanation:

The horizontal distance from the origin to point F is 3  units.
So, the x-coordinate is _3_.
The vertical distance from the origin to point F is __3 units.
So, the y-coordinate is _3__.
The ordered pair is _(3,3)_

Question 4.
Point J
Explanation:

The horizontal distance from the origin to point J is 1  units.
So, the x-coordinate is _1_.
The vertical distance from the origin to point J is __0_ units.
So, the y-coordinate is _0__.
The ordered pair is _(1,0)_.

Question 5.
Point G
Explanation:

The horizontal distance from the origin to point G is 7  units.
So, the x-coordinate is _7_.
The vertical distance from the origin to point G is __2_ units.
So, the y-coordinate is _2__.
The ordered pair is _(7,2)_.

Question 6.
Point K
Explanation:

The horizontal distance from the origin to point K is 8  units.
So, the x-coordinate is _8_.
The vertical distance from the origin to point K is __5_ units.
So, the y-coordinate is _5_.
The ordered pair is _(8,5)_.

Plot and label the point in the coordinate plane above.

Question 7.
Z(8, 0)
Answer: The Point can be labeled as (8,0)
Explanation:

Start the Origin. Move  Units right and  0 units up then plot and label the point.
The point can be labeled as Y,(8,0)

Question 8.
B(5, 5)
Answer: The Point can be labeled as (5,5)
Explanation:

Start the Origin. Move  Units right and  5 units up then plot and label the point.
The point can be labeled as Y,(5,5)

Question 9.
M(1, 2)
Answer: The Point can be labeled as (1,2)
Explanation:

Start the Origin. Move  Units right and  2 units up then plot and label the point.
The point can be labeled as Y,(1,2)

Name the point for the ordered pair.

Question 10.
(5, 4)
Explanation:

Start the Origin. Move Units right and units up then label the point.
The point can be Named as Q(5,4)

Question 11.
(0, 8)
Explanation:

Start the Origin. Move Units right and  8units up then label the point.
The point can be Named as P(0,8)

Question 12.
(3, 1)
Explanation:

Start the Origin. Move Units right and  1 units up then label the point.
The point can be Named as N(3,1)

Question 13.
Open-Ended
Use the coordinate plane above. Point T is 3 units from point M. Name two possible ordered pairs for point T.
Explanation:

The horizontal distance from the origin to point M is 4  units.
So, the x-coordinate is _4_.
The vertical distance from the origin to point M is _5_ units.
So, the y-coordinate is _5_.
The ordered pair is _(4,5)_.
Point T is 3 units from point M
The horizontal distance from  point M is 4  units to add
So, the x-coordinate is _4+3=7_.
The vertical distance from the point M is 5  units to add 3 units.
So, the y-coordinate is _5+3=8_.
The ordered pair is _(7,8)_.
The two possible ordered pairs for point T is (0,7)& (8,0).

Question 14.
Writing
Explain why the order of the x- and y-coordinates is important when identifying or plotting points in a coordinate plane.
Answer: Locations on the coordinate plane are described as ordered pairs. An ordered pair tells you the location of a point by relating the point’s location along the x-axis (the first value of the ordered pair) and along the y-axis (the second value of the ordered pair).

Question 15.
To get from the school to the arcade, you walk 4 blocks east and 3 blocks north. To get from the school to the skate park, your friend walks 2 blocks east and twice as many blocks north as you. Plot and label the locations of the arcade and the skate park.

Question 16.
DIG DEEPER!
Which building is closer to the bus station, the library or the post office? Explain.

Review & Refresh

Multiply.

Question 17.

Explanation: First Multiply with the whole number(1) with the denominator(4) then add numerator (4+1=5)
and denominator is same as original. i.e, 5/4
then same as next mixed fraction Multiply with the whole number(1) with the denominator(3) then add numerator (3+1=4)
and denominator is same as original. i.e, 4/3
Multiply 5/4*4/3=20/12
This fraction can be reduced by dividing both the numerator and denominator by the Greatest Common Factor of 20 and 12 using 4 i.e, 5/3=1 2/3

Question 18.

Explanation: First Multiply with the whole number(1) with the denominator(5) then add numerator (5+2=7)
and denominator is same as original. i.e, 7/5
then same as next mixed fraction Multiply with the whole number(2) with the denominator(2) then add numerator (4+1=5)
and denominator is same as original. i.e, 5/2
Multiply 7/5*5/2=35/10
This fraction can be reduced by dividing both the numerator and denominator by the Greatest Common Factor of 35 and 10 using 5 i.e, 7/2=3 1/2

Question 19.

Explanation: First Multiply with the whole number(2) with the denominator(6) then add numerator (12+5=17)
and denominator is same as original. i.e, 17/6
then same as next mixed fraction Multiply with the whole number(3) with the denominator(8) then add numerator (24+7=31)
and denominator is same as original. i.e, 31/8
Multiply 17/6*31/8= 527/48.
therefore we can write it as 10 47/48.

### Lesson 12.2 Relate Points in a Coordinate Plane

Plot and label the points in the coordinate plane.

Draw a line segment to connect each pair of points.

Plot and label more points that lie on the line segments you drew. What do you notice about the coordinates?
A and C
Explanation:

In A and C line segment we observed that the y- Coordinate of A and C is same.

B and G
Explanation:

In B and G line segment we observed that the X- Coordinate of B and G is same.
D and E
Explanation:

In D and E line segment we observed that the Y- Coordinate of D and E is same.
F and H
Explanation:

In F and H line segment we observed that the X- Coordinate of F and H is same.

Construct Arguments
How can you find the distance between each pair of points? Explain your reasoning.

Think and Grow: Relate Points in a Coordinate Plane

Key Idea
Points on a horizontal line have the same-coordinates. Points on a vertical line have the same x-coordinates.

You can count units or use subtraction to find the distance between two points when they lie on the same horizontal line or vertical line.

Example
Find the distance between points G and H.

One Way: Count units.
Step 1: Identify the locations of the points: Point G is located at (2, 3). Point H is located at (8, 3).
Step 2: Draw a line segment to connect the points.
Step 3: Count horizontal units: There are __ units between points G and H.
So, the distance between points G and H is ___.

Another Way: Use subtraction

Points G and H have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
8 – 2 = _6__
So, the distance between points G and H is 6___.

Show and Grow

Find the distance between the points in the coordinate plane above.

Question 1.
E and F
Explanation

Points E and F have the same X-coordinates. They lie on a Vertical line. Subtract the Y-coordinates to find the distance.
5 – 1 = __4_
So, the distance between points E and F is _4__ units

Question 2.
P and Q
Explanation:

Points P and Q have the same Y-coordinates. They lie on a Vertical line. Subtract the X-coordinates to find the distance.
3 – 0 = __3_
So, the distance between points P and Q is _3__ units.

Question 3.
S and T
Explanation:

Points S and T have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
7 – 0 = __7_
So, the distance between points S and T is _7_ Units.

Apply and Grow: Practice

Find the distance between the points in the coordinate plane.

Question 4.
E and F
Explanation:

Points E and F have the same X-coordinates. They lie on a Vertical line. Subtract the Y-coordinates to find the distance.
8 – 3 = __5
So, the distance between points E and F is _5_ units

Question 5.
J and G
Explanation:

Points J and G have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
8 – 0 = __8_
So, the distance between points J and G is _8_ Units.

Question 6.
F and K
Explanation:

Points F and K have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
8 – 5 = __3_
So, the distance between points F and K is _3_ Units.

Question 7.
Which is longer, $$\overline{J M}$$ or $$\overline{H R}$$?

Find the distance between the points.

Question 8.
(1, 7) and (7, 7)
Explanation:

Lets take point A is (1,7) and Point B is (7,7)
Points A and B have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
7 – 1 = __6_
So, the distance between points A and B is _6_ Units.

Question 9.
(0, 1) and (3, 1)
Explanation:

Lets take point C is (0,1) and Point D is (3,1)
Points C and D have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
3 – 0 = __3_
So, the distance between points C and D is _3_ Units.

Question 10.
(0, 0) and (6, 0)
Explanation:

Lets take point E is (0,0) and Point F is (6,0)
Points E and F have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
6 – 0 = __6_
So, the distance between points E and F is _6_ Units.

A line passes through the given points. Name two other points that lie on the line.

Question 11.
(0, 6) and (5, 6)
Explanation:
Lets take point A is (0,6) and Point B is (5,6)
Points A and B have the same y-coordinates. They lie on a horizontal line.
So the other points lie on the line are (1,6),(2,6),(3,6),(4,6)

Question 12.
(4, 2) and (4, 8)
Explanation:
Lets take point C is (4,2) and Point D is (4,8)
Points C and D have the same X-coordinates. They lie on a Vertical line.
So the other points lie on the line are (4,3),(4,4),(4,5),(4,6) etc

Question 13.
(3, 3) and (3, 6)
Explanation:
Lets take point E is (3,3) and Point F is (3,6)
Points E and F have the same X-coordinates. They lie on a Vertical line.
So the other points lie on the line are (3,4),(3,5)

Question 14.
YOU BE THE TEACHER
Newton plots the points (2, 7) and (6, 7) and connects them with a line segment. Descartes says that (10, 7) also lies on the line segment. Is he correct? Explain.
Explanation:
Points on a Vertical line have the same-coordinates. They lie on a horizontal line.

Question 15.
DIG DEEPER!
Which pair of points does not lie on a line that is parallel to x-axis? Explain.

Explanation:
Points on a Vertical line have the same-coordinates. They lie on a horizontal line. Except (1,2) and (1,6) , these two
Points on a horizontal line have the same-coordinates. They lie on a Vertical line.

Think and Grow: Modeling Real Life

Example
An archaeologist uses rope to section off a rectangular dig site. How many meters of rope does the archaeologist use?

To find how many meters of rope the archaeologist uses, find the perimeter of the rectangular dig site.

Use a formula to find the perimeter of the site.

So, the archaeologist uses __ meters of rope.

Show and Grow

Question 16.
The owner of an animal shelter uses fencing to create a rectangular dog pen. How many yards of fencing does the owner use?

yards of fencing the owner use
To find how many yards of fencing the owner use, to find the perimeter of the rectangular dog pen.

Question 17.
DIG DEEPER!
You run 5 laps around the edges of the volleyball court. How far do you run in feet? in yards?

10 × 6 = 60 feet
Convert from foot to yards
1 feet = 0.33 yards
60 feet = 20 yards

### Relate Points in a Coordinate Plane Homework & Practice 12.2

Find the distance between the points in the coordinate plane.

Question 1.
P and M
Explanation:

Points P and M have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
7 – 1 = __6_
So, the distance between points P and M is _6_ Units.

Question 2.
B and Z
Explanation:

Points B and Z have the same X-coordinates. They lie on a Vertical line. Subtract the Y-coordinates to find the distance.
3 – 0 = _0
So, the distance between points B and Z is _0_ units

Question 3.
K and T
Explanation:

Points K and T have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
6 – 1 = __5_
So, the distance between points K and T is _5_ Units.

Question 4.
Which is longer, $$\overline{C D}$$ or $$\overline{K P}$$?

Answer: $$\overline{C D}$$ longer than $$\overline{K P}$$

Find the distance between the points.

Question 5.
(1, 5) and (6, 5)
Explanation:

Lets take point A is (1,5) and Point B is (6,5)
Points A and B have the same y-coordinates. They lie on a horizontal line. Subtract the x-coordinates to find the distance.
6 – 1 = _5_
So, the distance between points A and B is _5_ Units.

Question 6.
(3, 4) and (3, 6)
Explanation:

Lets take point C is (3,4) and Point D is (3,6)
Points C and D have the same X-coordinates. They lie on a Vertical line. Subtract the Y-coordinates to find the distance.
6 – 4 = _2
So, the distance between points C and D is _2_ Units.

Question 7.
(0, 2) and (0, 9)
Explanation:

Lets take point E is (0,2) and Point F is (0,9)
Points E and F have the same X-coordinates. They lie on a Vertical line. Subtract the Y-coordinates to find the distance.
9 – 2 = _7
So, the distance between points E and F is _7_ Units.

A line passes through the given points. Name two other points that lie on the line.

Question 8.
(6, 0) and (6, 7)
Explanation:
Lets take point A is (6,0) and Point B is (6,7)
Points A and B have the same X-coordinates. They lie on a Vertical line.
So the other points lie on the line are (6,1),(6,2),(6,3)

Question 9.
(5, 3) and (1, 3)
Explanation:
Lets take point C is (5,3) and Point D is (1,3)
Points C and D have the same y-coordinates. They lie on a horizontal line.
So the other points lie on the line are (2,3),(3,3)(4,3)

Question 10.
(2, 2) and (2, 9)
Explanation:
Lets take point E is (2,2) and Point F is (2,9)
Points E and F have the same X-coordinates. They lie on a Vertical line.
So the other points lie on the line are (2,3),(2,4),(2,5)

Question 11.
Structure
Name four different points that are 3 units away from (5, 4).

A distance of 5 units from the origin represents the hypotenuse of a right triangle with sides 3- 4 – 5
If the x-coordinate is 4 then the y-coordinate is | 3 | = ± 3
the coordinates are (4, 3 ) and (4, – 3 )

Question 12.
Number Sense
Which point is farther from (3, 4)? Explain.

Question 13.
Modeling Real Life
A farmer builds a coop for his chickens. He uses poultry netting to enclose the coop. How many feet of netting does he use?

Question 14.
Modeling Real Life
A giant chessboard is painted on the ground in a park. How many square yards of space does the chessboard occupy?

Review & Refresh

Question 15.
23.6 ÷ 4 = ___

Divide 23.6 by 4
We get
23.6/4 = 5.9
Thus the quotient is 5.9

Question 16.
36.9 ÷ 3 = ___

Divide the two numbers 36.9 and 3.
36.9 ÷ 3 = 12.3

Question 17.
114.87 ÷ 7 = ___

Divide the two numbers 114.87 and 7
114.87 ÷ 7 = 16.41

### Lesson 12.3 Draw polygons in a Coordinate Plane

Explore and Grow

Plot and label three points in which two of the ordered pairs have the same x-coordinates and two of the ordered pairs have the same y-coordinates.

The points represent the vertices of a polygon. Describe the polygon.

Structure
Explain how you can plot another point above to form a rectangle.

Think and Grow: Draw Polygons in a Coordinate Plane

Key Idea
You can use ordered pairs to represent vertices of polygons. To draw a polygon in a coordinate plane, plot and connect the vertices.
Example
The vertices of a polygon are A (2, 2), B(3, 5), C(6, 6), and D(6, 2). Draw the polygon in a coordinate plane. Then identify it.
Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.

Show and Grow

Draw the polygon with the given vertices in a coordinate plane. Then identify it.

Question 1.
J(0, 8), K(4, 7), L(5, 0)

Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon J,K,L is a Triangle.

Question 2.
P(1, 4), Q(2, 7), R(6, 7), S(7, 4), T(4, 1)

Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon P,Q,R,S,T is a Pentagon.

Apply and Grow: Practice

Draw the polygon with the given vertices in a coordinate plane. Then identify it.

Question 3.
C(1, 6), D(4, 6), E(4, 1), F(1, 1)

Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon C,D,E,F is a Rectangle.

Question 4.
J(2, 2), K(2, 4), L(4, 6), M(6, 4), N(6, 2), P(4, 1)

Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon J,K,L,M,N,P is a Hexagon.

Identify the polygon with the given vertices.

Question 5.
A(2, 6), B(6, 2), C(3, 2)
Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon A ,B , C is a  Tringle.

Question 6.
G(0, 3), H(6, 3), I(4, 1), J(2, 1)
Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon G,H,I,J is a Isosceles Trapezoid .

Question 7.
P(1, 1), Q(1, 6), R(6, 6), S(6, 1)
Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon P, Q ,R, S is a Rectangle.

Question 8.
X(0, 0), Y(0, 7), Z(2, 0)
Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon X ,Y, Z is a Triangle.

Plot (6, 3), (6, 8), and (9, 3) in a coordinate plane. Plot another point to form the given quadrilateral. Name the point.

Question 9.
rectangle
Explanation:

Question 10.
trapezoid
Explanation:

Question 11.
Open-Ended
Write four ordered pairs that represent the vertices of a square

(2, 2), (4, 2), (2, 4), (4, 4)

Question 12.
YOU BE THE TEACHER
Your friend draws the polygon shown. She names the polygon. Is your friend correct? Explain.

No your friend is incorrect because the given figure has 4 sides so it is a quadrilateral not a polygon.

Think and Grow: Modeling Real Life

Example
You and a friend use computer software to create a symmetric company logo using a coordinate plane. Your friend completes one half of the logo as shown. Draw the other half. Then list the vertices of the logo.

Step 1: Plot the vertices for the other half of the logo on the opposite side of the line of symmetry.
Step 2: Draw line segments to connect the points.

Show and Grow

Draw the other half of the symmetric logo. Then list its vertices.

Question 13.

Question 14.

Question 15.
DIG DEEPER!
One half of the design for a symmetric flower garden is shown in the coordinate plane. The line of symmetry is represented by the walkway. Draw the other half of the design for the flower garden. Then list its vertices.

### Draw polygons in a Coordinate Plane Homework & Practice 12.3

Draw the polygon with the given vertices in a coordinate plane. Then identify it.

Question 1.
A(2, 3), B(2, 6), C(5, 6), D(5, 3)

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon A, B, C, D is a Trapezoid..

Question 2.
J(3, 2), K(3, 5), L(6, 5)

Explanation:

Step 1: Plot and label the vertices.
Step 2: Draw line segments to connect the points. Be sure to connect the points in order to draw the polygon.
Polygon J, K, L is a Triangle.

Identify the polygon with the given vertices.

Question 3.
M(2, 6), N(4, 4), P(4, 0), Q(2, 2)
Explanation:

Question 4.
A(1, 2), B(1, 6), C(4, 6), D(6, 4), E(4, 2)
Explanation:

Question 5.
P(4, 1), Q(0, 1), R(1, 4), S(5, 5)
Explanation:

Question 6.
E(1, 2), F(1, 3), G(6, 3), H(6, 2)

Plot (1, 2), (4, 2), and (3, 4) in a coordinate plane. Plot another point to form the given quadrilateral. Name the point.

Question 7.
trapezoid

Question 8.
parallelogram

Question 9.
Open-Ended
Write the coordinates of the vertices of a rectangle that has a perimeter of 12 units and an area of 5 square units.

Question 10.
Reasoning
Five ordered pairs represent the vertices of a polygon. Will the polygon always be a pentagon?

Question 11.
Modeling Real Life
Draw the other half of the symmetric logo. Then list its vertices.

Question 12.
DIG DEEPER!
You complete one fourth of an image with graphic design software. The computer generates the rest of the image with the two lines of symmetry. Draw the rest of the image.

Review & Refresh

Estimate the sum or difference.

Question 13.

Question 14.

The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
77/80

Question 15.

The fractions have, unlike denominators. First, find the Least Common Denominator and rewrite the fractions with the common denominator.
= 3/40

### Lesson 12.4 Graph Data

Explore and Grow

The table shows the amount of snow that falls each day for 7 days. Show how you can use ordered pairs in the coordinate plane to represent this information. Explain.

What conclusions can you make from your data display?

Explanation:

Lets take A(1,2), B(2,4), C(3,6), D(4,6), E(,5,8), F(6,10), G(7,14)
The horizontal distance from the origin to points A.B.C.D.E.F.G.H  are 1,2,3,4,5,6,7_ units respectively. So, the x-coordinates are 1,2,3,4,5,6,7 .
The vertical distance from the origin to points  A.B.C.D.E.F.G.H are 2,4,6,6,8,10,14 _ units. So, the y-coordinate are __2,4,6,6,8,10,14_. The ordered pair is _A(1,2), B(2,4), C(3,6), D(4,6), E(,5,8), F(6,10), G(7,14)_.
Reasoning
On Day 8, 1 inch of snow falls. How can you represent this information in the coordinate plane?

Explanation:
Start the Origin. Move  Units right and  16 units up. label the point.
The point can be labeled as H,(8,16).

Think and Grow: Graph Data

Key Idea
Data are values collected from observations or measurements. You can use a coordinate plane to graph and interpret two categories of related data.
Example
The table shows how many gold bars you collect at each level of a video game. Graph the data in a coordinate plane. In how many levels do you collect more than 30 gold bars?

Step 1: Write the ordered pairs from the table.

Step 2: For each axis, choose appropriate numbers to represent the data in the table.
Step 3: Write a title for the graph and label each axis.
Step 4: Plot a point for each ordered pair.
Three points are above the grid line that represents 30 bars. So, you collect more than 30 gold bars in __ levels.

Show and Grow

Question 1.
The table shows the water levels of a portion of a river during a flood. Graph the data.

What does the point (5, 7) represent?
Answer: In 5 hours time span 7 feet water level during flood.
Explanation:

Apply and Grow: Practice

Question 2.
The table shows how many cars a salesman sells in each of 6 months. Graph the data.

What does the point (1, 7) represent?

What is the difference of the greatest number of cars sold and the least number of cars sold? Explain.

Use the graph.
Answer: in month of span salesman sells 7 cars.
Explanation:

Question 3.
The graph shows how many receiving yards a football player has in each of seven games. How many receiving yards does he have in Game 3?

How many times as many receiving yards does he have in Game 4 as in Game 2?
In how many games does he have more than 40 receiving yards?
Explanation:
In each game receiving yards are lets label it as A,B,C,D,E,F,G.
A(1,30), B(2,20), C(3,50), D(4,80), E(5,60), F(6,40), G(7,100).
receiving yards a football player has in each of game 30,20,50,80,60,40,100.
In Game 3Receiving yards are 50
So, the distance between points Game 4 and game 2 is 80-60=60
60 times as many receiving yards does he have in Game 4 as in Game 2.
There are 4 games more than 40 receiving yards.

Question 4.
DIG DEEPER!
The player has 75 receiving yards in Game 8. The player has $$\frac{1}{5}$$ of this number of receiving yards in Game 9. Graph the data in the coordinate plane above.

Think and Grow: Modeling Real Life

Example
The table shows the ages of eight students and the time they spend on the Internet for 1 week. Graph the data. Of the students who spend more than 15 hours on the Internet, how many are older than 10?

Step 1: Write the ordered pairs from the table.

Step 2: For each axis, choose appropriate numbers to represent the data in the table.

Step 3: Write a title for the graph and label each axis.
Step 4: Plot a point for each ordered pair.
Five points are above the grid line that represents points represents 15 hours. Of those, ___ points represents students older than 10.
So, __ students older than 10 spend more than 15 hours on the Internet.

Show and Grow

Question 5.
The table show much five students sleep the night before a quiz and their quiz scores. Graph the data. Of the students who sleep more than 7 hours, how many score higher than 8 points?

Explanation:

The students who sleep more than 7 hours are 3(B,C,D)
3 members score higher than 8 points

### Graph Data Homework & Practice 12.4

Question 1.
The table shows how many students are in a choir club in each of 6 years. Graph the data.

What does the point (2, 20) represent?

What is the difference of the greatest number of students and least number of students? Explain.
Answer: In 2 years 20 students are joined in a choir club
Explanation:

What is the difference of the greatest number of students and least number of students
Grates number of students joined in the year of 6th that is 35 members
least number of students joined in the year of 1st that is 15 members
The difference of the greatest number(6,65) of students and least number of students(1,15) is 35-15=20

Use the graph.

Question 2.
The graph shows how many students earn an A on each of seven tests. How many students earn an A on Test 4?

How many times as many students earn an A on Test 6 as on Test 2?
On how many tests do fewer than 20 students earn an A? more than 20 students?

Answer: students earn an A on Test 4 is 18 students
students earn an A on Test 6 as on Test 2 is 10
5 students  are fewer than 20students earn an A
1 test only More than 20 students earn A.

Question 3.
DIG DEEPER!
Twenty-five students take Test 1. How many students do not earn an A on the test?
25 – 12 = 13

Question 4.
Modeling Real Life
The table shows the ages of five students and how many baby teeth each of them has lost. Graph the data. Of the students who are older than 10 years, how many lost more than 18 baby teeth?

Baby teeth lost from each team A,B ,C,D.E respectively 18,20,18,14,20
B,C,E are older than 10 years
2 (B&E) students are lost more than 18 baby teeth.

Review & Refresh

Question 5.
5 ÷ 0.8 = ___

Divide two numbers 5 and 0.8
5 ÷ 0.8 = 6.25

Question 6.
91.2 ÷ 15 = __

Divide two numbers 91.2 and 15
91.2 ÷ 15 = 6.08

Question 7.
14.4 ÷ 3.2 = ___

Divide two numbers 14.4 and 3.2
14.4 ÷ 3.2 = 4.5

### Lesson 12.5 Make and Interpret Line Graphs

Explore and Grow

The table shows the heights of a bamboo plant over several days. Show how you can use a coordinate plane to represent this information. Explain.

How can you use your graph to estimate the height of the plant on Day 4? Explain.

Reasoning
What could the height of the bamboo plant be on Day 10? Explain your reasoning.

Think and Grow: Make and Interpret Line Graphs

Key Idea
A line graph is a graph that uses line segments to show how data values change over time.
Example
The table shows the weights of a dog over 6 months. Make a line graph of the data. Between which two months of age does the dog gain the most weight?

Step 1: Write the ordered pairs from the table.
(1, 10), (2, 20), (3, 30), (4, 50), (5, 55), (6, 58)
Step 2: For each axis, choose appropriate numbers to represent the data in the table.
Step 3: Write a title for the graph and label each axis.
Step 4: Plot a point for each ordered pair. Then connect the points with line segments.

The greatest difference in weights occurs between the points (___, ___ ) and (___, ___ ).
So, the dog gains the most weight between ___ and ___ months of age.
Explanation:

The greatest difference in weights occurs between the points (_3__, __30_ ) and (__4_, __50_ ).
So, the dog gains the most weight between __3_ and _4__ months of age.

Show and Grow

Use the graph above.

Question 1.
Between which two months of age does the dog gain the least amount of weight? Explain.
Explanation:
The Least difference in weights occurs between the points (5__, __55_ ) and (__6_,  _58_ ).
So, the dog gains the least amount of weight __5 and _6_ months of age.

Question 2.
How much do you think the dog weighs when it is 7 months of age? Explain your reasoning.

Apply and Grow: Practice

Use the graph.

Question 3.
The table shows the height of a seedling over 7 days. Make a line graph of the data.

Between which two days did the seedling grow the most? Explain.

How tall do you think the seedling will be after 8 days? Explain.
Explanation

The greatest difference in seedling grow occurs between the points (_3__, __8_ ) and (__4_, _18_ ).
So, the Seeding grow between __3_ and _4__ days.
After 8 days Seeding may decrease.

Question 4.
Reasoning
Interpret the point (0, 0) in the context of the situation.

Use the graph.

Question 5.
The graph shows the total numbers of likes a social media page has over 8 days. How many likes does the page have after 4 days?

What is the difference of likes on Day 7 and Day 3?
Answer: After 4 days Social media likes are 225.
Difference between 7th day Social media likes and 3rd day Social media likes are=175.
Explanation:
Given that 8th day Social media likes are 300
4th day Social media likes are 75
Difference between 8th day Social media likes and 4th day Social media likes =300-75=225
After 4 days Social media likes are 225.
Given that 7th day Social media likes are 225
3rd day Social media likes are 50
Difference between 7th day Social media likes and 3rd day Social media likes are =225-50=175.

Question 6.
DIG DEEPER!
You track the likes between Days 7 and 8 by each hour. Does the total number of likes at every hour fall between 225 and 300? Explain.
Explanation:
The graph shows between 7th and 8th day the social media likes are increased 225 to 300.
So, the total number of likes at every hour fall between 225 and 300.

Think and Grow: Modeling Real Life

Example
The table shows your heart rate during an exercise routine. Make a line graph of the data. Use the graph to estimate your heart rate after exercising for 15 minutes.

Step 1: Write the ordered pairs from the table.
(0, 80), (10, 110), (20, 140), (30, 148), (35, 135)
Step 2: For each axis, choose appropriate numbers to represent the data in the table. You can show a break in the vertical axis between 0 and 80 because there are no data values between 0 and 80.

Step 3: Write a title for the graph and label each axis.
Step 4: Plot a point for each ordered pair. Then connect the points with line segments. Use the line segment that connects (10, 110) and (20, 140) to estimate your heart rate after exercising for 15 minutes.
After exercising for 15 minutes, your heart rate is about __ beats per minute.

Show and Grow

Question 7.
The table shows how many views a video has over several hours. Make a line graph of the data. Use the graph to estimate the total number of views the video has after 2 hours.

### Make and Interpret Line Graphs Homework & Practice 12.5

Example
The table shows the temperatures of a city over several hours during a snowstorm. Make a line graph of the data.

Between which two hours does the temperature decrease the most? Explain.

The greatest difference in temperatures occurs between the points (2, 28) and (3, 24). So, the temperature decreases the most between Hours 2 and 3.
Estimate the temperature at 4 hours and 30 minutes.
23 degrees Fahrenheit

Question 1.
The table shows the total number of pieces of beach glass you find during an hour at the beach. Make a line graph of the data.

Between which two times did you find the most pieces of beach glass? Explain.

Estimate how many pieces you had after 25 minutes.

Use the graph.

Question 2.
The graph shows the total amounts of money your class raises over 8 days. How much money does your class raise after 6 days?
How much money does your class raise between Days 2 and 7?

120 – 30 = 90

Question 3.
Logic
Your friend says that your class raises $115 after 9 days. Explain why your friend’s statement does not make sense. Answer Your friend’s statement makes sense. Use the graph. Question 4. Modeling Real Life The table shows a bald eagle’s heights above the ground after several seconds. Make a line graph of the data. Use the graph to estimate the eagle’s height above the ground after 6 seconds. Question 5. DIG DEEPER! The eagle flies past her nest, which is 120 feet above the ground. After how many seconds do you think the eagle flies past her nest? Explain. Review & Refresh Question 6. Answer: 8 Question 7. Answer: 12 Question 8. Answer: 50 ### Lesson 12.6 Numerical Paterns Explore and Grow Newton saves$10 each month. Descartes saves $30 each month. Complete each table. What patterns do you notice? Newton: Answer: Descartes: Answer: Repeated Reasoning How much will Newton have saved when Descartes has saved$300? Explain your reasoning.

Think and Grow: Numerical Paterns

Example
You use 2 pounds of beef to make a batch of empanadas. Each batch makes eight servings. Complete the rule that relates the number of servings to the number of pounds of beef.

Step 1: Create each pattern and complete the table.
Use the rule “Add __” to find the number of pounds of beef.
0, 2, 4, ___, ___, ___
Use the rule “Add __” to find the number of servings.
0, 8, 16, ___, ___, ___
Step 2: Write ordered pairs that relate the number of servings to the number of pounds of beef.
(0, 0), (8, 2), (16, 4), ___, ___, ___
Step 3: Write a rule. As you make each batch, the number of pounds of beef is always __ as much as the number of servings.

So, divide the number of servings by __ to find the number of pounds of beef.

Step 1: Create each pattern and complete the table.
Use the rule “Add __” to find the number of pounds of beef.
0, 2, 4, 6, 8, 10
Use the rule “Add __” to find the number of servings.
0, 8, 16, 24, 32, 40
Step 2: Write ordered pairs that relate the number of servings to the number of pounds of beef.
(0, 0), (8, 2), (16, 4), (24, 6), (32, 8), (40, 10)
Step 3: Write a rule. As you make each batch, the number of pounds of beef is always __ as much as the number of servings.

Show and Grow

Question 1.
Use the given rules to complete the table. Then complete the rule that relates the number of hours worked to the amount earned.

Multiply the number of hours worked by __ to find the amount earned.

Multiply the number of hours worked by 8 to find the amount earned.

Apply and Grow: Practice

Use the given rules to complete the table. Then complete the rule.

Question 2.

Divide the number of cups of water __ by to find the number of cups of lemon juice.

Divide the number of cups of water 7 by to find the number of cups of lemon juice.

Question 3.

Multiply the number of push-ups by __ to find the number of sit-ups.

Multiply the number of push-ups by 2 to find the number of sit-ups.

Question 4.
Complete the rule. Then use the rule to complete the table.
Multiply the amount of money that Newton saves by __ to find the amount of money that Descartes saves.

Multiply the amount of money that Newton saves by 3 to find the amount of money that Descartes saves.

Question 5.
Structure
The ordered pairs (3, 2), (6, 4), and (9, 6) relate the number of avocados to the number of plum tomatoes in a guacamole recipe. Use the relationship to complete the table.

Think and Grow: Modeling Real Life

Example
For each $1 bill you pay, you get 4 tokens and can play 2 games. You have 60 tokens. How many games can you play? Think: What do you know? What do you need to find? How will you solve? Use a rule to create each pattern. Use a table to organize the information. Write ordered pairs that relate the number of tokens to the number of games you can play. (4, 2), ___, ___, ___ Write a rule. The number of games you can play is always ___ as much as the number of tokens. So, divide the number of tokens by __ to find the number of games you can play. 60 ÷ __ = ___ So, you can play games. Answer: (4, 2), (8,2), (12,6), (16,8) Write a rule. The number of games you can play is always half as much as the number of tokens. So, divide the number of tokens by 2 to find the number of games you can play. 60 ÷ 2= 30 So, you can play 30 games. Show and Grow Question 6. Each day, you read 33 pages and your friend reads 11 pages. How many pages does your friend read when you read 396 pages? Answer: 132 pages Explanation: Given that, Each day, you read 33 pages and your friend reads 11 pages. 33/11 = 3 396/3 = 132 Thus your friend read 132 pages when you read 396 pages. Question 7. DIG DEEPER! Each pack of trading cards has 1 hero card, 5 combination cards, and 30 action cards. You buy packs of trading cards and get 35 combination cards. How many hero cards and action cards do you get? Answer: 7 hero cards ### Numerical Paterns Homework & Practice 12.6 Question 1. Use the given rules to complete the table. Then complete the rule. Multiply the number of candles sold by __ to find the amount of money raised. Answer: Multiply the number of candles sold by 8 to find the amount of money raised. Question 2. Multiply the number of servings by __ to find the number of pretzels. Answer: Multiply the number of servings by 20 to find the number of pretzels. Question 3. Complete the rule. Then use the rule to complete the table. Divide the number of contestants by __ to find the number of winners. Answer: Divide the number of contestants by 4 to find the number of winners. Question 4. DIG DEEPER! Draw Figure 4. How many red squares are in Figure 8? How many yellow squares are in Figure 8? Explain your reasoning. Answer: Question 5. Modeling Real Life Each person at a baseball game receives 3 raffle tickets and a$2 certificate for the team store. A group of people receives 39 raffle tickets. How much money in certificates does the group receive?

Given,
Each person at a baseball game receives 3 raffle tickets and a $2 certificate for the team store. A group of people receives 39 raffle tickets. 39 × 2 =$78

Question 6.
DIG DEEPER!
Write a rule that relates the number of months to the cost of a gym membership. What is the cost of a 1-year membership?

Given,
1 month $15 12 months = 15 × 12 =$180

Review & Refresh

Convert the mass.

Question 7.
7 g = __ mg

Question 8.
92 g = ___ kg

convert from grams to kgs.
92 grams = 0.92 kg

Convert the capacity

Question 9.
800 mL = __ L

Convert from ml to l
1 ml = 0.001
800 ml = 0.8 liters

Question 10.
3 L = __ mL

Convert from liters to ml
1 liter = 1000 ml
3 liter = 3 × 1000 ml = 3000 ml

### Lesson 12.7 Graph and Analyze Relationships

Explore and Grow

Complete each table and graph the data in the coordinate plane. What do you notice about the points?

Structure
How can you use the graphs to find the number of feet in 7 yards and the number of pints in 6 gallons? Explain your reasoning.

Think and Grow: Graph and Analyze Relationships

Example
For each glass of iced tea Newton makes, he uses 2 spoonfuls of iced-tea mix and 10 fluid ounces of water. Newton uses 16 spoonfuls of iced-tea mix. How many fluid ounces of water does he use?
Step 1: Find the first several numbers in the numerical patterns for the amounts of iced-tea mix and water.

Step 2: Write the ordered pairs from the table.

Step 3: Plot the ordered pairs. Connect the points with line segments.

Because the ordered pairs follow a pattern, you can extend the line to the point where the x-coordinateis 16.
When the x-coordinate is 16, the y-coordinate is ___
So, Newton uses __ fluid ounces of water.

(2, 10), (4, 20), (6, 30), (8, 40), (10, 50), (12, 60), (14, 70), (16, 80)
Because the ordered pairs follow a pattern, you can extend the line to the point where the x-coordinate is 16.
When the x-coordinate is 16, the y-coordinate is 80
So, Newton uses 80 fluid ounces of water.

Show and Use

Question 1.
Use the graph above. Newton uses 18 spoonfuls of iced-tea mix. How many fluid ounces of water does he use? Explain your reasoning

When the x-coordinate is 18, the y-coordinate is 100
So, Newton uses 100 fluid ounces of water.

Apply and Grow: Practice

Use the given information to complete the table. Describe the relationship between the two numerical patterns and plot the points.

Question 2.
A slime recipes calls for 120 milliliters of vegetable oil and 30 grams of cornstarch. You measure 600 milliliters of vegetable oil. How many grams of cornstarch do you need?

Question 3.
A sponsor donates $5 for every 4 laps walked around a track. How much money does the sponsor donate for 28 laps walked? Answer: Question 4. Writing How can you use the graph to determine the number of cups in 4 gallons? Answer: (4, 64) Question 5. Number Sense What does the ordered pair (0, 0) represent in the graph? Answer: Origin Question 6. DIG DEEPER! Use the graph to determine the number of cups in 2$$\frac{1}{2}$$ gallons. Answer: 40 cups Think and Grow: Modeling Real Life Example Some friends plan to go to a trampoline park for 2 hours. They want to go to the park that costs less money. Which park should they choose? What is the cost for each person? Graph the relationship between time and cost at both parks. Park A has been done for you. Step 1: Make a table for time and cost at Park B. Step 2: Write the ordered pairs from the table. Step 3: Plot the ordered pairs. Connect the points with line segments. Use the graph to compare the costs for 2 hours at the parks. So, the group of friends should choose Trampoline Park __ The cost for each person is$ ___.

So, the group of friends should choose Trampoline Park A.
The cost for each person is $15. Show and Grow Question 7. On your map, every centimeter represents 20 kilometers. On your friend’s map, every 2 centimeters represents 50 kilometers. On whose map does 6 centimeters represent a greater distance? How much greater? Explain. Answer: Given that, On your map, every centimeter represents 20 kilometers. On your friend’s map, every 2 centimeters represents 50 kilometers. 4 cm = 100 kilometers 6 cm = 150 kilometers ### Graph and Analyze Relationships Homework & Practice 12.7 Question 1. Use the graph above. You plan to park your car for 140 minutes. How much money do you put into the meter? Answer:$35 for 140 minutes

Question 2.
A boxer exercises by jumping rope. He completes 150 repetitions every minute. He completes 750 repetitions. For how many minutes does he jump rope?

Question 3.
YOU BE THE TEACHER
Your friend says a baker makes 60 plain bagels in 5 hours. Is your friend correct? Explain.

Question 4.
Modeling Real Life
Some friends plan to rent bicycles for 6 hours. They want to choose the option that costs less money. Which option should they choose? What is the cost for each person?

Answer: They should choose option B.

Review & Refresh

Convert the length.

Question 5.

Convert from feet to inches
3.3 feet = 39.6 inches

Question 6.
6 mi = __ yd

Convert from miles to yards
1 mile = 1760 yards
6 miles = 10560 yards

### Patterns in the Coordinate Plane Performance Task 12

You use a series of commands on an app to create an animation of Descartes dancing and jumping.

Question 1.
a. Complete the animation commands to moveDescartes. Plot the points to show his movement.

b. Connect the points. Describe the animation in your own words.

Question 2.
You play the animation commands to make Descartes dance.
a. It takes 4 seconds for Descartes to move through the animation commands 1 time. Complete the table and graph the data in the coordinate plane.

b. You want Descartes to dance for an exact number of seconds. How can you find the number of times to play the animation commands? Use an example to justify your reasoning.

### Patterns in the Coordinate Plane Treasure Hunt

Directions:

1. Each player arranges four Treasure Hunt Gold Bars on the My Treasure coordinate plane, horizontally or vertically.
2. On your turn, name an ordered pair in the coordinate plane. If your partner says you found part of a gold bar, then plot the ordered pair in red. Otherwise, plot the ordered pair in black. Your turn is over.
3. On your partner’s turn, if your partner finds part of a gold bar, then plot a red on the ordered X pair in the coordinate plane. Tell your partner when all parts of a gold bar have been found.
4. The first player to find all parts of the partner’s gold bars wins!

### Patterns in the Coordinate Plane Chapter Practice 12

12.1 Plot Points in a Coordinate Plane

Use the coordinate plane to write the ordered pair corresponding to the point.

Question 1.
Point A

Answer: The ordered pair of Point A is (4, 6)

Question 2.
Point D

Answer: The ordered pair of Point B is (0, 0)

Question 3.
Point B

Answer: The ordered pair of Point B is (2, 3)

Question 4.
Point E

Answer: The ordered pair of Point E is (2, 0)

Question 5.
Point C

Answer: The ordered pair of Point C is (8, 3)

Question 6.
Point F

Answer: The ordered pair of Point F is (8, 7)

Plot and label the point in the coordinate plane above.

Question 7.
N(3, 0)

Question 8.
P(1, 5)

Question 9.
R(2, 1)

Name the point for the ordered pair.

Question 10.
(0, 8)

Answer: G is the point for the ordered pair (0, 8)

Question 11.
(5, 6)

Answer: J is the point for the ordered pair (5, 6)

Question 12.
(4, 2)

Answer: H is the point for the ordered pair (4, 2)

Question 13.
Open-Ended
Use the coordinate plane above. Point S is 2 units from point J. Name two possible ordered pairs for point S.

Answer: The two possible ordered pairs for point S are (6, 2) or (4, 4)

12.2 Relate Points in a Cooordinate Plane

Find the distance between the points in the coordinate grid.

Question 14.
A and B

The distance between the two points A and B is 2 units.

Question 15.
E and F

Answer: The distance between the two points E and F is 3 units.

Question 16.
C and D

Answer: The distance between the two points C and D is 7 units.

Question 17.
Which is longer, $$\overline{A C}$$ or $$\overline{G E}$$ ?

Answer: $$\overline{A C}$$ is longer than $$\overline{G E}$$.

Find the distance between the points.

Question 18.
(0, 0) and (0, 4)

The formula for distance between the points is √(x2 – x1)² + (y2 – y1)²
= √(0 – 0)² + (4 – 0)²
= √16
= 4
Thus the distance between the points is 4.

Question 19.
(3, 2) and (3, 9)

The formula for distance between the points is √(x2 – x1)² + (y2 – y1)²
= √(3 – 3)² + (9 – 2)²
= √49
= 7
Thus the distance between the points is 7.

Question 20.
(0, 5) and (7, 5)

The formula for distance between the points is √(x2 – x1)² + (y2 – y1)²
= √(7 – 0)² + (5 – 5)²
= √49
= 7
Thus the distance between the points is 7.

A line passes through the given points. Name two other points that lie on the line.

Question 21.
(0, 1) and (0, 7)

Answer: (0, 5) and (0, 6) these two points lie on the same line.

Question 22.
(5, 2) and (5, 8)

Answer: (5, 3) and (5, 7) these two points lie on the same line.

Question 23.
(6, 3) and (0, 3)

Answer: (4, 3) and (3, 3) these two points lie on the same line.

12.3 Draw Polygons in a Coordinate Plane

Draw the polygon with the given vertices in a coordinate plane. Then identify it.

Question 24.
A(2, 5), B(5, 5), C(5, 0), D(2, 0)

Question 25.
D(1, 3), E(1, 5), F(3, 6), G(5, 5), H(5, 3), J(3, 2)

12.4 Graph Data

Question 26.
The table shows how many home runs your team scores in each of six kickball games. Graph the data.

What does the point (6, 3) represent? What is the difference of the greatest number of home runs and the least number of home runs? Explain.

(2, 1) is the least number of home runs.

12.5 Make an Interpret Line Graphs

Question 27.
The table shows the total numbers of coupon books you sell over 7 days. Make a line graph of the data.

On which day do you sell the most books? Explain.

How many books do you think you sell after 9 days? Explain.

12.7 Numerical Patterns

Question 28.
Use the given rules to complete the table. Then complete the rule.

Multiply the number of ounces of pudding mix by __ to find the number of ounces of pumpkin.

12.7 Graph and Analyze Relationships

Question 29.
An employee earns $80 every 8 hours. How much money does she earn after 40 hours? Answer: Question 30. Modeling Real Life A group of friends wants to play laser tag for 60 minutes. They want to go to the facility that costs less money. Which facility should they choose? What is the cost for each person? Answer: They need to choose Facility B. Final Words: Learn the Big Ideas Math Book 5th Grade Solution Key Chapter 12 Patterns in the Coordinate Plane provided and improve your math as well as problem-solving skills. You can achieve greater heights and fall in love with Math with our Big Ideas Math Answers Grade 5 Chapter 12 Patterns in the Coordinate Plane. Bookmark our ccssmathanswers.com to get the solutions of Big Ideas Math Grade 5 Chapters from 1 to 13. ## Go Math Grade 8 Answer Key Chapter 1 Real Numbers The Solutions for Go Math Grade 8 Answer Key Chapter 1 Real Numbers are given in detail here. Get the step by step explanations for all the question in Go Math Grade 8 Chapter 1 Real Numbers Answer Key and start your practice today. You can find the best ways to learn maths by using Go Math Grade 8 Answer Key. So, Download Go Math Grade 8 Chapter 1 Real Numbers Solution Key and make use of the given resources. ## Go Math Grade 8 Chapter 1 Real Numbers Answer Key It is essential for the students to choose the best material to practice the questions. Because by practicing only you can score the highest marks in the exams. Download HMH Go Math Grade 8 Answer Key Chapter 1 Real Numbers PDF for free. Quick learning is possible with our Go Math Grade 8 Chapter 1 Real Numbers Answer Key. Find a better way to make your learning simple by clicking on the below-provided links. Lesson 1: Rational and Irrational Numbers Lesson 2: Sets of real Numbers Lesson 3: Ordering Real Numbers Model Quiz Mixed Review ### Guided Practice – Rational and Irrational Numbers – Page No. 12 Write each fraction or mixed number as a decimal. Question 1. $$\frac{2}{5}$$ = Answer: 0.4 Explanation: $$\frac{2}{5}$$ = $$\frac{2 × 2}{5 × 2}$$ = $$\frac{4}{10}$$ = 0.4 Question 2. $$\frac{8}{9}$$ = Answer: 0.88 Explanation: $$\frac{8}{9}$$ = $$\frac{8 × 10}{9 × 10}$$ = $$\frac{80}{9 × 10}$$ = $$\frac{8.88}{10}$$ = 0.88 Question 3. 3 $$\frac{3}{4}$$ = Answer: 3.75 Explanation: 3 $$\frac{3}{4}$$ =$$\frac{15}{4}$$ = 3.75 Question 4. $$\frac{7}{10}$$ = Answer: 0.7 Explanation: $$\frac{7}{10}$$ = 0.7 Question 5. 2 $$\frac{3}{8}$$ = Answer: 2.375 Explanation: 2 $$\frac{3}{8}$$ = $$\frac{19}{8}$$ = 2.375 Question 6. $$\frac{5}{6}$$ = Answer: 0.833 Explanation: $$\frac{5}{6}$$ = $$\frac{5 × 10}{6 × 10}$$ = $$\frac{50}{6 × 10}$$ = $$\frac{8.33}{10}$$ = 0.833 Write each decimal as a fraction or mixed number in simplest form Question 7. 0.675 $$\frac{□}{□}$$ Answer: $$\frac{27}{40}$$ Explanation: $$\frac{0.675 × 1000}{1 × 1000}$$ = $$\frac{675}{1000}$$ = $$\frac{675/25}{1000/25}$$ = $$\frac{27}{40}$$ Question 8. 5.6 ______ $$\frac{□}{□}$$ Answer: 5 $$\frac{3}{5}$$ Explanation: $$\frac{5.6 × 10}{10}$$ = $$\frac{56}{10}$$ = 5 $$\frac{6}{10}$$ = 5 $$\frac{6/2}{10/2}$$ = 5 $$\frac{3}{5}$$ Question 9. 0.44 $$\frac{□}{□}$$ Answer: $$\frac{11}{25}$$ Explanation: $$\frac{0.44 × 100}{1 × 100}$$ = $$\frac{44}{100}$$ = $$\frac{44/4}{100/4}$$ = $$\frac{11}{25}$$ Question 10. 0.$$\bar{4}$$ $$\frac{□}{□}$$ Answer: $$\frac{4}{9}$$ Explanation: Let x = 0.$$\bar{4}$$ Now, 10x = 4.$$\bar{4}$$ 10x – x = 4.$$\bar{4}$$ – 0.$$\bar{4}$$ 9x = 4 x = $$\frac{4}{9}$$ Question 11. 0.$$\overline { 26 }$$ $$\frac{□}{□}$$ Answer: $$\frac{26}{99}$$ Explanation: Let x = 0.$$\overline {26}$$ Now, 100x = 26.$$\overline{26}$$ 100x – x = 26.$$\overline{26}$$ – 0.$$\overline {26}$$ 99x = 26 x = $$\frac{26}{99}$$ Question 12. 0.$$\overline { 325 }$$ $$\frac{□}{□}$$ Answer: $$\frac{325}{999}$$ Explanation: Let x = 0.$$\overline {325}$$ Now, 1000x = 325.$$\overline{325}$$ 1000x – x = 325.$$\overline{325}$$ – 0.$$\overline {325}$$ 999x = 325 x = $$\frac{325}{999}$$ Solve each equation for x Question 13. x2 = 144 ± ______ Answer: x=±12 Explanation: x2 = 144 Taking square roots on both the sides x2=±144 x = ±12 Question 14. x2 = $$\frac{25}{289}$$ ± $$\frac{□}{□}$$ Answer: x = ±$$\frac{5}{17}$$ Explanation: x2 = $$\frac{25}{289}$$ Taking square roots on both the sides x2=±√$$\frac{25}{289}$$ x = ±$$\frac{5}{17}$$ Question 15. x3 = 216 ______ Answer: x = 6 Explanation: x3 = 216 Taking cube roots on both the sides 3x3= 3√216 x = 6 Approximate each irrational number to two decimal places without a calculator. Question 16. $$\sqrt { 5 }$$ ≈ ______ Answer: 2.236 Explanation: x = $$\sqrt { 5 }$$ The 5 is in between 4 and 6 Take square root of each year √4 < √5 < √6 2 < √5 < 3 √5 = 2.2 (2.2)² = 4.84 (2.25)² = 5.06 (2.5)³ = 5.29 A good estimate for √5 is 2.25 Question 17. $$\sqrt { 3 }$$ ≈ ______ Answer: 1.75 Explanation: $$\sqrt { 3 }$$ 1 < 3 < 4 √1 < √3 < √4 1 < √3 < 2 √3 = 1.6 (1.65)² = 2.72 (1.7)² = 2.89 (1.75)² = 3.06 A good estimate for √3 is 1.75 Question 18. $$\sqrt { 10 }$$ ≈ ______ Answer: 3.15 Explanation: $$\sqrt { 10 }$$ 9 < 10 < 16 √9 < √10 < √16 3 < √10 < 4 √10 = 3.1 (3.1)² = 9.61 (3.15)² = 9.92 (3.2)² = 10.24 A good estimate for √10 is 3.15 Question 19. What is the difference between rational and irrational numbers? Type below: _____________ Answer: Rational number can be expressed as a ration of two integers such as 5/2 Irrational number cannot be expressed as a ratio of two integers such as √13 Explanation: A rational number is a number that can be express as the ratio of two integers. A number that cannot be expressed that way is irrational. ### 1.1 Independent Practice – Rational and Irrational Numbers – Page No. 13 Question 20. A $$\frac{7}{16}$$-inch-long bolt is used in a machine. What is the length of the bolt written as a decimal? ______ -inch-long Answer: 0.4375 inch Explanation: The length of the bolt is $$\frac{7}{16}$$-inch Let, x = $$\frac{7}{16}$$ Multiplying by 125 on both nominator and denominator x = $$\frac{7×125}{16×125}$$ = $$\frac{875}{2000}$$ =$$\frac{437.5}{1000}$$ = 0.4375 Question 21. The weight of an object on the moon is $$\frac{1}{6}$$ its weight on Earth. Write $$\frac{1}{6}$$ as a decimal. ______ Answer: 0.1666 Explanation: The weight of the object on the moon is $$\frac{1}{6}$$ Let, x = $$\frac{1}{6}$$ Multiplying by 100 on both nominator and denominator x = $$\frac{1×100}{6×100}$$ = $$\frac{16.6}{100}$$ =0.166 Question 22. The distance to the nearest gas station is 2 $$\frac{4}{5}$$ kilometers. What is this distance written as a decimal? ______ Answer: 2.8 Explanation: The distance of the nearest gas station is 2 $$\frac{4}{5}$$ Let, x = 2 $$\frac{4}{5}$$ Multiplying by 100 on both nominator and denominator x = 2 $$\frac{4×100}{5×100}$$ = $$\frac{80}{100}$$ =0.8 Question 23. A baseball pitcher has pitched 98 $$\frac{2}{3}$$ innings. What is the number of innings written as a decimal? ______ Answer: 98.6 Explanation: A baseball pitcher has pitched 98 $$\frac{2}{3}$$ innings. 98 $$\frac{2}{3}$$ = 98 + 2/3 = (294/3) + (2/3) 296/3 98.6 Question 24. A heartbeat takes 0.8 second. How many seconds is this written as a fraction? $$\frac{□}{□}$$ Answer: $$\frac{4}{5}$$ Explanation: A heartbeat takes 0.8 seconds. 0.8 There are 8 tenths. 8/10 = 4/5 Question 25. There are 26.2 miles in a marathon. Write the number of miles using a fraction. $$\frac{□}{□}$$ Answer: 26$$\frac{1}{5}$$ Explanation: There are 26.2 miles in a marathon. 26.2 miles 262/10 131/5 26 1/5 miles Question 26. The average score on a biology test was 72.$$\bar{1}$$. Write the average score using a fraction. $$\frac{□}{□}$$ Answer: 80 $$\frac{1}{9}$$ Explanation: The average score on a biology test was 72.$$\bar{1}$$. 72.$$\bar{1}$$ Let x = 72.$$\bar{1}$$ 10x = 10(72.$$\bar{1}$$) 10x = 721.1 -x = -0.1 9x = 721 x = 721/9 x = 80 1/9 Question 27. The metal in a penny is worth about 0.505 cent. How many cents is this written as a fraction? $$\frac{□}{□}$$ Answer: $$\frac{101}{200}$$ Explanation: The metal in a penny is worth about 0.505 cent. 0.505 cent 505 thousandths 505/1000 101/200 cents Question 28. Multistep An artist wants to frame a square painting with an area of 400 square inches. She wants to know the length of the wood trim that is needed to go around the painting. a. If x is the length of one side of the painting, what equation can you set up to find the length of a side? x2 = ______ Answer: x² = 400 Explanation: The area of a square is the square of its equal side, x x² = 400 Question 28. b. Solve the equation you wrote in part a. How many solutions does the equation have? x = ± ______ Answer: x = ± 20 Explanation: Take the square root on both sides. Solve x = ± 20 Question 28. c. Do all of the solutions that you found in part b make sense in the context of the problem? Explain. Type below: _____________ Answer: No. Both values of x do not make sense. Explanation: The length cannot be negative, hence negative value does not make sense. No. Both values of x do not make sense. Question 28. d. What is the length of the wood trim needed to go around the painting? P = ______ inches Answer: Length P = 20 + 2y ### Rational and Irrational Numbers – Page No. 14 Question 29. Analyze Relationships To find $$\sqrt { 15 }$$, Beau found 32 = 9 and 42 = 16. He said that since 15 is between 9 and 16, $$\sqrt { 15 }$$ must be between 3 and 4. He thinks a good estimate for $$\sqrt { 15 }$$ is $$\frac { 3+4 }{ 2 }$$ = 3.5. Is Beau’s estimate high, low, or correct? Explain. _____________ Answer: 3.85 Explanation: 15 is closer to 16 √15 is closer to √16 Beau’s estimate is low. (3.8)² = 14.44 (3.85)² = 14.82 (3.9)² = 15.21 √15 is 3.85 Question 30. Justify Reasoning What is a good estimate for the solution to the equation x3 = 95? How did you come up with your estimate? x ≈ ______ Answer: x ≈ 4.55 Explanation: 3√x = 95 x = 3√95 64 < 95 < 125 Take the cube root of each number 3√64 < 3√95 < 3√125 4 < 3√95 < 5 3√95 = 4.6 (4.5)³ = 91.125 (4.55)³ = 94.20 (4.6)³ = 97.336 3√95 = 4.55 Question 31. The volume of a sphere is 36π ft3. What is the radius of the sphere? Use the formula V = $$\frac { 4 }{ 3 }$$πr3 to find your answer. r = ______ Answer: r = 3 Explanation: V = 4/3 πr³ 36π = 4/3 πr³ r³ = 36π/π . 3/4 r³ = 27 r = 3√27 r = 3 FOCUS ON HIGHER ORDER THINKING Question 32. Draw Conclusions Can you find the cube root of a negative number? If so, is it positive or negative? Explain your reasoning. _____________ Answer: Yes Explanation: Yes. The cube root of a negative number would be negative. Because the product of three negative signs is always negative. Question 33. Make a Conjecture Evaluate and compare the following expressions. $$\sqrt { \frac { 4 }{ 25 } }$$ and $$\frac { \sqrt { 4 } }{ \sqrt { 25 } }$$ $$\sqrt { \frac { 16 }{ 81 } }$$ and $$\frac { \sqrt { 16 } }{ \sqrt { 81 } }$$ $$\sqrt { \frac { 36 }{ 49 } }$$ and$$\frac { \sqrt { 36 } }{ \sqrt { 49 } }$$ Use your results to make a conjecture about a division rule for square roots. Since division is multiplication by the reciprocal, make a conjecture about a multiplication rule for square roots. Expressions are: _____________ Answer: Evaluating and comparing √4/25 = 2/5 √16/81 = 4/9 √36/49 = 6/7 Conjecture about a division rule for square roots √a/√b = √(a/b) Conjecture about a multiplication rule for square roots √a × √b Question 34. Persevere in Problem Solving The difference between the solutions to the equation x2 = a is 30. What is a? Show that your answer is correct. _____ Answer: 30 Explanation: x2 = a x = ±√a √a – (-√a) = 30 √a + √a = 30 2√a = 30 √a = 15 a = 225 x2 = 225 x = ±225 x = ±15 15 – (-15) = 15 + 15 = 30 ### Guided Practice – Sets of real Numbers – Page No. 18 Write all names that apply to each number. Question 1. $$\frac{7}{8}$$ Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers Question 2. $$\sqrt { 36 }$$ Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers Explanation: $$\sqrt { 36 }$$ = 6 Question 3. $$\sqrt { 24 }$$ Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers e. Irrational Numbers Question 4. 0.75 Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers Question 5. 0 Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers Question 6. −$$\sqrt { 100 }$$ Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers c. Integers Explanation: −$$\sqrt { 100 }$$ = – 10 Question 7. 5.$$\overline { 45 }$$ Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers Question 8. −$$\frac{18}{6}$$ Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers c. Integers Explanation: −$$\frac{18}{6}$$ = -3 Tell whether the given statement is true or false. Explain your choice. Question 9. All whole numbers are rational numbers. i. True ii. False Answer: i. True Explanation: All whole numbers are rational numbers. Whole numbers are a subset of the set of rational numbers and can be written as ratio of the whole number to 1. Question 10. No irrational numbers are whole numbers. i. True ii. False Answer: i. True Explanation: True. Whole numbers are ration numbers. Identify the set of numbers that best describes each situation. Explain your choice. Question 11. the change in the value of an account when given to the nearest dollar Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: c. Integer Numbers Explanation: The change can be a whole dollar amount and can be positive, negative or zero. Question 12. the markings on a standard ruler Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: b. Rational Numbers Explanation: The ruler is marked every 1/16t inch. ESSENTIAL QUESTION CHECK-IN Question 13. What are some ways to describe the relationships between sets of numbers? Answer: There are two ways that we have been using until now to describe the relationships between sets of numbers • Using a scheme or a diagram as the one on page 15. • Verbal description, for example, “All irrational numbers are real numbers.” ### 1.2 Independent Practice – Sets of real Numbers – Page No. 19 Write all names that apply to each number. Then place the numbers in the correct location on the Venn diagram. Question 14. $$\sqrt { 9 }$$ Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers Explanation: $$\sqrt { 9 }$$ = 3 Question 15. 257 Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers Question 16. $$\sqrt { 50 }$$ Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers e. Irrational Numbers Question 17. 8 $$\frac{1}{2}$$ Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers Question 18. 16.6 Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers Question 19. $$\sqrt { 16 }$$ Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers Explanation: $$\sqrt { 16 }$$ = 4 Identify the set of numbers that best describes each situation. Explain your choice. Question 20. the height of an airplane as it descends to an airport runway Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: d. Whole Numbers Explanation: Whole. The height of an airplane as it descents to an airport runway is a whole number greater than 0 Question 21. the score with respect to par of several golfers: 2, – 3, 5, 0, – 1 Options: a. Real Numbers b. Rational Numbers c. Integer Numbers d. Whole Numbers e. Irrational Numbers Answer: c. Integer Numbers Explanation: Integers. The scores are counting numbers, their opposites, and zero. Question 22. Critique Reasoning Ronald states that the number $$\frac{1}{11}$$ is not rational because, when converted into a decimal, it does not terminate. Nathaniel says it is rational because it is a fraction. Which boy is correct? Explain. i. Ronald ii. Nathaniel Answer: ii. Nathaniel Explanation: Nathaniel is correct. A fraction is a rational real number, even if it is not a terminating decimal. ### Sets of real Numbers – Page No. 20 Question 23. Critique Reasoning The circumference of a circular region is shown. What type of number best describes the diameter of the circle? Explain your answer. Options: a. Real Numbers b. Rational Numbers c. Irrational Numbers d. Integers e. Whole Numbers Answer: e. Whole Numbers Explanation: Circumference of the circle A = 2πr π = 2πr Diameter is twice the radius 2r = 1 Whole Question 24. Critical Thinking A number is not an integer. What type of number can it be? Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: b. Rational Numbers e. Irrational Numbers Question 25. A grocery store has a shelf with half-gallon containers of milk. What type of number best represents the total number of gallons? Options: a. Real Numbers b. Rational Numbers c. Integers d. Whole Numbers e. Irrational Numbers Answer: b. Rational Numbers FOCUS ON HIGHER ORDER THINKING Question 26. Explain the Error Katie said, “Negative numbers are integers.” What was her error? Type below: _______________ Answer: Her error is that she stated that all negative numbers are integers. Some negative numbers are integers such as -4 but some are not such an -0.8 Question 27. Justify Reasoning Can you ever use a calculator to determine if a number is rational or irrational? Explain. Type below: _______________ Answer: Not always. Explanation: Not always. If the calculator shows a terminating decimal, the number is rational but otherwise, it is not possible as you can only see a few digits. Question 28. Draw Conclusions The decimal 0.$$\bar{3}$$ represents $$\frac{1}{3}$$. What type of number best describes 0.$$\bar{9}$$ , which is 3 × 0.$$\bar{3}$$? Explain. Type below: _______________ Answer: 1 Explanation: let x = 0.9999999 10x = 9.99999999 10x = 9 + 0.999999999 10x = 9 + x 9x = 9 x=1. Question 29. Communicate Mathematical Ideas Irrational numbers can never be precisely represented in decimal form. Why is this? Answer: Because irrational numbers are nonrepeating, otherwise they could be represented as a fraction. Although a potential counter-example to this claim is that some irrational numbers can only be represented in decimal form, for example, 0.1234567891011121314151617…, 0.24681012141618202224…, 0.101101110111101111101111110… are all irrational numbers. ### Guided Practice – Ordering Real Numbers – Page No. 24 Compare. Write <, >, or =. Question 1. $$\sqrt { 3 }$$ + 2 ________ $$\sqrt { 3 }$$ + 3 Answer: $$\sqrt { 3 }$$ + 2 < $$\sqrt { 3 }$$ + 3 Explanation: $$\sqrt { 3 }$$ is between 1 and 2 $$\sqrt { 3 }$$ + 2 is between 3 and 4 $$\sqrt { 3 }$$ + 3 is between 4 and 5 $$\sqrt { 3 }$$ + 2 < $$\sqrt { 3 }$$ + 3 Question 2. $$\sqrt { 11 }$$ + 15 _______ $$\sqrt { 8 }$$ + 15 Answer: $$\sqrt { 11 }$$ + 15 > $$\sqrt { 8 }$$ + 15 Explanation: $$\sqrt { 11 }$$ is between 3 and 4 $$\sqrt { 8 }$$ is between 2 and 3 $$\sqrt { 11 }$$ + 15 is between 18 and 19 $$\sqrt { 8 }$$ + 15 is between 17 and 18 $$\sqrt { 11 }$$ + 15 > $$\sqrt { 8 }$$ + 15 Question 3. $$\sqrt { 6 }$$ + 5 _______ 6 + $$\sqrt { 5 }$$ Answer: $$\sqrt { 6 }$$ + 5 < 6 + $$\sqrt { 5 }$$ Explanation: $$\sqrt { 6 }$$ is between 2 and 3 $$\sqrt { 5 }$$ is between 2 and 3 $$\sqrt { 6 }$$ is between 7 and 8 $$\sqrt { 5 }$$ is between 8 and 9 $$\sqrt { 6 }$$ + 5 < 6 + $$\sqrt { 5 }$$ Question 4. $$\sqrt { 9 }$$ + 3 _______ 9 + $$\sqrt { 3 }$$ Answer: $$\sqrt { 9 }$$ + 3 < 9 + $$\sqrt { 3 }$$ Explanation: $$\sqrt { 9 }$$ + 3 9 + $$\sqrt { 3 }$$ $$\sqrt { 3 }$$ is between 1 and 2 $$\sqrt { 9 }$$ + 3 = 3 + 3 = 6 9 + $$\sqrt { 3 }$$ is between 10 and 11 $$\sqrt { 9 }$$ + 3 < 9 + $$\sqrt { 3 }$$ Question 5. $$\sqrt { 17 }$$ – 3 _______ -2 + $$\sqrt { 5 }$$ Answer: $$\sqrt { 17 }$$ – 3 > -2 + $$\sqrt { 5 }$$ Explanation: $$\sqrt { 17 }$$ is between 4 and 5 $$\sqrt { 5 }$$ is between 2 and 3 $$\sqrt { 17 }$$ – 3 is between 1 and 2 -2 + $$\sqrt { 5 }$$ is between 0 and 1 $$\sqrt { 17 }$$ – 3 > -2 + $$\sqrt { 5 }$$ Question 6. 10 – $$\sqrt { 8 }$$ _______ 12 – $$\sqrt { 2 }$$ Answer: 10 – $$\sqrt { 8 }$$ < 12 – $$\sqrt { 2 }$$ Explanation: $$\sqrt { 8 }$$ is between 2 and 3 $$\sqrt { 2 }$$ is between 1 and 2 10 – $$\sqrt { 8 }$$ is between 8 and 7 12 – $$\sqrt { 2 }$$ is between 11 and 10 10 – $$\sqrt { 8 }$$ < 12 – $$\sqrt { 2 }$$ Question 7. $$\sqrt { 7 }$$ + 2 _______ $$\sqrt { 10 }$$ – 1 Answer: $$\sqrt { 7 }$$ + 2 > $$\sqrt { 10 }$$ – 1 Explanation: $$\sqrt { 7 }$$ is between 2 and 3 $$\sqrt { 10 }$$ is between 3 and 4 $$\sqrt { 7 }$$ + 2 is between 4 and 5 $$\sqrt { 10 }$$ – 1 is between 2 and 3 $$\sqrt { 7 }$$ + 2 > $$\sqrt { 10 }$$ – 1 Question 8. $$\sqrt { 17 }$$ + 3 _______ 3 + $$\sqrt { 11 }$$ Answer: $$\sqrt { 17 }$$ + 3 > 3 + $$\sqrt { 11 }$$ Explanation: $$\sqrt { 17 }$$ is between 4 and 5 $$\sqrt { 11 }$$ is between 3 and 4 $$\sqrt { 17 }$$ + 3 is between 7 and 8 3 + $$\sqrt { 11 }$$ is between 6 and 7 $$\sqrt { 17 }$$ + 3 > 3 + $$\sqrt { 11 }$$ Question 9. Order $$\sqrt { 3 }$$, 2 π, and 1.5 from least to greatest. Then graph them on the number line. $$\sqrt { 3 }$$ is between _________ and _____________ , so $$\sqrt { 3 }$$ ≈ ____________. π ≈ 3.14, so 2 π ≈ _______________. From least to greatest, the numbers are ______________, _____________________ ,_________________. Type below: ___________ Answer: 1.5, $$\sqrt { 3 }$$, 2 π Explanation: $$\sqrt { 3 }$$ is between 1.7 and 1.75 π = 3.14; 2 π = 6.28 1.5, $$\sqrt { 3 }$$, 2 π Question 10. Four people have found the perimeter of a forest using different methods. Their results are given in the table. Order their calculations from greatest to least. Type below: ___________ Answer: $$\sqrt { 17 }$$ – 2, 1+ π/2, 2.5, 12/5 Explanation: $$\sqrt { 17 }$$ – 2 $$\sqrt { 17 }$$ is between 4 and 5 Since, 17 is closer to 16, the estimated value is 4.1 1+ π/2 1 + (3.14/2) = 2.57 12/5 = 2.4 2.5 $$\sqrt { 17 }$$ – 2, 1+ π/2, 2.5, 12/5 ESSENTIAL QUESTION CHECK-IN Question 11. Explain how to order a set of real numbers. Type below: ___________ Answer: Evaluate the given numbers and write in decimal form. Plot on number line and arrange the numbers accordingly. ### Independent Practice – Ordering Real Numbers – Page No. 25 Order the numbers from least to greatest. Question 12. $$\sqrt { 7 }$$, 2, $$\frac { \sqrt { 8 } }{ 2 }$$ Type below: ____________ Answer: $$\frac { \sqrt { 8 } }{ 2 }$$, 2, $$\sqrt { 7 }$$ Explanation: $$\sqrt { 7 }$$, 2, $$\frac { \sqrt { 8 } }{ 2 }$$ $$\sqrt { 7 }$$ is between 2 and 3 Since 7 is closer to 9, (2.65)² = 7.02, hence the estimated value is 2.65 $$\frac { \sqrt { 8 } }{ 2 }$$ $$\sqrt { 8 }$$ is between 2 and 3 Since 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85 2.85/2 = 1.43 $$\frac { \sqrt { 8 } }{ 2 }$$, 2, $$\sqrt { 7 }$$ Question 13. $$\sqrt { 10 }$$, π, 3.5 Type below: ____________ Answer: π, $$\sqrt { 10 }$$, 3.5 Explanation: $$\sqrt { 10 }$$, π, 3.5 $$\sqrt { 10 }$$ is between 3 and 4 Since, 10 is closer to 9, (3.15)² = 9.92, hence the estimated value is 3.15 π = 3.14 3.5 π, $$\sqrt { 10 }$$, 3.5 Question 14. $$\sqrt { 220 }$$, −10, $$\sqrt { 100 }$$, 11.5 Type below: ____________ Answer: -10, √100, 11.5, √220 Explanation: $$\sqrt { 220 }$$, −10, $$\sqrt { 100 }$$, 11.5 196 < 220 < 225 √196 < √220 < √225 14 < √220 < 15 √220 = 14.5 √100 = 10 -10, √100, 11.5, √220 Question 15. $$\sqrt { 8 }$$, −3.75, 3, $$\frac{9}{4}$$ Type below: ____________ Answer: −3.75, $$\frac{9}{4}$$, $$\sqrt { 8 }$$ Explanation: $$\sqrt { 8 }$$, −3.75, 3, $$\frac{9}{4}$$ $$\sqrt { 8 }$$ is between 2 and 3 Since, 8 is closer to 9, (2.85)² = 8.12, hence the estimated value is 2.85 -3.75 = 3 9/4 = 2.25 −3.75, $$\frac{9}{4}$$, $$\sqrt { 8 }$$ Question 16. Your sister is considering two different shapes for her garden. One is a square with side lengths of 3.5 meters, and the other is a circle with a diameter of 4 meters. a. Find the area of the square. _______ m2 Answer: (3.5)² = 12.25 Explanation: Area of the square = x² Area = (3.5)² = 12.25 Question 16. b. Find the area of the circle. _______ m2 Answer: π(2)² = 12.56 Explanation: Area of the circle = πr² where r = d/2 = 4/2 = 2 Area = π(2)² = 12.56 Question 16. c. Compare your answers from parts a and b. Which garden would give your sister the most space to plant? ___________ Answer: 12.25 < 12.56 The circle will give more space Question 17. Winnie measured the length of her father’s ranch four times and got four different distances. Her measurements are shown in the table. a. To estimate the actual length, Winnie first approximated each distance to the nearest hundredth. Then she averaged the four numbers. Using a calculator, find Winnie’s estimate. ______ Answer: 7.4815 Explanation: $$\sqrt { 60 }$$ = 7.75 58/8 = 7.25 7.3333 7 3/5 = 7.60 Average = (7.75 + 7.25 + 7.33 + 7.60)/4 = 7.4815 Question 17. b. Winnie’s father estimated the distance across his ranch to be $$\sqrt { 56 }$$ km. How does this distance compare to Winnie’s estimate? ____________ Answer: They are nearly identical Explanation: $$\sqrt { 56 }$$ = 7.4833 They are nearly identical Give an example of each type of number. Question 18. a real number between $$\sqrt { 13 }$$ and $$\sqrt { 14 }$$ Type below: ____________ Answer: A real number between $$\sqrt { 13 }$$ and $$\sqrt { 14 }$$ Example: 3.7 Explanation: $$\sqrt { 13 }$$ = 3.61 $$\sqrt { 13 }$$ = 3.74 A real number between $$\sqrt { 13 }$$ and $$\sqrt { 14 }$$ Example: 3.7 Question 19. an irrational number between 5 and 7 Type below: ____________ Answer: An irrational number between 5 and 7 Example: $$\sqrt { 29 }$$ Explanation: 5² = 25 and 7² = 49 An irrational number between 5 and 7 Example: $$\sqrt { 29 }$$ ### Ordering Real Numbers – Page No. 26 Question 20. A teacher asks his students to write the numbers shown in order from least to greatest. Paul thinks the numbers are already in order. Sandra thinks the order should be reversed. Who is right? _____________ Answer: Neither are correct Explanation: $$\sqrt { 115 }$$, 115/11, 10.5624 $$\sqrt { 115 }$$ is between 10 and 11 Since, 115 is closer to 121, (10.7)² = 114.5, hence the estimated value is 10.7 115/11 = 10.4545 10.5624 Neither are correct Question 21. Math History There is a famous irrational number called Euler’s number, symbolized with an e. Like π, its decimal form never ends or repeats. The first few digits of e are 2.7182818284. a. Between which two square roots of integers could you find this number? Type below: _____________ Answer: The square of e lies between 7 and 8 2.718281828 (2.72)² = 7.3984 Hence, it lies between $$\sqrt { 7 }$$ = 2.65 and $$\sqrt { 8 }$$ = 2.82 Question 21. b. Between which two square roots of integers can you find π? Type below: _____________ Answer: 3.142 (3.14)² = 9.8596 Hence. it lies between $$\sqrt { 9 }$$ = 3 and $$\sqrt { 10 }$$ = 3.16 H.O.T. FOCUS ON HIGHER ORDER THINKING Question 22. Analyze Relationships There are several approximations used for π, including 3.14 and $$\frac{22}{7}$$. π is approximately 3.14159265358979 . . . a. Label π and the two approximations on the number line. Type below: _____________ Answer: Question 22. b. Which of the two approximations is a better estimate for π? Explain. Type below: _____________ Answer: As we can see from the number line, 22/7 is closer to π, so we can conclude that 22/7 is a better estimation for π. Question 22. c. Find a whole number x so that the ratio $$\frac{x}{113}$$ is a better estimate for π than the two given approximations. Type below: _____________ Answer: 355/113 is a better estimation for π, because 355/113 = 3.14159292035 = 3.14159265358979 = π Question 23. Communicate Mathematical Ideas What is the fewest number of distinct points that must be graphed on a number line, in order to represent natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers? Explain. _______ points Answer: 2 points Explanation: There need to be plotting of at least 2 points because a rational number can never be equal to an irrational number. So let’s say 5 points are the same among six but the 6th will be different as there both rational numbers and irrational numbers included. Question 24. Critique Reasoning Jill says that 12.$$\bar{6}$$ is less than 12.63. Explain her error. Type below: _____________ Answer: 12.$$\bar{6}$$ = 12.666 12.$$\bar{6}$$ > 12.63 ### 1.1 Rational and Irrational Numbers – Model Quiz – Page No. 27 Write each fraction as a decimal or each decimal as a fraction. Question 1. $$\frac{7}{20}$$ _______ Answer: 0.35 Explanation: $$\frac{7}{20}$$ = 0.35 Question 2. 1.$$\overline { 27}$$ ______ $$\frac{□}{□}$$ Answer: 1$$\frac{28}{99}$$ Explanation: 1.$$\overline { 27}$$ x = 1.$$\overline { 27}$$ 100x = 100(1.$$\overline { 27}$$) 100x = 127($$\overline { 27}$$) x = .$$\overline { 27}$$ 99x = 127 x = 127/99 x = 1 28/99 Question 3. 1 $$\frac{7}{8}$$ ______ Answer: 1.875 Explanation: 1 $$\frac{7}{8}$$ 1 + 7/8 8/8 + 7/8 15/8 = 1.875 Solve each equation for x. Question 4. x2 = 81 ± ______ Answer: ± 9 Explanation: x2 = 81 x = ± 81 x = ± 9 Question 5. x3 = 343 ______ Answer: x = 7 Explanation: x3 = 343 x = 7 Question 6. x2 = $$\frac{1}{100}$$ ± $$\frac{□}{□}$$ Answer: ± $$\frac{1}{10}$$ Explanation: x2 = $$\frac{1}{100}$$ x = ± $$\frac{1}{10}$$ Question 7. A square patio has an area of 200 square feet. How long is each side of the patio to the nearest 0.05? ______ feet Answer: 14.15 feet Explanation: The area of a square is found by multiplying the side of the square by itself. Therefore, to find the side of the square, we have to take the square root of the area. Let’s denote with A the area of the patio and with s each side of the square. We have: A = 200 A = s.s s = $$\sqrt { A }$$ = $$\sqrt { 200 }$$ Following the steps as in “Explore Activity” on page 9, we can make an estimation for the irrational number: 196 < 200 < 225 $$\sqrt { 196 }$$ < $$\sqrt { 200 }$$ < $$\sqrt { 225 }$$ 14 < $$\sqrt { 200 }$$ < 15 We see that 200 is much closer to 196 than to 225, therefore the square root of it should be between 14 and 14.5. To make a better estimation, we pick some numbers between 14 and 14.5 and calculate their squares: (14.1)² = 198.81 (14.2)² = 201.64 14.1 < $$\sqrt { 200 }$$ < 14.2 $$\sqrt { 200 }$$ = 14.15 We see that 200 is much closer to 14.1 than to 14.2, therefore the square root of it should be between 14.1 and 14.15. If we round to the nearest 0.05, we have: s = 14.15 1.2 Sets of Real Numbers Write all names that apply to each number. Question 8. $$\frac { 121 }{ \sqrt { 121 } }$$ Type below: ___________ Answer: Rational, whole, integer, real numbers Explanation: $$\frac { 121 }{ \sqrt { 121 } }$$ 121/11 = 11 Question 9. $$\frac{π}{2}$$ Type below: ___________ Answer: Irrational, real numbers Question 10. Tell whether the statement “All integers are rational numbers” is true or false. Explain your choice. ___________ Answer: True Explanation: “All integers are rational numbers” is true, because every integer can be expressed as a fraction with a denominator equal to 1. The set of integer A a subset of rational numbers. 1.3 Ordering Real Numbers Compare. Write <, >, or =. Question 11. $$\sqrt { 8 }$$ + 3 _______ 8 + $$\sqrt { 3 }$$ Answer: $$\sqrt { 8 }$$ + 3 < 8 + $$\sqrt { 3 }$$ Explanation: 4 < 8 < 9 $$\sqrt { 4 }$$ < $$\sqrt { 8 }$$ < $$\sqrt { 9 }$$ 2 < $$\sqrt { 8 }$$ < 3 1 < 3 < 4 $$\sqrt { 1 }$$ < $$\sqrt { 3 }$$ < $$\sqrt { 4 }$$ 1 < $$\sqrt { 3 }$$ < 2 $$\sqrt { 8 }$$ + 3 is between 5 and 6 8 + $$\sqrt { 3 }$$ is between 9 and 10 $$\sqrt { 8 }$$ + 3 < 8 + $$\sqrt { 3 }$$ Question 12. $$\sqrt { 5 }$$ + 11 _______ 5 + $$\sqrt { 11 }$$ Answer: $$\sqrt { 5 }$$ + 11 > 5 + $$\sqrt { 11 }$$ Explanation: $$\sqrt { 5 }$$ lies in between 2 and 3 $$\sqrt { 11 }$$ lies in between 3 and 4 $$\sqrt { 5 }$$ + 11 lies in between 13 and 14 5 + $$\sqrt { 11 }$$ lies in between 8 and 9 $$\sqrt { 5 }$$ + 11 > 5 + $$\sqrt { 11 }$$ Order the numbers from least to greatest. Question 13. $$\sqrt { 99 }$$, π2, 9.$$\bar { 8 }$$ Type below: _______________ Answer: π2, 9.$$\bar { 8 }$$, $$\sqrt { 99 }$$ Explanation: $$\sqrt { 99 }$$, π2, 9.$$\bar { 8 }$$ 99 lies between 9² and 10² 99 is closer to 100, hence $$\sqrt { 99 }$$ is closer to 10 (9.9)² = 98.01 (9.95)² = 99.0025 (10)² = 100 $$\sqrt { 99 }$$ = 9.95 π² = 9.86 9.88888 = 9.89 π2, 9.$$\bar { 8 }$$, $$\sqrt { 99 }$$ Question 14. $$\sqrt { \frac { 1 }{ 25 } }$$, $$\frac{1}{4}$$, 0.$$\bar { 2 }$$ Type below: ____________ Answer: $$\sqrt { \frac { 1 }{ 25 } }$$, 0.$$\bar { 2 }$$, $$\frac{1}{4}$$ Explanation: $$\sqrt { \frac { 1 }{ 25 } }$$, $$\frac{1}{4}$$, 0.$$\bar { 2 }$$ $$\sqrt { \frac { 1 }{ 25 } }$$ = 1/5 = 0.2 1/4 = 0.25 0.$$\bar { 2 }$$ = 0.222 = 0.22 $$\sqrt { \frac { 1 }{ 25 } }$$, 0.$$\bar { 2 }$$, $$\frac{1}{4}$$ Essential Question Question 15. How are real numbers used to describe real-world situations? Type below: _______________ Answer: In real-world situations, we use real numbers to count or make measurements. They can be seen as a convention for us to quantify things around, for example, the distance, the temperature, the height, etc. ### Selected Response – Mixed Review – Page No. 28 Question 1. The square root of a number is 9. What is the other square root? Options: a. -9 b. -3 c. 3 d. 81 Answer: a. -9 Explanation: We know that every positive number has two square roots, one positive and one negative. We are given the principal square root (9), so the other square root would be its negative (-9). To prove that, we square both numbers and we compare the results: 9 • 9 = 81 (-9). (-9)= 81 Question 2. A square acre of land is 4,840 square yards. Between which two integers is the length of one side? Options: a. between 24 and 25 yards b. between 69 and 70 yards c. between 242 and 243 yards d. between 695 and 696 yards Answer: b. between 69 and 70 yards Explanation: The area of a square is found by multiplying the side of the square by itself. Therefore, to Bud the side of the square, we have to take the square root of the area. Let’s denote with A the area of the land and with each side of the square. We have: A = 4840 A = s . s A = s² s = √A = √4840 Following the steps as in °Explore Activity on page 9, we can make an estimation for the irrational number: 4761 < 4840 < 4900 $$\sqrt { 4761 }$$ < $$\sqrt { 4840 }$$ < $$\sqrt { 4900 }$$ 69 < $$\sqrt { 4840 }$$ < 70 Each side of the land is between 69 and 70 yards. Question 3. Which of the following is an integer but not a whole number? Options: a. -9.6 b. -4 c. 0 d. 3.7 Answer: b. -4 Explanation: Whole numbers are not negative -4 is an integer but not a whole number Question 4. Which statement is false? Options: a. No integers are irrational numbers. b. All whole numbers are integers. c. No real numbers are irrational numbers. d. All integers greater than 0 are whole numbers. Answer: c. No real numbers are irrational numbers. Explanation: Rational and irrational numbers are real numbers. Question 5. Which set of numbers best describes the displayed weights on a digital scale that shows each weight to the nearest half pound? Options: a. whole numbers b. rational numbers c. real numbers d. integers Answer: b. rational numbers Explanation: The scale weighs nearest to 1/2 pound. Question 6. Which of the following is not true? Options: a. π2 < 2π + 4 b. 3π > 9 c. $$\sqrt { 27 }$$ + 3 > 172 d. 5 – $$\sqrt { 24 }$$ < 1 Answer: c. $$\sqrt { 27 }$$ + 3 > 172 Explanation: a. π2 < 2π + 4 (3.14)² < 2(3.14) + 4 9.86 < 10.28 True b. 3π > 9 9.42 > 9 True c. $$\sqrt { 27 }$$ + 3 > 172 5.2 + 3 > 8.5 8.2 > 8.5 False d. 5 – $$\sqrt { 24 }$$ < 1 5 – 4.90 < 1 0.1 < 1 True Question 7. Which number is between $$\sqrt { 21 }$$ and $$\frac{3π}{2}$$ ? Options: a. $$\frac{14}{3}$$ b. 2 $$\sqrt { 6 }$$ c. 5 d. π + 1 Answer: Explanation: a. $$\sqrt { 21 }$$ and $$\frac{3π}{2}$$ $$\sqrt { 21 }$$ = 4.58 $$\frac{3π}{2}$$ = 4.71 14/3 = 4.67 b. 2$$\sqrt { 6 }$$ = 4.90 c. 5 d. π + 1 = 3.14 + 1 = 4.14 Question 8. What number is shown on the graph? Options: a. π+3 b. $$\sqrt { 4 }$$ + 2.5 c. $$\sqrt { 20 }$$ + 2 d. 6.$$\overline { 14 }$$ Answer: c. $$\sqrt { 20 }$$ + 2 Explanation: 6.48 a. π+3 = 3.14 + 3 = 6.14 b. $$\sqrt { 4 }$$ + 2.5 = 2 + 2.5 = 4.5 c. $$\sqrt { 20 }$$ + 2 = 4.47 + 2 = 6.47 d. 6.$$\overline { 14 }$$ = 6.1414 Question 9. Which is in order from least to greatest? Options: a. 3.3, $$\frac{10}{3}$$, π, $$\frac{11}{4}$$ b. $$\frac{10}{3}$$, 3.3, $$\frac{11}{4}$$, π c. π, $$\frac{10}{3}$$, $$\frac{11}{4}$$, 3.3 d. $$\frac{11}{4}$$, π, 3.3, $$\frac{10}{3}$$ Answer: d. $$\frac{11}{4}$$, π, 3.3, $$\frac{10}{3}$$ Explanation: 10/3 = 3.3333333 11/4 = 2.75 Mini-Task Question 10. The volume of a cube is given by V = x3, where x is the length of an edge of the cube. The area of a square is given by A = x2, where x is the length of a side of the square. A given cube has a volume of 1728 cubic inches. a. Find the length of an edge. ______ inches Answer: 12 inches Explanation: V = x3 A = x2 1728 = x3 x = 12 The length of an edge = 12 in Question 10. b. Find the area of one side of the cube. ______ in2 Answer: 144 in2 Explanation: A = (12)² = 144 Area of the side of the cube = 144 in2 Question 10. c. Find the surface area of the cube. ______ in2 Answer: 864 in2 Explanation: SA = 6 (12)² = 864 The surface area of the cube = 864 in2 Question 10. d. What is the surface area in square feet? ______ ft2 Answer: 6 ft2 Explanation: SA = 864/144 = 6 The surface area of the cube = 6 ft2 ### Conclusion: We hope the details prevailed in this Grade 8 Go Math Answer Key Chapter 1 Real Numbers is helpful for you guys. Make use of the above links and try to solve all the problems. This HMH Go Math Grade 8 Answer Key also helps to complete the homework within the time without any mistakes. Stick to our ccssmathanswers.com site to get the pdf links of all the chapters. Feel free to post your comments in the below box. We will try to clarify your doubts as early as possible. All the Best Guys!!! ## Go Math Grade 6 Answer Key Chapter 5 Model Percents Go Math Grade 6 Answer Key Chapter 5 Model Percents Pdf is available here. So, the pupils who are in search of the solutions of Chapter 5 Model Percents can get them on this page along with images. Relate the questions in real-time and make your practice best. Students who are preparing for exams must have the best material. Our team will provide step by step explanations for all the questions on Go Math Grade 6 Answer Key. ## Go Math Grade 6 Chapter 5 Model Percents Answer Key Make yourself comfortable by using HMH Go math Grade 6 Answer Key Chapter 5 Model Percents. So, make use of the resources of Go Math Answer Key to score good marks in the exams. Test your skills by solving the problems given at the end of the chapter. Just click on the links and start solving the problems. Lesson 1: Investigate • Model Percents Lesson 2: Write Percents as Fractions and Decimals Lesson 3: Write Fractions and Decimals as Percents Mid-Chapter Checkpoint Lesson 4: Percent of a Quantity Lesson 5: Problem Solving • Percents Lesson 6: Find the Whole from a Percent Chapter 5 Review/Test ### Share and Show – Page No. 271 Write a ratio and a percent to represent the shaded part. Question 1. Type below: _____________ Answer: 53% and $$\frac{53}{100}$$ Explanation: 53 squares are shaded out of 100. So, 53% and 35/100 are the answers. Question 2. Type below: _____________ Answer: 1% and $$\frac{100}{100}$$ Explanation: 100 out of 100 squares are shaded So, So, 1% and 100/100 are the answers. Question 3. Type below: _____________ Answer: 40% and $$\frac{40}{100}$$ Explanation: 40 squares are shaded out of 100. So, 40% and 40/100 are the answers. Model the percent and write it as a ratio. Question 4. 30% $$\frac{□}{□}$$ Answer: Explanation: 30% is 30 out of 100 30 out of 100 squares is 30/100 30% = $$\frac{30}{100}$$ Question 5. 5% $$\frac{□}{□}$$ Answer: Explanation: 5% is 5 out of 100 5 out of 100 squares is 5/100 5% = $$\frac{5}{100}$$ Question 6. 75% $$\frac{□}{□}$$ Answer: Explanation: 75% is 75 out of 100 75 out of 100 squares is 75/100 75% = $$\frac{75}{100}$$ Problem Solving + Applications Question 7. Use a Concrete Model Explain how to model 32% on a 10-by-10 grid. How does the model represent the ratio of 32 to 100? Type below: _____________ Answer: Question 8. A floor has 100 tiles. There are 24 black tiles and 35 brown tiles. The rest of the tiles are white. What percent of the tiles are white? _______ % Answer: 41% Explanation: A floor has 100 tiles. There are 24 black tiles and 35 brown tiles. 24 + 35 = 59 100 – 59 = 41 tiles are white 41 tiles out of 100 are white tiles ### Pose a Problem – Page No. 272 Question 9. Javier designed a mosaic wall mural using 100 tiles in 3 different colors: yellow, blue, and red. If 64 of the tiles are yellow, what percent of the tiles are either red or blue? To find the number of tiles that are either red or blue, count the red and blue squares. Or subtract the number of yellow squares, 64, from the total number of squares, 100. 36 out of 100 tiles are red or blue. The ratio of red or blue tiles to all tiles is $$\frac{36}{100}$$. So, the percent of the tiles that are either red or blue is 36%. Write another problem involving a percent that can be solved by using the mosaic wall mural. Type below: _____________ Answer: Sam designed a mosaic wall mural using 100 squares using two colors. She represented the squares with red and blue colors. She has 54 red tiles. What percent of other tiles she can use with blue color? 100 – 54 = 46 blue tiles. Question 10. Select the 10-by-10 grids that model 45%. Mark all that apply. Options: a. b. c. d. e. Answer: a. c. e. ### Model Percents – Page No. 273 Write a ratio and a percent to represent the shaded part. Question 1. Type below: _____________ Answer: 31% and $$\frac{31}{100}$$ Explanation: 31 squares are shaded out of 100. So, 31% and 31/100 are the answers. Question 2. Type below: _____________ Answer: 70% and $$\frac{70}{100}$$ Explanation: 70 squares are shaded out of 100. So, 70% and 70/100 are the answers. Question 3. Type below: _____________ Answer: 48% and $$\frac{48}{100}$$ Explanation: 48 squares are shaded out of 100. So, 48% and 48/100 are the answers. Model the percent and write it as a ratio. Question 4. 97% $$\frac{□}{□}$$ Answer: Explanation: 97% is 97 out of 100 97 out of 100 squares is 97/100 97% = $$\frac{97}{100}$$ Question 5. 24% $$\frac{□}{□}$$ Answer: Explanation: 24% is 24 out of 100 24 out of 100 squares is 24/100 24% = $$\frac{24}{100}$$ Question 6. 50% $$\frac{□}{□}$$ Answer: Explanation: 50% is 50 out of 100 50 out of 100 squares is 50/100 50% = $$\frac{50}{100}$$ Problem Solving The table shows the pen colors sold at the school supply store one week. Write the ratio comparing the number of the given color sold to the total number of pens sold. Then shade the grid. Question 7. Black $$\frac{□}{□}$$ Answer: $$\frac{49}{100}$$ Explanation: The total number of pens sold = 36 + 49 + 15 = 100 Black : total number of pens sold = 49:100 49 out of 100 squares need to shade the grid Question 8. Not Blue $$\frac{□}{□}$$ Answer: $$\frac{64}{100}$$ Explanation: Not Blue = Black + Red = 49 + 15 = 64 Question 9. Is every percent a ratio? Is every ratio a percent? Explain. Type below: _____________ Answer: Every percent is a ratio but not all ratios are percent. All ratios can be expressed as percents, decimals, or fractions or in ratio form. ### Lesson Check – Page No. 274 Question 1. What percent of the large square is shaded? _______ % Answer: 63% Explanation: 63 squares are shaded out of 100. So, 63% and 63/100 are the answers. Question 2. Write a ratio to represent the shaded part. $$\frac{□}{□}$$ Answer: $$\frac{10}{100}$$ Explanation: 63 squares are shaded out of 100. 63/100 is the answer. Spiral Review Question 3. Write a number that is less than −2 $$\frac{4}{5}$$ and greater than −3 $$\frac{1}{5}$$. Type below: _____________ Answer: -2.9, -3.0, -3.1 Explanation: −2 $$\frac{4}{5}$$ = -14/5 = -2.8 −3 $$\frac{1}{5}$$ = -16/5 = -3.2 -2.9, -3.0, -3.1 are the numbers less than −2 $$\frac{4}{5}$$ and greater than −3 $$\frac{1}{5}$$ Question 4. On a coordinate grid, what is the distance between (2, 4) and (2, –3)? _______ units Answer: 7 units Explanation: |-3| = 3 4+ 0 = 4; 0 + 3 = 3 4 + 3 = 7 Question 5. Each week, Diana spends 4 hours playing soccer and 6 hours babysitting. Write a ratio to compare the time Diana spends playing soccer to the time she spends babysitting. $$\frac{□}{□}$$ Answer: $$\frac{2}{3}$$ Explanation: Each week, Diana spends 4 hours playing soccer and 6 hours babysitting. The ratio to compare the time Diana spends playing soccer to the time she spends babysitting is 4:6 or 4/6 = 2/3 Question 6. Antwone earns money at a steady rate mowing lawns. The points (1, 25) and (5, 125) appear on a graph of the amount earned versus number of lawns mowed. What are the coordinates of the point on the graph with an x-value of 3? Type below: _____________ Answer: (3, 75) Explanation: y2-y1/x2-x1. Y2 is 125, Y1 is 25, X2 is 5, and X1 is 1. You then plug the numbers in, 125-25=100. 5-1=4. Then you divide 100/4, in which you get 25. So you time 25 by 3, getting 75. ### Share and Show – Page No. 277 Write the percent as a fraction. Question 1. 80% $$\frac{□}{□}$$ Answer: $$\frac{80}{100}$$ Explanation: 80% is 80 out of 100 80 out of 100 squares is 80/100 Question 2. 150% ______ $$\frac{□}{□}$$ Answer: 1$$\frac{1}{2}$$ Explanation: 150% is 150 out of 100 150 out of 100 squares is 150/100 = 3/2 = 1 1/2 Question 3. 0.2% $$\frac{□}{□}$$ Answer: $$\frac{2}{1,000}$$ Explanation: 0.2% is 0.2 out of 100 0.2 out of 100 squares is 0.2/100 = 2/1,000 Write the percent as a decimal. Question 4. 58% ______ Answer: 0.58 Explanation: 58% is 58 out of 100 58 out of 100 squares is 58/100 58/100 = 0.58 Question 5. 9% ______ Answer: 0.09 Explanation: 9% is 9 out of 100 9 out of 100 squares is 9/100 9/100 = 0.09 On Your Own Write the percent as a fraction or mixed number. Question 6. 17% $$\frac{□}{□}$$ Answer: $$\frac{17}{100}$$ Explanation: 17% is 17 out of 100 17 out of 100 squares is 17/100 Question 7. 20% $$\frac{□}{□}$$ Answer: $$\frac{1}{5}$$ Explanation: 20% is 20 out of 100 20 out of 100 squares is 20/100 = 2/10 = 1/5 Question 8. 125% ______ $$\frac{□}{□}$$ Answer: 1$$\frac{1{4}$$ Explanation: 125% is 125 out of 100 125 out of 100 squares is 125/100 = 1 1/4 Question 9. 355% ______ $$\frac{□}{□}$$ Answer: 3$$\frac{11}{20}$$ Explanation: 355% is 355 out of 100 355 out of 100 squares is 355/100 = 3 11/20 Question 10. 0.1% $$\frac{□}{□}$$ Answer: $$\frac{1}{1,000}$$ Explanation: 0.1% is 0.1 out of 100 0.1 out of 100 squares is 0.1/100 = 1/1,000 Question 11. 2.5% $$\frac{□}{□}$$ Answer: $$\frac{1}{40}$$ Explanation: 2.5% is 2.5 out of 100 2.5 out of 100 squares is 2.5/100 = 25/1,000 = 1/40 Write the percent as a decimal. Question 12. 89% ______ Answer: 0.89 Explanation: 89% is 89 out of 100 89 out of 100 squares is 89/100 89/100 = 0.89 Question 13. 30% ______ Answer: 0.3 Explanation: 30% is 30 out of 100 30 out of 100 squares is 30/100 30/100 = 0.3 Question 14. 2% ______ Answer: 0.02 Explanation: 2% is 2 out of 100 2 out of 100 squares is 2/100 2/100 = 0.02 Question 15. 122% ______ Answer: 1.22 Explanation: 122% is 122 out of 100 122 out of 100 squares is 122/100 122/100 = 1.22 Question 16. 3.5% ______ Answer: 0.035 Explanation: 3.5% is 3.5 out of 100 3.5 out of 100 squares is 3.5/100 3.5/100 = 0.035 Question 17. 6.33% ______ Answer: 0.0633 Explanation: 6.33% is 6.33 out of 100 6.33 out of 100 squares is 6.33/100 6.33/100 = 0.0633 Question 18. Use Reasoning Write <, >, or =. 21.6% ______ $$\frac{1}{5}$$ Answer: 21.6% > $$\frac{1}{5}$$ Explanation: 1/5 × 100/100 = 100/500 = 0.2/100 = 0.2% 21.6% > 0.2% Question 19. Georgianne completed 60% of her homework assignment. Write the portion of her homework that she still needs to complete as a fraction. $$\frac{□}{□}$$ Answer: $$\frac{2}{5}$$ Explanation: Georgianne completed 60% of her homework assignment. 60/100 She needs to complete 40% of her homework = 40/100 = 2/5 ### Problem Solving + Applications – Page No. 278 Use the table for 20 and 21. Question 20. What fraction of computer and video game players are 50 years old or more? $$\frac{□}{□}$$ Answer: $$\frac{13}{50}$$ Explanation: computer and video game players, 50 or more are of 26% = 26/100 = 13/50 Question 21. What fraction of computer and video game players are 18 years old or more? $$\frac{□}{□}$$ Answer: $$\frac{49}{100}$$ Explanation: 18 years old or more are of 49% = 49/100 Question 22. Box A and Box B each contain black tiles and white tiles. They have the same total number of tiles. In Box A, 45% of the tiles are black. In Box B, $$\frac{11}{20}$$ of the tiles are white. Compare the number of black tiles in the boxes. Explain your reasoning. Type below: _____________ Answer: In Box A, 45% of the tiles are black. In Box B, $$\frac{11}{20}$$ of the tiles are white. 11/20 = 0.55 = 55/100 = 55% 100 – 55 = 45% Both Box A and Box B have an equal number of black tiles Question 23. Mr. Truong is organizing a summer program for 6th grade students. He surveyed students to find the percent of students interested in each activity. Complete the table by writing each percent as a fraction or decimal. Type below: _____________ Answer: Sports = 48% = 48/100 = 0.48 Cooking = 23% = 23/100 Music = 20% = 20/100 Art = 9% = 9/100 = 0.09 ### Write Percents as Fractions and Decimals – Page No. 279 Write the percent as a fraction or mixed number. Question 1. 44% $$\frac{□}{□}$$ Answer: $$\frac{11}{25}$$ Explanation: 44% is 44 out of 100 44 out of 100 squares is 44/100 = 11/25 Question 2. 32% $$\frac{□}{□}$$ Answer: $$\frac{8}{25}$$ Explanation: 32% is 32 out of 100 32 out of 100 squares is 32/100 = 8/25 Question 3. 116% ______ $$\frac{□}{□}$$ Answer: 1 $$\frac{4}{25}$$ Explanation: 116% is 116 out of 100 116 out of 100 squares is 116/100 = 1 4/25 Question 4. 250% ______ $$\frac{□}{□}$$ Answer: 2$$\frac{1}{2}$$ Explanation: 250% is 250 out of 100 250 out of 100 squares is 250/100 = 2 1/2 Question 5. 0.3% $$\frac{□}{□}$$ Answer: $$\frac{3}{1,000}$$ Explanation: 0.3% is 0.3 out of 100 0.3 out of 100 squares is 0.3/100 3/1,000 Question 6. 0.4% $$\frac{□}{□}$$ Answer: $$\frac{1}{250}$$ Explanation: 0.4% is 0.4 out of 100 0.4 out of 100 squares is 0.4/100 = 4/1,000 = 1/250 Question 7. 1.5% $$\frac{□}{□}$$ Answer: $$\frac{3}{200}$$ Explanation: 1.5% is 1.5 out of 100 1.5 out of 100 squares is 1.5/100 = 15/1,000 = 3/200 Question 8. 12.5% $$\frac{□}{□}$$ Answer: $$\frac{1}{8}$$ Explanation: 12.5% is 12.5 out of 100 12.5 out of 100 squares is 12.5/100 = 125/1,000 = 25/200 = 5/40 = 1/8 Write the percent as a decimal. Question 9. 63% ______ Answer: 0.63 Explanation: 63% is 63 out of 100 63 out of 100 squares is 63/100 63/100 = 0.63 Question 10. 110% ______ Answer: 1.1 Explanation: 110% is 110 out of 100 110 out of 100 squares is 110/100 = 1.1 Question 11. 42.15% ______ Answer: 0.4215 Explanation: 42.15% is 42.15 out of 100 42.15 out of 100 squares is 42.15/100 = 0.4215 Question 12. 0.1% ______ Answer: 0.001 Explanation: 0.1% is 0.1 out of 100 0.1 out of 100 squares is 0.1/100 = 0.001 Problem Solving Question 13. An online bookstore sells 0.8% of its books to foreign customers. What fraction of the books are sold to foreign customers? $$\frac{□}{□}$$ Answer: $$\frac{1}{125}$$ Explanation: An online bookstore sells 0.8% of its books to foreign customers. 0.8% = 0.8/100 = 8/1,000 = 1/125 Question 14. In Mr. Klein’s class, 40% of the students are boys. What decimal represents the portion of the students that are girls? ______ Answer: 0.4 Explanation: In Mr. Klein’s class, 40% of the students are boys. 40/100 = 0.4 Question 15. Explain how percents, fractions, and decimals are related. Use a 10-by-10 grid to make a model that supports your explanation. Type below: _____________ Answer: 53 squares are shaded out of 100. 53% or $$\frac{53}{100}$$ or 0.53 ### Lesson Check – Page No. 280 Question 1. The enrollment at Sonya’s school this year is 109% of last year’s enrollment. What decimal represents this year’s enrollment compared to last year’s? ______ Answer: 1.09 represents this year’s enrollment compared to last year’s Explanation: The enrollment at Sonya’s school this year is 109% of last year’s enrollment. 109% = 109/100 = 1.09 Question 2. An artist’s paint set contains 30% watercolors and 25% acrylics. What fraction represents the portion of the paints that are watercolors or acrylics? Write the fraction in simplest form. $$\frac{□}{□}$$ Answer: $$\frac{11}{20}$$ Explanation: An artist’s paint set contains 30% watercolors and 25% acrylics. 30 + 25 = 55% = 55/100 = 11/20 Spiral Review Question 3. Write the numbers in order from least to greatest. -5.25 1.002 -5.09 Type below: _____________ Answer: -5.25, -5.09, 1.002 Question 4. On a coordinate plane, the vertices of a rectangle are (2, 4), (2, −1), (−5, −1), and ( −5, 4). What is the perimeter of the rectangle? ______ units Answer: 24 units Explanation: (2, 4) to (2, −1) is 4 + 1 = 5 (2, −1) to (−5, −1) is 2 + 5 = 7 5 + 7 + 5 + 7 = 24 Question 5. The table below shows the widths and lengths, in feet, for different playgrounds. Which playgrounds have equivalent ratios of width to length? Type below: _____________ Answer: 12/20 and 16.5/27.5 are equal Explanation: 12/20 = 0.6 15/22.5 = 0.666 20/25 = 0.8 16.5/27.5 = 0.6 Question 6. What percent represents the shaded part? _______ % Answer: 85% Explanation: 85 squares are shaded out of 100. 85% ### Share and Show – Page No. 283 Write the fraction or decimal as a percent. Question 1. $$\frac{3}{25}$$ _______ % Answer: 12% Explanation: 3/25 ÷ 25/25 = 0.12/1 = 12/100 = 12% Question 2. $$\frac{3}{10}$$ _______ % Answer: 30% Explanation: 3/10 ÷ 10/10 = 0.3 = 0.3 × 100/100 = 30/100 = 30% Question 3. 0.717 _______ % Answer: 71.7% Explanation: 0.717 = 717/100 = 71.7% Question 4. 0.02 _______ % Answer: 2% Explanation: 0.02 = 2/100 = 2% On Your Own Write the number in two other forms ( fraction, decimal, or percent). Write the fraction in simplest form. Question 5. 0.01 Type below: _____________ Answer: 1% and $$\frac{1}{100}$$ Explanation: 0.01 as a fraction 1/100 0.01 as percent 1% Question 6. $$\frac{13}{40}$$ Type below: _____________ Answer: 0.325 and 32.5% Explanation: $$\frac{13}{40}$$ as decimal 0.325 $$\frac{13}{40}$$ as percent 32.5/100 = 32.5% Question 7. $$\frac{6}{5}$$ Type below: _____________ Answer: 1.2 and 120% Explanation: $$\frac{6}{5}$$ as decimal 1.2 $$\frac{6}{5}$$ as percent 120/100 = 120% Question 8. 0.08 Type below: _____________ Answer: 8% and $$\frac{8}{100}$$ Explanation: 0.08 as a fraction 8/100 0.08 as percent 8% The table shows the portion of Kim’s class that participates in each sport. Use the table for 9–10. Question 9. Do more students take part in soccer or in swimming? Explain your reasoning. Type below: _____________ Answer: Soccer = 1/5 = 0.2 Swimming = 0.09 0.2 > 0.09 more students take part in Soccer Question 10. Explain What percent of Kim’s class participates in one of the sports listed? Explain how you found your answer _______ % Answer: 23% Explanation: Kim’s class participates in Baseball that is mentioned with 23% Question 11. For their reading project, students chose to either complete a character study, or write a book review. $$\frac{1}{5}$$ of the students completed a character study, and 0.8 of the students wrote a book review. Joia said that more students wrote a book review than completed a character study. Do you agree with Joia? Use numbers and words to support your answer Type below: _____________ Answer: 1/5 = 0.2 0.2 < 0.8 More students completed writing a book review. I agree with Joia ### Sand Sculptures – Page No. 284 Every year, dozens of teams compete in the U.S. Open Sandcastle Competition. Recent winners have included complex sculptures in the shape of flowers, elephants, and racing cars. Teams that participate in the contest build their sculptures using a mixture of sand and water. Finding the correct ratios of these ingredients is essential for creating a stable sculpture. The table shows the recipes that three teams used. Which team used the greatest percent of sand in their recipe? Convert to percents. Then order from least to greatest. From least to greatest, the percents are 75%, 84%, 95%. So, Team B used the greatest percent of sand. Solve. Question 12. Which team used the greatest percent of water in their recipe? Type below: _____________ Answer: Team A used the greatest percent of water in their recipe Explanation: Team A, 10/10+30 = 10/40 = 0.25 = 25% Team B, 1/20 × 5/5 = 5/100 = 5% Team C, 0.16 = 16% Question 13. Some people say that the ideal recipe for sand sculptures contains 88.9% sand. Which team’s recipe is closest to the ideal recipe? Type below: _____________ Answer: Team C Question 14. Team D used a recipe that consists of 20 cups of sand, 2 cups of flour, and 3 cups of water. How does the percent of sand in Team D’s recipe compare to that of the other teams? Type below: _____________ Answer: Total number of cups together = 20 + 2+ 3 =25 cups 20/25 × 100 = 80/100 = 80% ### Write Fractions and Decimals as Percents – Page No. 285 Write the fraction or decimal as a percent. Question 1. $$\frac{7}{20}$$ _______ % Answer: 35% Explanation: 7/20 = 0.35 = 35% Question 2. $$\frac{3}{50}$$ _______ % Answer: 6% Explanation: 3/50 = 0.06 = 6% Question 3. $$\frac{1}{25}$$ _______ % Answer: 4% Explanation: 1/25 = 0.04 = 4% Question 4. $$\frac{5}{5}$$ _______ % Answer: 0.01% Explanation: 5/5 = 1 = 0.01% Question 5. 0.622 _______ % Answer: 6.22% Explanation: 0.622 = 6.22/100 = 6.22% Question 6. 0.303 _______ % Answer: 3.03% Explanation: 0.303 = 3.03/100 = 3.03% Question 7. 0.06 _______ % Answer: 6% Explanation: 0.06 = 6/100 = 6% Question 8. 2.45 _______ % Answer: 245% Explanation: 2.45 × 100/100 = 245/100 = 245% Write the number in two other forms (fraction, decimal, or percent). Write the fraction in simplest form Question 9. $$\frac{19}{20}$$ Type below: _____________ Answer: 0.95 and 95% Explanation: $$\frac{19}{20}$$ as a decimal 0.95 $$\frac{19}{20}$$ as a percentage 95% Question 10. $$\frac{9}{16}$$ Type below: _____________ Answer: 0.5625 and 56.25% Explanation: $$\frac{9}{16}$$ as a decimal 0.5625 $$\frac{9}{16}$$ as a percentage 56.25% Question 11. 0.4 Type below: _____________ Answer: $$\frac{2}{5}$$ and 40% Explanation: 0.4 as a fraction 2/5 0.4 as a percentage 40/100 = 40% Question 12. 0.22 Type below: _____________ Answer: $$\frac{11}{50}$$ and 22% Explanation: 0.22 as a fraction 11/50 0.22 as a percentage 22/100 = 22% Problem Solving Question 13. According to the U.S. Census Bureau, $$\frac{3}{25}$$ of all adults in the United States visited a zoo in 2007. What percent of all adults in the United States visited a zoo in 2007? _______ % Answer: 12% Explanation: According to the U.S. Census Bureau, $$\frac{3}{25}$$ of all adults in the United States visited a zoo in 2007. $$\frac{3}{25}$$ = 0.12 = 12% Question 14. A bag contains red and blue marbles. Given that $$\frac{17}{20}$$ of the marbles are red, what percent of the marbles are blue? _______ % Answer: 15% Explanation: The total number of marbles = 20 If 17 marbles are red, the remaining 3 marbles out of 20 are blue marbles 3/20 = 0.15 = 15% Question 15. Explain two ways to write $$\frac{4}{5}$$ as a percent. Type below: _____________ Answer: Decimal =0.8. Percentage =80% Explanation: 4/5 = 0.8 = 80/100 = 80% ### Lesson Check – Page No. 286 Question 1. The portion of shoppers at a supermarket who pay by credit card is 0.36. What percent of shoppers at the supermarket do NOT pay by credit card? _______ % Answer: 36% Explanation: The portion of shoppers at a supermarket who pay by credit card is 0.36. 0.36 = 0.36 × 100/100 = 36/100 = 36% Question 2. About $$\frac{23}{40}$$ of a lawn is planted with Kentucky bluegrass. What percent of the lawn is planted with Kentucky bluegrass? _______ % Answer: 57.5% Explanation: About $$\frac{23}{40}$$ of a lawn is planted with Kentucky bluegrass. 23/40 = 0.575 = 0.575 × 100/100 = 57.5/100 = 57.5% Spiral Review Question 3. A basket contains 6 peaches and 8 plums. What is the ratio of peaches to total pieces of fruit? Type below: _____________ Answer: 6:14 Explanation: total pieces of fruit 6 + 8 = 14 the ratio of peaches to total pieces of fruit is 6:14 Question 4. It takes 8 minutes for 3 cars to move through a car wash. At the same rate, how many cars can move through the car wash in 24 minutes? _______ cars Answer: 9 cars Explanation: It takes 8 minutes for 3 cars to move through a car wash. 3/8 × 24 = 9 cars Question 5. A 14-ounce box of cereal sells for$2.10. What is the unit rate?
$_______ per ounce Answer:$0.15 per ounce

Explanation:
$2.10/14 × 14/14 =$0.15 per ounce

Question 6.
A model railroad kit contains curved tracks and straight tracks. Given that 35% of the tracks are curved, what fraction of the tracks are straight? Write the fraction in simplest form.
$$\frac{□}{□}$$

$$\frac{7}{20}$$

Explanation:
A model railroad kit contains curved tracks and straight tracks. Given that 35% of the tracks are curved,
35% = 35/100 = 7/20

### Vocabulary – Page No. 287

Choose the best term from the box to complete the sentence.

Question 1.
A _____ is a ratio that compares a quantity to 100.
Type below:
_____________

percent

Concepts and Skills

Write a ratio and a percent to represent the shaded part.

Question 2.

Type below:
_____________

17% and $$\frac{17}{100}$$

Explanation:
17 squares are shaded out of 100.
So, 17% and 17/100 are the answers.

Question 3.

Type below:
_____________

60% and $$\frac{60}{100}$$

Explanation:
60 squares are shaded out of 100.
So, 60% and 60/100 are the answers.

Question 4.

Type below:
_____________

7% and $$\frac{7}{100}$$

Explanation:
7 squares are shaded out of 100.
So, 7% and 7/100 are the answers.

Question 5.

Type below:
_____________

11% and $$\frac{11}{100}$$

Explanation:
11 squares are shaded out of 100.
So, 11% and 11/100 are the answers.

Question 6.

Type below:
_____________

82% and $$\frac{82}{100}$$

Explanation:
82 squares are shaded out of 100.
So, 82% and 82/100 are the answers.

Question 7.

Type below:
_____________

36% and $$\frac{36}{100}$$

Explanation:
36 squares are shaded out of 100.
So, 36% and 36/100 are the answers.

Write the number in two other forms (fraction, decimal, or percent).

Write the fraction in simplest form.

Question 8.
0.04
Type below:
_____________

$$\frac{1}{25}$$ and 4%

Explanation:
0.04 as a fraction 4/100 = 1/25
0.04 as a decimal 0.04 × 100/100 = 4/100 = 4%

Question 9.
$$\frac{3}{10}$$
Type below:
_____________

0.3 and 30%

Explanation:
$$\frac{3}{10}$$ as a decimal 0.3
$$\frac{3}{10}$$ as a percentage 0.3 × 100/100 = 30/100 = 30%

Question 10.
1%
Type below:
_____________

$$\frac{1}{100}$$ and 0.01

Explanation:
1% as a fraction 1/100
1% as a decimal 1/100 = 0.01

Question 11.
1 $$\frac{1}{5}$$
Type below:
_____________

1.2 and 120%

Explanation:
1 $$\frac{1}{5}$$ as a decimal = 6/5 = 1.2
1 $$\frac{1}{5}$$ as a percentage 1.2 × 100/100 = 120/100 = 120%

Question 12.
0.9
Type below:
_____________

$$\frac{90}{100}$$ and 90%

Explanation:
0.9 as a fraction 0.9 × 100/100 = 90/100 = 90%

Question 13.
0.5%
Type below:
_____________

$$\frac{5}{1,000}$$ and 0.005

Explanation:
0.5% as a fraction = 0.5/100 = 5/1,000
0.5% as a decimal = 0.5/100 = 0.005

Question 14.
$$\frac{7}{8}$$
Type below:
_____________

0.875 and 87.5%

Explanation:
$$\frac{7}{8}$$ as a decimal 0.875
$$\frac{7}{8}$$ as a percentage 87.5/100 = 87.5%

Question 15.
355%
Type below:
_____________

$$\frac{71}{20}$$ and 35.5

Explanation:
355% as a decimal 355/100 = 71/20 = 35.5

### Page No. 288

Question 16.
About $$\frac{9}{10}$$ of the avocados grown in the United States are grown in California. About what percent of the avocados grown in the United States are grown in California?
_______ %

90%

Explanation:
About $$\frac{9}{10}$$ of the avocados grown in the United States are grown in California.
9/10 × 10/10 = 90/100 = 90%

Question 17.
Morton made 36 out of 48 free throws last season. What percent of his free throws did Morton make?
_______ %

75%

Explanation:
Morton made 36 out of 48 free throws last season.
36/48 = 0.75 = 75/100 = 75%

Question 18.
Sarah answered 85% of the trivia questions correctly. What fraction describes this percent?
$$\frac{□}{□}$$

$$\frac{17}{20}$$

Explanation:
Sarah answered 85% of the trivia questions correctly.
85% = 85/100 = 17/20

Question 19.
About $$\frac{4}{5}$$ of all the orange juice in the world is produced in Brazil. About what percent of all the orange juice in the world is produced in Brazil?
_______ %

80%

Explanation:
About $$\frac{4}{5}$$ of all the orange juice in the world is produced in Brazil.
4/5 = 0.8 × 100/100 = 80/100 = 80%

Question 20.
If you eat 4 medium strawberries, you get 48% of your daily recommended amount of vitamin C. What fraction of your daily amount of vitamin C do you still need?
$$\frac{□}{□}$$

$$\frac{13}{25}$$

Explanation:
If you eat 4 medium strawberries, you get 48% of your daily recommended amount of vitamin C.
48% = 48/100
100 – 48 = 52
52% = 52/100 = 13/25 of your daily amount of vitamin C do you still need

### Share and Show – Page No. 290

Find the percent of the quantity.

Question 1.
25% of 320
_______

80

Explanation:
Write the percent as a rate per 100
25% = 25/100
25/100 × 320 = 80

Question 2.
80% of 50
_______

40

Explanation:
Write the percent as a rate per 100
80% = 80/100
80/100 × 50 = 40

Question 3.
175% of 24
_______

42

Explanation:
Write the percent as a rate per 100
175% = 175/100
175/100 × 24 = 42

Question 4.
60% of 210
_______

126

Explanation:
Write the percent as a rate per 100
60% = 60/100
60/100 × 210 = 126

Question 5.
A jar contains 125 marbles. Given that 4% of the marbles are green, 60% of the marbles are blue, and the rest are red, how many red marbles are in the jar?
_______ marbles

45 marbles

Explanation:
A jar contains 125 marbles.
4% of the marbles are green = 125 × 4/100 = 5
60% of the marbles are blue = 125 × 60/100 = 75
Red Marbles = Total Number of Marbles -[Number of Green Marbles + Number of Blue Marbles]
Red Marbles = 125 – (5 + 75) = 125 – 80 = 45

Question 6.
There are 32 students in Mr. Moreno’s class and 62.5% of the students are girls. How many boys are in the class?
_______ students

12 students

Explanation:
There are 32 students in Mr. Moreno’s class
62.5% of the students are girls = 32 × 62.5/100 = 20
boys = 32 – 20 = 12

### On Your Own – Page No. 291

Find the percent of the quantity.

Question 7.
60% of 90
_______

54

Explanation:
Write the percent as a rate per 100
60% = 60/100
60/100 × 90 = 54

Question 8.
25% of 32.4
_______

8.1

Explanation:
Write the percent as a rate per 100
25% = 25/100
25/100 × 32.4 = 8.1

Question 9.
110% of 300
_______

330

Explanation:
Write the percent as a rate per 100
110% = 110/100
110/100 × 300 = 330

Question 10.
0.2% of 6500
_______

13

Explanation:
Write the percent as a rate per 100
0.2% = 0.2/100
0.2/100 × 6500 = 13

Question 11.
A baker made 60 muffins for a cafe. By noon, 45% of the muffins were sold. How many muffins were sold by noon?
_______ muffins

27 muffins

Explanation:
A baker made 60 muffins for a cafe. By noon, 45% of the muffins were sold.
60 × 45%
60 × 45/100 = 27

Question 12.
There are 30 treasures hidden in a castle in a video game. LaToya found 80% of them. How many of the treasures did LaToya find?
_______ treasures

24 treasures

Explanation:
There are 30 treasures hidden in a castle in a video game.
LaToya found 80% of them.
30 × 80/100 = 24

Question 13.
A school library has 260 DVDs in its collection. Given that 45% of the DVDs are about science and 40% are about history, how many of the DVDs are about other subjects?
_______ DVDs

39 DVDs

Explanation:
A school library has 260 DVDs in its collection.
45% of the DVDs are about science = 260 × 45/100 = 117
40% are about history = 260 × 40/100 = 104
other subjects = 260 – (117 + 104) = 260 – 221 = 39

Question 14.
Mitch planted cabbage, squash, and carrots on his 150-acre farm. He planted half the farm with squash and 22% with carrots. How many acres did he plant with cabbage?
_______ acres

Explanation:
Mitch planted cabbage, squash, and carrots on his 150-acre farm.
He planted half the farm with squash 150/2 = 75
22% with carrots = 150 × 22/100 = 33
cabbage = 150 – (75 + 33) = 150 – 108 = 42

Question 15.
45% of 60 _______ 60% of 45

45% of 60 = 60% of 45

Explanation:
45% of 60
45/100 × 60 = 27
60% of 45
60/100 × 45 = 27
45% of 60 = 60% of 45

Question 16.
10% of 90 _______ 90% of 100

10% of 90 _______ 90% of 100

Explanation:
10% of 90
10/100 × 90 = 9
90% of 100
90/100 × 100 = 90
10% of 90 < 90% of 100

Question 17.
75% of 8 _______ 8% of 7.5

75% of 8 > 8% of 7.5

Explanation:
75% of 8
75/100 × 8 = 6
8% of 7.5
8/100 × 7.5 = 0.6
75% of 8 > 8% of 7.5

Question 18.
Sarah had 12 free throw attempts during a game and made at least 75% of the free throws. What is the greatest number of free throws Sarah could have missed during the game?
_______ free throws

3 free throws

Explanation:
Sarah had 12 free throw attempts during a game and made at least 75% of the free throws.
So, she missed 25% of the free throws.
12 × 25/100 = 3

Question 19.
Chrissie likes to tip a server in a restaurant a minimum of 20%. She and her friend have a lunch bill that is $18.34. Chrissie says the tip will be$3.30. Her friend says that is not a minimum of 20%. Who is correct? Explain.
Type below:
_____________

100% = $18.34 10% =$18.34 / 10 = 1.834
20% = 1.834 × 2 = 3.66800 = $3.70 Her friend is correct because$3.70 is more than $3.30. ### Unlock The Problem – Page No. 292 Question 20. One-third of the juniors in the Linwood High School Marching Band play the trumpet. The band has 50 members and the table shows what percent of the band members are freshmen, sophomores, juniors, and seniors. How many juniors play the trumpet? a. What do you need to find? Type below: _____________ Answer: The percent of the band members are freshmen, sophomores, juniors, and seniors. How many juniors play the trumpet Question 20. b. How can you use the table to help you solve the problem? Type below: _____________ Answer: percent of the band members that are Juniors: 24% In 50 members of the band, 50×24/100 = 12 are Juniors. One-third of them play the trumpet, which makes 12×(1/3) = 4 members. Question 20. c. What operation can you use to find the number of juniors in the band? Type below: _____________ Answer: percent of the band members that are Juniors: 24% In 50 members of the band, 50×24/100 = 12 are Juniors. Explanation: Question 20. d. Show the steps you use to solve the problem. Type below: _____________ Answer: percent of the band members that are Juniors: 24% In 50 members of the band, 50×24/100 = 12 are Juniors. One-third of them play the trumpet, which makes 12×(1/3) = 4 members. Question 20. e. Complete the sentences. The band has _____ members. There are _____ juniors in the band. The number of juniors who play the trumpet is _____. Type below: _____________ Answer: The band has 50 members. There are 12 juniors in the band. The number of juniors who play the trumpet is 4. Question 21. Compare. Circle <, >, or =. a. 25% of 44 Ο 20% of 50 b. 10% of 30 Ο 30% of 100 c. 35% of 60 Ο 60% of 35 25% of 44 _____ 20% of 50 10% of 30 _____ 30% of 100 35% of 60 _____ 60% of 35 Answer: 25% of 44 > 20% of 50 10% of 30 < 30% of 100 35% of 60 = 60% of 35 Explanation: 25% of 44 = 25/100 × 44 = 11 20% of 50 = 20/100 × 50 = 1000/100 = 10 25% of 44 > 20% of 50 10% of 30 = 10/100 × 30 = 3 30% of 100 = 30/100 × 100 = 30 10% of 30 < 30% of 100 35% of 60 = 35/100 × 60 = 21 60% of 35 = 60/100 × 35 = 21 35% of 60 = 60% of 35 ### Percent of a Quantity – Page No. 293 Find the percent of the quantity. Question 1. 60% of 140 _____ Answer: 84 Explanation: 60% of 140 60/100 × 140 = 84 Question 2. 55% of 600 _____ Answer: 330 Explanation: 55% of 600 55/100 × 600 = 330 Question 3. 4% of 50 _____ Answer: 2 Explanation: 4% of 50 4/100 × 50 = 2 Question 4. 10% of 2,350 _____ Answer: 235 Explanation: 10% of 2,350 10/100 × 2,350 = 235 Question 5. 160% of 30 _____ Answer: 48 Explanation: 160% of 30 160/100 × 30 = 48 Question 6. 105% of 260 _____ Answer: 273 Explanation: 105% of 260 105/100 × 260 = 273 Question 7. 0.5% of 12 _____ Answer: 0.06 Explanation: 0.5% of 12 0.5/100 × 12 = 0.06 Question 8. 40% of 16.5 _____ Answer: 6.6 Explanation: 40% of 16.5 40/100 × 16.5 = 6.6 Problem Solving Question 9. The recommended daily amount of vitamin C for children 9 to 13 years old is 45 mg. A serving of a juice drink contains 60% of the recommended amount. How much vitamin C does the juice drink contain? _____ mg Answer: 27 mg Explanation: The recommended daily amount of vitamin C for children 9 to 13 years old is 45 mg. A serving of a juice drink contains 60% of the recommended amount. 45% of 60 = 45/100 × 60 = 27 Question 10. During a 60-minute television program, 25% of the time is used for commercials and 5% of the time is used for the opening and closing credits. How many minutes remain for the program itself? _____ minutes Answer: 42 minutes Explanation: 60 minutes of tv 25% + 5% = 30% 30%= 0.30 60 times 0.30= 18 60-18=42 inly 42 minutes are used for the program itself Question 11. Explain two ways you can find 35% of 700. Type below: _____________ Answer: First way 700 : 100 = x : 35 x = 700 × 35 : 100 x = 245 Second way 700 : 100 × 35 = 245 ### Lesson Check – Page No. 294 Question 1. A store has a display case with cherry, peach, and grape fruit chews. There are 160 fruit chews in the display case. Given that 25% of the fruit chews are cherry and 40% are peach, how many grape fruit chews are in the display case? _____ grape fruit chews Answer: 56 grape fruit chews Explanation: A store has a display case with cherry, peach, and grape fruit chews. There are 160 fruit chews in the display case. Given that 25% of the fruit chews are cherry and 40% are peach, 25% + 40% +?% = 100% 65% + ?% = 100% ?% = 35% .35×160 = 56 Question 2. Kelly has a ribbon that is 60 inches long. She cuts 40% off the ribbon for an art project. While working on the project, she decides she only needs 75% of the piece she cut off. How many inches of ribbon does Kelly end up using for her project? _____ inches Answer: 18 inches Explanation: Length of ribbon = 60 inches Part of ribbon cut off for an art project = 40% So, the Length of the ribbon remains is given by 40% of 60 = 40/100 × 60 = 24 Part of a piece she only needs from cut off = 75% so, the Length of ribbon she need end up using in her project is given by 75/100 × 24 = 18 Spiral Review Question 3. Three of the following statements are true. Which one is NOT true? |−12| > 1 |0| > −4 |20| > |−10| 6 < |−3| Type below: _____________ Answer: |−12| > 1 12 > 1; True |0| > −4 0 > -4; True |20| > |−10| 20 > 10; True 6 < |−3| 6 < 3; False Question 4. Miyuki can type 135 words in 3 minutes. How many words can she expect to type in 8 minutes? _____ words Answer: 360 words Explanation: Miyuki can type 135 words in 3 minutes. 135/3 = 45 45 × 8 = 360 Question 5. Which percent represents the model? _____ % Answer: 63% Explanation: 63 squares are shaded out of 100 63% Question 6. About $$\frac{3}{5}$$ of the students at Roosevelt Elementary School live within one mile of the school. What percent of students live within one mile of the school? _____ % Answer: 60% Explanation: About $$\frac{3}{5}$$ of the students at Roosevelt Elementary School live within one mile of the school. 3/5 × 100/100 = 60/100 = 60% ### Share and Show – Page No. 297 Question 1. A geologist visits 40 volcanoes in Alaska and California. 15% of the volcanoes are in California. How many volcanoes does the geologist visit in California and how many in Alaska? Type below: _____________ Answer: 40 volcanoes = 100% of them 100 – 15% = 85% Number of volcanoes in California = 15% of 40 volcanoes = 0.15 x 40 = 6 Number of volcanoes in Alaska = 85% of 40 volcanoes 0.85 x 40 = 34 Question 2. What if 30% of the volcanoes were in California? How many volcanoes would the geologist have visited in California and how many in Alaska? Type below: _____________ Answer: Number of volcanoes in California = 30% of 40 = 30/100 x 40 = 12 Number of volcanoes in Alaska = 70% of 40 = 70/100 x 40 = 28 Question 3. Ricardo has$25 to spend on school supplies. He spends 72% of the money on a backpack and the rest on a large binder. How much does he spend on the backpack? How much does he spend on the binder?
Type below:
_____________

$18 on Backpack$7 on binder.
If you turn the percent into a decimal .72 and multiply .72 by 25 you get 18 which is the cost of the backpack.
subtract 18 from 25 and you get $7 left meaning the binder was$7

Question 4.
Kevin is hiking on a trail that is 4.2 miles long. So far, he has hiked 80% of the total distance. How many more miles does Kevin have to hike in order to complete the trail?
Type below:
_____________

0.84 miles

Explanation:
Kevin is hiking on a trail that is 4.2 miles long. So far, he has hiked 80% of the total distance.
80% of 4.2 = 80/100 x 4.2 = 3.36
4.2 – 3.36 = 0.84 miles

### On Your Own – Page No. 298

Question 5.
Jordan takes 50% of the cherries from a bowl. Then Mei takes 50% of the remaining cherries. Finally, Greg takes 50% of the remaining cherries. There are 3 cherries left. How many cherries were in the bowl before Jordan arrived?
_____ cherries

24 cherries

Explanation:
Let total cherries in a bowl=x
Jordan takes cherries=50% of x = 50x/100
Remaining cherries = x – 50x/100 = x/2
Mei takes cherries=50% of 50x/100 = x/4
remaining cherries= x/2 – x/4 = x/4
Greg takes cherries=50% of x/4 = x/8
remaining cherries = x/4 – x/8 = x/8
Now,remaining cherries in a bowl=3
x/8 =3
x = 8 × 3 = 24

Question 6.
Each week, Tasha saves 65% of the money she earns babysitting and spends the rest. This week she earned $40. How much more money did she save than spend this week?$ _____

Tasha saved $26 and spent$14

Explanation:
Since 65% of 40 is 26, that’s how much Tasha saves. Then do 40 – 26 to get 14, which is how much she spends.
So Tasha saved $26 and spent$14.

Question 7.
An employee at a state park has 53 photos of animals found at the park. She wants to arrange the photos in rows so that every row except the bottom row has the same number of photos. She also wants there to be at least 5 rows. Describe two different ways she can arrange the photos
Type below:
_____________

5 rows of 10 photos and last row with 3 photos,
6 rows of 8 photos and last row with 5 photos,
7 rows of 7 photos and last row with 4 photos,
Also, reverse the rows and photos in each row (ex 5 rows 10 photos=10 rows 5 photos) to get another 3 sets.

Question 8.
Explain a Method Maya wants to mark a length of 7 inches on a sheet of paper, but she does not have a ruler. She has pieces of wood that are 4 inches, 5 inches, and 6 inches long. Explain how she can use these pieces to mark a length of 7 inches.
Type below:
_____________

Maya can put the 5 and 6-inch pieces together to get 11 inches. She can then subtract the length of the 4-inch piece to get 7 inches.

Question 9.
Pierre’s family is driving 380 miles from San Francisco to Los Angeles. On the first day, they drive 30% of the distance. On the second day, they drive 50% of the distance. On the third day, they drive the remaining distance and arrive in Los Angeles. How many miles did Pierre’s family drive each day? Write the number of miles in the correct box.

Type below:
_____________

76 miles

Explanation:
Pierre’s family is driving 380 miles from San Francisco to Los Angeles.
On the first day, they drive 30% of the distance. 380 × 30/100 = 114
On the second day, they drive 50% of the distance. 380 × 50/100 = 190
They traveled 80%.
On the third day, they drive the remaining distance and arrive in Los Angeles.
380 × 20/100 = 76 miles

### Problem Solving Percents – Page No. 299

Question 1.
On Saturday, a souvenir shop had 125 customers. Sixty-four percent of the customers paid with a credit card. The other customers paid with cash. How many customers paid with cash?T
_____ costumers

45 costumers

Explanation:
On Saturday, a souvenir shop had 125 customers. Sixty-four percent of the customers paid with a credit card.
125 × 64/100 = 80
100 – 64 = 36
125 × 36/100 = 45

Question 2.
A carpenter has a wooden stick that is 84 centimeters long. She cuts off 25% from the end of the stick. Then she cuts the remaining stick into 6 equal pieces. What is the length of each piece?
_____ cm

10 1/2 cm

Explanation:
A carpenter has a wooden stick that is 84 centimeters long. She cuts off 25% from the end of the stick. Then she cuts the remaining stick into 6 equal pieces.
84 × 75/100 = 63
63/6 = 10 1/2

Question 3.
A car dealership has 240 cars in the parking lot and 17.5% of them are red. Of the other 6 colors in the lot, each color has the same number of cars. If one of the colors is black, how many black cars are in the lot?
_____ black cars

33 black cars

Explanation:
number of red cars 17.5% × 240 = 42
number of cars of other colors = 240 – 42 = 198
number of black cars 1/6 × 198 = 33

Question 4.
The utilities bill for the Millers’ home in April was $132. Forty-two percent of the bill was for gas, and the rest was for electricity. How much did the Millers pay for gas, and how much did they pay for electricity? Type below: _____________ Answer: Amount of money paid for gas = 132 * (42/100) dollars = 5544/100 dollars = 55.44 dollars Then The amount of money paid for electricity = (132 – 55.44) dollars = 76.56 dollars So the Millers paid 55.44 dollars for gas and 76.56 dollars for electricity in the month of April. Question 5. Andy’s total bill for lunch is$20. The cost of the drink is 15% of the total bill and the rest is the cost of the food. What percent of the total bill did Andy’s food cost? What was the cost of his food?
Type below:
_____________

$17 Explanation: Andy paid$20 total for his lunch (100%).
15% is for drink.
Therefore, 100 – 15 = 85% is the percent that was constituted by the food.
85% of 20 is equal to 0.85 × 20 is equal to:
17 × 20/20 = 17
Andy’s food cost $17. Question 6. Write a word problem that involves finding the additional amount of money needed to purchase an item, given the cost and the percent of the cost already saved. Type below: _____________ Answer: Each week, Tasha saves 65% of the money she earns babysitting and spends the rest. This week she earned$40. How much more money did she save than spend this week?
Tasha saved $26 and spent$14

### Lesson Check – Page No. 300

Question 1.
Milo has a collection of DVDs. Out of 45 DVDs, 40% are comedies and the remaining are action-adventures. How many actionadventure DVDs does Milo own?
_____ DVDs

27 DVDs

Explanation:
100%-40%=60%
60/100*45=27

Question 2.
Andrea and her partner are writing a 12-page science report. They completed 25% of the report in class and 50% of the remaining pages after school. How many pages do Andrea and her partner still have to write?
_____ pages

9 pages

Explanation:
first 50% + 25% = 75%
then you can do 75% of 12
75% = 0.75
of = multiplication
0.75 • 12 which should equal 9
so they have 9 pages left

Spiral Review

Question 3.
What is the absolute value of $$\frac{-4}{25}$$?
$$\frac{□}{□}$$

$$\frac{4}{25}$$

Explanation:
|$$\frac{-4}{25}$$| = 4/25

Question 4.
Ricardo graphed a point by starting at the origin and moving 5 units to the left. Then he moved up 2 units. What is the ordered pair for the point he graphed?
Type below:
_____________

(-5, 2)

Explanation:
In a coordinate system, the coordinates of the origin are (0, 0).
If he moves 5 units to the left, he is moving in the negative direction along the x-axis, and x takes the value -5.
If he moves up 2 units, he is moving in the positive direction along the y-axis, and y takes the value 2.
The ordered pair (x, y) is (-5, 2).

Question 5.
The population of birds in a sanctuary increases at a steady rate. The graph of the population over time has the points (1, 105) and (3, 315). Name another point on the graph.
Type below:
_____________

You could do (2, 210) or (4, 420) or (5, 525)

Question 6.
Alicia’s MP3 player contains 1,260 songs. Given that 35% of the songs are rock songs and 20% of the songs are rap songs, how many of the songs are other types of songs?
_____ songs

567 songs

Explanation:
Since 55% of the songs are rock and rap, 45% of the songs are other.
To find 45% of 1260 we multiply by the decimal:
1260 x 0.45 = 567
Therefore 567 of the songs are other.

### Share and Show – Page No. 303

Find the unknown value.

Question 1.
9 is 25% of _____.
_____

36

Explanation:
25/100 ÷ 25/25 = 1/4
1/4 = 9/s
1/4 × 9/9 = 9/36
the unknown value is 36

Question 2.
14 is 10% of _____.
_____

140

Explanation:
10/100 ÷ 10/10 = 1/10
1/10 = 14/s
1/10 × 14/14 = 14/140
the unknown value is 140

Question 3.
3 is 5% of _____.
_____

6

Explanation:
5/10 ÷ 5/5 = 1/2
1/2 × 3/3 = 3/6
the unknown value is 6

Question 4.
12 is 60% of _____.
_____

20

Explanation:
60/100 ÷ 60/60 = 60/100
60/100 ÷ 5/5 = 12/20
the unknown value is 20

Find the unknown value.

Question 5.
16 is 20% of _____.
_____

80

Explanation:
20/100 ÷ 20/20 = 1/5
1/5 × 16/16 = 16/80
the unknown value is 80

Question 6.
42 is 50% of _____.
_____

84

Explanation:
50/100 ÷ 50/50 = 1/2
1/2 × 42/42 = 42/84
the unknown value is 84

Question 7.
28 is 40% of _____.
_____

70

Explanation:
40/100 ÷ 40/40 = 1/2.5
1/2.5 × 28/28 = 28/70
the unknown value is 70

Question 8.
60 is 75% of _____.
_____

80

Explanation:
75/100 ÷ 75/75 = 60/s
60 × 100 = 6000/75 = 80
the unknown value is 80

Question 9.
27 is 30% of _____.
_____

90

Explanation:
30/100 ÷ 30/30 = 3/10
3/10 × 9/9 = 27/90
the unknown value is 90

Question 10.
21 is 60% of _____.
_____

35

Explanation:
60/100 ÷ 60/60 = 3/5
3/5 × 7/7 = 21/35
the unknown value is 35

Question 11.
12 is 15% of _____.
_____

80

Explanation:
15/100 ÷ 15/15 = 3/20
3/20 × 4/4 = 12/80
the unknown value is 80

Solve.

Question 12.
40% of the students in the sixth grade at Andrew’s school participate in sports. If 52 students participate in sports, how many sixth graders are there at Andrew’s school?
_____ students

130 students

Explanation:
52/s = 40%
52/s = 40/100
s = 40/100 × 52 = 130

Question 13.
There were 136 students and 34 adults at the concert. If 85% of the seats were filled, how many seats are in the auditorium?
_____ seats

80 seats

Explanation:
There are 170 seats filled total. 170 is 85% of 200. There are 200 seats in the auditorium.
If you were to solve for x in the equation 40% = 32/x, you would get x = 80.

Use Reasoning Algebra Find the unknown value.

Question 14.
40% = $$\frac{32}{?}$$
_____

80

Explanation:
40/100 = 32/?
40/100 ÷ 40/40 = 2/5
2/5 × 16/16 = 32/80
the unknown value is 80

Question 15.
65% = $$\frac{91}{?}$$
_____

140

Explanation:
65/100 = 91/?
65/100 ÷ 65/65 = 13/20
13/20 × 7/7 = 91/140
the unknown value is 140

Question 16.
45% = $$\frac{54}{?}$$
_____

120

Explanation:
45/100 ÷ 45/45 = 9/20
9/20 × 6/6 = 54/120

### Problem Solving + Applications – Page No. 304

Question 17.
Corey spent 20% of his savings on a printer at Louie’s Electronics. How much did Corey have in his savings account before he bought the printer?
$_____ Answer:$800

Explanation:
(printer cost) = 0.20 * (savings)
(printer cost)/0.20 = (savings)
savings = 5*(printer cost)
Corey’s savings was 5 times that amount.
savings = 5 × 160 = 800

Question 18.
Kai spent 90% of his money on a laptop that cost $423. Does he have enough money left to buy a scanner? Explain. Type below: _____________ Answer:$42.3

Explanation:
He spent 90% of his money. So, he left 10% of money with him.
423 × 10/100 = 42.3 left to buy a scanner

Question 19.
Maurice has completed 17 pages of the research paper he is writing. That is 85% of the required length of the paper. What is the required length of the paper?
_____ pages

20 pages

Explanation:
Maurice has completed 17 pages of the research paper he is writing. That is 85% of the required length of the paper.
100multiplied by 17 divided by 85% =20

Question 20.
Of 250 seventh-grade students, 175 walk to school. What percent of seventh-graders do not walk to school?
_____ %

30%

Explanation:
it’s either 30 percent or 70. 70 percent walks to school and 30 percent DO NOT walk to school

Question 21.
What’s the Error? Kate has made 20 free throws in basketball games this year. That is 80% of the free throws she has attempted. To find the total number of free throws she attempted, Kate wrote the equation $$\frac{80}{100}=\frac{?}{20}$$. What error did Kate make?
Type below:
_____________

20 free throws is 80% of the total attempted
80% to decimal is:
80/100 = 0.8
If total attempted is x, we can say:
20 is 80% (0.8) of x
We can now write an algebraic equation:
20 = 0.8x
We simply solve this for x, that is the number of free throws she attempted:
20 = 0.8x
x = 20/0.8 = 25

Question 22.
Maria spent 36% of her savings to buy a smart phone. The phone cost $90. How much money was in Maria’s savings account before she purchased the phone? Find the unknown value.$ _____

$250 Explanation: let her savings be A A/Q- 36% of A =$90
36/100 of A = $90 A = 90×100/36 A=$ 250

### Find the Whole from a Percent – Page No. 305

Find the Whole from a Percent

Question 1.
9 is 15% of _____.
_____

60

Explanation:
15/100 ÷ 15/15 = 3/20
3/20 × 3/3 = 9/60
the unknown value is 60

Question 2.
54 is 75% of _____.
_____

72

Explanation:
75/100 ÷ 75/75 = 3/4
3/4 × 18/18 = 54/72
the unknown value is 72

Question 3.
12 is 2% of _____.
_____

600

Explanation:
2/100 = 1/50
1/50 × 12/12 = 12/600
the unknown value is 600

Question 4.
18 is 50% of _____.

36

Explanation:
50/100 = 1/2
1/2 × 18/18 = 18/36
the unknown value is 36

Question 5.
16 is 40% of _____.
_____

40

Explanation:
40/100 = 2/5
2/5 × 8/8 = 16/40
the unknown value is 40

Question 6.
56 is 28% of _____.
_____

200

Explanation:
28/100 = 14/50 = 7/25
7/25 × 8/8 = 56/200
the unknown value is 200

Question 7.
5 is 10% of _____.
_____

50

Explanation:
10/100 = 1/10
1/10 × 5/5 = 5/50
the unknown value is 50

Question 8.
24 is 16% of _____.
_____

150

Explanation:
16/100 = 4/25
4/25 × 6/6 = 24/150
the unknown value is 150

Question 9.
15 is 25% of _____.
_____

60

Explanation:
25/100 = 1/4
1/4 × 15/15 = 15/60
the unknown value is 60

Problem Solving

Question 10.
Michaela is hiking on a weekend camping trip. She has walked 6 miles so far. This is 30% of the total distance. What is the total number of miles she will walk?
_____ miles

20 miles

Explanation:
Since 6mi=30%,
You should find ten percent.
This is how, divide both sides by 3, and this gives you
2m=10% (2m being 2 miles)
So, to find 100%, you need to multiply both sides by 10
20m=100%
So now, Michaela will walk 20 miles this weekend

Question 11.
A customer placed an order with a bakery for muffins. The baker has completed 37.5% of the order after baking 81 muffins. How many muffins did the customer order?
_____ muffins

216 muffins

Explanation:
A customer placed an order with a bakery for muffins. The baker has completed 37.5% of the order after baking 81 muffins.
37.5/100=0.375 and 81/0.375=216

Question 12.
Write a question that involves finding what number is 25% of another number. Solve using a double number line and check using equivalent ratios. Compare the methods.
Type below:
_____________

25% of 15 = 25/100 × 15 = 375/100 = 3.75

### Lesson Check – Page No. 306

Question 1.
Kareem saves his coins in a jar. 30% of the coins are pennies. If there are 24 pennies in the jar, how many coins does Kareem have?
_____ coins

80 coins

Explanation:
24=30%
find 100%
24=30%
diivde by 3
8=10%
multiply 10
80=100%
80 coins

Question 2.
A guitar shop has 19 acoustic guitars on display. This is 19% of the total number of guitars. What is the total number of guitars the shop has?
_____ guitars

100 guitars

Explanation:
Let’s find out how much 1% is worth first.
19 guitars = 19%
therefore 19 ÷ 19 = [ 1 guitar = 1% ]
The total number of guitars is going to be 100%,
so if 1% × 100 = 100%, then 1 guitar × 100 = 100 guitars total.

Spiral Review

Question 3.
On a coordinate grid, in which quadrant is the point (−5, 4) located?
Type below:
_____________

Explanation:
(-5, 4)
-5 is the negative point of the x coordinate
4 is the positive point of the y coordinate

Question 4.
A box contains 16 cherry fruit chews, 15 peach fruit chews, and 12 plum fruit chews. Which two flavors are in the ratio 5 to 4?
Type below:
_____________

peach fruit chews and plum fruit chews are in the ratio 5 to 4

Explanation:
15 peach fruit chews, and 12 plum fruit chews
15/12 = 5/4

Question 5.
During basketball season, Marisol made $$\frac{19}{25}$$ of her free throws. What percent of her free throws did Marisol make?
_____ %

76%

Explanation:
During the basketball season, Marisol made $$\frac{19}{25}$$ of her free throws.
(19 ÷ 25) × 100 = 76%. Marisol made 76% of her free throws.

Question 6.
Landon is entering the science fair. He has a budget of $115. He has spent 20% of the money on new materials. How much does Landon have left to spend?$ _____

$92 Explanation: Landon has$92 left because if you divide 115/.20 you get 23 and then you subtract 115-23=92 or $92. ### Chapter 5 Review/Test – Page No. 307 Question 1. What percent is represented by the shaded part? Options: a. 46% b. 60% c. 64% d. 640% Answer: c. 64% Explanation: 64 squares are shaded out of 100. So, 64% and 64/100 are the answers. Question 2. Write a percent to represent the shaded part. _____ % Answer: 42% Explanation: 42 squares are shaded out of 100. So, 42% and 42/100 are the answers. Question 3. Rosa made a mosaic wall mural using 42 black tiles, 35 blue tiles and 23 red tiles. Write a percent to represent the number of red tiles in the mural. _____ % Answer: 23% Explanation: 42+35+23= 100 So plug it in. 23/100 23% Your answer is 23%. Question 4. Model 39%. Type below: _____________ Answer: Explanation: 39 squares out of 100 need to shaded ### Page No. 308 Question 5. For 5a–5d, choose Yes or No to indicate whether the percent and the fraction represent the same amount. 5a. 50% and $$\frac{1}{2}$$ 5b. 45% and $$\frac{4}{5}$$ 5c. $$\frac{3}{8}$$ and 37.5% 5d. $$\frac{2}{10}$$ and 210% 5a. _____________ 5b. _____________ 5c. _____________ 5d. _____________ Answer: 5a. Yes 5b. No 5c. Yes 5d. No Explanation: 1/2 = 0.5 × 100/100 = 50/100 = 50% 4/5 = 0.8 × 100/100 = 80/100 = 80% 3/8 = 0.375 × 100/100 = 37.5/100 = 37.5% 2/10 = 0.2 × 100/100 = 20/100 = 20% Question 6. The school orchestra has 25 woodwind instruments, 15 percussion instruments, 30 string instruments, and 30 brass instruments. Select the portion of the instruments that are percussion. Mark all that apply. Options: a. 15% b. 1.5 c. $$\frac{3}{20}$$ d. 0.15 Answer: a. 15% c. $$\frac{3}{20}$$ d. 0.15 Explanation: 25 + 15 + 30 + 30 = 100 15 percussion instruments = 15/100 = 15% = 0.15 Question 7. For a science project, $$\frac{3}{4}$$ of the students chose to make a poster and 0.25 of the students wrote a report. Rosa said that more students made a poster than wrote a report. Do you agree with Rosa? Use numbers and words to support your answer Type below: _____________ Answer: Yes, because 3/4 is equal to 0.75 and 0.75 > 0.25 Or 0.25 is equal to 1/4, and 1/4 < 3/4 Question 8. Select other ways to write 0.875. Mark all that apply. Options: a. 875% b. 87.5% c. $$\frac{7}{8}$$ d. $$\frac{875}{100}$$ Answer: c. $$\frac{7}{8}$$ Explanation: 0.875 = 8.75/100 = 8.75% ### Page No. 309 Question 9. There are 88 marbles in a bin and 25% of the marbles are red. There are _____________ red marbles in the bin. Answer: There are 22 red marbles in the bin. Explanation: 88 × 25% = 88 × 25/100 = 22 Question 10. Harrison has 30 CDs in his music collection. If 40% of the CDs are country music and 30% are pop music, how many CDs are other types of music? _____ CDs Answer: 9 CDs Explanation: Harrison has 30 CDs in his music collection. If 40% of the CDs are country music and 30% are pop music, 40 + 30 = 70 100 – 70 = 30% 30 × 30/100 = 9 Question 11. For numbers 11a–11b, choose <, >, or =. 11a. 30% of 90 Ο 35% of 80 11b. 25% of 16 Ο 20% of 25 30% of 90 _____ 35% of 80 25% of 16 _____ 20% of 25 Answer: 30% of 90 < 35% of 80 25% of 16 < 20% of 25 Explanation: 30% of 90 = 30/100 × 90 = 27 35% of 80 = 35/100 × 80 = 28 30% of 90 < 35% of 80 25% of 16 = 25/100 × 16 = 4 20% of 25 = 20/100 × 25 = 5 25% of 16 < 20% of 25 Question 12. There were 200 people who voted at the town council meeting. Of these people, 40% voted for building a new basketball court in the park. How many people voted against building the new basketball court? Use numbers and words to explain your answer. Type below: _____________ Answer: There were 200 people who voted at the town council meeting. Of these people, 40% voted for building a new basketball court in the park. 100 – 40% = 60% 200 × 60/100 = 120 people ### Page No. 310 Question 13. James and Sarah went out to lunch. The price of lunch for both of them was$20. They tipped their server 20% of that amount. How much did each person pay if they shared the price of lunch and the tip equally?
$_____ Answer:$12

Explanation:
James and Sarah went out to lunch. The price of lunch for both of them was $20. They tipped their server 20% of that amount. 20% of 20 = 20/100 × 20 = 4 20 + 4 = 24 24/2 = 12$12

Question 14.
A sandwich shop has 30 stores and 60% of the stores are in California. The rest of the stores are in Nevada.
Part A
How many stores are in California and how many are in Nevada?
Type below:
_____________

30 × 60/100 = 18 stores in California
30 – 18 = 12 stores in Nevada

Question 14.
Part B
The shop opens 10 new stores. Some are in California, and some are in Nevada. Complete the table.

Type below:
_____________

Explanation:
100 – 45 = 55%
55% of 40 = 55/100 × 40 = 22
45% of 40 = 45/100 × 40 = 18

Question 15.
Juanita has saved 35% of the money that she needs to buy a new bicycle. If she has saved $63, how much money does the bicycle cost? Use numbers and words to explain your answer$ _____

$180 Explanation: Juanita has saved 35% of the money that she needs to buy a new bicycle. If she has saved$63,
35/100 = 7/20
7/20 × 9/9 = 63/180
The bicycle cost is $180 ### Page No. 311 Question 16. For 16a–16d, choose Yes or No to indicate whether the statement is correct. 16a. 12 is 20% of 60. 16b. 24 is 50% of 48. 16c. 14 is 75% of 20. 16d. 9 is 30% of 30. 16a. _____________ 16b. _____________ 16c. _____________ 16d. _____________ Answer: 16a. Yes 16b. Yes 16c. No 16d. Yes Explanation: 20% of 60 = 20/100 × 60 = 12 50% of 48 = 50/100 × 48 = 24 75% of 20 = 75/100 × 20 = 15 30% of 30 = 30/100 × 30 = 9 Question 17. Heather and her family are going to the grand opening of a new amusement park. There is a special price on tickets this weekend. Tickets cost$56 each. This is 70% of the cost of a regular price ticket
Part A
What is the cost of a regular price ticket? Show your work.
$_____ Answer:$80

Explanation:
70/100 = 56/s
s = 56 × 100/70 = 80

Question 17.
Part B
Heather’s mom says that they would save more than $100 if they buy 4 tickets for their family on opening weekend. Do you agree or disagree with Heather’s mom? Use numbers and words to support your answer. If her statement is incorrect, explain the correct way to solve it. Type below: _____________ Answer: 80 × 4 = 320 56 × 4 = 224 320 – 224 = 96$96

Question 18.
Elise said that 0.2 equals 2%. Use words and numbers to explain her mistake.
Type below:
_____________

0.2 × 100/100 = 20/100 = 2%

### Page No. 312

Question 19.
Write 18% as a fraction.
$$\frac{□}{□}$$

$$\frac{9}{50}$$

Explanation:
18% = 18/100 = 9/50

Question 20.
Noah wants to put a variety of fish in his new fish tank. His tank is large enough to hold a maximum of 70 fish.
Part A
Complete the table.

Type below:
_____________

Explanation:
70 × 20/100 = 14
70 × 40/100 = 28
70 × 30/100 = 21

Question 20.
Part B
Has Noah put the maximum number of fish in his tank? Use numbers and words to explain how you know. If he has not put the maximum number of fish in the tank, how many more fish could he put in the tank?
Type below:
_____________

No, since 20% + 40% + 30% = 90%, he can add 10% in the tank.

### Conclusion:

Test your knowledge by solving the problems from Go Math Grade 6 Answer Key Chapter 5 Model Percents. Get the solutions for Mid Chapter Checkpoint and Review Test along with the exercise problems in Go Math Grade 6 Chapter 5 Model Percents Solution Key. Quick learning and best practice come in a single hand with our Go Math Grade 6 Solution Key Chapter 5 Model Percents @ ccssmathanswers.com

## Go Math Grade 8 Answer Key Chapter 8 Solving Systems of Linear Equations

Students of Grade 8 can get a detailed explanation for all the questions in Go Math Answer Key Chapter 8 Solving Systems of Linear Equations. In addition to the exercise problems we also provide the solutions for the review test. So, go through all the answers and explanations provided by the math experts in Go Math Grade 8 Chapter 8 Solving Systems of Linear Equations Answer Key. Our aim is to provide easy and simple tricks to solve the problems in Go Math Grade 8 Answer Key Chapter 8 Solving Systems of Linear Equations.

Students who are interested to secure the highest marks in the exams are suggested to download the Go Math Grade 8 Answer Key Chapter 8 Solving Systems of Linear Equations pdf. All the solutions are provided in the pdf format as per the list of the chapters provided in the latest edition. Hence refer to Go Math 8th Grade Solution Key to learning the easy way of maths practice. Check the list of the topics covered in Chapter 8 Solving Systems of Linear Equations from the following section.

Lesson 1: Solving Systems of Linear Equations by Graphing

Lesson 2: Solving Systems by Substitution

Lesson 3: Solving Systems by Elimination

Lesson 4: Solving Systems by Elimination with Multiplication

Lesson 5: Solving Solving Special Systems

Model Quiz

Review

### Guided Practice – Solving Systems of Linear Equations by Graphing – Page No. 232

Solve each system by graphing.

Question 1.
$$\left\{\begin{array}{l}y=3 x-4 \\y=x+2\end{array}\right.$$

Type below:
______________

Explanation:
y = 3x – 4
y = x + 2
The solution of thr linear system of equations is the intersection point of the two equations.
(3, 5) is the solution of the system of equations.
If x = 3, y = 3(3) – 4 = 9 – 4 = 5; y = 3 + 2 = 5
5 = 5; True

Question 2.
$$\left\{\begin{array}{l}x-3 y=2 \\-3x+9y=-6\end{array}\right.$$

Type below:
______________

Infinitely many solutions

Explanation:
x – 3y = 2
-3x + 9y = -6
x – 3y – x = -x + 2
-3y = -x + 2
y = 1/3 . x – 2/3
-3x + 9y + 3x = 3x – 6
9y = 3x – 6
y = 3/9 . x – 6/9
y = 1/3 . x – 2/3
The solution of the linear system of equations is the intersection of the two equations.
Infinitely many solutions

Question 3.
Mrs. Morales wrote a test with 15 questions covering spelling and vocabulary. Spelling questions (x) are worth 5 points and vocabulary questions (y) are worth 10 points. The maximum number of points possible on the test is 100.
a. Write an equation in slope-intercept form to represent the number of questions on the test.

Type below:
______________

y = -x + 15

Explanation:
Mrs. Morales wrote a test with 15 questions covering spelling and vocabulary. Spelling questions (x) are worth 5 points and vocabulary questions (y) are worth 10 points.
x + y = 15
x + y – x = -x + 15
y = -x + 15

Question 3.
b. Write an equation in slope-intercept form to represent the total number of points on the test.
Type below:
______________

y = -1/2 . x + 10

Explanation:
The total number of points on test is 100
5x + 10y = 100
5x + 10y – 5x = -5x + 100
10y = -5x + 100
y = -5/10 . x + 100/10
y = -1/2 . x + 10

Question 3.
c. Graph the solutions of both equations.
Type below:
______________

Question 3.
d. Use your graph to tell how many of each question type are on the test.
_________ spelling questions
_________ vocabulary questions

10 spelling questions
5 vocabulary questions

ESSENTIAL QUESTION CHECK-IN

Question 4.
When you graph a system of linear equations, why does the intersection of the two lines represent the solution of the system?
Type below:
______________

To solve a system of linear equations means finding the solutions that satisfy all the equations of that system. When we graph a system of linear equations, the intersection point lies on the line of each equation, which means that satisfies all the equations. Therefore, it is considered to be the solution to that system.

### Solving Systems of Linear Equations by Graphing – Page No. 233

Question 5.
Vocabulary
A_________________ is a set of equations that have the same variables.
______________

system of equations

Explanation:
A system of equations is a set of equations that have the same variables.

Question 6.
Eight friends started a business. They will wear either a baseball cap or a shirt imprinted with their logo while working. They want to spend exactly $36 on the shirts and caps. Shirts cost$6 each and caps cost $3 each. a. Write a system of equations to describe the situation. Let x represent the number of shirts and let y represent the number of caps. ______________ Answer: 6x + 3y = 36 Explanation: The sum of caps and shirts is 8. The total cost of caps and shirts is$36.
x + y = 8
6x + 3y = 36

Question 6.
b. Graph the system. What is the solution and what does it represent?

Type below:
______________

The solution is (4, 4)

Explanation:
x + y – x = -x + 8
y = -x + 8
6x + 3y – 6x = -6x + 36
3y = -6x + 36
y = -6/2 . x + 36/3
y = -2x + 12
(4, 4). They should order 4 shirts and 4 caps.

Question 7.
Multistep The table shows the cost for bowling at two bowling alleys.

a. Write a system of equations, with one equation describing the cost to bowl at Bowl-o-Rama and the other describing the cost to bowl at Bowling Pinz. For each equation, let x represent the number of games played and let y represent the total cost.
Type below:
______________

y = 2.5x + 2
y = 2x + 4

Explanation:
Cost at Bowl-o-Rama => y = 2.5x + 2
Cost at Bowling Pinz => y = 2x + 4

Question 7.
b. Graph the system. What is the solution and what does it represent?

Type below:
______________

Explanation:
The solution of the linear system of equations is the intersection of the two equations.
(4, 12)
When 4 games are played, the total cost is $12. ### Solving Systems of Linear Equations by Graphing – Page No. 234 Question 8. Multi-Step Jeremy runs 7 miles per week and increases his distance by 1 mile each week. Tony runs 3 miles per week and increases his distance by 2 miles each week. In how many weeks will Jeremy and Tony be running the same distance? What will that distance be? Type below: ______________ Answer: After 4 weeks Jeremy and Tony will be running the same distance and that distance would be 11 miles. Explanation: Multi-Step Jeremy runs 7 miles per week and increases his distance by 1 mile each week. y = x + 7 Tony runs 3 miles per week and increases his distance by 2 miles each week. y = 2x + 3 The solution of the system of linear equation is (4, 11) which means that after 4 weeks Jeremy and Tony will be running the same distance and that distance would be 11 miles. Question 9. Critical Thinking Write a real-world situation that could be represented by the system of equations shown below. $$\left\{\begin{array}{l}y=4 x+10 \\y=3x+15\end{array}\right.$$ Type below: ______________ Answer: The entry fee of the first gym is$10 and for every hour that you spend there, you pay an extra $4. If we denote with x the number of hours that somebody spends at the gym and with y the total cost is y = 4x + 10 The entry fee of the second gym is$15 and for every hour that you spend there, you pay an extra $3. If we denote with x the number of hours that somebody spends at the gym and with y the total cost is y = 3x + 15 y = 4x + 10 y = 3x + 15 FOCUS ON HIGHER ORDER THINKING Question 10. Multistep The table shows two options provided by a high-speed Internet provider. a. In how many months will the total cost of both options be the same? What will that cost be? ________ months$ ________

5 months
$200 Explanation: Let y be the total cost after x month y = 30x + 50 Let y be the total cost after x month y = 40x Substitute y = 40x in y = 30x + 50 40x = 30x + 50 40x – 30x = 50 10x = 50 x = 50/10 x = 5 The total cost of both options will be the same after 5 months. Total cost would be y = 40(5) =$200.

Question 10.
b. If you plan to cancel your Internet service after 9 months, which is the cheaper option? Explain.
______________

When x = 9 months
y = 30(9) + 50 = $320 y = 40(9) =$360
$320 <$360
Option 1 is cheaper as the total cost is less for option 1

Question 11.
Draw Conclusions How many solutions does the system formed by x − y = 3 and ay − ax + 3a = 0 have for a nonzero number a? Explain.
Type below:
______________

x – y = 3
ay – ax + 3a =0
ay – ax + 3a – 3a = 0 – 3a
ay – ax = – 3a
a(y – x) = -3a
y – x = -3
x – y = 3
Both equations are the same. The system of linear equations have infinitely many solutions.

### Guided Practice – Solving Systems by Substitution – Page No. 240

Solve each system of linear equations by substitution.

Question 1.
$$\left\{\begin{array}{l}3x-2y=9 \\y=2x-7\end{array}\right.$$
x = ________
y = ________

x = 5
y = 3

Explanation:
$$\left\{\begin{array}{l}3x-2y=9 \\y=2x-7\end{array}\right.$$
Substitute 2x – 7 in 3x – 2y = 9
3x – 2(2x – 7) = 9
3x – 4x + 14 = 9
-x + 14 = 9
-x + 14 – 14 = 9 – 14
-x = -5
x = -5/-1 = 5
y = 2(5) – 7 = 3
Solution is (5, 3)

Question 2.
$$\left\{\begin{array}{l}y=x-4 \\2x+y=5\end{array}\right.$$
x = ________
y = ________

x = 3
y = -1

Explanation:
$$\left\{\begin{array}{l}y=x-4 \\2x+y=5\end{array}\right.$$
2x + x – 4 = 5
3x – 4 = 5
3x – 4 + 4 = 5 + 4
3x = 9
x = 9/3 = 3
y = 3 – 4 = -1
The solution is (3, -1)

Question 3.
$$\left\{\begin{array}{l}x+4y=6 \\y=-x+3\end{array}\right.$$
x = ________
y = ________

x = 2
y = 1

Explanation:
$$\left\{\begin{array}{l}x+4y=6 \\y=-x+3\end{array}\right.$$
Substitute y = -x + 3 in x + 4y = 6
x + 4(-x + 3) = 6
x – 4x + 12 = 6
-3x + 12 = 6
-3x + 12 – 12 = 6 – 12
-3x = -6
x = -6/-3 = 2
y = -2 + 3 = 1
The solution is (2, 1)

Question 4.
$$\left\{\begin{array}{l}x+2y=6 \\x-y=3\end{array}\right.$$
x = ________
y = ________

x = 4
y = 1

Explanation:
$$\left\{\begin{array}{l}x+2y=6 \\x-y=3\end{array}\right.$$
y = x – 3
Substitute y = x – 3 in x + 2y = 6
x + 2(x – 3) = 6
x + 2x – 6 = 6
3x = 12
x = 12/3
x = 4
4 – y = 3
-y = 3 – 4
-y = -1
y = 1
The solution is (4, 1)

Solve each system. Estimate the solution first.

Question 5.
$$\left\{\begin{array}{l}6x+y=4 \\x-4y=19\end{array}\right.$$
Estimate ______________
Solution ______________
Type below:
______________

Estimate (2, -5)
Solution (1.4, -4.4)

Explanation:
$$\left\{\begin{array}{l}6x+y=4 \\x-4y=19\end{array}\right.$$
Let’s find the estimation by graphing the equations
Estimate: (2, -5)

x = 4y + 19
6(4y + 19) + y = 4
24y + 114 + y = 4
25y + 114 = 4
25y = 4 – 114
25y = -110
y = -110/25
y = -4.4
x + 4(-4.4) = 19
x + 17.6 = 19
x = 19 – 17.6
x = 1.4
The solution is (1.4, -4.4)

Question 6.
$$\left\{\begin{array}{l}x+2y=8 \\3x+2y=6\end{array}\right.$$
Estimate ______________
Solution ______________
Type below:
______________

Estimate (-1, 5)
Solution (-1, 4.5)

Explanation:
$$\left\{\begin{array}{l}x+2y=8 \\3x+2y=6\end{array}\right.$$
Let’s find the estimation by graphing the equations
Estimate: (-1, 5)

x = -2y + 8
Substitute the equation x = -2y + 8 in 3x + 2y = 6
3(-2y + 8) + 2y = 6
-6y + 24 + 2y = 6
-4y = 6 – 24
-4y = -18
y = -18/-4
y = 4.5
x + 2(4.5) = 8
x + 9 = 8
x = 8 – 9
x = -1
The solution is (-1, 4.5)

Question 7.
$$\left\{\begin{array}{l}3x+y=4 \\5x-y=22\end{array}\right.$$
Estimate ______________
Solution ______________
Type below:
______________

Estimate (3, -6)
Solution (3.25, -5.75)

Explanation:
$$\left\{\begin{array}{l}3x+y=4 \\5x-y=22\end{array}\right.$$
Find the Estimation using graphing the equations.
Estimate: (3, -6)

y = -3x + 4
Substitute y = -3x + 4 in 5x – y = 22
5x – (-3x + 4) = 22
5x + 3x -4 = 22
8x = 26
x = 26/8
x = 3.25
3(3.25) + y = 4
9.75 + y = 4
y = 4 – 9.75
y = -5.75
The solution is (3.25, -5.75)

Question 8.
$$\left\{\begin{array}{l}2x+7y=2 \\x+y=-1\end{array}\right.$$
Estimate ______________
Solution ______________
Type below:
______________

Estimate (-2, 1)
Solution (-1.8, 0.8)

Explanation:
$$\left\{\begin{array}{l}2x+7y=2 \\x+y=-1\end{array}\right.$$
Find the Estimation using graphing the equations.
Estimate: (-2, 1)

y = -x -1
Substitute y = -x – 1 in 2x + 7y = 2
2x + 7(-x – 1) = 2
2x – 7x -7 = 2
-5x = 2 + 7
-5x = 9
x = -9/5
x = -1.8
-1.8 + y = -1
y = -1 + 1.8
y = 0.8
The solution is (-1.8, 0.8)

Question 9.
Adult tickets to Space City amusement park cost x dollars. Children’s tickets cost y dollars. The Henson family bought 3 adult and 1 child tickets for $163. The Garcia family bought 2 adult and 3 child tickets for$174.
a. Write equations to represent the Hensons’ cost and the Garcias’ cost.
Hensons’ cost: ________________
Garcias’ cost:__________________
Type below:
______________

Hensons’ cost: 3x + y = 163
Garcias’ cost: 2x + 3y = 174

Explanation:
Henson’s cost
3x + y = 163
Garcia’s cost
2x + 3y = 174

Question 9.
b. Solve the system.
adult ticket price: $_________ Garcias’ cost:$ _________

adult ticket price: $45 Garcias’ cost:$ 28

Explanation:
y = -3x + 163
Substitute y = -3x + 163 in 2x + 3y = 174
2x + 3(-3x + 163) = 174
2x -9x + 489 = 174
-7x = -315
x = -315/-7 = 45
3(45) + y = 163
135 + y = 163
y = 163 – 135
y = 28
adult ticket price: $45 Garcias’ cost:$ 28

ESSENTIAL QUESTION CHECK-IN

Question 10.
How can you decide which variable to solve for first when you are solving a linear system by substitution?
Type below:
______________

The variable with the unit coefficient should be solved first when solving a linear system by substitution.

### 8.2 Independent Practice – Solving Systems by Substitution – Page No. 241

Question 11.
Check for Reasonableness Zach solves the system
$$\left\{\begin{array}{l}x+y=-3 \\x-y=1\end{array}\right.$$
and finds the solution (1, -2). Use a graph to explain whether Zach’s solution is reasonable.

Type below:
______________

Explanation:
$$\left\{\begin{array}{l}x+y=-3 \\x-y=1\end{array}\right.$$
The x coordinate of the solution is negative, hence Zach’s solution is not reasonable.

Represent Real-World Problems Angelo bought apples and bananas at the fruit stand. He bought 20 pieces of fruit and spent $11.50. Apples cost$0.50 and bananas cost $0.75 each. a. Write a system of equations to model the problem. (Hint: One equation will represent the number of pieces of fruit. A second equation will represent the money spent on the fruit.) Type below: ______________ Answer: x + y = 20 0.5x + 0.75y = 11.5 Explanation: x + y = 20 0.5x + 0.75y = 11.5 where c is the number of Apples and y is the number of Bananas. Question 12. b. Solve the system algebraically. Tell how many apples and bananas Angelo bought. ________ apples ________ bananas Answer: 14 apples 6 bananas Explanation: y = -x + 20 Substitute y = -x + 20 in 0.5x + 0.75y = 11.5 0.5x + 0.75(-x + 20) = 11.5 0.5x – 0.75x + 15 = 11.5 -0.25x + 15 = 11.5 -0.25x = 11.5 – 15 -0.25x = -3.5 x = -3.5/-0.25 x = 14 14 + y = 20 y = 6 Angelo bought 14 apples and 6 bananas. Question 13. Represent Real-World Problems A jar contains n nickels and d dimes. There is a total of 200 coins in the jar. The value of the coins is$14.00. How many nickels and how many dimes are in the jar?
________ nickels
________ dimes

120 nickels
80 dimes

Explanation:
A jar contains n nickels and d dimes. There is a total of 200 coins in the jar. The value of the coins is $14.00.$14 = 1400 cents
n + d = 200
5n + 10d = 1400
d = -n + 200
5n + 10(-n + 200) = 1400
5n – 10n + 2000 = 1400
-5n = -600
n = -600/-5
n = 120
120 + d = 200
d = 200 – 120
d = 80
There are 120 nickles and 80 dimes in the jar.

Question 14.
Multistep The graph shows a triangle formed by the x-axis, the line 3x−2y=0, and the line x+2y=10. Follow these steps to find the area of the triangle.
a. Find the coordinates of point A by solving the system
$$\left\{\begin{array}{l}3x-2y=0 \\x-2y=10\end{array}\right.$$
Point A: ____________________

Type below:
______________

Point A: (2.5, 3.75)Coordinate of A is (2.5, 3.75)

Explanation:
$$\left\{\begin{array}{l}3x-2y=0 \\x-2y=10\end{array}\right.$$
x = -2y + 10
Substitute x = -2y + 10 in 3x – 2y = 0
3(-2y + 10) -2y = 0
-6y + 30 – 2y = 0
-8y = -30
y = -30/-8 = 3.75
x + 2(3.75) = 10
x + 7.5 = 10
x = 10 – 7.5
x = 2.5
Coordinate of A is (2.5, 3.75)

Question 14.
b. Use the coordinates of point A to find the height of the triangle.
height:__________________
height: $$\frac{□}{□}$$ units

height: 3.75
height: $$\frac{15}{4}$$ units

Explanation:
Height of the triangle is the y coordinate of A
Height = 3.75

Question 14.
c. What is the length of the base of the triangle?
base:________________
base: ______ units

base: 10 units

Explanation:
Length of the base = 10

Question 14.
d. What is the area of the triangle?
A = ______ $$\frac{□}{□}$$ square units

A = 18.75 square units
A = 18 $$\frac{3}{4}$$ square units

Explanation:
Area of the triangle = 1/2 . Height . Base
Area = 1/2 . 3.75 . 10 = 18.75

### Solving Systems by Substitution – Page No. 242

Question 15.
Jed is graphing the design for a kite on a coordinate grid. The four vertices of the kite are at A(−$$\frac{4}{3}$$, $$\frac{2}{3}$$), B($$\frac{14}{3}$$, −$$\frac{4}{3}$$), C($$\frac{14}{3}$$, −$$\frac{16}{3}$$), and D($$\frac{2}{3}$$, −$$\frac{16}{3}$$). One kite strut will connect points A and C. The other will connect points B and D. Find the point where the struts cross.

Type below:
______________

The struts cross as (8/3, 10/3)

Explanation:
1. From AC
Slope = (y2 – y1)/(x2 – x1) = [(-16/3)-(2/3)] ÷ [(14/3) – (-4/3)] = (-18/3) ÷ (18/3) = -1
y = mx + b
2/3 = -1(-4/3) + b
2/3 = 4/3 + b
1. From BD
Slope = (y2 – y1)/(x2 – x1) = [(-16/3)-(-4/3)] ÷ [(2/3) – (144/3)] = (-12/3) ÷ (-12/3) = 1
y = mx + b
-4/3 = 1(14/3) + b
-4/3 = 14/3 + b
-18/3 = b
-6 = b
y = mx + b
y = x -6
3. y = -x -2/3
y = x – 6
4. y = -x – 2/3
x – 6 = -x – 2/3
x = -x – 2/3 + 6
x = – x + 16/3
2x = 16/3
x = 16/6
x = 8/3
then y = x – 6
y = 8/3 – 18/3
y = -10/3
The struts cross as (8/3, 10/3)

FOCUS ON HIGHER ORDER THINKING

Question 16.
Analyze Relationships Consider the system
$$\left\{\begin{array}{l}6x-3y=15 \\x+3y=-8\end{array}\right.$$
Describe three different substitution methods that can be used to solve this system. Then solve the system.
Type below:
______________

Explanation:
As there are three different substitution methods, we can write
Solve for y in the first equation, then substitute that value into the second equation.
Solve for x in the second equation, then substitute that value into the first equation.
Solve either equation for 3y, then substitute that value into the other equation.
From the Second method,
x + 3y = -8
x = -3y – 8
6x – 3y = 15
6 (-3y – 8) -3y = 15
-18y – 48 -3y = 15
-21y – 48 = 15
-21y = 63
y = -3
x + 3y = -8
x + 3(-3) = -8
x – 9 = -8
x = 1

Question 17.
Communicate Mathematical Ideas Explain the advantages, if any, that solving a system of linear equations by substitution has over solving the same system by graphing.
Type below:
______________

The advantage of solving a system of linear equations by graphing is that it is relatively easy to do and requires very little algebra.

Question 18.
Persevere in Problem Solving Create a system of equations of the form
$$\left\{\begin{array}{l}Ax+By=C \\Dx+Ey=F\end{array}\right.$$
that has (7, −2) as its solution. Explain how you found the system.
Type below:
______________

x + y = 5
x – y = 9
solves in :
x = (5+9)/2 = 7
y = 5-9)/2 = -2
A=1, B=2, C= 5
D=1, E= -1, F=9
x = 7
y = -2
IS a system (even if it is a trivial one) of equations so this answer would be acceptable.
The target for a system is to find it SOLUTION SET and not to conclude with x=a and y=b

### Guided Practice – Solving Systems by Elimination – Page No. 248

Question 1.
Solve the system
$$\left\{\begin{array}{l}4x+3y=1 \\x-3y=-11\end{array}\right.$$

Type below:
______________

4x + 3y = 1
x – 3y = -11
4x + 3y = 1
+(x – 3y = -11)
Add to eliminate the variable y
5x + 0y = -10
Simplify and solve for x
5x = -10
Divide both sided by 5
x = -10/5 = -2
Substitute into one of the original equations and solve for y.
4(-2) + 3y = 1
-8 + 3y = 1
3y = 9
y = 9/3 = 3
So, (-2, 3) is the solution of the system.

Solve each system of equations by adding or subtracting.

Question 2.
$$\left\{\begin{array}{l}x+2y=-2 \\-3x+2y=-10\end{array}\right.$$
x = ________
y = ________

x = 2
y = -2

Explanation:
$$\left\{\begin{array}{l}x+2y=-2 \\-3x+2y=-10\end{array}\right.$$
Subtract the equations
x + 2y = -2
-(-3x + 2y = -10)
y is eliminated as it has reversed coefficients. Solve for x
x + 2y + 3x – 2y = -2 + 10
4x = 8
x = 8/4 = 2
Substituting x in either of the equation to find y
2 + 2y = -2
2 + 2y -2 = -2 -2
2y = -4
y = -4/2 = -2

Question 3.
$$\left\{\begin{array}{l}3x+y=23 \\3x-2y=8\end{array}\right.$$
(________ , ________)

(6, 5)

Explanation:
$$\left\{\begin{array}{l}3x+y=23 \\3x-2y=8\end{array}\right.$$
Subtract the equations
3x + y = 23
-(3x – 2y = 8)
x is eliminated as it has reversed coefficients. Solve for y
3x + y – 3x + 2y = 23 – 8
3y = 15
y = 15/3 = 5
Substituting y in either of the equation to find x
3x + 5 = 23
3x + 5 – 5 = 23 – 5
3x = 18
x = 18/3 = 6
Solution is (6, 5)

Question 4.
$$\left\{\begin{array}{l}-4x-5y=7 \\3x+5y=-14\end{array}\right.$$
(________ , ________)

(7, -7)

Explanation:
$$\left\{\begin{array}{l}-4x-5y=7 \\3x+5y=-14\end{array}\right.$$
-4x – 5y = 7
+(3x + 5y = -14)
y is eliminated as it has reversed coefficients. Solve for x
-4x -5y +3x + 5y = 7 -14
-x = -7
x = -7/-1 = 7
Substituting x in either of the equation to find y
3(7) + 5y = -14
21 + 5y -21 = -14 -21
5y = -35
y = -35/5 = -7

Question 5.
$$\left\{\begin{array}{l}x-2y=-19 \\5x+2y=1\end{array}\right.$$
(________ , ________)

(-3, 8)

Explanation:
$$\left\{\begin{array}{l}x-2y=-19 \\5x+2y=1\end{array}\right.$$
x – 2y = -19
+(5x + 2y = 1)
y is eliminated as it has reversed coefficients. Solve for x
x – 2y + 5x + 2y = -19 + 1
6x = -18
x = -18/6 = -3
Substituting x in either of the equation to find y
-3 -2y = -19
-3 -2y + 3 = -19 + 3
-2y = -16
y = -16/-2 = 8

Question 6.
$$\left\{\begin{array}{l}3x+4y=18 \\-2x+4y=8\end{array}\right.$$
(________ , ________)

(2, 3)

Explanation:
$$\left\{\begin{array}{l}3x+4y=18 \\-2x+4y=8\end{array}\right.$$
Subtract the equations
3x + 4y = 18
-(-2x + 4y = 8)
y is eliminated as it has reversed coefficients. Solve for x
3x + 4y + 2x – 4y = 18 – 8
5x = 10
x = 10/5 = 2
Substituting x in either of the equation to find y
3(2) + 4y = 18
6 + 4y – 6 = 18 – 6
4y = 12
y = 12/4 =3
Solution is (2, 3)

Question 7.
$$\left\{\begin{array}{l}-5x+7y=11 \\-5x+3y=19\end{array}\right.$$
(________ , ________)

(-5, -2)

Explanation:
$$\left\{\begin{array}{l}-5x+7y=11 \\-5x+3y=19\end{array}\right.$$
Subtract the equations
-5x + 7y = 11
-(-5x + 3y = 19)
x is eliminated as it has reversed coefficients. Solve for y
-5x + 7y + 5x – 3y = 11 – 19
4y = -8
y = -8/4 = -2
Substituting y in either of the equation to find x
-5x + 7(-2) = 11
-5x -14 + 14 = 11 + 14
-5x = 25
x = 25/-5 = -5
Solution is (-5, -2)

Question 8.
The Green River Freeway has a minimum and a maximum speed limit. Tony drove for 2 hours at the minimum speed limit and 3.5 hours at the maximum limit, a distance of 355 miles. Rae drove 2 hours at the minimum speed limit and 3 hours at the maximum limit, a distance of 320 miles. What are the two speed limits?
a. Write equatios to represent Tony’s distance and Rae’s distance.
Type below:
______________

Tony’s distance: 2x + 3.5y = 355
Rae’s distance: 2x + 3y = 320
where x is the minimum speed and y is the maximum speed.

Question 8.
b. Solve the system.
minimum speed limit:______________
maximum speed limit______________
minimum speed limit: ________ mi/h
maximum speed limit: ________ mi/h

minimum speed limit:55
maximum speed limit70
minimum speed limit: 55mi/h
maximum speed limit: 70mi/h

Explanation:
Subtract the equations
2x + 3.5y = 355
-(2x + 3y = 320)
x is eliminated as it has reversed coefficients. Solve for y
2x + 3.5y – 2x – 3y = 355 – 320
0.5y = 35
y = 35/0.5 = 70
Substituting y in either of the equation to find x
2x + 3(70) = 320
2x + 210 – 210 = 320 – 210
2x = 110
x = 110/2 = 55
Minimum speed limit: 55 miles per hour
Maximum speed limit: 70 miles per hour

ESSENTIAL QUESTION CHECK-IN

Question 9.
Can you use addition or subtraction to solve any system? Explain.
________

No. One of the variables should have the same coefficient in order to add or subtract the system.

### 8.3 Independent Practice – Solving Systems by Elimination – Page No. 249

Question 10.
Represent Real-World Problems Marta bought new fish for her home aquarium. She bought 3 guppies and 2 platies for a total of $13.95. Hank also bought guppies and platies for his aquarium. He bought 3 guppies and 4 platies for a total of$18.33. Find the price of a guppy and the price of a platy.

Guppy: $________ Platy:$ ________

Guppy: $3.19 Platy:$ 2.19

Explanation:
3x + 2y = 13.95
3x + 4y = 18.33
where x is the unit price of guppy and y is the unit price of platy
Subtract the equations
3x + 2y = 13.95
-(3x + 4y = 18.33)
x is eliminated as it has reversed coefficients. Solve for y
3x + 2y – 3x – 4y = 13.95 – 18.33
-2y = -4.38
y = -4.38/-2 = 2.19
Substituting y in either of the equation to find x
3x + 2(2.19) = 13.95
3x + 4.38 – 4.38 = 13.95 – 4.38
3x = 9.57
x = 9.57/3 = 3.19
The price of a guppy is $3.19 and price of a platy is$2.19

Question 11.
Represent Real-World Problems The rule for the number of fish in a home aquarium is 1 gallon of water for each inch of fish length. Marta’s aquarium holds 13 gallons and Hank’s aquarium holds 17 gallons. Based on the number of fish they bought in Exercise 10, how long is a guppy and how long is a platy?
Length of a guppy = ________ inches
Length of a platy = ________ inches

Length of a guppy = 3 inches
Length of a platy = 2 inches

Explanation:
3x + 2y = 13
3x + 4y = 17
where x is the length of guppy and y is the length of a platy
Subtract the equations
3x + 2y = 13
-(3x + 4y = 17)
x is eliminated as it has reversed coefficients. Solve for y
3x + 2y – 3x – 4y = 13 – 17
-2y = -4
y = -4/-2 = 2
Substituting y in either of the equation to find x
3x + 2(2) = 13
3x + 4 – 4 = 13 – 4
3x = 9
x = 9/3 = 3
The length of a guppy is 3 inches and price of a platy is 2 inches

Question 12.
Line m passes through the points (6, 1) and (2, -3). Line n passes through the points (2, 3) and (5, -6). Find the point of intersection of these lines.
Type below:
________________

The intersection of these lines is (3.5, -1.5)

Explanation:
Find the slope of line m = (y2 – y1)/(x2 – x1) where (x2, y2) = (2, -3) and (x1, y1) = (6, 1)
Slope = (-3 -1)/(2 – 6) = -4/-4 = 1
Substitute the value of m and any of the given ordered pair (x, y) in point-slope form of equation: y – y1 = m(x – x1)
y – 1 = 1(x – 6)
y – 1 = x – 6
y = x – 6 + 1
x – y = 5
Find the slope of line n = (y2 – y1)/(x2 – x1) where (x2, y2) = (5, -6) and (x1, y1) = (2, 3)
Slope = (-6 -3)/(5 – 2) = -9/3 = -3
Substitute the value of m and any of the given ordered pair (x, y) in point-slope form of equation: y – y1 = m(x – x1)
y – 3 = -3(x – 2)
y – 3 = -3x + 6
y = -3x + 6 + 3
3x + y = 9
x – y = 5
+(3x + y = 9)
y is eliminated as it has reversed coefficients. Solve for x
x – y + 3x + y = 5 + 9
4x = 14
x = 14/4 = 3.5
Substituting x in either of the equation to find y
3.5 – y = 5
3.5 – y – 3.5 = 5 – 3.5
-y = 1.5
y = -1.5
The intersection of these lines is (3.5, -1.5)

Question 13.
Represent Real-World Problems Two cars got an oil change at the same auto shop. The shop charges customers for each quart of oil plus a flat fee for labor. The oil change for one car required 5 quarts of oil and cost $22.45. The oil change for the other car required 7 quarts of oil and cost$25.45. How much is the labor fee and how much is each quart of oil?
Labor fee: $________ Quart of oil:$ ________

Labor fee: $14.95 Quart of oil:$ 1.5

Explanation:
5x + y = 22.45
7x + y = 25.45
where x is the unit cost of quarts of oil and y is the flat fee for labor
Subtract the equations
5x + y = 22.45
-(7x + y = 25.45)
y is eliminated as it has reversed coefficients. Solve for x
5x + y – 7x – y = 22.45 – 25.45
-2x = -3
x = -3/-2 = 1.5
Substituting x in either of the equation to find y
5(1.5) + y = 22.45
7.5 + y – 7.5 = 22.45 – 7.5
y = 14.95
Labor fee is $14.95 and unit cost of quart of oil is$1.5

Question 14.
Represent Real-World Problems A sales manager noticed that the number of units sold for two T-shirt styles, style A and style B, was the same during June and July. In June, total sales were $2779 for the two styles, with A selling for$15.95 per shirt and B selling for $22.95 per shirt. In July, total sales for the two styles were$2385.10, with A selling at the same price and B selling at a discount of 22% off the June price. How many T-shirts of each style were sold in June and July combined?
________ T-shirts of style A and style B were sold in June and July.

15.95x + 22.95y = 2779
15.95x + 17.9y = 2385.10
where x is number of style A shirt and y is the number of style B shirt
In July, the price of style B shirt is 22% of the price of style B shirt in June, hence 0.78(22.95) = 17.90
Subtract the equations
15.95x + 22.95y = 2779
-(15.95x + 17.9y = 2385.10)
x is eliminated as it has reversed coefficients. Solve for y
15.95x + 22.95 – 15.95x – 17.9y = 2779 – 2385.10
5.05y = 393.9
y = 393.9/5.05 = 78
Substituting y in either of the equation to find x
15.95x +22.95(78) = 2779
15.95x + 1790.1 – 1790.1 = 2779 – 1790.1
15.95x = 988.9
x = 988.9/15.95 = 62
The number of style A T shirt sold in June is 62.
Since the number of T-shirts sold in both numbers is the same, the total number = 2. 62 = 124.
The number of style B T-shirts sold in June is 78.
Since the number of T-shirts sold in both numbers is the same, the total number = 2. 78 = 156.

Question 15.
Represent Real-World Problems Adult tickets to a basketball game cost $5. Student tickets cost$1. A total of $2,874 was collected on the sale of 1,246 tickets. How many of each type of ticket were sold? img 14 ________ adult tickets ________ student tickets Answer: 407 adult tickets 839 student tickets Explanation: x + y = 1246 5x + y = 2874 where x is the number of adult tickets sold and y is the number of student tickets sold. Subtract the equations x + y = 1246 -(5x + y = 2874) y is eliminated as it has reversed coefficients. Solve for x x + y – 5x – y = 1246 – 2874 -4x = -1628 x = -1628/-4 = 407 Substituting x in either of the equation to find y 407 + y = 1246 407 + y – 407 = 1246 – 407 y = 839 The number of adult tickets sold is 407 and student tickets sold is 839. ### FOCUS ON HIGHER ORDER THINKING – Solving Systems by Elimination – Page No. 250 Question 16. Communicate Mathematical Ideas Is it possible to solve the system $$\left\{\begin{array}{l}3x-2y=10 \\x+2y=6\end{array}\right.$$ by using substitution? If so, explain how. Which method, substitution or elimination, is more efficient? Why? ________ Answer: The system can be solved by substitution as x in equation 2 can be isolated. 3x – 2y = 10 x + 2y = 6 Solve the equation for x in the equation. x = -2y + 6 Substitute the expression for x in the other equation and solve. 3(-2y + 6) -2y = 10 -6y + 18 – 2y = 10 -8y + 18 = 10 -8y = -8 y = -8/-8 = 1 Substitute the values of y into one of the equations and solve for the other variable x. x + 2(1) = 6 x = 4 The solution is (4, 1) As the cofficient if variable y is opposite, it will be eliminated and solved for x in less number of steps. Elimination would be more efficient. Question 17. Jenny used substitution to solve the system $$\left\{\begin{array}{l}2x+y=8 \\x-y=1\end{array}\right.$$. Her solution is shown below. Step 1: y = -2x + 8 Solve the first equation for y. Step 2: 2x + (-2x + 8) = 8 Substitute the value of y in an original equation. Step 3: 2x – 2x + 8 = 8 Use the Distributive Property. Step 4: 8 = 8 Simplify. a. Explain the Error Explain the error Jenny made. Describe how to correct it. Type below: ______________ Answer: 2x + y = 8 x – y = 1 Rewritten equation should be substituted in the other original equation Error is that Jenny solved for y in the first equation and substitute it in the original equation. x – (-2x + 8) = 1 3x – 8 = 1 3x = 9 x = 9/3 = 3 x = 3 Question 17. b. Communicate Mathematical Ideas Would adding the equations have been a better method for solving the system? If so, explain why. ________ Answer: Yes Explanation: As the coefficient, if variable y is the opposite, it will be eliminated and solved for x in less number of steps. ### Guided Practice – Solving Systems by Elimination with Multiplication – Page No. 256 Question 1. Solve the system $$\left\{\begin{array}{l}3x-y=8 \\-2x+4y=-12\end{array}\right.$$ by multiplying and adding. Type below: ______________ Answer: $$\left\{\begin{array}{l}3x-y=8 \\-2x+4y=-12\end{array}\right.$$ Multiply each term in the first equation by 4 to get opposite coefficients for the y-terms. 4(3x – y = 8) 12x – 4y = 32 Add the second equation to the new equation 12x – 4y = 32 +(-2x + 4y = -12) Add to eliminate the variable y 10x = 20 Divide both sides by 10 x = 20/10 = 2 Substitue into one of the original equations and solve for y y = 3(2) – 8 = -1 S0, (2, -2)is the solution of the system. Solve each system of equations by multiplying first. Question 2. $$\left\{\begin{array}{l}x+4y=2 \\2x+5y=7\end{array}\right.$$ (________ , ________ ) Answer: (6, -1) Explanation: x + 4y = 2 2x + 5y = 7 To eliminate x terms, multiply the 2nd equation by 2 2(x + 4y = 2) 2x + 8y = 4 Subtract the equations 2x + 8y = 4 -(2x + 5y = 7) x is eliminated as it has reversed coefficients. Solve for y 2x + 8y – 2x – 5y = 4 – 7 3y = -3 y = -3/3 = -1 Substituting y in either of the equation to find x x + 4(-1) = 2 x – 4 + 4 = 2 + 4 x = 6 Solution: (6, -1) Question 3. $$\left\{\begin{array}{l}3x+y=-1 \\2x+3y=18\end{array}\right.$$ (________ , ________ ) Answer: (-3, 8) Explanation: $$\left\{\begin{array}{l}3x+y=-1 \\2x+3y=18\end{array}\right.$$ To eliminate y terms, multiply the 1st equation by 3 3(3x + y = -1) 9x + 3y = -3 Subtract the equations 9x + 3y = -3 -(2x + 3y = 18) y is eliminated as it has reversed coefficients. Solve for x 9x + 3y – 2x – 3y = -3 -18 7x = -21 x = -21/7 x = -3 Substituting x in either of the equation to find y 3(-3) + y = -1 -9 + y + 9 = -1 + 9 y = 8 Solution: (-3, 8) Question 4. $$\left\{\begin{array}{l}2x+8y=21 \\6x-4y=14\end{array}\right.$$ Type below: ______________ Answer: The soultion is (3.5, 1.75) Explanation: $$\left\{\begin{array}{l}2x+8y=21 \\6x-4y=14\end{array}\right.$$ To eliminate y terms, multiply the 2nd equation by 2 2(6x – 4y = 14) 2x + 8y = 21 Add the equations 2x + 8y = 21 +(12x – 8y = 28) y is eliminated it has reversed coefficients. Solve for x 2x + 8y + 12x – 8y = 21 + 28 14x = 49 x = 49/14 = 3.5 Substituting x in either of the equation to find y 6(3.5) – 4y = 14 21 – 4y – 21 = 14 – 21 -4y = -7 y = -7/-4 = 1.75 The soultion is (3.5, 1.75) Question 5. $$\left\{\begin{array}{l}2x+y=3 \\-x+3y=-12\end{array}\right.$$ (________ , ________ ) Answer: Explanation: $$\left\{\begin{array}{l}2x+y=3 \\-x+3y=-12\end{array}\right.$$ To eliminate x terms, multiply the 2nd equation by 2 2(-x + 3y = -12) -2x + 6y = -24 Add the equations 2x + y = 3 +(-2x + 6y = -24) x is eliminated it has reversed coefficients. Solve for y 2x + y – 2x + 6y = 3 – 24 7y = -21 y = -21/7 = -3 Substituting y in either of the equation to find x -x + 3(-3) = -12 -x -9 + 9 = -12 + 9 -x = -3 x = 3 The soultion is (3, -3) Question 6. $$\left\{\begin{array}{l}6x+5y=19 \\2x+3y=5\end{array}\right.$$ (________ , ________ ) Answer: The soultion is (4, -1) Explanation: $$\left\{\begin{array}{l}6x+5y=19 \\2x+3y=5\end{array}\right.$$ To eliminate x terms, multiply the 2nd equation by 3 3(2x + 3y = 5) 6x + 9y = 15 Subtract the equations 6x + 5y = 19 -(6x + 9y = 15) x is eliminated it has reversed coefficients. Solve for y 6x + 5y – 6x – 9y = 19 – 15 -4y = 4 y = 4/-4 = -1 Substituting y in either of the equation to find x 2x + 3(-1) = 5 2x – 3 + 3 = 5 + 3 2x = 8 x = 8/2 = 4 The soultion is (4, -1) Question 7. $$\left\{\begin{array}{l}2x+5y=16 \\-4x+3y=20\end{array}\right.$$ (________ , ________ ) Answer: The soultion is (-2, 4) Explanation: $$\left\{\begin{array}{l}2x+5y=16 \\-4x+3y=20\end{array}\right.$$ To eliminate x terms, multiply the 1st equation by 2 2(2x + 5y = 16) 4x + 10y = 32 Add the equations 4x + 10y = 32 +(-4x + 3y = 20) x is eliminated it has reversed coefficients. Solve for y 10y + 3y = 32 + 20 13y = 52 y = 52/13 = 4 Substituting y in either of the equation to find x 2x + 5(4) = 16 2x + 20 – 20 = 16 – 20 2x = -4 x = -4/2 = -2 The soultion is (-2, 4) Question 8. Bryce spent$5.26 on some apples priced at $0.64 each and some pears priced at$0.45 each. At another store he could have bought the same number of apples at $0.32 each and the same number of pears at$0.39 each, for a total cost of $3.62. How many apples and how many pears did Bryce buy? a. Write equations to represent Bryce’s expenditures at each store First store: _____________ Second store: _____________ Type below: _____________ Answer: First store: 0.64x + 0.45y = 5.26 Second store: 0.32x + 0.39y = 3.62 Explanation: First store = 0.64x + 0.45y = 5.26 Second store = 0.32x + 0.39y = 3.62 where x is the number of apples and y is the number of pears. Question 8. b. Solve the system. Number of apples: _______ Number of pears: _______ Answer: Number of apples: 4 Number of pears: 6 Explanation: First store = 0.64x + 0.45y = 5.26 Second store = 0.32x + 0.39y = 3.62 Multiply by 100 64x + 45y = 526 32x + 39y = 362 To eliminate x terms, multiply the 2nd equation by 2 2(32x + 39y = 362) 64x + 45y = 526 Subtract the equations 64x + 45y = 526 -(64x + 78y = 724) x is eliminated it has reversed coefficients. Solve for y 64x + 45y – 64x – 78y = 526 – 724 -33y = -198 y = -198/-33 = 6 Substituting y in either of the equation to find x 32x + 39(6) = 362 32x + 234 – 234 = 362 – 234 32x = 128 x = 128/32 = 4 He bought 4 apples and 6 pears. ESSENTIAL QUESTION CHECK-IN Question 9. When solving a system by multiplying and then adding or subtracting, how do you decide whether to add or subtract? Type below: _____________ Answer: If the variable with the same coefficient but reversed sign, we add and if they have the same sign, we subtract. ### Solving Systems by Elimination with Multiplication – Page No. 257 Question 10. Explain the Error Gwen used elimination with multiplication to solve the system $$\left\{\begin{array}{l}2x+6y=3 \\x-3y=-1\end{array}\right.$$ Her work to find x is shown. Explain her error. Then solve the system. 2(x − 3y) = -1 2x − 6y = -1 +2x + 6y = 3 _____________ 4x + 0y = 2 x = $$\frac{1}{2}$$ Type below: ____________ Answer: 2x + 6y = 3 x – 3y = -1 To eliminate x terms, multiply the 2nd equation by 2 2(x – 3y = -1) 2x – 6y = -2 Error is the Gnew did not multiply the entire expression with 2. Add the equations 2x + 6y = 3 +(2x – 6y = -2) y is eliminated it has reversed coefficients. Solve for x 2x + 6y + 2x – 6y = 3 – 2 4x = 1 x = 1/4 Substituting x in either of the equation to find y x – 3y = -1 1/4 – 3y – 1/4 = -1 -1/4 -3y = -5/4 y = -5/4(-3) = 5/12 Question 11. Represent Real-World Problems At Raging River Sports, polyester-fill sleeping bags sell for$79. Down-fill sleeping bags sell for $149. In one week the store sold 14 sleeping bags for$1,456.
a. Let x represent the number of polyester-fill bags sold and let y represent the number of down-fill bags sold. Write a system of equations you can solve to find the number of each type sold.

Type below:
____________

x + y = 14
79x + 149y = 1456
where x is the polyster-fill bags and y is the number of down-fill bags

Question 11.
b. Explain how you can solve the system for y by multiplying and subtracting.
Type below:
____________

x + y = 14
79x + 149y = 1456
Multiply the second equation by 79. Subtract the new equation from the first equation and solve the resulting equation for y.

Question 11.
c. Explain how you can solve the system for y using substitution.
Type below:
____________

Solve the second equation for x. Substitute the expression for x , in the first equation and solve the resulting equation for y.

Question 11.
d. How many of each type of bag were sold?
_______ polyester-fill
_______ down-fill

9 polyester-fill
5 down-fill

Explanation:
x + y = 14
79x + 149y = 1456
To eliminate x terms, multiply the 2nd equation by 2
79(x + y = 14)
79x + 149y = 1456
Subtract the equations
79x + 79y = 1106
-(79x + 149y = 1456)
x is eliminated it has reversed coefficients. Solve for y
79x + 79y – 79x – 149y = 1106 – 1456
-70y = -350
y = -350/-70 = 5
Substituting y in either of the equation to find x
x + 5 = 14
x = 14 – 5
x = 9
There were 9 polyster-fill bags and 5 down-fill bags sold.

Question 12.
Twice a number plus twice a second number is 310. The difference between the numbers is 55. Find the numbers by writing and solving a system of equations. Explain how you solved the system.
x = _______
y = _______

x = 105
y = 50

Explanation:
2x + 2y = 310
x – y = 55
To eliminate y terms, multiply the 2nd equation by 2
2(x – y = 55)
2x – 2y = 110
2x + 2y = 310
+ (2x – 2y = 110)
y is eliminated it has reversed coefficients. Solve for x
2x + 2y + 2x – 2y = 310 + 110
4x = 420
x = 420/4 = 105
Substituting x in either of the equation to find y
105 – y = 55
y = 105 – 55
y = 50
The solution is (105, 50)

### Solving Systems by Elimination with Multiplication – Page No. 258

Question 13.
Represent Real-World Problems A farm stand sells apple pies and jars of applesauce. The table shows the number of apples needed to make a pie and a jar of applesauce. Yesterday, the farm picked 169 Granny Smith apples and 95 Red Delicious apples. How many pies and jars of applesauce can the farm make if every apple is used?

_______ pies
_______ jars of applesauce

21 pies
16 jars of applesauce

Explanation:
5x + 4y = 169
3x + 2y = 95
where x is the number of apples needed for pie and y is the number of apples for jar of applesauce
To eliminate y terms, multiply the 2nd equation by 2
2(3x + 2y = 95)
6x + 4y = 190
Subtract the equations
5x + 4y = 169
– (6x + 4y = 190)
y is eliminated it has reversed coefficients. Solve for x
5x + 4y – 6x – 4y = 169 – 190
-x = -21
x = -21/-1 = 21
Substituting x in either of the equation to find y
5(21) + 4y = 169
105 + 4y – 105 = 169 – 105
4y = 64
y = 64/4 = 16
The number of apples needed for pie is 21 and the number of apples for jar of applesauce is 16.

FOCUS ON HIGHER ORDER THINKING

Question 14.
Make a Conjecture Lena tried to solve a system of linear equations algebraically and in the process found the equation 5 = 9. Lena thought something was wrong, so she graphed the equations and found that they were parallel lines. Explain what Lena’s graph and equation could mean.
Type below:
____________

Lena’s graph is a parallel line which means the graph does not intersect each other, hence they have no solutions. Equation 5 = 9 means variables are eliminated and this statement is not true. This linear system has no solution.

Question 15.
Consider the system
$$\left\{\begin{array}{l}2x+3y=6 \\3x+7y=-1\end{array}\right.$$
a. Communicate Mathematical Ideas Describe how to solve the system by multiplying the first equation by a constant and subtracting. Why would this method be less than ideal?
Type below:
____________

Multiplying the first equation by a constant and subtracting
2x + 3y = 6
3x + 7y = -1
Multiply the first equation by 1.5 and subtract. This would be less than ideal because you would introduce decimals into the solution process.

Question 15.
b. Draw Conclusions Is it possible to solve the system by multiplying both equations by integer constants? If so, explain how.
Type below:
____________

Yes

Explanation:
Multiply the first equation by 3 and the second equation by 2. Both x-term coefficients would be 6. Solve by eliminating the x-terms using subtraction.

Question 15.
(_______ , _______)

(9, -4)

Explanation:
2x + 3y = 6
3x + 7y = -1
Multiply the first equation by 3 and the second equation by 2.
3(2x + 3y = 6)
2(3x + 7y = -1)
Subtract the equations
6x + 9y = 18
-(6x + 14y = -2)
x is eliminated it has reversed coefficients. Solve for y
6x + 9y – 6x – 14y = 18 + 2
-5y = 20
y = 20/-5 = -4
Substituting y in either of the equation to find x
2x + 3(-4) = 6
2x = 18
x = 18/2 = 9
The solution is (9, -4)

### Guided Practice – Solving Solving Special Systems – Page No. 262

Use the graph to solve each system of linear equations

Question 1.
A. $$\left\{\begin{array}{l}4x-2y=-6 \\2x-y=4\end{array}\right.$$
B. $$\left\{\begin{array}{l}4x-2y=-6 \\x+y=6\end{array}\right.$$
C. $$\left\{\begin{array}{l}2x-y=4 \\6x-3y=-12\end{array}\right.$$
STEP 1 Decide if the graphs of the equations in each system intersect, are parallel, or are the same line.

System A: The graphs __________
System B: The graphs __________
System C: The graphs __________

System A: The graphs are parallel
System B: The graphs are intersecting
System C: The graphs are  the same line

Explanation:
System A: 4x – 2y = -6
2x – y = 4
System B: 4x – 2y = -6
x + y = 6
System C: 2x – y = 4
6x – 3y = 12

Question 1.
STEP 2 Decide how many points the graphs have in common.
a. Intersecting lines have _______________ point(s) in common.
b. Parallel lines have _______________ point(s) in common.
c. The same lines have ___________ point(s) in common.
a. __________
b. __________
c. __________

a. Intersecting lines have one point(s) in common.
b. Parallel lines have no point(s) in common.
c. The same lines have infinitely many points (s) in common.

Explanation:
From the graphs,
Intersecting lines have one point(s) in common
Parallel lines have no point(s) in common
The same lines have infinitely many points (s) in common

Question 1.
STEP 3 Solve each system.
System A has __________ points in common, so it has __________ solution.
System B has __________ point in common. That point is the solution, __________.
System C has __________ points in common. ________ ordered pairs on the line will make both equations true.
Type below:
___________

System A has no points in common, so it has no solution. System B has one point in common. That point is the solution, (1,5). System C has an infinite number of points in common. All ordered pairs on the line will make both equations true.

Explanation:
Number of solutions for each system
System A has no points in common, so it has no solution. System B has one point in common. That point is the solution, (1,5). System C has an infinite number of points in common. All ordered pairs on the line will make both equations true.

Solve each system. Tell how many solutions each system has.

Question 2.
$$\left\{\begin{array}{l}x-3y=4 \\-5x+15y=-20\end{array}\right.$$
___________

infinitely many solutions

Explanation:
x – 3y = 4
-5x + 15y = -20
To eliminate y terms, multiply the 1st equation by 5
5(x – 3y = 4)
5x – 15y = 20
5x – 15y = 20
+(-5x + 15y = -20)
x and y is eliminated as it has reversed coefficients.
5x – 15y – 5x + 15y = 20 – 20
0 = 0
The statement is true, hence the solution has infinitely many solutions.

Question 3.
$$\left\{\begin{array}{l}6x+2y=-4 \\3x+y=4\end{array}\right.$$
___________

no solution

Explanation:
6x + 2y = -4
3x + y = 4
To eliminate y terms, multiply the 2nd equation by 5
2(3x + y = 4)
6x + 2y = 8
Subtract the equations
6x + 2y = -4
-(6x + 2y = 8)
x and y is eliminated as it has reversed coefficients.
6x + 2y – 6x – 2y = -4 -8
0 = -12
The statement is false, hence the solution has no solution.

Question 4.
$$\left\{\begin{array}{l}6x-2y=-10 \\3x+4y=-25\end{array}\right.$$
___________

one solution

Explanation:
6x – 2y = -10
3x + 4y = -25
To eliminate y terms, multiply the 1st equation by 2
2(6x – 2y = -10)
12x – 4y = -20
12x – 4y = -20
+(3x + 4y = -25)
y is eliminated as it has reversed coefficients. Solve for x.
12x – 4y + 3x + 4y = -20 – 25
15x = -45
x = -45/15 = -3
Substitute x in any one of the original equations and solve for y
3(-3) + 4y = -25
-9 + 4y + 9 = -25 + 9
4y = -16
y = -16/4
y = -4
There is one solution, (-3, -4)

ESSENTIAL QUESTION CHECK-IN

Question 5.
When you solve a system of equations algebraically, how can you tell whether the system has zero, one, or an infinite number of solutions?
Type below:
___________

When x and y are eliminated and the statement is true, the system has infinitely many solutions.
When x and y are eliminated and the statement is false, the system has no solutions.
When the system has one solution by solving, the system has one solution.

### 8.5 Independent Practice – Solving Solving Special Systems – Page No. 263

Question 6.
$$\left\{\begin{array}{l}-2x+6y=12 \\x-3y=3\end{array}\right.$$

Solution: ______________
___________

$$\left\{\begin{array}{l}-2x+6y=12 \\x-3y=3\end{array}\right.$$
Graph the equations on same coordinate plane
No solution as equations are parallel

To eliminate y terms, multiply the 2nd equation by 2
2(x – 3y = 3)
2x – 6y = 6
-2x + 6y = 12
2x – 6y = 6
x and y is eliminated as it has reversed coefficients.
-2x + 6y + 2x – 6y = 12 + 6
0 = 18
The statement is false, hence the system has no solution.

Question 7.
$$\left\{\begin{array}{l}15x+5y=5 \\3x+y=1\end{array}\right.$$

Solution: ______________
___________

$$\left\{\begin{array}{l}15x+5y=5 \\3x+y=1\end{array}\right.$$
Graph the equations on same coordinate plane

Infinitely many solutions as equations are overlapping
To eliminate y terms, multiply the 2nd equation by 5
5(3x + y = 1)
15x + 5y = 5
Subtarct the equations
15x + 5y = 5
-(15x + 5y = 5)
x and y is eliminated as it has reversed coefficients.
15x + 5y -15x – 5y = 5 – 5
0 = 0
The statement is true, hence the system has infinitely many solutions.

For Exs. 8–

14, state the number of solutions for each system of linear equations

Question 8.
a system whose graphs have the same slope but different y-intercepts
___________

No solutions

Explanation:
Equations are parallel
No solutions

Question 9.
a system whose graphs have the same y-intercepts but different slopes
___________

One solution

Explanation:
Equations are intersecting
One solution

Question 10.
a system whose graphs have the same y-intercepts and the same slopes
___________

Infinitely many solutions

Explanation:
Equations are overlapping
Infinitely many solutions

Question 11.
a system whose graphs have different y-intercepts and different slopes
___________

One solution

Explanation:
Equations are intersecting
One solution

Question 12.
the system
$$\left\{\begin{array}{l}y=2 \\y=-3\end{array}\right.$$
___________

No solutions

Explanation:
Equations are parallel
No solutions

Question 13.
the system
$$\left\{\begin{array}{l}y=2 \\y=-3\end{array}\right.$$
___________

One solution

Explanation:
Equations are intersecting
One solution

Question 14.
the system whose graphs were drawn using these tables of values:

___________

No solutions

Explanation:
Equations are parallel The slope is the same for both equations but the y-intercept is different.
No solutions

Question 15.
Draw Conclusions The graph of a linear system appears in a textbook. You can see that the lines do not intersect on the graph, but also they do not appear to be parallel. Can you conclude that the system has no solution? Explain.
___________

No; although the lines do not intersect on the graph, they intersect at a point that is not on the graph. To prove that a system has no solution, you must do so algebraically

### Solving Solving Special Systems – Page No. 264

Question 16.
Represent Real-World Problems Two school groups go to a roller skating rink. One group pays $243 for 36 admissions and 21 skate rentals. The other group pays$81 for 12 admissions and 7 skate rentals. Let x represent the cost of admission and let y represent the cost of a skate rental. Is there enough information to find values for x and y? Explain.

___________

36x + 21y = 243
12x + 7y = 81
where x is the cost of admission and y is the cost of stake rentals.
Although the information can be used to develop a system of linear equation, where each equation has two variables when the system is solved, the number of solutions is infinite, Mee the values of x and y cannot be determined.
No

Question 17.
Represent Real-World Problems Juan and Tory are practicing for a track meet. They start their practice runs at the same point, but Tory starts 1 minute after Juan. Both run at a speed of 704 feet per minute. Does Tory catch up to Juan? Explain.
___________

No; Both Juan and Tory-run at the same rate, so the lines representing the distances each has run are parallel. There is no solution to the system

FOCUS ON HIGHER ORDER THINKING

Question 18.
Justify Reasoning A linear system with no solution consists of the equation y = 4x − 3 and a second equation of the form y = mx + b. What can you say about the values of m and b? Explain your reasoning.
Type below:
___________

y = 4x – 3
y = mx + b
Since the system has no solutions, the two equations are parallel. The value of the slope, m would be the same i.e. 4. The value of y-intercept, b can be any number except -3 as b is different for parallel lines.

Question 19.
Justify Reasoning A linear system with infinitely many solutions consists of the equation 3x + 5 = 8 and a second equation of the form Ax + By = C. What can you say about the values of A, B, and C? Explain your reasoning.
Type below:
___________

3x + 5 = 8
Ax + By = C
Since the system has infinitely many solutions, the values of A, B, and C must all be the same multiple of 3, 5, and 8, respectively. The two equations represent a single line, so the coefficients and constants of one equation must be a multiple of the other.

Question 20.
Draw Conclusions Both the points (2, -2) and (4, -4) are solutions of a system of linear equations. What conclusions can you make about the equations and their graphs?
Type below:
___________

If a system has more than one solution, the equations represent the same line and have infinitely many solutions.

### Ready to Go On? – Model Quiz – Page No. 265

8.1 Solving Systems of Linear Equations by Graphing

Solve each system by graphing.

Question 1.
$$\left\{\begin{array}{l}y=x-1 \\y=2x-3\end{array}\right.$$

(________ , ________)

(2, 1)

Explanation:
y = x – 1
y = 2x – 3
Graph the equations on the same coordinate plane

The solution of the system is the point of intersection
The solution is (2, 1)

Question 2.
$$\left\{\begin{array}{l}x+2y=1 \\-x+y=2\end{array}\right.$$

(________ , ________)

(-1, 1)

Explanation:
x + 2y = 1
-x + y = 2
Graph the equations on same coordinate plane

The solution of the system is the point of intersection
The solution is (-1, 1)

8.2 Solving Systems by Substitution

Solve each system of equations by substitution.

Question 3.
$$\left\{\begin{array}{l}y=2x \\x+y=-9\end{array}\right.$$
(________ , ________)

(-3, -6)

Explanation:
y = 2x
x + y = -9
Substitute y from equation 1 in the other equation.
x + 2x = -9
3x = -9
x = -9/3
x = -3
Then, y = 2(-3) = -6
The Solution is (-3, -6)

Question 4.
$$\left\{\begin{array}{l}3x-2y=11 \\x+2y=9\end{array}\right.$$
(________ , ________)

(5, 2)

Explanation:
3x – 2y = 11
x + 2y = 9
Solve for x in equation 2
x = – 2y + 9
Substitute x from equation 2 in the other equation
3(-2y + 9) – 2y = 11
-6y + 27 -2y = 11
-8y = -16
y = -16/-8 = 2
Substitute y in any of the equations to find x
x + 2(2) = 9
x + 4 – 4 = 9 – 4
x = 5
The solution is (5, 2)

8.3 Solving Systems by Elimination

Solve each system of equations by adding or subtracting.

Question 5.
$$\left\{\begin{array}{l}3x+y=9 \\2x+y=5\end{array}\right.$$
(________ , ________)

(4, -3)

Explanation:
$$\left\{\begin{array}{l}3x+y=9 \\2x+y=5\end{array}\right.$$
Subtract the equations
3x + y = 9
-(2x + y = 5)
y is eliminated as it has reversed coefficients. Solve for x
3x + y – 2x – y = 9 – 5
x = 4
Substituting x in either of the equation to find y
2(4) + y = 5
8 + y – 8 = 5 – 8
y = -3
The solution is (4, -3)

Question 6.
$$\left\{\begin{array}{l}-x-2y=4 \\3x+2y=4\end{array}\right.$$
(________ , ________)

(4, -4)

Explanation:
$$\left\{\begin{array}{l}-x-2y=4 \\3x+2y=4\end{array}\right.$$
-x – 2y = 4
+(3x + 2y = 4)
y is eliminated as it has reversed coefficients. Solve for x
-x – 2y + 3x + 2y = 4 + 4
2x = 8
x = 8/2 = 4
Substituting x in either of the equation to find y
3(4) + 2y = 4
12 + 2y – 12 = 4 – 12
2y = -8
y = -8/2 = -4
The solution is (4, -4)

8.4 Solving Systems by Elimination with Multiplication

Solve each system of equations by multiplying first.

Question 7.
$$\left\{\begin{array}{l}x+3y=-2 \\3x+4y=-1\end{array}\right.$$
(________ , ________)

(1, -1)

Explanation:
$$\left\{\begin{array}{l}x+3y=-2 \\3x+4y=-1\end{array}\right.$$
Subtract the equations
3x + 9y = -6
-(3x + 4y = -1)
x is eliminated as it has reversed coefficients. Solve for y
3x + 9y – 3x – 4y = -6 + 1
5y = -5
y = -5/5
y = -1
Substituting y in either of the equation to find x
x + 3(-1) = -2
x – 3 = -2
x = -2 + 3
x = 1
The solution is (1, -1)

Question 8.
$$\left\{\begin{array}{l}2x+8y=22 \\3x-2y=5\end{array}\right.$$
(________ , ________)

(3, 2)

Explanation:
$$\left\{\begin{array}{l}2x+8y=22 \\3x-2y=5\end{array}\right.$$
Multiply equation 2 by 4 so that y can be eliminated
4(3x – 2y = 5)
12x – 8y = 20
2x + 8y = 22
+(12x – 8y = 20)
y is eliminated as it has reversed coefficients. Solve for x
2x + 8y + 12x – 8y = 22 + 20
14x = 42
x = 42/14
x = 3
Substituting y in either of the equation to find x
2(3) + 8y = 22
6 + 8y = 22
8y = 22 – 6
8y = 16
y = 16/8
y = 2
The solution is (3, 2)

8.5 Solving Special Systems

Solve each system. Tell how many solutions each system has.

Question 9.
$$\left\{\begin{array}{l}-2x+8y=5 \\x-4y=-3\end{array}\right.$$
_____________

no solution

Explanation:
$$\left\{\begin{array}{l}-2x+8y=5 \\x-4y=-3\end{array}\right.$$
Multiply equation 2 by 2 so that y can be eliminated
2(x – 4y = -3)
2x – 8y = -6
-2x + 8y = 5
+(2x – 8y = -6)
x and y is eliminated
-2x + 8y + 2x – 8y = 5 – 6
0 = -1
The statement is false. Hence, the system has no solution.

Question 10.
$$\left\{\begin{array}{l}6x+18y=-12 \\x+3y=-2\end{array}\right.$$
_____________

infinitely many solutions

Explanation:
$$\left\{\begin{array}{l}6x+18y=-12 \\x+3y=-2\end{array}\right.$$
Multiply equation 2 by 6 so that x can be eliminated
6(x + 3y = -2)
6x + 18y = -12
Subtract the equations
6x + 18y = -12
-(6x + 18y = -12)
x and y is eliminated
6x + 18y -6x -18y = -12 + 12
0 = 0
The statement is true. Hence, the system has infinitely many solutions.

ESSENTIAL QUESTION

Question 11.
What are the possible solutions to a system of linear equations, and what do they represent graphically?
Type below:
___________

System of linear equations can have no solution, which is represented by parallel lines; one solution, which is represented by intersecting lines; and infinitely many solutions, which is represented by overlapping lines.

### Selected Response – Mixed Review – Page No. 266

Question 1.
The graph of which equation is shown?

Options:
A. y = −2x + 2
B. y = −x + 2
C. y = 2x + 2
D. y = 2x + 1

C. y = 2x + 2

Explanation:
Option A and B are eliminated as the slope of the graph is 2.
Option D is eliminated as the y-intercept from the graph should be 2.
Option C is the equation of the graph

Question 2.
Which best describes the solutions to the system
$$\left\{\begin{array}{l}x+y=-4 \\-2x-2y=0\end{array}\right.$$
Options:
A. one solution
B. no solution
C. infinitely many
D. (0, 0)

B. no solution

Explanation:
$$\left\{\begin{array}{l}x+y=-4 \\-2x-2y=0\end{array}\right.$$
Multply equation 1 by 2 so that x can be eliminated
2(x + y = -4)
2x + 2y = -8
2x + 2y = -8
-2x – 2y = 0
x and y is eliminated
2x + 2y – 2x -2y = -8 + 0
0 = -8
The statement is false. Hence, the system has no solution.

Question 3.
Which of the following represents 0.000056023 written in scientific notation?
Options:
A. 5.6023 × 105
B. 5.6023 × 104
C. 5.6023 × 10-4
D. 5.6023 × 10-5

D. 5.6023 × 10-5

Explanation:
Move the decimal 5 points right to get the equation.
D. 5.6023 × 10-5

Question 4.
Which is the solution to
$$\left\{\begin{array}{l}2x-y=1 \\4x+y=11\end{array}\right.$$
Options:
A. (2, 3)
B. (3, 2)
C. (-2, 3)
D. (3, -2)

A. (2, 3)

Explanation:
$$\left\{\begin{array}{l}2x-y=1 \\4x+y=11\end{array}\right.$$
2x – y = 1
4x + y = 11
y is eliminated as it has reversed coefficients. Solve for x.
2x – y + 4x + y = 1 + 11
6x = 12
x = 12/6 = 2
Substituting x in either of the equation to find y
4(2) + y = 11
8 + y = 11
y = 11 – 8
y = 3
The solution is (2, 3)

Question 5.
Which expression can you substitute in the indicated equation to solve
$$\left\{\begin{array}{l}3x-y=5 \\x+2y=4\end{array}\right.$$
Options:
A. 2y – 4 for x in 3x – y = 5
B. 4 – x for y in 3x – y = 5
C. 3x – 5 for y in 3x – y = 5
D. 3x – 5 for y in x + 2y = 4

D. 3x – 5 for y in x + 2y = 4

Explanation:
$$\left\{\begin{array}{l}3x-y=5 \\x+2y=4\end{array}\right.$$
Solve for y in equation 1
y = 3x – 5
Substitute in other equation x + 2y = 4

Question 6.
What is the solution to the system of linear equations shown on the graph?

Options:
A. -1
B. -2
C. (-1, -2)
D. (-2, -1)

C. (-1, -2)

Explanation:
The point of intersection is (-1, -2), which is the solution of the system

Question 7.
Which step could you use to start solving
$$\left\{\begin{array}{l}x-6y=8 \\2x-5y=3\end{array}\right.$$
Options:
A. Add 2x – 5y = 3 to x – 6y = 8.
B. Multiply x – 6y = 8 by 2 and add it to 2x – 5y = 3.
C. Multiply x – 6y = 8 by 2 and subtract it from 2x – 5y = 3.
D. Substitute x = 6y – 8 for x in 2x – 5y = 3.

C. Multiply x – 6y = 8 by 2 and subtract it from 2x – 5y = 3.

Explanation:
x – 6y = 8
2x – 5y = 3
Multiply the 1st equation by 2 so that the coefficient of variable x is the same in both equations
Subtract the equations as x has the same sign.

Question 8.
A hot-air balloon begins rising from the ground at 4 meters per second at the same time a parachutist’s chute opens at a height of 200 meters. The parachutist descends at 6 meters per second.
a. Define the variables and write a system that represents the situation.
Type below:
_____________

y represents the distance from the ground and x represents the time in seconds
y = 4x
y = -6x + 200

Question 8.
b. Find the solution. What does it mean?
Type below:
_____________

Substitute y from the equation 1 in the equation 2
4x = -6x + 200
4x + 6x = -6x + 200 + 6x
10x = 200
x = 200/10 = 20
Substitute x in any one of the equations and solve for x
y = 4(20) = 80
The solution is (20, 80)
The ballon and parachute meets after 20sec at 80m from the ground.

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Lesson 1: Lines, Rays, and Angles

Lesson 2: Classify Triangles by Angles

Lesson 3: Parallel Lines and Perpendicular Lines

Mid-Chapter Checkpoint

Lesson 5: Line Symmetry

Lesson 6: Find and Draw Lines of Symmetry

Lesson 7: Problem Solving • Shape Patterns

Review/Test

### Common Core – New – Page No. 553

Lines, Rays, and Angles

Draw and label an example of the figure.

Question 1.
obtuse ∠ABC

Think: An obtuse angle is greater than a right angle. The middle letter, B, names the vertex of the angle.
_________

Explanation:
An obtuse angle is greater than a right angle. The middle letter, B, names the vertex of the angle.

Question 2.
$$\overrightarrow{G H}$$
_________

Explanation:
GH is a ray that has one endpoint and continues without an end in one direction.

Question 3.
acute ∠JKL
_________

Explanation:
Angle JKL is an acute angle that is less than a right angle.

Question 4.
$$\overline{B C}$$

Explanation:
BC is a line that continues without an end in both directions.

Use the figure for 5–8.

Question 5.
Name a line segment.

line segment EF

Explanation:

EF line is a straight path of points that continues without an end in both directions.

Question 6.
Name a right angle.
∠ _____

∠EJF

Explanation:
EJF is a right angle that forms a square corner.

Question 7.
Name an obtuse angle.
obtuse ∠ _____

∠CEJ

Explanation:
CEJ is an obtuse angle that is greater than a right angle.

Question 8.
Name a ray.

Ray JD

Explanation:
JD is a ray that has one endpoint and continues without an end in one direction.

Problem Solving

Use the figure at the right for 9–11.

Question 9.
Classify ∠AFD
_________

Obtuse Angle

Explanation:
AFD is an obtuse angle that is greater than a right angle.

Question 10.
Classify ∠CFE.
_________

Right Angle

Explanation:
∠CFE is a right angle that forms a square corner.

Question 11.
Name two acute angles.
acute ∠ _____ acute ∠ _____

∠AFB and ∠DFE

Explanation:
∠AFB and ∠DFE are two acute angles with less than a right angle.

### Common Core – New – Page No. 554

Lesson Check

Question 1.
The hands of a clock show the time 12:25.

Which best describes the angle between the hands of the clock?
Options:
a. acute
b. right
c. obtuse
d. straight

c. obtuse

Explanation:
The hands of the time 12:25 are forming greater than a right angle. So, the answer is the Obtuse angle.

Question 2.
Which of the following name two different figures?
Options:
a. $$\overline{A B} \text { and } \overline{B A}$$
b. $$\stackrel{\longleftrightarrow}{A B}$$ and $$\stackrel{\longleftrightarrow}{B A}$$
c. $$\overrightarrow{A B} \text { and } \overrightarrow{B A}$$
d. ∠ABC and ∠CBA

c. $$\overrightarrow{A B} \text { and } \overrightarrow{B A}$$

Explanation:
In $$\overrightarrow{A B}$$, A is an end point and B continues without end in one direction.
In $$\overrightarrow{B A}$$, B is an end point and A continues without an end in one direction.

Spiral Review

Question 3.
Jan’s pencil is 8.5 cm long. Ted’s pencil is longer. Which could be the length of Ted’s pencil?
Options:
a. 0.09 cm
b. 0.8 cm
c. 8.4 cm
d. 9.0 cm

d. 9.0 cm

Explanation:
9 ones is greater than 8 ones. So, 9.0 cm > 8.5 cm

Question 4.
Kayla buys a shirt for $8.19. She pays with a$10 bill. How much change should she receive?
Options:
a. $1.81 b.$1.89
c. $2.19 d.$2.81

a. $1.81 Explanation: Kayla buys a shirt for$8.19. She pays with a $10 bill. To find the change she received,$10 – $8.19 = 1.81 Question 5. Sasha donated $$\frac{9}{100}$$ of her class’s entire can collection for the food drive. Which decimal is equivalent to $$\frac{9}{100}$$ ? Options: a. 9 b. 0.99 c. 0.9 d. 0.09 Answer: d. 0.09 Explanation: $$\frac{9}{100}$$ is 9 hundredths. So, the decimal is 0.09. Question 6. Jose jumped 8 $$\frac{1}{3}$$ feet. This was 2 $$\frac{2}{3}$$ feet farther than Lila jumped. How far did Lila jump? Options: a. 5 $$\frac{1}{3}$$ b. 5 $$\frac{2}{3}$$ c. 6 $$\frac{1}{3}$$ d. 11 Answer: b. 5 $$\frac{2}{3}$$ Explanation: Jose jumped 8 $$\frac{1}{3}$$ feet. This was 2 $$\frac{2}{3}$$ feet farther than Lila jumped. 8 $$\frac{1}{3}$$ – 2 $$\frac{2}{3}$$ = $$\frac{25}{3}$$ – $$\frac{8}{3}$$ = $$\frac{7}{3}$$ = 5 $$\frac{2}{3}$$ ### Page No. 557 Question 1. Name the triangle. Tell whether each angle is acute, right, or obtuse. A name for the triangle is __________ . Name: ∠F is _________ ∠G is _________ ∠H is _________ Answer: Right Triangle; Triangle FGH; ∠F and ∠H are acute angles. ∠G is Right angle Explanation: ∠F and ∠H are acute angles with less than a right angle. ∠G is the Right angle that forms a square corner. A triangle that has one right angle is called a right triangle. Classify each triangle. Write acute, right, or obtuse. Question 2. _____ Answer: Obtuse triangle; Angle B and Angle C are both acute. Angle A is obtuse. Explanation: From triangle ABC, Angle B, and Angle C are both acute with less than a right angle. Angle A is obtuse angle that is greater than a right angle. Question 3. _____ Answer: Obtuse triangle; Angle A and Angle C are both acute. Angle B is obtuse. Explanation: From triangle ABC, Angle A, and Angle C are both acute with less than a right angle. Angle B is an obtuse angle that is greater than a right angle. A triangle with an obtuse angle is called an obtuse triangle. Question 4. _____ Answer: Acute triangle; Angle A, Angle B, and Angle C are acute angles. Explanation: From triangle ABC, Angle A, Angle B, and Angle C are acute angles with less than a right angle. A triangle with three acute angles called an acute triangle. So, the given triangle is an acute triangle. Question 5. _____ Answer: Right Triangle; Triangle ABC; ∠A and ∠C are acute angles. ∠B is Right angle Explanation: ∠A and ∠C are acute angles with less than a right angle. ∠B is the Right angle that forms a square corner. A triangle that has one right angle is called a right triangle. Question 6. _____ Answer: Acute triangle; Angle A, Angle B, and Angle C are acute angles. Explanation: From triangle ABC, Angle A, Angle B, and Angle C are acute angles with less than a right angle. A triangle with three acute angles called an acute triangle. So, the given triangle is an acute triangle. Question 7. _____ Answer: Right Triangle; ∠A and ∠C are acute angles. ∠B is Right angle Explanation: ∠A and ∠C are acute angles with less than a right angle. ∠B is the Right angle that forms a square corner. A triangle that has one right angle is called a right triangle. Question 8. Cross out the figure that does not belong. Explain. Type below: _________ Answer: Explanation: From the given image, 1, 3, and 4 have two acute angles, and one obtuse angle. 2 have three acute angles. ### Page No. 558 Use the Venn diagram for 9–10. Question 9. Which triangles do NOT have an obtuse angle? Explain. _______ triangles Answer: 4 triangles; Triangle DEF, Triangle SPN, Triangle ABC, and Triangle GHP are don’t have an obtuse angle. Triangle DEF, Triangle SPN are acute angles. An acute triangle is a triangle with three acute angles. Triangle ABC, and Triangle GHP are right angles. A right triangle is a triangle with one right angle. The sum of the triangle is 180 degrees. A right triangle has 90 degrees. So, the remaining angles must be acute angles. Question 10. How many triangles have at least two acute angles? Explain. _______ triangles Answer: 4 triangles; Triangle DEF, Triangle SPN, Triangle ABC, and Triangle GHP at least two acute angles. Triangle DEF, Triangle SPN are acute angles. An acute triangle is a triangle with three acute angles. Triangle ABC, and Triangle GHP are right angles. A right triangle is a triangle with one right angle and two acute angles. Question 11. Use the square shown at the right. Draw a line segment from point M to point P. Name and classify the triangles formed by the line segment. Type below: _________ Answer: Angle MNP and Angle MQP Explanation: The line segment from M to P forms Angle MNP and Angle MQP. Question 12. Write the letter of the triangle under its correct classification. Type below: _________ Answer: Explanation: Triangle A and triangle B have three acute angles. So, they are acute triangles. Triangle D and triangle F have one obtuse angle. So, they are obtuse triangles. Triangle C and triangle E have one right angle. So, they are right triangles. ### Common Core – New – Page No. 559 Classify Triangles Classify each triangle. Write acute, right, or obtuse. Question 1. Think: Angles A and C are both acute. Angle B is obtuse. Answer: Obtuse triangle; Angle A and Angle C are both acute. Angle B is obtuse. Explanation: From triangle ABC, Angle A, and Angle C are both acute with less than a right angle. Angle B is an obtuse angle that is greater than a right angle. Question 2. _________ Answer: Right Triangle; Triangle DEF; ∠D and ∠F are acute angles. ∠E is Right angle Explanation: ∠D and ∠F are acute angles with less than a right angle. ∠E is the Right angle that forms a square corner. A triangle that has one right angle is called a right triangle. Question 3. _________ Answer: Acute triangle; Angle G, Angle J, and Angle H are acute angles. Explanation: From triangle GJH, Angle G, Angle J, and Angle H are acute angles with less than a right angle. A triangle with three acute angles called an acute triangle. So, the given triangle is an acute triangle. Question 4. _________ Answer: Obtuse triangle; Angle L and Angle N are both acute. Angle M is obtuse. Explanation: From triangle LMN, Angle L and Angle N are both acute with less than a right angle. Angle M is an obtuse angle that is greater than a right angle. A triangle with an obtuse angle is called an obtuse triangle. Problem Solving Question 5. Use figure ABCD below. Draw a line segment from point B to point D. Name and classify the triangles formed. Two _________ triangles △ _________ △ _________ Answer: Two Acute triangles. △ ABD △ BCD Explanation: If we draw a line segment from point B to point D, then there are two traingles formed with less than right angles. They are △ ABD and △ BCD. Question 6. Use figure ABCD below. Draw a line segment from point A to point C. Name and classify the triangles formed. Two _________ triangles △ _________ △ _________ Answer: Two Acute triangles. △ ABC △ ADC Explanation: If we draw a line segment from point A to point C, then there are two traingles formed with less than right angles. They are △ ABC and △ ADC. ### Common Core – New – Page No. 560 Lesson Check Question 1. Stephen drew this triangle. How many obtuse angles does the triangle have? Options: a. 0 b. 1 c. 2 d. 3 Answer: a. 0 Explanation: The given image has three acute angles. So, there are 0 obtuse angles. Question 2. Joan was asked to draw a right triangle. How many right angles are in a right triangle? Options: a. 0 b. 1 c. 2 d. 3 Answer: b. 1 Explanation: A right triangle has only one right angle. Spiral Review Question 3. Oliver drew the figure below to show light traveling from the sun to Earth. Name the figure he drew. Options: a. segment SE b. ray SE c. line SE d. ray ES Answer: b. ray SE Explanation: SE is a ray that has one endpoint and continues without an end in one direction. Question 4. Armon added $$\frac{1}{10}$$ and $$\frac{8}{100}$$. Which is the correct sum? Options: a. $$\frac{18}{10}$$ b. $$\frac{9}{10}$$ c. $$\frac{9}{100}$$ d. $$\frac{18}{100}$$ Answer: d. $$\frac{18}{100}$$ Explanation: $$\frac{1 X 10}{10 X 10}$$ + $$\frac{8}{100}$$ = $$\frac{10}{100}$$ + $$\frac{8}{100}$$ = $$\frac{18}{100}$$ Question 5. Sam counted out loud by 6s. Jorge counted out loud by 8s. What are the first three numbers both students said? Options: a. 8, 16, 24 b. 14, 28, 42 c. 24, 48, 72 d. 48, 96, 144 Answer: c. 24, 48, 72 Explanation: Sam counted out loud by 6s = 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, 72. Jorge counted out loud by 8s = 8, 16, 24, 32, 40, 48, 56, 64, 72, 80. Both students said the first three numbers are 24, 48, 72. Question 6. A basketball team averaged 105 points per game. How many points did the team score in 6 games? Options: a. 605 points b. 630 points c. 900 points d. 6,030 points Answer: b. 630 points Explanation: A basketball team averaged 105 points per game. They score in 6 games = 6 x 105 = 630 points. ### Page No. 563 Question 1. Draw and label $$\overline{Q R} \| \overline{S T}$$. Think: Parallel lines never intersect. Parallel line segments are parts of parallel lines. Type below: _________ Answer: $$\overline{Q R} \| \overline{S T}$$ Explanation: Parallel lines never intersect. Parallel line segments are parts of parallel lines. Use the figure for 2 and 3. Question 2. Name two line segments that appear to be parallel. Type below: _________ Answer: Line Segment CB and Line Segment DF. Explanation: Line Segment CB and Line Segment DF are parallel lines. The both lines never intersect and are always the same distance apart. Question 3. Name two line segments that appear to be perpendicular. Type below: _________ Answer: Line Segment CB and Line Segment BF are perpendicular lines. Explanation: Line Segment CB and Line Segment DF are perpendicular lines. Both lines intersect to form four right angles. Use the figure for 4–5. Question 4. Name a pair of lines that are perpendicular. Type below: _________ Answer: FJ and HG are perpendicular lines. Explanation: FJ and HG lines intersect each other and form four right angles. Question 5. Name a pair of lines that appear to be parallel. Type below: _________ Answer: DC and FJ are parallel lines. Explanation: DC and FJ are never intersected and are always the same distance apart. Question 6. $$\overline{R S} \| \overline{T U}$$ Type below: _________ Answer: $$\overline{R S} \| \overline{T U}$$ Explanation: Parallel lines never intersect. Parallel line segments are parts of parallel lines. Question 7. $$\overrightarrow{K L} \text { and } \overrightarrow{K M}$$ Type below: _________ Answer: Explanation: KL and KM are two rays and start at the same point K. Question 8. $$\overline{C D} \perp \overline{D E}$$ Type below: _________ Answer: $$\overline{C D} \perp \overline{D E}$$ Explanation: $$\overline{C D} \perp \overline{D E}$$ are two lines. They are intersect each other and form four right angles. Question 9. $$\overset { \longleftrightarrow }{ JK }$$ ⊥ $$\overset { \longleftrightarrow }{ LM }$$ Type below: _________ Answer: $$\overset { \longleftrightarrow }{ JK }$$ ⊥ $$\overset { \longleftrightarrow }{ LM }$$ Explanation: JK and LM are two lines and intersected each other to form right angles. Question 10. $$\overset { \longleftrightarrow }{ ST }$$ intersecting $$\overset { \longleftrightarrow }{ UV }$$ at point X Type below: _________ Answer: Explanation: ST and UV are two lines intersecting at point X. Question 11. $$\overset { \longleftrightarrow }{ AB }$$ || $$\overset { \longleftrightarrow }{ FG }$$ Type below: _________ Answer: $$\overset { \longleftrightarrow }{ AB }$$ || $$\overset { \longleftrightarrow }{ FG }$$ Explanation: Parallel lines never intersect. Parallel line segments are parts of parallel lines. Use the figure for 12–13. Question 12. Dan says that $$\overset { \longleftrightarrow }{ HL }$$ is parallel to $$\overset { \longleftrightarrow }{ IM }$$. Is Dan correct? Explain. _____ Answer: No; HL and IM are not parallel lines. Parallel lines are always the same distance apart. But from the given image, the H and I may intersect if the line is extended. Question 13. Name two intersecting line segments that are not perpendicular. Type below: _________ Answer: JM and KG are two intersecting line segments and also not perpendicular. ### Page No. 564 Use the house plan at the right for 14–16. Question 14. What geometric term describes a corner of the living room? _________ Answer: The corner of the living room is a vertex. Corners of any shape are vertexes. Question 15. Name three parts of the plan that show line segments. _________ Answer: Kitchen, Living Room, and Master Bedroom. Question 16. Name a pair of line segments that appear to be parallel _________ Answer: Two sidelines of Living Room are parallel. Two sidelines of Master Bedroom are parallel. Use the map at the right for 17–19. Question 17. Name a street that appears to be parallel to S 17th Street. _________ Answer: S 18th Street Explanation: S 18th Street is parallel to S 17th Street. They never meet each other and are always the same distance apart. Question 18. Use Diagrams Name a street that appears to be parallel to Vernon Street. _________ Answer: Perry Street Explanation: Perry Street is parallel to Vernon Street. They never meet each other and are always the same distance apart. Question 19. Name a street that appears to be perpendicular to S 19th Street. _________ Answer: Austin Street Explanation: Austin Street is perpendicular to S 19th Street. They are intersect with each other and form four right angles. Question 20. Choose the labels to make a true statement. Type below: _________ Answer: Line AB is perpendicular to Line EF. ### Common Core – New – Page No. 565 Parallel Lines and Perpendicular Lines Use the figure for 1–3. Question 1. Name a pair of lines that appear to be perpendicular. Think: Perpendicular lines form right angles. $$\overset { \longleftrightarrow }{ AB }$$ and $$\overset { \longleftrightarrow }{ EF }$$ appear to form right angles. $$\overset { \longleftrightarrow }{ AB }$$ and $$\overset { \longleftrightarrow }{ EF }$$ Answer: $$\overset { \longleftrightarrow }{ AB }$$ and $$\overset { \longleftrightarrow }{ EF }$$ Explanation: Perpendicular lines form right angles. $$\overset { \longleftrightarrow }{ AB }$$ and $$\overset { \longleftrightarrow }{ EF }$$ appear to form right angles. $$\overset { \longleftrightarrow }{ AB }$$ and $$\overset { \longleftrightarrow }{ EF }$$ Question 2. Name a pair of lines that appear to be parallel. _____ and _____ Answer: $$\overset { \longleftrightarrow }{ AB }$$ and $$\overset { \longleftrightarrow }{ CD}$$ Explanation: Parallel lines never interest each other. $$\overset { \longleftrightarrow }{ AB }$$ and $$\overset { \longleftrightarrow }{ CD}$$ are parallel lines. Question 3. Name another pair of lines that appear to be perpendicular. _____ and _____ Answer: $$\overset { \longleftrightarrow }{ CD }$$ and $$\overset { \longleftrightarrow }{ EF }$$ Explanation: Perpendicular lines form right angles. $$\overset { \longleftrightarrow }{ CD }$$ and $$\overset { \longleftrightarrow }{ EF }$$ appear to form right angles. $$\overset { \longleftrightarrow }{ CD }$$ and $$\overset { \longleftrightarrow }{ EF }$$ Draw and label the figure described. Question 4. $$\overset { \longleftrightarrow }{ MN }$$ and $$\overset { \longleftrightarrow }{ PQ }$$ intersecting at point R Answer: Explanation: MN and PQ are two lines and interesting at point R. Question 5. $$\overset { \longleftrightarrow }{ WX }$$ || $$\overset { \longleftrightarrow }{ YZ }$$ Answer: Explanation: WX and YZ are parallel lines and they never intersect with each other. Question 6. $$\overset { \longleftrightarrow }{ FH }$$ ⊥ $$\overset { \longleftrightarrow }{ JK }$$ Answer: Explanation: FH and JK are two lines and intersecting each other to form four right angles. Problem Solving Use the street map for 7–8. Question 7. Name two streets that intersect but do not appear to be perpendicular. Type below: _________ Answer: Maple and Oak or Oak and Birch Explanation: Maple and Oak or Oak and Birch; They are intersecting with each other and not perpendicular. Question 8. Name two streets that appear to be parallel to each other. Type below: _________ Answer: Maple and Birch Explanation: Maple and Birch are streets and not intersect with each other. They appear to be parallel to each other. ### Common Core – New – Page No. 566 Lesson Check Question 1. Which capital letter appears to have perpendicular line segments? Options: a. N b. O c. T d. V Answer: c. T Explanation: T has two lines and interesting to form four right angles. Question 2. In the figure, which pair of line segments appear to be parallel? Options: a. $$\overline{F G} \text { and } \overline{G H}$$ b. $$\overline{F J} \text { and } \overline{G H}$$ c. $$\overline{F G} \text { and } \overline{J H}$$ d. $$\overline{J H} \text { and } \overline{F J}$$ Answer: c. $$\overline{F G} \text { and } \overline{J H}$$ Explanation: $$\overline{F G} \text { and } \overline{J H}$$ are parallel lines that never intersect Spiral Review Question 3. Nolan drew a right triangle. How many acute angles did he draw? Options: a. 0 b. 1 c. 2 d. 3 Answer: c. 2 Explanation: A triangle with one right angle will have two acute angles. Question 4. Mike drank more than half the juice in his glass. What fraction of the juice could Mike have drunk? Options: a. $$\frac{1}{3}$$ b. $$\frac{2}{5}$$ c. $$\frac{3}{6}$$ d. $$\frac{5}{8}$$ Answer: d. $$\frac{5}{8}$$ Explanation: Mike drank more than half the juice in his glass. He drunk $$\frac{5}{8}$$ of the juice. Question 5. A school principal ordered 1,000 pencils. He gave an equal number to each of 7 teachers until he had given out as many as possible. How many pencils were left? Options: a. 2 b. 4 c. 6 d. 142 Answer: c. 6 Explanation: A school principal ordered 1,000 pencils. He gave an equal number to each of 7 teachers until he had given out as many as possible. He shared 142 pencils for each of 7 teachers. So, 142 X 7 = 994. The remaining pencils are 6. Question 6. A carton of juice contains 64 ounces. Ms. Wilson bought 6 cartons of juice. How many ounces of juice did she buy? Options: a. 364 ounces b. 370 ounces c. 384 ounces d. 402 ounces Answer: c. 384 ounces Explanation: A carton of juice contains 64 ounces. Ms. Wilson bought 6 cartons of juice. 64 X 6 = 384 ounces juice she can buy. ### Page No. 569 Question 1. Tell whether the quadrilateral is also a trapezoid, parallelogram, rhombus, rectangle, or square. Think: ____ pairs of parallel sides ____ sides of equal length ____ right angles Quadrilateral ABCD is also a __________ __________ Answer: 2 pairs of parallel sides 4 sides of equal length 0 right angles. Quadrilateral ABCD is also a Rhombus Explanation: A Rhombus is a quadrilateral that has 2 pairs of parallel sides and 4 sides of equal lengths. Classify each figure as many ways as possible. Write quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square. Question 2. __________ Answer: Quadrilateral Explanation: 0 pairs of parallel sides 0 sides of equal length 0 right angles. The given image is quadrilateral. The quadrilateral doesn’t have a name because it has 0 pairs of parallel sides, 0 sides of equal length, and 0 right angles. Question 3. _________ _________ _________ Answer: Quadrilateral, Rectangle, and Parallelogram Explanation: 2 pairs of parallel sides 2 pairs of sides of equal length 4 right angles. Given quadrilateral is Rectangle and Parallelogram. A Rectangle is a quadrilateral that has 2 pairs of parallel sides and 2 pairs of sides of equal lengths, and 4 right angles. Question 4. _________ _________ _________ Answer: Quadrilateral, Parallelogram, and Rhombus Explanation: 2 pairs of parallel sides 4 sides of equal length 0 right angles. Given quadrilateral is Rhombus and Parallelogram. A Rhombus is a quadrilateral that has 2 pairs of parallel sides and 4 sides of equal lengths, and 0 right angles. Question 5. _________ _________ Answer: Quadrilateral and Parallelogram Explanation: 2 pairs of parallel sides 2 pairs of sides of equal length 0 right angles. Given quadrilateral is Parallelogram. A Parallelogram is a quadrilateral that has 2 pairs of parallel sides and 2 pairs of sides of equal lengths, and 0 right angles. Question 6. _________ _________ _________ Answer: Quadrilateral and Square Explanation: 2 pairs of parallel sides 4 sides of equal length 4 right angles. Given quadrilateral is Square. A Square is a quadrilateral that has 2 pairs of parallel sides and 4 sides of equal lengths, and 4 right angles. Question 7. _________ _________ Answer: Quadrilateral and Trapezoid Explanation: 1 pair of parallel sides 0 sides of equal length 0 right angles. Given quadrilateral is Trapezoid. A Square is a quadrilateral that has 1 pair of parallel sides and 0 sides of equal lengths, and 0 right angles. ### Page No. 570 Question 8. Explain how a rhombus and square are alike, and how they are different. Type below: _________ Answer: The rhombus and square have 2 pairs of parallel sides and 4 sides of equal length. But the rhombus has 0 right angles and the square has 4 right angles. Question 9. Classify the figure. Select all that apply. Options: a. quadrilateral b. trapezoid c. parallelogram d. rectangle e. rhombus f. square Answer: a. quadrilateral b. trapezoid c. parallelogram Explanation: A Parallelogram is a quadrilateral that has 2 pairs of parallel sides and 2 pairs of sides of equal lengths, and 0 right angles. The Louvre Museum is located in Paris, France. Architect I. M. Pei designed the glass and metal structure at the main entrance of the museum. This structure is called the Louvre Pyramid. Below is a diagram of part of the entrance to the Louvre Pyramid. Question 10. Describe the quadrilaterals you see in the diagram. _________ _________ Answer: Trapezoid and Rhombus Explanation: There are 2 quadrilaterals available in the given image. One is Trapezoid with 1 pair of parallel sides. Another one is Rhombus is with 2 pairs of parallel sides and 4 sides of equal lengths, and 0 right angles. Question 11. How many triangles do you see in the diagram? Explain. ______ triangles Answer: 11 triangles Explanation: The given image has 11 triangles ### Common Core – New – Page No. 571 Classify Quadrilaterals Classify each figure as many ways as possible. Write quadrilateral, trapezoid, parallelogram, rhombus, rectangle, or square. Question 1. Think: 2 pairs of parallel sides 4 sides of equal length 0 right angles quadrilateral, parallelogram, rhombus Answer: Quadrilateral, Parallelogram, and rhombus. Explanation: 2 pairs of parallel sides 4 sides of equal length 0 right angles Quadrilateral, Parallelogram, and rhombus. Question 2. Type below: _________ Answer: Quadrilateral, Parallelogram, Rectangle Explanation: 2 pairs of parallel sides 2 pairs of sides of equal length 4 right angles Quadrilateral, Parallelogram, Rectangle Question 3. Type below: _________ Answer: Explanation: 1 pair of parallel sides 2 sides of equal length 0 right angles Quadrilateral, Trapezoid Question 4. Type below: _________ Answer: Quadrilateral Explanation: 0 pair of parallel sides 0 sides of equal length 0 right angles Quadrilateral Question 5. Type below: _________ Answer: Quadrilateral, Parallelogram, and rhombus Explanation: 2 pairs of parallel sides 4 sides of equal length 0 right angles Quadrilateral, Parallelogram, and rhombus Question 6. Type below: _________ Answer: Explanation: 1 pair of parallel sides 0 sides of equal length 2 right angles Quadrilateral, Trapezoid Question 7. Type below: _________ Answer: Explanation: 2 pairs of parallel sides 2 pairs of sides of equal length 0 right angles Quadrilateral, Parallelogram Problem Solving Question 8. Alan drew a polygon with four sides and four angles. All four sides are equal. None of the angles are right angles. What figure did Alan draw _________ Answer: Quadrilateral or rhombus Explanation: Alan drew a polygon with four sides and four angles. All four sides are equal. None of the angles are right angles. Alan drew Quadrilateral or rhombus Question 9. Teresa drew a quadrilateral with 2 pairs of parallel sides and 4 right angles. What quadrilateral could she have drawn? _________ Answer: square or rectangle Explanation: 2 pairs of parallel sides and 4 right angles. she could draw a square or rectangle. ### Common Core – New – Page No. 572 Lesson Check Question 1. Joey is asked to name a quadrilateral that is also a rhombus. What should be his answer? Options: a. square b. rectangle c. parallelogram d. trapezoid Answer: a. square Explanation: The quadrilateral square is also called a rhombus. Both square and rhombus have 2 pairs of parallel sides and 4 sides of equal length. Question 2. Which quadrilateral has exactly one pair of parallel sides? Options: a. square b. rhombus c. parallelogram d. trapezoid Answer: d. trapezoid Explanation: A trapezoid has exactly one pair of parallel sides. Spiral Review Question 3. Terrence has 24 eggs to divide into equal groups. What are all the possible numbers of eggs that Terence could put in each group? Options: a. 1, 2, 3, 4 b. 2, 4, 6, 8, 12 c. 1, 2, 3, 4, 6, 8, 12, 24 d. 24, 48, 72, 96 Answer: c. 1, 2, 3, 4, 6, 8, 12, 24 Explanation: Terrence has 24 eggs to divide into equal groups. Terence could put in each group in 1, 2, 3, 4, 6, 8, 12, 24 ways. Question 4. In a line of students, Jenna is number 8. The teacher says that a rule for a number pattern is add 4. The first student in line says the first term, 7. What number should Jenna say? Options: a. 31 b. 35 c. 39 d. 43 Answer: b. 35 Explanation: In a line of students, Jenna is number 8. The teacher says that a rule for a number pattern is add 4. The first student in line says the first term, 7. 7 + 4 = 11 11 + 4 = 15 15 + 4 = 19 19 + 4 = 23 23 + 4 = 27 27 + 4 = 31 31 + 4 = 35. Jenna says 35. Question 5. Lou eats $$\frac{6}{8}$$ of a pizza. What fraction of the pizza is left over? Options: a. $$\frac{1}{8}$$ b. $$\frac{1}{4}$$ c. $$\frac{1}{2}$$ d. $$\frac{3}{4}$$ Answer: b. $$\frac{1}{4}$$ Explanation: Lou eats $$\frac{6}{8}$$ of a pizza. So, 6 parts of pizza is finished and remaining 2 parts of pizza is remained. So, the left over pizza is $$\frac{2}{8}$$ = $$\frac{1}{4}$$. Question 6. Which capital letter appears to have parallel lines? Options: a. D b. L c. N d. T Answer: c. N Explanation: N has two parallel lines and never intersect each other. ### Page No. 573 Choose the best term from the box to complete the sentence. Question 1. A _______ is part of a line between two endpoints. _________ Answer: line segment Question 2. A _______ forms a square corner. _________ Answer: Right angle Question 3. An _______ is greater than a right angle and less than a straight angle. _________ Answer: Obtuse angle Question 4. The two-dimensional figure that has one endpoint is a ________. _________ Answer: ray Question 5. An angle that forms a line is called a _______. _________ Answer: straight line Question 6. On the grid below, draw a polygon that has 2 pairs of parallel sides, 2 pairs of sides equal in length, and 2 acute and 2 obtuse angles. Tell all the possible names for the figure. Type below: _________ Answer: Parallelogram Explanation: The possible polygon that has 2 pairs of parallel sides, 2 pairs of sides equal in length, and 2 acute and 2 obtuse angles is Parallelogram. Draw the figure. Question 7. parallel lines Type below: _________ Answer: Explanation: QR and ST are two parallel lines. they never intersect each other. Question 8. obtuse ∠ABC Type below: _________ Answer: Explanation: From triangle, ABC, Angle A, and Angle C are both acute with less than a right angle. Angle B is an obtuse angle that is greater than a right angle. Question 9. intersecting lines that are not perpendicular Type below: _________ Answer: Explanation: ST and UV are two lines intersecting at point X. Question 10. acute ∠RST Type below: _________ Answer: ### Page No. 574 Question 11. Which triangle has one right angle? _________ Answer: A right triangle has one right angle. Question 12. Which figure has 2 pairs of parallel sides, 2 pairs of sides of equal length, and 4 right angles? _________ Answer: A Rectangle has 2 pairs of parallel sides, 2 pairs of sides of equal length, and 4 right angles. Question 13. Which quadrilateral can have 2 pairs of parallel sides, all sides with equal length, and no right angles? _________ Answer: Rhombus can have 2 pairs of parallel sides, all sides with equal length, and no right angles. Question 14. What is the correct name of the figure shown? _________ Answer: Ray Explanation: EF is a ray that has one endpoint and continues without an end in one direction. Question 15. Describe the angles of an obtuse triangle. Type below: _________ Answer: An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. ### Page No. 577 Tell whether the parts on each side of the line match. Is the line a line of symmetry? Write yes or no. Question 1. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 2. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 3. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 4. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Tell if the blue line appears to be a line of symmetry. Write yes or no. Question 5. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 6. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Question 7. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Question 8. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Tell if the blue line appears to be a line of symmetry. Write yes or no. Question 9. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Question 10. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 11. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Question 12. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 13. Which best describes the symmetry in the letter I? Type below: ________ Answer: The two parts of the folded I match exactly. The fold line is a line of symmetry. Explanation: Take the Horizontal line in the middle of the Letter I. Cut out the tracing. Fold the tracing over a horizontal line. The two parts of the folded I match exactly. The fold line is a line of symmetry. ### Page No. 578 Question 14. Which shape has a correctly drawn line of symmetry? a. What do you need to find? Type below: ________ Answer: Find the shape that has an exact line of symmetry. Question 14. b. How can you tell if the line of symmetry is correct? Type below: ________ Answer: If the two parts of the folded match exactly, then the line is a line of symmetry. Question 14. c. Tell how you solved the problem. Type below: ________ Answer: From fig 1 to 4, the fig 2 is has a line of symmetry that can exactly separate the two parts equally. Question 14. d. Circle the correct shape above. Type below: ________ Answer: Question 15. Reason Abstractly Draw a line of symmetry in the figure shown. Answer: Question 16. Evie’s birthday is on the 18th of May. Since May is the 5th month, Evie wrote the date as shown. Evie says all the numbers she wrote have line symmetry. Is she correct? Explain. Answer: No; The number 5 doesn’t have a line of symmetry. So, Evie explanation is wrong. ### Common Core – New – Page No. 579 Line Symmetry Tell if the dashed line appears to be a line of symmetry. Write yes or no. Question 1. yes Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 2. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Question 3. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 4. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Question 5. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Question 6. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Question 7. ____ Answer: No; Explanation: The line of symmetry divides a shape into two parts that are not with the same size and shape. Question 8. ____ Answer: Yes; Explanation: The line of symmetry divides a shape into two parts that are the same size and shape. Complete the design by reflecting over the line of symmetry. Question 9. Answer: Question 10. Answer: Problem Solving Question 11. Kara uses the pattern below to make paper dolls. The dashed line represents a line of symmetry. A complete doll includes the reflection of the pattern over the line of symmetry. Complete the design to show what one of Kara’s paper dolls looks like. Answer: ### Common Core – New – Page No. 580 Lesson Check Question 1. Which best describes the line of symmetry in the letter D? Options: a. horizontal b. vertical c. diagonal d. half turn Answer: a. horizontal Explanation: The horizontal line of symmetry in the letter D can exactly separate two parts equally. Question 2. Which shape has a correctly drawn line of symmetry? Options: a. b. c. d. Answer: b. Explanation: Image b has the line of symmetry that separates two parts equally. Spiral Review Question 3. The class has 360 unit cubes in a bag. Johnnie divides the unit cubes equally among 8 groups. How many unit cubes will each group get? Options: a. 40 b. 44 c. 45 d. 48 Answer: c. 45 Explanation: The class has 360 unit cubes in a bag. Johnnie divides the unit cubes equally among 8 groups. 360/8= 45. Question 4. There are 5,280 feet in one mile. How many feet are there in 6 miles? Options: a. 30,680 b. 31,260 c. 31,608 d. 31,680 Answer: d. 31,680 Explanation: There are 5,280 feet in one mile. So, for 6 miles = 6 x 5, 280 = 31,680. Question 5. Sue has 4 pieces of wood. The lengths of her pieces of wood are $$\frac{1}{3}$$ foot, $$\frac{2}{5}$$ foot, $$\frac{3}{10}$$ foot, and $$\frac{1}{4}$$ foot. Which piece of wood is the shortest? Options: a. the $$\frac{1}{3}$$ foot piece b. the $$\frac{2}{5}$$ foot piece c. the $$\frac{3}{10}$$ foot piece d. the $$\frac{1}{4}$$ foot piece Answer: d. the $$\frac{1}{4}$$ foot piece Explanation: The lengths of $$\frac{1}{4}$$ foot piece is less compared to other lengths. Question 6. Alice has $$\frac{1}{5}$$ as many miniature cars as Sylvester has. Sylvester has 35 miniature cars. How many miniature cars does Alice have? Options: a. 7 b. 9 c. 40 d. 175 Answer: a. 7 Explanation: Alice has $$\frac{1}{5}$$ as many miniature cars as Sylvester has. Sylvester has 35 miniature cars. Alice have $$\frac{1}{5}$$ X 35 = 7 miniature cars. ### Page No. 583 Question 1. The shape at the right has line symmetry. Draw the 2 lines of symmetry. Type below: _________ Answer: Tell whether the shape appears to have zero lines, 1 line, or more than 1 line of symmetry. Write zero, 1, or more than 1. Question 2. _________ Answer: more than 1 Explanation: There is more than 1 line of symmetries that separates two parts equally. Question 3. _________ Answer: more than 1 Explanation: There is more than 1 lines of symmetries that separates two parts equally. Question 4. _________ Answer: 1 line Explanation: There is 1 line of symmetry that separates two parts equally. Question 5. _________ Answer: zero lines Explanation: There is no line of symmetries that separates two parts equally. Tell whether the shape appears to have zero lines, 1 line, or more than 1 line of symmetry. Write zero, 1, or more than 1. Question 6. _________ Answer: more than 1 Explanation: There is more than 1 lines of symmetries that separate two parts equally. Question 7. _________ Answer: zero lines Explanation: There is no line of symmetries that separates two parts equally. Question 8. _________ Answer: zero lines Explanation: There is no line of symmetries that separates two parts equally. Question 9. _________ Answer: 1 line Explanation: There is 1 line of symmetry that separates two parts equally. Practice: Copy and Solve Does the design have line symmetry? Write yes or no. If your answer is yes, draw all lines of symmetry. Question 10. ____ Answer: Yes; Question 11. _____ Answer: No; Question 12. _____ Answer: Yes; Question 13. _____ Answer: No; Question 14. Draw a figure that has 5 sides and exactly 1 line of symmetry. Type below: _________ Answer: Explanation: the above 5 sides shape has only 1 line symmetry ### Page No. 584 Use the chart for 15–17. Question 15. Which letters appear to have only 1 line of symmetry? Type below: _________ Answer: A, B, C, D, E, T, U, V, W Explanation: The letters A, B, C, D, E, T, U, V, W have only 1 line of symmetry. Question 16. Which letters appear to have zero lines of symmetry? Type below: _________ Answer: J, N, S Explanation: The letters J, N, S have only zero lines of symmetry. Question 17. The letter C has horizontal symmetry. The letter A has vertical symmetry. Which letters appear to have both horizontal and vertical symmetry? Type below: _________ Answer: H and I Explanation: The letters H and I have both horizontal and vertical symmetry. Question 18. Verify the Reasoning of Others Jeff says that the shape has only 2 lines of symmetry. Does his statement make sense? Explain. Type below: _________ Answer: No; Jeff’s explanation is wrong. Because the given shape has only 2 lines of symmetry. Question 19. Match each figure with the correct number of lines of symmetry it has. Type below: _________ Answer: ### Common Core – New – Page No. 585 Find and Draw Lines of Symmetry Tell whether the shape appears to have zero lines, 1 line, or more than 1 line of symmetry. Write zero, 1, or more than 1. Question 1. 1 Answer: more than 1 Explanation: There is more than 1 line of symmetry that separates two parts equally. Question 2. ________ Answer: more than 1 Explanation: There is more than 1 line of symmetry that separates two parts equally. Question 3. ________ Answer: Zero Explanation: There are 0 lines of symmetries. Question 4. ________ Answer: more than 1 Explanation: There is more than 1 line of symmetry that separates two parts equally. Does the design have line symmetry? Write yes or no. If your answer is yes, draw all lines of symmetry. Question 5. _____ Answer: Yes; Question 6. _____ Answer: Yes; _____ Answer: No; Question 8. ______ Answer: Yes; Draw a shape for the statement. Draw the line or lines of symmetry. Question 9. zero lines of symmetry Answer: Question 10. 1 line of symmetry Answer: Question 11. 2 lines of symmetry Answer: Problem Solving Use the chart for 12–13. Question 12. Which number or numbers appear to have only 1 line of symmetry? _____ Answer: 3 Explanation: The number 3 has only 1 line of symmetry. Question 13. Which number or numbers appear to have 2 lines of symmetry? _____ Answer: 0 and 8 Explanation: The numbers 0 and 8 appear to have 2 lines of symmetry. ### Common Core – New – Page No. 586 Lesson Check Question 1. How many lines of symmetry does this shape appear to have? Options: a. 0 b. 2 c. 6 d. 12 Answer: c. 6 Explanation: The given shape has 6 lines of symmetries. Question 2. Which of the following shapes appears to have exactly 1 line of symmetry? Options: a. b. c. d. Answer: d. Explanation: The trapezoid has exactly 1 line of symmetry. Spiral Review Question 3. Richard practiced each of 3 piano solos for $$\frac{5}{12}$$ hour. How long did he practice in all? Options: a. $$\frac{2}{3}$$ hours b. 1 $$\frac{1}{4}$$ hours c. 1 $$\frac{1}{3}$$ hours d. 1 $$\frac{5}{12}$$ hours Answer: b. 1 $$\frac{1}{4}$$ hours Explanation: Richard practiced each of 3 piano solos for $$\frac{5}{12}$$ hour. $$\frac{5}{12}$$ hour = 1 $$\frac{1}{4}$$ hours hours. Question 4. Which of the following decimals is equivalent to three and ten hundredths? Options: a. 0.30 b. 0.31 c. 3.01 d. 3.1 Answer: d. 3.1 Explanation: three and ten hundredths = 310 hundredths = 3.1 Question 5. Lynne used $$\frac{3}{8}$$ cup of flour and $$\frac{1}{3}$$ cup of sugar in a recipe. Which number below is a common denominator for $$\frac{3}{8}$$ and $$\frac{1}{3}$$? Options: a. 8 b. 12 c. 16 d. 24 Answer: d. 24 Explanation: Lynne used $$\frac{3}{8}$$ cup of flour and $$\frac{1}{3}$$ cup of sugar in a recipe. To find the common denominator for $$\frac{3}{8}$$ and $$\frac{1}{3}$$, multiply 8 X3 and 3 X 8 = 24. Question 6. Kevin draws a figure that has four sides. All sides have the same length. His figure has no right angles. What figure does Kevin draw? Options: a. square b. trapezoid c. rhombus d. rectangle Answer: c. rhombus Explanation: ### Page No. 589 Question 1. Marisol is making a pattern with blocks. What might the missing shape be? First, look at the blocks. Next, describe the pattern. Type below: _________ Answer: The first image 1 has three line segments. The next shape will have four line segments. Question 1. Finally, draw the missing shape. Type below: _________ Answer: Question 2. Use the shapes to write a number pattern. Then describe the pattern in the numbers. Answer: Shape 1: Triangle Shape 2: Square Shape 3: Pentagon Shape 4: Hexagon Shape 5: Heptagon Question 3. What if the pattern continued? Write an expression to describe the number of sides the sixth shape has in Marisol’s pattern. Type below: _________ Answer: If the pattern continued, then the next shape will have one more extra line segment to it. The sixth shape will become the octagon. Shape 6: Octagon Question 4. Sahil made a pattern using circles. The first nine circles are shown. Describe the pattern. If Sahil continues the pattern, what might the next three circles be? Type below: _________ Answer: The pattern is repeated for every three circles. One big circle followed by two small circles. ### Page No. 590 Use the toy quilt designs for 5–6. Question 5. Lu is making a quilt that is 20 squares wide and has 24 rows. The border of the quilt is made by using each toy design equally as often. Each square can hold one design. How many of each design does she use for the border? ______ times Answer: The border will have 20 squares two times, and 24 squares two times as well, that is the perimeter or the border, because a quilt has 4 sides: 20 X 2 + 24 X 2 = 40 + 48 = 88 So, the border will have 88 squares in total. So if the border can have only one design, Lu can use any toy design 88 times if she wants the border to have the same toy design in it. Question 6. Communicate Starting in the first square of her quilt, Lu lined up her toy designs in this order: plane, car, fire truck, helicopter, crane, and wagon. Using this pattern unit, which design will Lu place in the fifteenth square? Explain how you found your answer. _________ Answer: The answer is fire truck. As the pattern repeats, the fifteenth square will fire truck. Question 7. Missy uses 1 hexagonal, 2 rectangular, and 4 triangular pieces of fabric to make 1 bug design for a quilt. If she uses 70 pieces in all to make bug designs, how many of each shape does she use? Hexagonal: _________ shapes Rectangular: _________ shapes Triangular: _________ shapes Answer: Hexagonal: 10 shapes Rectangular: 10 shapes Triangular: 10 shapes (1 x 10) + (2 x 10) + (4 x 10) = 10 + 20 + 40 = 70 pieces in all. Question 8. Norris drew the pattern shown. Label the circles to show the colors in the fourth figure of the pattern. Type below: _________ Answer: ### Common Core – New – Page No. 591 Problem Solving Shape Patterns Solve each Problem. Question 1. Marta is using this pattern to decorate a picture frame. Describe the pattern. Draw what might be the next three figures in the pattern. Possible answer: the pattern repeats: one trangle followed by two squares. Answer: The pattern repeats one triangle followed by two squares. Question 2. Describe the pattern. Draw what might be the next three figures in the pattern. How many circles are in the sixth figure in the pattern? _____ circles Answer: Add one more column with 1 more circle than in the previous column; 21. Question 3. Larry stencils this pattern to make a border at the top of his bedroom walls. Describe the pattern. Draw what might be the missing figure in the pattern. Answer: 2 triangles placed side to side followed by 2 sets of 2 triangles placed vertex to vertex ### Common Core – New – Page No. 592 Lesson Check Question 1. What might be the next three figures in this pattern? Options: a. b. c. d. Answer: a. Explanation: the pattern has odd numbers of up arrows then even number of down arrows. So, the next three figures are . Question 2. Which might be the missing figure in the following pattern? Options: a. b. c. d. Answer: a. Explanation: From the pattern, the missing image will have vertical rectangle with the circle and X mark in it. Spiral Review Question 3. Chad has two pieces of wood. One piece is $$\frac{7}{12}$$ foot long. The second piece is $$\frac{5}{12}$$ foot longer than the first piece. How long is the second piece? Options: a. $$\frac{2}{12}$$ foot b. $$\frac{1}{2}$$ foot c. $$\frac{12}{18}$$ foot d. 1 foot Answer: d. 1 foot Explanation: $$\frac{7}{12}$$ + $$\frac{5}{12}$$ = $$\frac{12}{12}$$ = 1 foot. Question 4. Olivia finished a race in 40.64 seconds. Patty finished the race in 40.39 seconds. Miguel finished the race in 41.44 seconds. Chad finished the race in 40.46 seconds. Who finished the race in the least time? Options: a. Olivia b. Patty c. Miguel d. Chad Answer: b. Patty Explanation: Patty finished the race in 40.39 seconds that is the least time compared to others. Question 5. Justin bought 6 ribbons for an art project. Each ribbon is $$\frac{1}{4}$$ yard long. How many yards of ribbon did Justin buy? Options: a. $$\frac{2}{3}$$ yard b. 1 $$\frac{1}{4}$$ yards c. 1 $$\frac{1}{2}$$ yards d. 1 $$\frac{3}{4}$$ yards Answer: c. 1 $$\frac{1}{2}$$ yards Explanation: Justin bought 6 ribbons for an art project. Each ribbon is $$\frac{1}{4}$$ yard long. So, 6 X $$\frac{1}{4}$$ = $$\frac{3}{2}$$ = 1 $$\frac{1}{2}$$ yards. Question 6. Kyle and Andrea were asked to make a list of prime numbers. Kyle: 1, 3, 7, 19, 23 Andrea: 2, 3, 5, 7, 11 Whose list is correct? Options: a. Only Kyle’s list b. Only Andrea’s list c. Both lists are correct. d. Neither list is correct. Answer: b. Only Andrea’s list Explanation: 1 is not a prime number. So, the answer is Only Andrea’s list is correct. ### Page No. 593 Question 1. Gavin is designing a kite. He sketched a picture of the kite. How many right angles does the kite appear to have? _____ right angles Answer: 0 right angles Explanation: There is no right angles in the given shape. Question 2. Write the letter of the triangle under its correct classification. Answer: Explanation: C and F are Acute angles with less than right angles. B and D are Obtuse Angles with more than right angles. A and E are Right Angles. Question 3. Select the angles that identify an obtuse triangle. Mark all that apply. Options: a. acute, acute, acute b. acute, acute, obtuse c. right, acute, acute d. obtuse, right, acute Answer: b. acute, acute, obtuse Explanation: An obtuse triangle will have one obtuse angle and two acute angles. ### Page No. 594 Question 4. Write the word that describes the part of Figure A written below. $$\overline{E B}$$ _________ $$\overset { \longleftrightarrow }{ AB }$$ _________ $$\overrightarrow{G A}$$ _________ ∠EBG _________ ∠CGB _________ Answer: $$\overline{E B}$$ line segment. $$\overset { \longleftrightarrow }{ AB }$$ Line. $$\overrightarrow{G A}$$ Ray. ∠EBG right angle. ∠CGB acute angle. Explanation: $$\overline{E B}$$ is a line segment that has two endpoints connected to form a line. $$\overset { \longleftrightarrow }{ AB }$$ is a Line that continues without an end in both directions. $$\overrightarrow{G A}$$ is a Ray that has one endpoint and continues without an end in one direction. ∠EBG right angle. ∠CGB is an acute angle with less than the right angle. Question 5. What term best describes the figure shown below? Answer: perpendicular lines Explanation: The lines are forming four right angles they form squares. So, the both lines are perpendicular lines. Question 6. Naomi leaves for her trip to Los Angeles on the 12th day of August. Since August is the 8th month, Naomi wrote the date as shown. Naomi says all the numbers she wrote have line symmetry. Is she correct? Explain your thinking. _______ Answer: Naomi is incorrect. The number 2 does not have a line of symmetry because if it were cut out, there would be no way to fold it in half so that the two parts matched exactly. ### Page No. 595 Question 7. Max made a pennant that looks like a triangle. How can you classify the triangle based upon its angles? The triangle is a(n) ____________ triangle. Answer: The triangle is an acute triangle. Explanation: The triangle is an acute triangle. because it has angles with less than right angles. Question 8. Choose the labels to make a true statement. _____ is parallel to ______ Answer: Line AB is parallel to line CD. Explanation: From the given image, Line AB is parallel to line CD. Question 9. Classify the figure. Select all that apply. Options: a. quadrilateral b. trapezoid c. parallelogram d. rectangle e. rhombus f. square Answer: a. quadrilateral b. trapezoid c. parallelogram d. rectangle Explanation: The given image has 2 parallel sides, 2 pairs of sides of length, and four right angles. So, the possible answers are quadrilateral, trapezoid, parallelogram, and rectangle. Question 10. Lily designed a deck in her backyard that looks like a quadrilateral that has only 1 pair of parallel sides. How can you classify the figure? The quadrilateral is a ________ Answer: The quadrilateral is a trapezoid Explanation: Lily designed a deck in her backyard that looks like a quadrilateral that has only 1 pair of parallel sides. So, the answer is a trapezoid. ### Page No. 596 Question 11. Match each figure with the correct number of lines of symmetry it has. Answer: Explanation: Image 1: 1 line of symmetry Image 2: 2 lines of symmetry Image 3: 0 lines of symmetry Image 4: More than 2 lines of symmetry. Question 12. Barb drew the pattern shown. Use the square shown to draw the missing pattern. □ Answer: Explanation: The fourth shape must consist of one extra square box in the top line and bottom line. Question 13. Claudia drew the figure below. Draw a line of symmetry on Claudia’s figure. Answer: Explanation: The image can have one line symmetry. Question 14. Write the word or words that best describe this figure. _________ Answer: Ray Explanation: The ray that has one endpoint and continues without an end in one direction. Question 15. How many acute angles does a right triangle have? A right triangle has ____ acute angles. Answer: A right triangle has 2 acute angles. ### Page No. 597 Question 16. Mike drew a figure with opposite sides parallel. Write the pairs of parallel sides. What figure is it? Answer: Line DG is parallel to Line FE and Line DE is parallel to Line GF; the figure is a parallelogram. Question 17. Circle the letter that does not have line symmetry. Answer: Explanation: The S does not have line symmetry. Question 18. Joseph made a pattern using ovals and rectangles. The first four figures of his pattern are shown. Draw the next figure in the pattern. Answer: Question 19. Jeremy drew Figure 1 and Louisa drew Figure 2. Part A Jeremy says both figures are rectangles. Do you agree with Jeremy? Support your answer. _____ Answer: Yes; both figures have 2 pairs of parallel sides, opposite sides that are equal in length, and 4 right angles. Question 19. Part B Louisa says both figures are rhombuses. Do you agree with Louisa? Support your answer. _____ Answer: No; figure 2 is a rhombus since it has 2 pairs of parallel sides and 4 sides of equal length. Figure 1 does not have 4 sides of equal length so it cannot be a rhombus. ### Page No. 598 Question 20. Veronica found the number of lines of symmetry for the figure below. How many lines of symmetry does it have? ______ lines of symmetry Answer: 2 lines of symmetry Explanation: the given shape can have 2 lines of symmetry. Question 21. Jordan drew the pattern below. Part A Describe the pattern. Answer: Each figure has 2 more squares than the preceding figure. Question 21. Part B Write a rule using numbers to find the number of squares in any figure in the pattern. Answer: multiply the figure number by 2. Question 21. Part C Draw Figure 5. Answer: ### Page No. 603 Tell what fraction of the circle the shaded angle represents. Question 1. $$\frac{□}{□}$$ Answer: $$\frac{1}{2}$$ Explanation: Half of the part is shaded out of the circle. So, the answer is 1/2. Question 2. $$\frac{□}{□}$$ Answer: $$\frac{1}{4}$$ Explanation: $$\frac{1}{4}$$ part of the circle is shaded out of the circle. Question 3. $$\frac{□}{□}$$ Answer: $$\frac{3}{4}$$ Explanation: 3 parts of the circle is shaded out of the circle Question 4. $$\frac{□}{□}$$ Answer: $$\frac{1}{1}$$ = 1 Explanation: The complete circle is shaded. So, the answer is 1. Question 5. $$\frac{□}{□}$$ Answer: $$\frac{1}{2}$$ Explanation: Half of the part is shaded out of the circle. So, the answer is 1/2. Question 6. $$\frac{□}{□}$$ Answer: $$\frac{1}{12}$$ Tell whether the angle on the circle shows a $$\frac{1}{4}, \frac{1}{2}, \frac{3}{4}$$, or 1 full turn clockwise or counterclockwise. Question 7. Type below: ________ Answer: $$\frac{3}{4}$$; counterclockwise Explanation: The image shows the counterclockwise and formed the fraction of $$\frac{3}{4}$$. Question 8. Type below: ________ Answer: $$\frac{1}{2}$$; clockwise Explanation: The image shows the clockwise and formed the fraction of $$\frac{1}{2}$$. Question 9. Type below: ________ Answer: $$\frac{1}{4}$$; clockwise Explanation: The image shows the clockwise and formed the fraction of $$\frac{1}{4}$$. Question 10. Susan watched the game from 1 p.m. to 1:30 p.m. Describe the turn the minute hand made. Type below: ________ Answer: The minute hand made a $$\frac{1}{2}$$ turn clockwise. Question 11. Compare the angles in Exercises 1 and 5. Does the position of the angle affect the size of the angle? Explain. _____ Answer: No; The size of the angle does not depend on the lengths of its sides. ### Page No. 604 Question 12. Malcolm drew this angle on the circle. Which of the following describes the angle? Mark all that apply. Options: a. $$\frac{3}{4}$$ turn b. $$\frac{1}{4}$$ turn c. clockwise d. counterclockwise Answer: a. $$\frac{3}{4}$$ turn d. counterclockwise Explanation: The image show the $$\frac{3}{4}$$ turn and also the counterclockwise. Sense or Nonsense? Question 13. Whose statement makes sense? Whose statement is nonsense? Explain your reasoning. Type below: __________ Answer: The girl’s statement makes sense. The boy’s statement makes non-sense. Because from the figure it is clearly shown that the shaded part is $$\frac{1}{4}$$ of the circle. There is no particular direction given to measure the shaded part. ### Conclusion: Finally, you will find detailed solutions to all questions that you’re looking for. So, you can attempt the exam with utmost confidence and secure good grades in the exams. The topics covered in Ch 10 Two-dimensional figures are such Lines, Rays, Angles, Classify Triangles by Angles, Parallel Lines and Perpendicular Lines, etc. Understand how to solve the problems of two-dimensional figures by accessing our Go Math Grade 4 Solution Key Chapter 10 Two-dimensional figures Homework Practice FL. ## Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation Get Free Access to Download Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation PDF from here. Start your preparation with the help of Go Math Grade 8 Answer Key. It is essential for all the students to learn the concepts of this chapter in-depth. So, make use of the Go Math Grade 8 Chapter 2 Exponents and Scientific Notation Solution Key links and go through the solutions. ## Go Math Grade 8 Chapter 2 Exponents and Scientific Notation Answer Key Check out the list of the topics before you start your preparation. You can step by step explanation for all the questions in HMH Go Math Grade 8 Answer Key Chapter 2 Exponents and Scientific Notation for free of cost. Quickly download Go Math Grade 8 Chapter 2 Answer Key PDF and fix the timetable to prepare. Lesson 1: Integer Exponents Lesson 2: Scientific Notation with Positive Powers of 10 Lesson 3: Scientific Notation with Negative Powers of 10 Lesson 4: Operations with Scientific Notation Model Quiz Mixed Review ### Guided Practice – Integer Exponents – Page No. 36 Find the value of each power. Question 1. 8−1 = $$\frac{□}{□}$$ Answer: $$\frac{1}{8}$$ Explanation: Base = 8 Exponent = 1 8−1 = (1/8)1 = 1/8 Question 2. 6−2 = $$\frac{□}{□}$$ Answer: $$\frac{1}{36}$$ Explanation: Base = 6 Exponent = 2 6−2 = (1/6)2 = 1/36 Question 3. 2560 = ______ Answer: 1 Explanation: 2560 Base = 256 Exponent = 0 Anything raised to the zeroth power is 1. 2560 = 1 Question 4. 102 = ______ Answer: 100 Explanation: Base = 10 Exponent = 2 102 = 10 × 10 = 100 Question 5. 54 = ______ Answer: 625 Explanation: Base = 5 Exponent = 4 54 = 5 × 5 × 5 × 5 = 625 Question 6. 2−5 = $$\frac{□}{□}$$ Answer: $$\frac{1}{32}$$ Explanation: Base = 2 Exponent = 5 2−5 = (1/2)5 = (1/2) × (1/2) × (1/2) × (1/2) × (1/2) = 1/32 Question 7. 4−5 = $$\frac{□}{□}$$ Answer: $$\frac{1}{1,024}$$ Explanation: Base = 4 Exponent = 5 4−5 = (1/4)5 = (1/4) × (1/4) × (1/4) × (1/4) × (1/4) = 1/1,024 Question 8. 890 = ______ Answer: 1 Explanation: 890 Base = 89 Exponent = 0 Anything raised to the zeroth power is 1. 890 = 1 Question 9. 11−3 = $$\frac{□}{□}$$ Answer: $$\frac{1}{1,331}$$ Explanation: Base = 11 Exponent = 3 11−3 = (1/11)3 = (1/11) × (1/11) × (1/11) = 1/1,331 Use properties of exponents to write an equivalent expression. Question 10. 4 ⋅ 4 ⋅ 4 = 4? Type below: _____________ Answer: 43 Explanation: The same number 4 is multiplying 3 times. The number of times a term is multiplied called the exponent. So the base is 4 and the exponent is 3 4 ⋅ 4 ⋅ 4 = 43 Question 11. (2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) = 2? ⋅ 2? = 2? Type below: _____________ Answer: 25 Explanation: The same number 2 is multiplying 5 times. The number of times a term is multiplied called the exponent. So the base is 2 and the exponent is 5 (2 ⋅ 2) ⋅ (2 ⋅ 2 ⋅ 2) = 22 ⋅ 23 = 25 Question 12. $$\frac { { 6 }^{ 7 } }{ { 6 }^{ 5 } }$$ = $$\frac{6⋅6⋅6⋅6⋅6⋅6⋅6}{6⋅6⋅6⋅6⋅6}$$ = 6? Type below: _____________ Answer: 62 Explanation: $$\frac { { 6 }^{ 7 } }{ { 6 }^{ 5 } }$$ = $$\frac{6⋅6⋅6⋅6⋅6⋅6⋅6}{6⋅6⋅6⋅6⋅6}$$ Cancel the common factors 6.6 Base = 6 Exponent = 2 62 Question 13. $$\frac { { 8 }^{ 12 } }{ { 8 }^{ 9 } }$$ = 8?-? = 8? Type below: _____________ Answer: 83 Explanation: $$\frac { { 8 }^{ 12 } }{ { 8 }^{ 9 } }$$ Bases are common. So, the exponents are subtracted 812-9 = 83 Question 14. 510 ⋅ 5 ⋅ 5 = 5? Type below: _____________ Answer: 512 Explanation: Bases are common and multiplied. So, the exponents are added Base = 5 Exponents = 10 + 1 + 1 = 12 512 Question 15. 78 ⋅ 75 = 7? Type below: _____________ Answer: 713 Explanation: Bases are common and multiplied. So, the exponents are added Base = 7 Exponents = 8 + 5 = 13 713 Question 16. (62)4 = (6 ⋅ 6)? = (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (? ⋅ ?) ⋅ ? = 6? Type below: _____________ Answer: 68 Explanation: (62)4 = (6 ⋅ 6)4 = (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (6 ⋅ 6) ⋅ (6 ⋅ 6) = 62 ⋅ 62 . 62 ⋅ 62 Bases are common and multiplied. So, the exponents are added = 62+2+2+2 68 Question 17. (33)3 = (3 ⋅ 3 ⋅ 3)3 = (3 ⋅ 3 ⋅ 3) ⋅ (? ⋅ ? ⋅ ?) ⋅ ? = 3? Type below: ______________ Answer: 39 Explanation: (3 ⋅ 3 ⋅ 3) ⋅ (3 ⋅ 3 ⋅ 3) ⋅ (3 ⋅ 3 ⋅ 3) = 33 ⋅ 33 ⋅ 33 Bases are common and multiplied. So, the exponents are added 33 + 3 + 3 39 Simplify each expression. Question 18. (10 − 6)3⋅42 + (10 + 2)2 ______ Answer: 1,168 Explanation: 4³. 4² + (12)² = 45 + (12)² = 45 + (12 . 12)² 45 + (144) = 1,024 + 144 = 1,168 Question 19. $$\frac { { (12-5) }^{ 7 } }{ { [(3+4)^{ 2 }] }^{ 2 } }$$ ________ Answer: 343 Explanation: 77 ÷ (7²)² = 77 ÷ 74 77-4 7 . 7 . 7 = 343 ESSENTIAL QUESTION CHECK-IN Question 20. Summarize the rules for multiplying powers with the same base, dividing powers with the same base, and raising a power to a power. Type below: ______________ Answer: The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents. The quotient rule tells us that we can divide two powers with the same base by subtracting the exponents. The “power rule” tells us that to raise a power to a power, just multiply the exponents. ### Independent Practice – Integer Exponents – Page No. 37 Question 21. Explain why the exponents cannot be added in the product 123 ⋅ 113. Type below: ______________ Answer: The exponent “product rule” tells us that, when multiplying two powers that have the same base, you can add the exponents. The bases are not the same in the given problem. => (12)³ x (11)³ If we solve this equation following the rule of exponent will get the correct answer: => (12 x 12 x 12) x (11 x 11 x 11) => 1728 X 1331 => the answer is 2 299 968 But if we add the exponent, the answer would be wrong => (12)³ x (11)³ => 132^6 => 5289852801024 which is wrong. Question 22. List three ways to express 35 as a product of powers. Type below: ______________ Answer: 3¹ . 34 3² . 33 3³ . 32 Question 23. Astronomy The distance from Earth to the moon is about 224 miles. The distance from Earth to Neptune is about 227 miles. Which distance is the greater distance and about how many times greater is it? _______ times Answer: (22)³ or 10,648 times Explanation: The distance from Earth to the moon is about 224 miles. The distance from Earth to Neptune is about 227 miles. 227 – 224 = (22)³ The greatest distance is from Earth to Neptune The distance from Earth to Neptune is greater by (22)³ or 10,648 miles Question 24. Critique Reasoning A student claims that 83 ⋅ 8-5 is greater than 1. Explain whether the student is correct or not. ______________ Answer: 83 ⋅ 8-5 is = 8-2 (1/8)² (1/8) . (1/8) = 1/64 = 0.015 The student is not correct. Find the missing exponent. Question 25. (b2)? = b-6 _______ Answer: (b2)-8 Explanation: (b2)? = b-6 (b-6) = b2-8 (b2-8) = b2 . b-8 (b2)-8 = b-6 Question 26. x? ⋅ x6 = x9 _______ Answer: Explanation: x? ⋅ x6 = x9 x9 = x3 + 6 x³ x6 Question 27. $$\frac { { y }^{ 25 } }{ { y }^{ ? } }$$ = y6 _______ Answer: y25 ÷ y16 Explanation: $$\frac { { y }^{ 25 } }{ { y }^{ ? } }$$ = y y6 = y25 – 16 y25 ÷ y16 Question 28. Communicate Mathematical Ideas Why do you subtract exponents when dividing powers with the same base? Type below: ______________ Answer: To divide exponents (or powers) with the same base, subtract the exponents. The division is the opposite of multiplication, so it makes sense that because you add exponents when multiplying numbers with the same base, you subtract the exponents when dividing numbers with the same base. Question 29. Astronomy The mass of the Sun is about 2 × 1027 metric tons, or 2 × 1030 kilograms. How many kilograms are in one metric ton? ________ kgs in one metric ton Answer: 1,000 kgs in one metric ton Explanation: The mass of the Sun is about 2 × 1027 metric tons, or 2 × 1030 kilograms. 2 × 1027 metric tons = 2 × 1030 ki 1 metric ton = 2 × 1030 ki ÷ 2 × 1027 = (10)³ = 1,000 kgs in one metric ton Question 30. Represent Real-World Problems In computer technology, a kilobyte is 210 bytes in size. A gigabyte is 230 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. What is the size of a terabyte? Type below: ______________ Answer: 240 bytes Explanation: In computer technology, a kilobyte is 210 bytes in size. A gigabyte is 230 bytes in size. The size of a terabyte is the product of the size of a kilobyte and the size of a gigabyte. terabyte = 210 bytes × 230 bytes = 210+30 bytes = 240 bytes ### Integer Exponents – Page No. 38 Question 31. Write equivalent expressions for x7 ⋅ x-2 and $$\frac { { x }^{ 7 } }{ { x }^{ 2 } }$$. What do you notice? Explain how your results relate to the properties of integer exponents. Type below: ______________ Answer: x^a * x^b = x^(a+b) and x^-a = 1/x^a Therefore, x^7 * x^-2 = x^7/x^2 = x^5 or x^7 * x^-2 = x^(7-2) = x^5 x^7 / x^2 = x^7 * x^-2 A toy store is creating a large window display of different colored cubes stacked in a triangle shape. The table shows the number of cubes in each row of the triangle, starting with the top row. Question 32. Look for a Pattern Describe any pattern you see in the table. Type below: ______________ Answer: As the number of rows increased, the number of cubes in each row by multiple of 3. Question 33. Using exponents, how many cubes will be in Row 6? How many times as many cubes will be in Row 6 than in Row 3? _______ times more cubes Answer: (33) times more cubes Explanation: For row 6, the number of cubes in each row = (36) (36) ÷ (33) = (36-3) = (33) (33) times more cubes Question 34. Justify Reasoning If there are 6 rows in the triangle, what is the total number of cubes in the triangle? Explain how you found your answer. ______ cubes Answer: 1,092 cubes Explanation: (31) + (32) + (33) + (34) + (35) + (36) 3 + 9 + 27 + 81 + 243 + 729 = 1,092 H.O.T. Focus on Higher Order Thinking Question 35. Critique Reasoning A student simplified the expression $$\frac { { 6 }^{ 2 } }{ { 36 }^{ 2 } }$$ as $$\frac{1}{3}$$. Do you agree with this student? Explain why or why not. ______________ Answer: $$\frac { { 6 }^{ 2 } }{ { 36 }^{ 2 } }$$ (62) ÷ (62 (62) ÷ (64) (62 – 4) (6-2) = 1/36 I don’t agree with the student Question 36. Draw Conclusions Evaluate –an when a = 3 and n = 2, 3, 4, and 5. Now evaluate (–a)n when a = 3 and n = 2, 3, 4, and 5. Based on this sample, does it appear that –an = (–a)n? If not, state the relationships, if any, between –an and (–a)n. Type below: ______________ Answer: –an when a = 3 and n = 2, 3, 4, and 5. -3n -(32 )= -9 (–a)n = -3 . -3 = 9 –an = (–a)n are not equal. Question 37. Persevere in Problem Solving A number to the 12th power divided by the same number to the 9th power equals 125. What is the number? _______ Answer: Let’s call our number a. (a12 ) ÷ (a9 ) (a12-9 ) = (a3 ) (a3 ) = 125 a = (125)1/3 a = 5 ### Guided Practice – Scientific Notation with Positive Powers of 10 – Page No. 42 Write each number in scientific notation. Question 1. 58,927 (Hint: Move the decimal left 4 places) Type below: ______________ Answer: 5.8927 × (10)4 Explanation: 58,927 Move the decimal left 4 places 5.8927 × (10)4 Question 2. 1,304,000,000 (Hint: Move the decimal left 9 places.) Type below: ______________ Answer: 1.304 × (10)9 Explanation: 1,304,000,000 Move the decimal left 9 places 1.304 × (10)9 Question 3. 6,730,000 Type below: ______________ Answer: Explanation: 6,730,000 Move the decimal left 6 places 6.73 × (10)6 Question 4. 13,300 Type below: ______________ Answer: Explanation: 13,300 Move the decimal left 4 places 1.33 × (10)4 Question 5. An ordinary quarter contains about 97,700,000,000,000,000,000,000 atoms. Type below: ______________ Answer: Explanation: 97,700,000,000,000,000,000,000 Move the decimal left 22 places 9.77 × (10)22 Question 6. The distance from Earth to the Moon is about 384,000 kilometers. Type below: ______________ Answer: 3.84 × (10)6 Explanation: 384,000 Move the decimal left 6 places 3.84 × (10)6 Write each number in standard notation. Question 7. 4 × 105 (Hint: Move the decimal right 5 places.) Type below: ______________ Answer: 400,000 Explanation: 4 × 105 Move the decimal right 5 places 400,000 Question 8. 1.8499 × 109 (Hint: Move the decimal right 9 places.) Type below: ______________ Answer: 1849900000 Explanation: 1.8499 × 109 Move the decimal right 9 places 1849900000 Question 9. 6.41 × 103 Type below: ______________ Answer: 6410 Explanation: 6.41 × 103 Move the decimal right 3 places 6410 Question 10. 8.456 × 107 Type below: ______________ Answer: 84560000 Explanation: 8.456 × 107 Move the decimal right 7 places 84560000 Question 11. 8 × 105 Type below: ______________ Answer: 800,000 Explanation: 8 × 105 Move the decimal right 5 places 800,000 Question 12. 9 × 1010 Type below: ______________ Answer: 90000000000 Explanation: 9 × 1010 Move the decimal right 10 places 90000000000 Question 13. Diana calculated that she spent about 5.4 × 104 seconds doing her math homework during October. Write this time in standard notation. Type below: ______________ Answer: 5400 Explanation: Diana calculated that she spent about 5.4 × 104 seconds doing her math homework during October. 5.4 × 104 Move the decimal right 4 places 5400 Question 14. The town recycled 7.6 × 106 cans this year. Write the number of cans in standard notation Type below: ______________ Answer: 7600000 Explanation: The town recycled 7.6 × 106 cans this year. 7.6 × 106 Move the decimal right 10 places 7600000 ESSENTIAL QUESTION CHECK-IN Question 15. Describe how to write 3,482,000,000 in scientific notation. Type below: ______________ Answer: 3.482 × (10)9 Explanation: 3,482,000,000 Move the decimal left 9 places 3.482 × (10)9 ### Independent Practice – Scientific Notation with Positive Powers of 10 – Page No. 43 Paleontology Use the table for problems 16–21. Write the estimated weight of each dinosaur in scientific notation. Question 16. Apatosaurus ______________ Type below: ______________ Answer: 6.6 × (10)4 Explanation: 66,000 Move the decimal left 4 places 6.6 × (10)4 Question 17. Argentinosaurus ___________ Type below: ______________ Answer: 2.2 × (10)5 Explanation: 220,000 Move the decimal left 5 places 2.2 × (10)5 Question 18. Brachiosaurus ______________ Type below: ______________ Answer: 1 × (10)5 Explanation: 100,000 Move the decimal left 5 places 1 × (10)5 Question 19. Camarasaurus ______________ Type below: ______________ Answer: 4 × (10)4 Explanation: 40,000 Move the decimal left 4 places 4 × (10)4 Question 20. Cetiosauriscus ____________ Type below: ______________ Answer: 1.985 × (10)4 Explanation: 19,850 Move the decimal left 4 places 1.985 × (10)4 Question 21. Diplodocus _____________ Type below: ______________ Answer: 5 × (10)4 Explanation: 50,000 Move the decimal left 4 places 5 × (10)4 Question 22. A single little brown bat can eat up to 1,000 mosquitoes in a single hour. Express in scientific notation how many mosquitoes a little brown bat might eat in 10.5 hours. Type below: ______________ Answer: 1.05 × (10)4 Explanation: (1000 x 10.5) = 10500. The little brown bat can eat 10500 mosquitoes in 10.5 hours. 1.05 × (10)4 Question 23. Multistep Samuel can type nearly 40 words per minute. Use this information to find the number of hours it would take him to type 2.6 × 105 words. Type below: ______________ Answer: Samuel can type 40 words per minute. Then how many hours will it take for him to type 2.6 words times 10 to the power of five words 2.6 words time 10 to the power of 5 2.6 × (10)4 2.6 x 100 000 = 260 000 words in all. Now, we need to find the number of words Samuel can type in an hour 40 words/minutes, in 1 hour there are 60 minutes 40 x 60 2,400 words /hour Now, let’s divide the total of words he needs to type to the number of words he can type in an hour 260 000 / 2 400 108.33 hours. Question 24. Entomology A tropical species of mite named Archegozetes longisetosus is the record holder for the strongest insect in the world. It can lift up to 1.182 × 103 times its own weight. a. If you were as strong as this insect, explain how you could find how many pounds you could lift. Type below: ______________ Answer: Number of pounds you can lift by 1.182 × 103 by your weight Question 24. b. Complete the calculation to find how much you could lift, in pounds, if you were as strong as an Archegozetes longisetosus mite. Express your answer in both scientific notation and standard notation. Type below: ______________ Answer: scientific notation: 1.182 × 105 standard notation: 118200 Explanation: 1.182 × 103 × 102 1.182 × 105 118200 Question 25. During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer. In scientific notation, express approximately how much the calves weighed all together. Type below: ______________ Answer: 4.6 × 103 Explanation: During a discussion in science class, Sharon learns that at birth an elephant weighs around 230 pounds. In four herds of elephants tracked by conservationists, about 20 calves were born during the summer. Total weight of the claves = 230 × 20 = 4600 Move the decimal left 3 places 4.6 × 103 Question 26. Classifying Numbers Which of the following numbers are written in scientific notation? 0.641 × 103 9.999 × 104 2 × 101 4.38 × 510 Type below: ______________ Answer: 0.641 × 103 4.38 × 510 ### Scientific Notation with Positive Powers of 10 – Page No. 44 Question 27. Explain the Error Polly’s parents’ car weighs about 3500 pounds. Samantha, Esther, and Polly each wrote the weight of the car in scientific notation. Polly wrote 35.0 × 102, Samantha wrote 0.35 × 104, and Esther wrote 3.5 × 104. a. Which of these girls, if any, is correct? ______________ Answer: None of the girls is correct Question 27. b. Explain the mistakes of those who got the question wrong. Type below: ______________ Answer: Polly did not express the number such first part is greater than or equal to 1 and less than 10 Samantha did not express the number such first part is greater than or equal to 1 and less than 10 Esther did not express the exponent of 10 correctly Question 28. Justify Reasoning If you were a biologist counting very large numbers of cells as part of your research, give several reasons why you might prefer to record your cell counts in scientific notation instead of standard notation. Type below: ______________ Answer: It is easier to comprehend the magnitude of large numbers when in scientific notation as multiple zeros in the number are removed and express as an exponent of 10. It is easier to compare large numbers when in scientific notation as numbers are be expressed as a product of a number greater than or equal to 1 and less than 10 It is easier to multiply the numbers in scientific notation. H.O.T. Focus on Higher Order Thinking Question 29. Draw Conclusions Which measurement would be least likely to be written in scientific notation: number of stars in a galaxy, number of grains of sand on a beach, speed of a car, or population of a country? Explain your reasoning. Type below: ______________ Answer: speed of a car Explanation: As we know scientific notation is used to express measurements that are extremely large or extremely small. The first two are extremely large, then, they could be expressed in scientific notation. If we compare the speed of a car and the population of a country, it is clear that the larger will be the population of a country. Therefore, it is more likely to express that in scientific notation, so the answer is the speed of a car. Question 30. Analyze Relationships Compare the two numbers to find which is greater. Explain how you can compare them without writing them in standard notation first. 4.5 × 106 2.1 × 108 Type below: ______________ Answer: 2.1 × 108 Explanation: 2.1 × 108 is greater because the power of 10 is greater in 2.1 × 108 Question 31. Communicate Mathematical Ideas To determine whether a number is written in scientific notation, what test can you apply to the first factor, and what test can you apply to the second factor? Type below: ______________ Answer: The first term must have one number before the decimal point the second term (factor) must be 10 having some power. ### Guided Practice – Scientific Notation with Negative Powers of 10 – Page No. 48 Write each number in scientific notation. Question 1. 0.000487 Hint: Move the decimal right 4 places. Type below: ______________ Answer: 4.87 × 10-4 Explanation: 0.000487 Move the decimal right 4 places 4.87 × 10-4 Question 2. 0.000028 Hint: Move the decimal right 5 places Type below: ______________ Answer: 2.8 × 10-5 Explanation: 0.000028 Move the decimal right 5 places 2.8 × 10-5 Question 3. 0.000059 Type below: ______________ Answer: 5.9 × 10-5 Explanation: 0.000059 Move the decimal right 5 places 5.9 × 10-5 Question 4. 0.0417 Type below: ______________ Answer: 4.17 × 10-2 Explanation: 0.0417 Move the decimal right 2 places 4.17 × 10-2 Question 5. Picoplankton can be as small as 0.00002 centimeters. Type below: ______________ Answer: 2 × 10-5 Explanation: 0.00002 Move the decimal right 5 places 2 × 10-5 Question 6. The average mass of a grain of sand on a beach is about 0.000015 gram. Type below: ______________ Answer: 1.5 × 10-5 Explanation: 0.000015 Move the decimal right 5 places 1.5 × 10-5 Write each number in standard notation. Question 7. 2 × 10-5 Hint: Move the decimal left 5 places. Type below: ______________ Answer: 0.00002 Explanation: 2 × 10-5 Move the decimal left 5 places 0.00002 Question 8. 3.582 × 10-6 Hint: Move the decimal left 6 places. Type below: ______________ Answer: 0.000003582 Explanation: 3.582 × 10-6 Move the decimal left 6 places 0.000003582 Question 9. 8.3 × 10-4 Type below: ______________ Answer: 0.00083 Explanation: 8.3 × 10-4 Move the decimal left 4 places 0.00083 Question 10. 2.97 × 10-2 Type below: ______________ Answer: 0.0297 Explanation: 2.97 × 10-2 Move the decimal left 2 places 0.0297 Question 11. 9.06 × 10-5 Type below: ______________ Answer: 0.0000906 Explanation: 9.06 × 10-5 Move the decimal left 5 places 0.0000906 Question 12. 4 × 10-5 Type below: ______________ Answer: 0.00004 Explanation: 4 × 10-5 Move the decimal left 5 places 0.00004 Question 13. The average length of a dust mite is approximately 0.0001 meters. Write this number in scientific notation. Type below: ______________ Answer: 1 × 10-4 Explanation: The average length of a dust mite is approximately 0.0001 meters. 0.0001 Move the decimal right 4 places 1 × 10-4 Question 14. The mass of a proton is about 1.7 × 10-24 grams. Write this number in standard notation. Type below: ______________ Answer: 0.000000000000000000000017 Explanation: The mass of a proton is about 1.7 × 10-24 grams. 1.7 × 10-24 Move the decimal left 24 places 0.000000000000000000000017 ESSENTIAL QUESTION CHECK-IN Question 15. Describe how to write 0.0000672 in scientific notation. Type below: ______________ Answer: 6.72 × 10-5 Explanation: 0.0000672 Move the decimal right 5 places 6.72 × 10-5 ### Independent Practice – Scientific Notation with Negative Powers of 10 – Page No. 49 Use the table for problems 16–21. Write the diameter of the fibers in scientific notation. Question 16. Alpaca _______ Type below: ______________ Answer: 2.77 × 10-3 Explanation: 0.00277 Move the decimal right 3 places 2.77 × 10-3 Question 17. Angora rabbit _____________ Type below: ______________ Answer: 1.3 × 10-3 Explanation: 0.0013 Move the decimal right 3 places 1.3 × 10-3 Question 18. Llama ____________ Type below: ______________ Answer: 3.5 × 10-3 Explanation: 0.0035 Move the decimal right 3 places 3.5 × 10-3 Question 19. Angora goat ____________ Type below: ______________ Answer: 4.5 × 10-3 Explanation: 0.0045 Move the decimal right 3 places 4.5 × 10-3 Question 20. Orb web spider ___________ Type below: ______________ Answer: 1.5 × 10-2 Explanation: 0.015 Move the decimal right 2 places 1.5 × 10-2 Question 21. Vicuña __________ Type below: ______________ Answer: 8 × 10-4 Explanation: 0.0008 Move the decimal right 4 places 8 × 10-4 Question 22. Make a Conjecture Which measurement would be least likely to be written in scientific notation: the thickness of a dog hair, the radius of a period on this page, the ounces in a cup of milk? Explain your reasoning. Type below: ______________ Answer: The ounces in a cup of milk would be least likely to be written in scientific notation. The ounces in a cup of milk is correct. Scientific notation is used for either very large or extremely small numbers. The thickness of dog hair is very small as the hair is thin. Hence can be converted to scientific notation. The radius of a period on this page is also pretty small. Hence can be converted to scientific notation. The ounces in a cup of milk. There are 8 ounces in a cup, so this is least likely to be written in scientific notation. Question 23. Multiple Representations Convert the length 7 centimeters to meters. Compare the numerical values when both numbers are written in scientific notation Type below: ______________ Answer: 7 centimeters convert to meters In every 1 meter, there are 100 centimeters = 7/100 = 0.07 Therefore, in 7 centimeters there are 0.07 meters. 7 cm is a whole number while 0.07 m is a decimal number Scientific Notation of each number 7 cm = 7 x 10° 7 m = 1 x 10¯² Scientific notation, by the way, is an expression used by the scientist to make a large number of very small number easy to handle. Question 24. Draw Conclusions A graphing calculator displays 1.89 × 1012 as 1.89E12. How do you think it would display 1.89 × 10-12? What does the E stand for? Type below: ______________ Answer: 1.89E-12. E= Exponent Explanation: Question 25. Communicate Mathematical Ideas When a number is written in scientific notation, how can you tell right away whether or not it is greater than or equal to 1? Type below: ______________ Answer: A number written in scientific notation is of the form a × 10-n where 1 ≤ a < 10 and n is an integer The number is greater than or equal to one if n ≥ 0. Question 26. The volume of a drop of a certain liquid is 0.000047 liter. Write the volume of the drop of liquid in scientific notation. Type below: ______________ Answer: 4.7 × 10-5 Explanation: The volume of a drop of a certain liquid is 0.000047 liter. Move the decimal right 5 places 4.7 × 10-5 Question 27. Justify Reasoning If you were asked to express the weight in ounces of a ladybug in scientific notation, would the exponent of the 10 be positive or negative? Justify your response. ______________ Answer: Negative Explanation: Scientific notation is used to express very small or very large numbers. Very small numbers are written in scientific notation using negative exponents. Very large numbers are written in scientific notation using positive exponents. Since a ladybug is very small, we would use the very small scientific notation, which uses negative exponents. ### Physical Science – Scientific Notation with Negative Powers of 10 – Page No. 50 The table shows the length of the radii of several very small or very large items. Complete the table. Question 28. Type below: ______________ Answer: 1.74 × (10)6 Explanation: The moon = 1,740,000 Move the decimal left 6 places 1.74 × (10)6 Question 29. Type below: ______________ Answer: 1.25e-10 Explanation: 1.25 × (10)-10 Move the decimal left 10 places 1.25e-10 Question 30. Type below: ______________ Answer: 2.8 × (10)3 Explanation: 0.0028 Move the decimal left 3 places 2.8 × (10)3 Question 31. Type below: ______________ Answer: 71490000 Explanation: 7.149 × (10)7 Move the decimal left 7 places 71490000 Question 32. Type below: ______________ Answer: 1.82 × (10)-10 Explanation: 0.000000000182 Move the decimal right 10 places 1.82 × (10)-10 Question 33. Type below: ______________ Answer: 3397000 Explanation: 3.397 × (10)6 Move the decimal left 6 places 3397000 Question 34. List the items in the table in order from the smallest to the largest. Type below: ______________ Answer: 1.82 × (10)-10 1.25 × (10)-10 2.8 × (10)3 1.74 × (10)6 3.397 × (10)6 7.149 × (10)7 H.O.T. Focus on Higher Order Thinking Question 35. Analyze Relationships Write the following diameters from least to greatest. 1.5 × 10-2m ; 1.2 × 102 m ; 5.85 × 10-3 m ; 2.3 × 10-2 m ; 9.6 × 10-1 m. Type below: ______________ Answer: 5.85 × 10-3 m, 1.5 × 10-2m, 2.3 × 10-2 m, 9.6 × 10-1 m, 1.2 × 102 m Explanation: 1.5 × 10-2m = 0.015 1.2 × 102 m = 120 5.85 × 10-3 m = 0.00585 2.3 × 10-2 m = 0.023 9.6 × 10-1 m = 0.96 0.00585, 0.015, 0.023, 0.96, 120 Question 36. Critique Reasoning Jerod’s friend Al had the following homework problem: Express 5.6 × 10-7 in standard form. Al wrote 56,000,000. How can Jerod explain Al’s error and how to correct it? Type below: ______________ Answer: Explanation: 5.6 × 10-7 in 0.000000056 Al wrote 56,000,000. AI wrote the zeroes to the right side of the 56 which is not correct. As the exponent of 10 is negative zero’s need to add to the left of the number. Question 37. Make a Conjecture Two numbers are written in scientific notation. The number with a positive exponent is divided by the number with a negative exponent. Describe the result. Explain your answer. Type below: ______________ Answer: When the division is performed, the denominator exponent is subtracted from the numerator exponent. Subtracting a negative value from the numerator exponent will increase its value. ### Guided Practice – Operations with Scientific Notation – Page No. 54 Add or subtract. Write your answer in scientific notation. Question 1. 4.2 × 106 + 2.25 × 105 + 2.8 × 106 4.2 × 106 + ? × 10 ? + 2.8 × 106 4.2 + ? + ? ? × 10? Type below: ______________ Answer: 4.2 × 106 + 0.225 × 10 × 105 + 2.8 × 106 Rewrite 2.25 = 0.225 × 10 (4.2 + 0.225 + 2.8) × 106 7.225 × 106 Question 2. 8.5 × 103 − 5.3 × 103 − 1.0 × 102 8.5 × 103 − 5.3 × 103 − ? × 10? ? − ? − ? ? × 10? Type below: ______________ Answer: 8.5 × 103 − 5.3 × 103 − 0.1 × 103 (8.5 − 5.3 − 0.1) × 103 (3.1) × 103 Question 3. 1.25 × 102 + 0.50 × 102 + 3.25 × 102 Type below: ______________ Answer: 1.25 × 102 + 0.50 × 102 + 3.25 × 102 (1.25 + 0.50 + 3.25) × 102 5 × 102 Question 4. 6.2 × 105 − 2.6 × 104 − 1.9 × 102 Type below: ______________ Answer: 6.2 × 105 − 2.6 × 104 − 1.9 × 102 6.2 × 105 − 0.26 × 105 − 0.0019 × 105 (6.2 – 0.26 – 0.0019) × 105 5.9381 × 105 Multiply or divide. Write your answer in scientific notation. Question 5. (1.8 × 109)(6.7 × 1012) Type below: ______________ Answer: 12.06 × 1021 Explanation: (1.8 × 109)(6.7 × 1012) 1.8 × 6.7 = 12.06 109+12 = 1021 12.06 × 1021 Question 6. $$\frac { { 3.46×10 }^{ 17 } }{ { 2×10 }^{ 9 } }$$ Type below: ______________ Answer: 1.73 × 108 Explanation: 3.46/2 = 1.73 1017/109 = 1017-9 = 108 1.73 × 108 Question 7. (5 × 1012)(3.38 × 106) Type below: ______________ Answer: 16.9 × 1018 Explanation: (5 × 1012)(3.38 × 106) 5 × 3.38 = 16.9 106+12 = 1018 16.9 × 1018 Question 8. $$\frac { { 8.4×10 }^{ 21 } }{ { 4.2×10 }^{ 14 } }$$ Type below: ______________ Answer: 2 × 107 Explanation: 8.4/4.2 = 2 1021/1014 = 1021-14 = 107 2 × 107 Write each number using calculator notation. Question 9. 3.6 × 1011 Type below: ______________ Answer: 3.6e11 Question 10. 7.25 × 10-5 Type below: ______________ Answer: 7.25e-5 Question 11. 8 × 10-1 Type below: ______________ Answer: 8e-1 Write each number using scientific notation. Question 12. 7.6E − 4 Type below: ______________ Answer: 7.6 × 10-4 Question 13. 1.2E16 Type below: ______________ Answer: 1.2 × 1016 Question 14. 9E1 Type below: ______________ Answer: 9 × 101 ESSENTIAL QUESTION CHECK-IN Question 15. How do you add, subtract, multiply, and divide numbers written in scientific notation? Type below: ______________ Answer: Numbers with exponents can be added and subtracted only when they have the same base and exponent. To multiply two numbers in scientific notation, multiply their coefficients and add their exponents. To divide two numbers in scientific notation, divide their coefficients, and subtract their exponents. ### Independent Practice – Operations with Scientific Notation – Page No. 55 Question 16. An adult blue whale can eat 4.0 × 107 krill in a day. At that rate, how many krill can an adult blue whale eat in 3.65 × 102 days? Type below: ______________ Answer: 14.6 × 109 Explanation: (4.0 × 107 )(3.65 × 102 ) 4.0 × 3.65 = 14.6 107+2 = 109 14.6 × 109 Question 17. A newborn baby has about 26,000,000,000 cells. An adult has about 4.94 × 1013 cells. How many times as many cells does an adult have than a newborn? Write your answer in scientific notation. Type below: ______________ Answer: 1.9 × 103 Explanation: 26,000,000,000 = 2.6 × 1010 4.94 × 1013 (4.94 × 1013 )/(2.6 × 1010 ) 1.9 × 103 Represent Real-World Problems The table shows the number of tons of waste generated and recovered (recycled) in 2010. Question 18. What is the total amount of paper, glass, and plastic waste generated? Type below: ______________ Answer: 11.388 × 107 Explanation: 7.131 × 107 + 1.153 × 107 + 3.104 × 107 11.388 × 107 Question 19. What is the total amount of paper, glass, and plastic waste recovered? Type below: ______________ Answer: 5.025 × 107 Explanation: 4.457 × 107 + 0.313 × 107 + 0.255 × 107 5.025 × 107 Question 20. What is the total amount of paper, glass, and plastic waste not recovered? Type below: ______________ Answer: 6.363 × 107 Explanation: (11.388 × 107 ) – (5.025 × 107) 6.363 × 107 Question 21. Which type of waste has the lowest recovery ratio? Type below: ______________ Answer: Plastics Explanation: 7.131 × 107 – 4.457 × 107 = 2.674 × 107 1.153 × 107 – 0.313 × 107 = 0.84 × 107 3.104 × 107 – 0.255 × 107 = 2.849 × 107 Plastics has the lowest recovery ratio Social Studies The table shows the approximate populations of three countries. Question 22. How many more people live in France than in Australia? Type below: ______________ Answer: 4.33 × 107 Explanation: (6.48 × 107 ) – (2.15× 107) 4.33 × 107 Question 23. The area of Australia is 2.95 × 106 square miles. What is the approximate average number of people per square mile in Australia? Type below: ______________ Answer: About 7 people per square mile Explanation: 2.95 × 106 square miles = (2.15× 107) 1 square mile = (2.15× 107)/(2.95 × 106) = 7.288 Question 24. How many times greater is the population of China than the population of France? Write your answer in standard notation. Type below: ______________ Answer: 20.52; there are about 20 people in china for every 1 person in France. Question 25. Mia is 7.01568 × 106 minutes old. Convert her age to more appropriate units using years, months, and days. Assume each month to have 30.5 days. Type below: ______________ Answer: 13 years 3 months 22.5 days Explanation: 7.01568 × 106 minutes (7.01568 × 106 minutes) ÷ (6 × 101)(2.4 × 101)(1.2 × 101)(3.05 × 101) = (1.331 × 101) = 13 years 3 months 22.5 days ### Operations with Scientific Notation – Page No. 56 Question 26. Courtney takes 2.4 × 104 steps during her a long-distance run. Each step covers an average of 810 mm. What total distance (in mm) did Courtney cover during her run? Write your answer in scientific notation. Then convert the distance to the more appropriate unit kilometers. Write that answer in standard form. ______ km Answer: 19.4 km Explanation: Courtney takes 2.4 × 104 steps during her a long-distance run. Each step covers an average of 810 mm. (2.4 × 104 steps) × 810mm (2.4 × 104 ) × (8.1 × 102 ) The total distance covered = (19.44 × 106 ) Convert to unit kilometers: (19.44 × 106 ) × (1 × 10-6 ) (1.94 × 101 ) 19.4 km Question 27. Social Studies The U.S. public debt as of October 2010 was$9.06 × 1012. What was the average U.S. public debt per American if the population in 2010 was 3.08 × 108 people?
$_______ Answer:$29,400 per American

Explanation:
($9.06 × 1012.)/(3.08 × 108 ) ($2.94 × 104.) = $29,400 per American H.O.T. Focus on Higher Order Thinking Question 28. Communicate Mathematical Ideas How is multiplying and dividing numbers in scientific notation different from adding and subtracting numbers in scientific notation? Type below: ______________ Answer: When you multiply or divide in scientific notation, you just add or subtract the exponents. When you add or subtract in scientific notation, you have to make the exponents the same before you can do anything else. Question 29. Explain the Error A student found the product of 8 × 106 and 5 × 109 to be 4 × 1015. What is the error? What is the correct product? Type below: ______________ Answer: The error student makes is he multiply the terms instead of addition. Explanation: product of 8 × 106 and 5 × 109 40 × 1015 4 × 1016 The student missed the 10 while multiplying the product of 8 × 106 and 5 × 109 Question 30. Communicate Mathematical Ideas Describe a procedure that can be used to simplify $$\frac { { (4.87×10 }^{ 12 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (6.1×10 }^{ 8 }) }$$. Write the expression in scientific notation in simplified form. Type below: ______________ Answer: $$\frac { { (4.87×10 }^{ 12 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (6.1×10 }^{ 8 }) }$$ $$\frac { { (487×10 }^{ 10 }) – { (7×10 }^{ 10 }) }{ { (3×10 }^{ 7 })-{ (61×10 }^{ 7 }) }$$ (480 × 1010 )/(64 × 107 ) 7.50 × 10³ ### 2.1 Integer Exponents – Model Quiz – Page No. 57 Find the value of each power. Question 1. 3-4 $$\frac{□}{□}$$ Answer: $$\frac{1}{81}$$ Explanation: Base = 3 Exponent = 4 3-4 = (1/3)4 = 1/81 Question 2. 350 ______ Answer: 1 Explanation: 350 Base = 35 Exponent = 0 Anything raised to the zeroth power is 1. 350 = 1 Question 3. 44 ______ Answer: 256 Explanation: Base = 4 Exponent = 4 44 = 4 . 4 . 4 . 4 = 2561 Use the properties of exponents to write an equivalent expression. Question 4. 83 ⋅ 87 Type below: ____________ Answer: 810 Explanation: 83 ⋅ 87 83+7 810 Question 5. $$\frac { 12^{ 6 } }{ 12^{ 2 } }$$ Type below: ____________ Answer: 124 Explanation: 126 ÷ 122 126-2 124 Question 6. (103)5 Type below: ____________ Answer: 108 Explanation: (103)5 (103+5) (108) 2.2 Scientific Notation with Positive Powers of 10 Convert each number to scientific notation or standard notation. Question 7. 2,000 Type below: ____________ Answer: 2 × (103) Explanation: 2 × 1,000 Move the decimal left 3 places 2 × (103) Question 8. 91,007,500 Type below: ____________ Answer: 9.10075 × (107) Explanation: 91,007,500 Move the decimal left 7 places 9.10075 × (107) Question 9. 1.0395 × 109 Type below: ____________ Answer: 1039500000 Explanation: 1.0395 × 109 Move the decimal right 9 places 1039500000 Question 10. 4 × 102 Type below: ____________ Answer: 400 Explanation: 4 × 102 Move the decimal right 2 places 400 2.3 Scientific Notation with Negative Powers of 10 Convert each number to scientific notation or standard notation. Question 11. 0.02 Type below: ____________ Answer: 2 × 10-2 Explanation: 0.02 Move the decimal right 2 places 2 × 10-2 Question 12. 0.000701 Type below: ____________ Answer: 7.01 × 10-4 Explanation: 0.000701 Move the decimal right 4 places 7.01 × 10-4 Question 13. 8.9 × 10-5 Type below: ____________ Answer: 0.000089 Explanation: 8.9 × 10-5 Move the decimal left 5 places 0.000089 Question 14. 4.41 × 10-2 Type below: ____________ Answer: 0.0441 Explanation: 4.41 × 10-2 Move the decimal left 2 places 0.0441 2.4 Operations with Scientific Notation Perform the operation. Write your answer in scientific notation. Question 15. 7 × 106 − 5.3 × 106 Type below: ____________ Answer: 1.7 × 106 Explanation: 7 × 106 − 5.3 × 106 (7 – 5.3) × 106 1.7 × 106 Question 16. 3.4 × 104 + 7.1 × 105 Type below: ____________ Answer: 7.44 × 104 Explanation: 3.4 × 104 + 7.1 × 105 0.34 × 105 + 7.1 × 105 (0.34 + 7.1) × 105 7.44 × 105 Question 17. (2 × 104)(5.4 × 106) Type below: ____________ Answer: 10.8 × 1010 Explanation: (2 × 104)(5.4 × 106) (2 × 5.4)(104 × 106) 10.8 × 1010 Question 18. $$\frac { 7.86×10^{ 9 } }{ 3×10^{ 4 } }$$ Type below: ____________ Answer: 2.62 × 105 Explanation: 7.86/3 = 2.62 109/104 = 105 2.62 × 105 Question 19. Neptune’s average distance from the Sun is 4.503×109 km. Mercury’s average distance from the Sun is 5.791 × 107 km. About how many times farther from the Sun is Neptune than Mercury? Write your answer in scientific notation. Type below: ____________ Answer: (0.7776 × 102 km) = 77.76 times Explanation: As Neptune’s average distance from the sun is 4.503×109 km and Mercury is 5.791 × 107 km (4.503×109 km)/(5.791 × 107 km) (0.7776 × 109-7 km) (0.7776 × 102 km) 77.76 times Essential Question Question 20. How is scientific notation used in the real world? Type below: ____________ Answer: Scientific notation is used to write very large or very small numbers using less digits. ### Selected Response – Mixed Review – Page No. 58 Question 1. Which of the following is equivalent to 6-3? Options: a. 216 b. $$\frac{1}{216}$$ c. −$$\frac{1}{216}$$ d. -216 Answer: b. $$\frac{1}{216}$$ Explanation: Base = 6 Exponent = 3 63 = (1/6)3 = 1/216 Question 2. About 786,700,000 passengers traveled by plane in the United States in 2010. What is this number written in scientific notation? Options: a. 7,867 × 105 passengers b. 7.867 × 102 passengers c. 7.867 × 108 passengers d. 7.867 × 109 passengers Answer: c. 7.867 × 108 passengers Explanation: 786,700,000 Move the decimal left 8 places 7.867 × 108 passengers Question 3. In 2011, the population of Mali was about 1.584 × 107 people. What is this number written in standard notation? Options: a. 1.584 people b. 1,584 people c. 15,840,000 people d. 158,400,000 people Answer: c. 15,840,000 people Explanation: 1.584 × 107 Move the decimal right 7 places 15,840,000 people Question 4. The square root of a number is between 7 and 8. Which could be the number? Options: a. 72 b. 83 c. 51 d. 66 Answer: c. 51 Explanation: 7²= 49 8²=64 (49+64)/2 56.5 Question 5. Each entry-level account executive in a large company makes an annual salary of$3.48 × 104. If there are 5.2 × 102 account executives in the company, how much do they make in all?
Options:
a. $6.69 × 101 b.$3.428 × 104
c. $3.532 × 104 d.$1.8096 × 107

d. $1.8096 × 107 Explanation: Each entry-level account executive in a large company makes an annual salary of$3.48 × 104. If there are 5.2 × 102 account executives in the company,
($3.48 × 104)( 5.2 × 102)$1.8096 × 107

Question 6.
Place the numbers in order from least to greatest.
0.24,4 × 10-2, 0.042, 2 × 10-4, 0.004
Options:
a. 2 × 10-4, 4 × 10-2, 0.004, 0.042, 0.24
b. 0.004, 2 × 10-4, 0.042, 4 × 10-2, 0.24
c. 0.004, 2 × 10-4, 4 × 10-2, 0.042, 0.24
d. 2 × 10-4, 0.004, 4 × 10-2, 0.042, 0.24

d. 2 × 10-4, 0.004, 4 × 10-2, 0.042, 0.24

Explanation:
2 × 10-4 = 0.0002
4 × 10-2 = 0.04

Question 7.
Guillermo is 5 $$\frac{5}{6}$$ feet tall. What is this number of feet written as a decimal?
Options:
a. 5.7 feet
b. 5.$$\bar{7}$$ feet
c. 5.83 feet
d. 5.8$$\bar{3}$$ feet

c. 5.83 feet

Question 8.
A human hair has a width of about 6.5 × 10-5 meters. What is this width written in standard notation?
Options:
a. 0.00000065 meter
b. 0.0000065 meter
c. 0.000065 meter
d. 0.00065 meter

c. 0.000065 meter

Explanation:
6.5 × 10-5 meter = 0.000065

Question 9.
Consider the following numbers: 7000, 700, 70, 0.7, 0.07, 0.007
a. Write the numbers in scientific notation.
Type below:
_____________

7000 = 7 × 10³
700 = 7 × 10²
70 = 7 × 10¹
0.7 = 7 × 10¯¹
0.07 = 7 × 10¯²
0.007 = 7 × 10¯³

Question 9.
b. Look for a pattern in the given list and the list in scientific notation. Which numbers are missing from the lists?
Type below:
_____________

In the given list the decimal is moving to the left by one place. From the scientific notation, numbers are decreasing by 10. The number missing is 7

Question 9.
c. Make a conjecture about the missing numbers.
Type below:
_____________

The numbers will continue to decrease by 10 in the given list.

### Conclusion:

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## Go Math Grade 3 Answer Key Chapter 12 Two-Dimensional Shapes Extra Practice

Go Math Grade 3 Answer Key Chapter 12 Two-Dimensional Shapes Extra Practice makes it easy for you to test your preparation level. Solve all the practice questions on Go Math Grade 3 Answer Key Chapter 12 Two-Dimensional Shapes Extra Practice. We even provided the Step by Step Solutions for all the 3rd Grade Go Math Answer Key Ch 12 Two-Dimensional Shapes by which you can verify your answers. HMH Go Math Grade 3 gives you a new way of problem-solving and makes it easy for you to get a good grip on the concepts underlying.

## Go Math Grade 3 Chapter 12 Two-Dimensional Shapes Extra Practice Answer Key

Download Go Math Grade 3 Answer Key Chapter 12 Two-Dimensional Shapes Extra Practice and prepare whenever you want. Grab the required knowledge and solve the problems of Grade 3 Chapter 12 Two-Dimensional Shapes Extra Practice on a regular basis. Check out the Step by Step Solutions provided Go Math Grade 3 Chapter 12 Two-Dimensional Shapes Extra Practice Answer Key and cross-check your answers.

### Common Core – Page No. 257000

Chapter 12 Extra Practice

Lessons 12.1–12.3

Name the polygon.

Question 1.

_________

Explanation:

Question 2.

_________

decagon

Explanation:

10 sides; 10 angles; decagon

Question 3.

_________

hexagon

Explanation:

6 sides; 6 angles; hexagon

Question 4.

_________

triangle

Explanation:

3 sides; 3 angles; triangle

Question 5.

_________

octagon

Explanation:

8 sides; 8 angles; octagon

Question 6.

_________

pentagon

Explanation:

5 sides; 5 angles; pentagon

Lesson 12.4

Look at the dashed sides of the polygon. Tell if they appear to be intersecting, perpendicular, or parallel. Write all the words that describe the sides.

Question 7.

_________
_________

perpendicular lines

Explanation:
The dashed sides are meeting to form a right angle. So, they are perpendicular lines.

Question 8.

_________

parallel lines

Explanation:
The dashed sides are not intersecting with each other. So, the given lines are parallel lines.

Question 9.

_________

intersecting lines

Explanation:
The dashed line segments meet and form an angle. So, they are intersecting lines.

Lesson 12.5

Circle all the words that describe the quadrilateral.

Question 10.

Options:
a. rhombus
b. trapezoid
c. rectangle

c. rectangle

Explanation:
The given shape has two pairs opposite with the same length. Also, all the angles are right angles. The given shape is a rectangle.

Question 11.

Options:
a. square
b. rhombus
c. trapezoid

a. square
b. rhombus

Explanation:
The given shape has 4 sides with equal lengths. Also, all the angles are right angles. So, a possible answer is a square and rhombus.

Question 12.

Options:
a. trapezoid
b. rectangle
c. rhombus

a. trapezoid

Explanation:
Even though the given shape has four sides, they are not equal. Also, it has only two right angles. The given shape is a trapezoid.

### Common Core – Page No. 258000

Lesson 12.6

Draw a quadrilateral that does not belong. Then explain why.

Question 1.

Type below:
_________

Explanation:
The shape is a trapezoid. Even though the given shape has four sides, they are not equal. Also, the angles are not right angles.

Lesson 12.7

Use the triangles for 1–2. Write A, B, or C.
Then complete the sentences.

Question 2.
Triangle ____ has 1 angle greater than a right angle and appears to have ____ sides of equal length.

Triangle C has 1 angle greater than a right angle and appears to have 0 sides of equal length.

Question 3.
Triangle____ has 1 right angle and appears to have ____ sides of equal length.

Triangle A has 1 right angle and appears to have 2 sides of equal length.

Lesson 12.8

Question 4.
What label could you use to describe Circle A?
Type below:
_________

All sides of Equal Lengths

Question 5.
What label could you use to describe Circle B?
Type below:
_________

Right Angle

Lesson 12.9

Draw lines to divide the shape into equal parts that show the fraction given.

Question 6.
$$\frac{1}{4}$$

Question 7.
$$\frac{1}{3}$$

### Conclusion

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## Go Math Grade 4 Answer Key Chapter 3 Multiply 2-Digit Numbers

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## Go Math Grade 4 Solution Key Pdf Chapter 3 Multiply 2-Digit Numbers

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Lesson 1: Multiply by Tens

Lesson 2: Estimate Products

Lesson 3: Investigate • Area Models and Partial Products

Lesson 4: Multiply Using Partial Products

Mid-Chapter Checkpoint

Lesson 5: Multiply with Regrouping

Lesson 6: Choose a Multiplication Method

Lesson 7: Problem Solving • Multiply 2-Digit Numbers

Review/Test

### Common Core – Page No. 149

Multiply by Tens

Choose a method. Then find the product.

Question 1.
16 × 60 = 960
Use the halving-and-doubling strategy.
Find half of 16: 16 ÷ 2 = 8.
Multiply this number by 60: 8 × 60 = 480
Double this result: 2 × 480 = 960

960

Explanation:
Use the halving-and-doubling strategy.
Find half of 16: 16 ÷ 2 = 8.
Multiply this number by 60: 8 × 60 = 480
Double this result: 2 × 480 = 960

Question 2.
80 × 22 = ______

1760

Explanation:
By using the place value method, Multiply 80 x 22
You can think of 80 as 8 tens
80 x 22 = (22 x 8) tens
= 176 tens
= 176 x 10 = 1760
80 x 22 = 1760

Question 3.
30 × 52 = ______

1560

Explanation:
Use the Associative Property
You can think of 30 as 3 x 10
30 x 52 = (3 x 10) x 52
= 3 x (10 x 52)
=  3 x 520
= 1560
30 x 52 = 1560

Question 4.
60 × 20 = ______

1200

Explanation:
60 x 20
Use the halving and doubling strategy
half of the 60 to make the problem simpler
60/ 2 = 30
Multiply 30 with 20
30 x 20 = 600
Double the 600
2 x 600= 1200
60 x 20 = 1200

Question 5.
40 × 35 = ______

1400

Explanation:
By using the Associative Property 40 x 35
You can think of 40 as 4 x 10
40 x 35 = (4 x 10) x 35
= 4  x (10 x 35)
= 4 x 350
= 1400
40 x 35 = 1400

Question 6.
10 × 90 = ______

900

Explanation:
By using the place value method, Multiply 10 x 90
You can think of 90 as 9 tens
10 x 90 = (10 x 9) tens
= 90 tens
= 10 x 90 = 900

Question 7.
31 × 50 = ______

1,550

Explanation:
Use the place value method to multiply 31 x 50
You can think of 50 as 5 tens
31 x 50 = 31 x 5 tens
= 155 tens
= 1,550
31 x 50 = 1,550

Problem Solving

Question 8.
Kenny bought 20 packs of baseball cards. There are 12 cards in each pack. How many cards did Kenny buy?
______ cards

240 cards

Explanation:
From the given data,
Kenny bought 20 packs of basketball cards
There are 12 cards in each pack = 12 x 20 cards
Use the associative property
You can write 20 as 2 x 10
12 x 20 = 12 x (2 x 10)
= (12 x 2) x 10
= (24) x 10
= 240 cards
Kenny bought 240 cards

Question 9.
The Hart family drove 10 hours to their vacation spot. They drove an average of 48 miles each hour. How many miles did they drive in all?
______ miles

480 miles

Explanation:
As per the given data,
Hart family drove 10 hours to their vacation spot
Average speed per each hour is = 48 miles
Total miles = 48 x 10
Use the halving and doubling strategy
Half of the 48 to make the problem simpler
48/ 2 = 24
Multiply 24 with 10 = 24x 10 = 240
Double the value = 2 x 240 = 480 miles
Total miles drove by hart family = 480 miles.

### Common Core – Page No. 150

Lesson Check

Question 1.
For the school play, 40 rows of chairs are set up. There are 22 chairs in each row. How many chairs are there in all?
Options:
a. 800
b. 840
c. 880
d. 8,800

c. 880

Explanation:
As per the given data
For the school play, 40 rows of chairs are available. 22 chairs are available in each row.
Then total chairs in school play are = 22 x 40
By using the place value method
You can think of 40 as 4 tens
22 x 40 = 22 x 4 tens
= 88 tens
= 880
Total chairs in school are = 880

Question 2.
At West School, there are 20 classrooms. Each classroom has 20 students. How many students are at West School?
Options:
a. 40
b. 400
c. 440
d. 4,000

b. 400

Explanation:
From the given data,
Total classrooms in west school = 20
Number of students per each classroom = 20
Then, total students at West School = 20 x 20
By using the associative property
You can think of 20 as 2 x 10
20 x 20 = 20 x (2 x 10)
= (20 x 2) x 10
=(40) x 10
=400
Total number of students at West School = 400

Spiral Review

Question 3.
Alex has 48 stickers. This is 6 times the number of stickers Max has. How many stickers does Max have?
Options:
a. 6
b. 7
c. 8
d. 9

c. 8

Explanation:
As per the give data,
Alex has 48 stickers
That means, X= 48
This is 6 times the number of stickers max has = Y = 6X = 48
Then, number of stickers with Max = Y = X = 48/6 = 8
Number of stickers with Max = Y = 8 Stickers.

Question 4.
Ali’s dog weighs 8 times as much as her cat. Together, the two pets weigh 54 pounds. How much does Ali’s dog weigh?
Options:
a. 6 pounds
b. 42 pounds
c. 46 pounds
d. 48 pounds

d. 48 pounds

Explanation:
From the given data,
Ali’s cat weight = X
Ali’s dog weight = 8 times as much as Ali’s cat = 8X
Together, the two pets weight = (X+8X) = 54 pounds
= 9X = 54 pounds
= X = 54/9 pounds = 6 pounds
Then, Ali’s dog weight = 8X =8 x 6 = 48 pounds.

Question 5.
Allison has 3 containers with 25 crayons in each. She also has 4 boxes of markers with 12 markers in each box. She gives 10 crayons to a friend. How many crayons and markers does Allison have now?
Options:
a. 34
b. 113
c. 123
d. 133

b. 113

Explanation:
As per the given data,
Allison has 3 containers with 25 crayons in each = X = 3 x 25 = 75
Allison has 4 boxes of markers with 12 markers in each box = Y = 4 x 12 = 48
Allison gives 10 crayons to a friend = Z = 75-10 = 65
Now, total number of crayons and markers with Allison = Y + Z = 48 + 65 = 113

Question 6.
The state of Utah covers 82,144 square miles. The state of Montana covers 145,552 square miles. What is the total area of the two states?
Options:
a. 63,408 square miles
b. 223,408 square miles
c. 227,696 square miles
d. 966,992 square miles

c. 227,696 square miles

Explanation:
From the given data,
The state of Utah covers 82,144 square miles
The state of Montana covers 145,552 square miles
Then, Total area of the two states = 82,144 + 145,552
The total area of two states = 227,696 square miles.

### Page No. 153

Question 1.
To estimate the product of 62 and 28 by rounding, how would you round the factors? What would the estimated product be?

1800

Explanation:
By using rounding and mental math
Estimate 62 x 28
Firstly, round each factor
62 x 28

60 x 30
Use mental math
6 x 3 = 18
60 x 30 = 1800
So, estimated product of 62 and 28 = 1800

Estimate the product. Choose a method.

Question 2.
96 × 34
Estimate: _____

3000

Explanation:
Use mental math and compatible numbers
96 x 34

100 x 30
Use mental math
1 x 30 = 30
100 x 30= 3000

Question 3.
47 × $39 Estimate:$ _____

2000

Explanation:
Round to the nearest ten
47 x $39 50 x$40
50 x $4 =$200
50 x $40 = 2000 Question 4. 78 × 72 Estimate: _____ Answer: 5600 Explanation: Use rounding and mental math Round each factor 78 x 72 80 x 70 Use mental math 8 x 7 = 56 80 x 70 = 5600 Question 5. 41 × 78 Estimate: _____ Answer: 3200 Explanation: Use compatible numbers and mental math 41 x 78 40 x 80 Use mental math 40 x 8 = 320 40 x 80 = 3200 Question 6. 51 × 73 Estimate: _____ Answer: 3500 Explanation: Round to the nearest ten 51 x 73 50 x 70 = 3500 Question 7. 34 × 80 Estimate: _____ Answer: 2400 Explanation: Round each factor 34 x 80 30 x 80 3 x 8 = 240 30 x 80 = 2400 Practice: Copy and Solve Estimate the product. Choose a method. Question 8. 61 × 31 Estimate: _____ Answer: 1800 Explanation: Round to the nearest ten 61 x 31 60 x 30 = 1800 Question 9. 52 × 68 Estimate: _____ Answer: 3500 Explanation: Round each factor 52 x 68 50 x 70 Use mental math 5 x 7 =35 50 x 70 = 3500 Question 10. 26 × 44 Estimate: _____ Answer: 1200 Explanation: Round to the nearest tens 26 x 44 30 x 40 = 1200 Question 11. 57 ×$69
Estimate: $_____ Answer:$4200

Explanation:
Round each factor
57 x $69 60 x$70
Use mental math
6 x $7 =$42
60 x $70 =$4200

Find two possible factors for the estimated product.

Question 12.
2,800
Type below:
___________

2800

Explanation:
Let us consider 7 x 4 = 28
70 x 40 = 2800

Question 13.
8,100
Type below:
___________

8,100

Explanation:
Let us take 9 x 9 = 81
90 x 90 = 8,100

Question 14.
5,600
Type below:
___________

5,600

Explanation:
Let us consider 7 x 8 = 56
70 x 80 = 5,600

Question 15.
2,400
Type below:
___________

2,400

Explanation:
Let us take 4 x 6 = 24
40 x 60 = 2400
Or 3 x 8 = 24
30 x 80 = 2,400

Question 16.
Mr. Parker jogs for 35 minutes each day. He jogs 5 days in week 1, 6 days in week 2, and 7 days in week 3. About how many minutes does he jog?

Explanation:
From the given data,
Mr. Parker jogs per day = 35 minutes
He jogs 5 days in week 1 = 5 x 35 = 175 minutes
6 days in week 2 = 6 x 35 = 210 minutes
7 days in week 3 = 7 x 35 = 245 minutes
Total minutes of jog by Mr. Parker = week 1 + week 2 + week 3
= 175 + 210 + 245
= 630 minutes
So, total minutes of jog by Mr. Parker = 630 minutes

Question 17.
There are 48 beads in a package. Candice bought 4 packages of blue, 9 packages of gold, 6 packages of red, and 2 packages of silver beads. About how many beads did Candice buy?

Explanation:
As per the given data,
48 beads are there in a package
Candice bought 4 packages of blue beads = 4 x 48 = 192
9 packages of gold beads = 9 x 48 = 432
6 packages of red beads = 6 x 48 = 288
2 packages of silver beads = 2 x 48 = 96
Total beads bought by Candice = 192 + 432 + 288 + 96
So, total beads bought by Candice = 1008.

### Page No. 154

Question 18.
On average, a refrigerator door is opened 38 times each day. Len has two refrigerators in his house. Based on this average, about how many times in a 3-week period are the refrigerator doors opened?

Explanation:
From the given data,
On average, a refrigerator door is opened per day = 38 times
3-week period = 7 x 3 = 21
Then, a refrigerator door is opened per 21 days = 21 x 38 = 798 times
Len has 2 refrigerators in his house
Then, two refrigerators door are opened per 21 days = 2 x 798
= 1596 times
So, in a 3 – week period refrigerator door is opened about 1600 times

Question 19.
The cost to run a refrigerator is about $57 each year. About how much will it have cost to run by the time it is 15 years old? about$ _____

1200

Explanation:
As per the data,
The cost to run a refrigerator per each year = $57 Cost to run a refrigerator by the time it is 15 years old =$57 * 15
Round to the nearest tens
$57 x 15$60 x 20
Use mental math
$6 x 2 = 12$ 60 x 20 = 1200

Question 20.
If Mel opens his refrigerator door 36 times every day, about how many times will it be opened in April? Will the exact answer be more than or less than the estimate? Explain.
Type below:
___________

1200

Explanation:
From the given data,
Mel opens his refrigerator door per day = 36 times
Number of days in April month = 30 days
Refrigerator door opened in April month = 36 * 30
Round the factors
36 x 30

40 x 30 = 1200

Question 21.
Represent a Problem What question could you write for this answer? The estimated product of two numbers, that are not multiples of ten, is 2,800.
Type below:
___________

2800

Explanation:
Let us take
1.
38 × 21
↓        ↓
40 × 20 = 800
2,800 = 42 x 68
↓    ↓
40 x  70 = 2800

Question 22.
Which is a reasonable estimate for the product? Write the estimate. An estimate may be used more than once.

26 × 48 __________
28 × 21 __________
21 × 22 __________
51 × 26 __________

25 x 50 = 1250
30 x 20 = 600
20 x 20 = 400
50 x 25 = 1250

Explanation:
26 x 48 -> 25 x 50 = 1250
28 x 21 -> 30 x 20 = 600
21 x 22 -> 20 x 20 = 400
51 x 26 -> 50 x 25 = 1250

### Common Core – Page No. 155

Estimate Products
Estimate the product. Choose a method.

Question 1.
38 × 21
38 × 21
↓       ↓
40 × 20
800

800

Explanation:
38 × 21
↓        ↓
40 × 20
800

Question 2.
63 × 19
Estimate: _____

1200

Explanation:
63 x 19

60 x 20 = 1200
Estimated product of 63 x 19 = 1200

Question 3.
27 × $42 Estimate:$ _____

$1000 Explanation: 27 ×$42

25 x $40 =$1000
Estimated Product of 25 x $42 =$1000

Question 4.
73 × 67
Estimate: _____

4900

Explanation:
73 × 67

70 x 70 = 4900
Estimated Product of 73 x 67 = 4900

Question 5.
37 × $44 Estimate:$ _____

$1600 Explanation: 37 ×$44

40 x $40 =$1600
Estimated Product of 37 x $44 =$1600

Question 6.
85 × 71
Estimate: _____

6300

Explanation:
85 × 71

90 x 70 = 6300
Estimated Product of 85 x 71 = 6300

Question 7.
88 × 56
Estimate: _____

4950

Explanation:
88 × 56

90 x 55 = 4950
Estimated Product of 90 x 55 = 4950

Question 8.
97 × 13
Estimate: _____

1,000

Explanation:
97 × 13

100 x 10 = 1,000

Question 9.
92 × 64
Estimate: _____

5850

Explanation:
92 × 64

90 x 65 = 5850

Problem Solving

Question 10.
A dime has a diameter of about 18 millimeters. About how many millimeters long would a row of 34 dimes be?

Explanation:
From the given data,
A dime has a diameter of about 18 millimeters
Then, 34 dimes diameter = 18 * 34
18 x 34

20 x 30 = 600
So, 34 dimes have a diameter of about 600 millimeters long

Question 11.
A half-dollar has a diameter of about 31 millimeters. About how many millimeters long would a row of 56 half-dollars be?

1800 millimeters

Explanation:
As per the given data,
A half – dollar has a diameter of about 31 millimeters
Then, 56 half-dollars diameter = 31 * 56
31 * 56

30 * 60
So, 56 half-dollars have a diameter of about 1800 millimeters long.

### Common Core – Page No. 156

Lesson Check

Question 1.
Which is the best estimate for the product
43 × 68?
Options:
a. 3,500
b. 2,800
c. 2,400
d. 280

b. 2,800

Explanation:
Round to the nearest tens
43 x 68

40 x 70
Use mental math
4 x 7 = 28
40 x 70 = 2800
Estimated product of 43 x 68 = 2800

Question 2.
Marissa burns 93 calories each time she plays fetch with her dog. She plays fetch with her dog once a day. About how many calories will Marissa burn playing fetch with her dog in 28 days?
Options:
a. 4,000
b. 2,700
c. 2,000
d. 270

b. 2,700

Explanation:
From the given data,
Marissa burned calories each time when she plays fetch with her dog= 93 calories
Then, Marissa burned calories in 28 days while playing fetch with her dog = 28 x 93
Round to the nearest tens
28 x 93

30 x 90
Then, estimated burned calories in 28 days by Marissa = 2700 calories

Spiral Review

Question 3.
Use the model to find 3 × 126.

Options:
a. 368
b. 378
c. 468
d. 478

b. 378

Explanation:
From the above Figure,
3 x 126 = 3 x 100 + 3 x 20 + 3 x 6
= 300 + 60 + 18
= 378
3 x 126 = 378

Question 4.
A store sells a certain brand of jeans for $38. One day, the store sold 6 pairs of jeans of that brand. How much money did the store make from selling the 6 pairs of jeans? Options: a.$188
b. $228 c.$248
d. $288 Answer: b.$228

Explanation:
As per the given data,
A store sells a certain brand of jeans for rupees = $38 One day, the store sold 6 pairs of jeans of that brand = 6 x$38
6 x $38 =$228
The total amount of 6 pairs of jeans = $228 Question 5. The Gateway Arch in St. Louis, Missouri, weighs about 20,000 tons. Which amount could be the exact number of tons the Arch weighs? Options: a. 31,093 tons b. 25,812 tons c. 17,246 tons d. 14,096 tons Answer: c. 17,246 tons Explanation: From the given data, The Gateway Arch in St.Louis, Missouri weight = about 20,000 tons From the available options, 17,246 tons is closer to 20,000 tons Then, the exact number of tons the Arch weighs = 17,246 tons Question 6. Which is another name for 23 ten thousands? Options: a. 23,000,000 b. 2,300,000 c. 230,000 d. 23,000 Answer: c. 230,000 Explanation: As per the data, Another name for 23 ten thousands = 23 x 10,000 = 230,000 Another name for 23 ten thousand = 2,30,000 ### Page No. 159 Find the product. Question 1. 16 × 19 16 × 19 = _____ Answer: 304 Explanation: 16 x 19 = 304 Question 2. 18 × 26 18 × 26 = _____ Answer: 468 Explanation: 200 + 160 + 60 + 48 = 468 Question 3. 27 × 39 27 × 39 = ______ Answer: 1,053 Explanation: 600 + 210 + 180 +63 = 1053 Draw a model to represent the product. Then record the product. Question 4. 14 × 16 = ______ Answer: 224 Explanation: 100 + 40 + 60 + 24 = 224 Question 5. 23 × 25 = ______ Answer: 575 Explanation: 400 + 60 + 100 + 15 = 575 Question 6. Explain how modeling partial products can be used to find the products of greater numbers. Type below: __________ Answer: You can use mental math to find the partial products and then find the sum of the partial products. Explanation: Question 7. Emma bought 16 packages of rolls for a party. There were 12 rolls in a package. After the party there were 8 rolls left over. How many rolls were eaten? Explain. ______ rolls Answer: 184 rolls were eaten Explanation: From the given data, Emma bought 16 packages of rolls for a party There were 12 rolls in a package Then, total rolls = 16 x 12 = 192 100 + 60 + 20 + 12 =192 After the party there were 8 rolls left over Then, total eaten rolls are = 192 – 8 = 184 ### Page No. 160 Question 8. Jamal and Kim used different ways to solve 12 × 15 by using partial products. Whose answer makes sense? Whose answer is nonsense? Explain your reasoning. Jamal’s Work 100 + 20 + 10 = 130 Kim’s Work 120 + 60 = 180 a. For the answer that is nonsense, write an answer that makes sense. Type below: __________ Answer: a. Jamal’s work makes nonsense. 100 + 20 + 50 + 10 = 180 it makes sense Question 8. b. Look at Kim’s method. Can you think of another way Kim could use the model to find the product? Explain. Type below: __________ Answer: Other method: 12 x 15 10 x 12 = 120 5 x 12 = 60 120 + 60 = 180. Explanation: Kim follows another method to find 12 x 15 That is, 100 + 50 = 150 20 + 10 = 30 Then, 150 + 30 =180 12 x 15 = 180 Question 9. Look at the model in 8b. How would the partial products change if the product was 22 × 15? Explain why you think the products changed. Type below: __________ Answer: 330 Explanation: Following the 8b method 22 x 15 =330 200 + 100 = 300 20 + 10 = 30 Now, 300 + 30 = 330 Finally, 22 x 15 = 330 The factor of 15 is increased in present problem. So, the product also increases for 15 x 22. ### Common Core – Page No. 161 Area Models and Partial Products Draw a model to represent the product. Then record the product. Question 1. 13 × 42 Answer: Question 2. 18 × 34 = ______ Answer: 300 + 40 + 240 + 32 = 612 Question 3. 22 × 26 = ______ Answer: 400 + 120 + 40 + 12 = 572 Question 4. 1 5 × 33 = ______ Answer: 300 + 30 + 150 + 15 = 495 Question 5. 23 × 29 = ______ Answer: 400 + 180 + 60 + 27 = 667 Question 6. 19 × 36 = ______ Answer: 300 + 60 + 270 + 54 = 684 Problem Solving Question 7. Sebastian made the following model to find the product 17 × 24. Is his model correct? Explain. a. yes b. no Answer: b. no Explanation: 200 + 40 + 140 + 28 = 408 Question 8. Each student in Ms. Sike’s kindergarten class has a box of crayons. Each box has 36 crayons. If there are 18 students in Ms. Sike’s class, how many crayons are there in all? ______ crayons Answer: 648 crayons Explanation: From the given information, Each student in Ms.Sike’s kindergarten class has a box of crayons Crayons in each box = 36 Crayons Number of students in Mr.Sike’s class = 18 students Total crayons = 18 x 36 300 + 60 + 240 + 48 = 648 ### Common Core – Page No. 162 Lesson Check Question 1. Which product does the model below represent? Options: a. 161 b. 230 c. 340 d. 391 Answer: d. 391 Explanation: 200 + 30 + 140 + 21 = 391 17 x 23 = 391 Question 2. Which product does the model below represent? Options: a. 219 b. 225 c. 244 d. 275 Answer: b. 225 Explanation: 130 + 20 + 65 + 10 = 225 15 x 15 = 225 Spiral Review Question 3. Mariah builds a tabletop using square tiles. There are 12 rows of tiles and 30 tiles in each row. How many tiles in all does Mariah use? Options: a. 100 b. 180 c. 360 d. 420 Answer: c. 360 Explanation: From the given data, Mariah builds a tabletop using square tiles Square contains 12 rows of tiles and 30 tiles in each row = 12 x 30 12 x 30 = 360 tiles Total tiles used by Mariah = 360 tiles Question 4. Trevor bakes 8 batches of biscuits, with 14 biscuits in each batch. He sets aside 4 biscuits from each batch for a bake sale and puts the rest in a jar. How many biscuits does Trevor put in the jar? Options: a. 112 b. 80 c. 50 d. 32 Answer: b. 80 Explanation: As per the given data, Number of biscuits baked by Trevor = 8 batches Number of biscuits in each batch = 14 biscuits So, total biscuits = 14 x 8 = 112 Trevor sets aside 4 biscuits from each batch for a bake = 8*4 = 32 biscuits are aside for a bake Trevor kept rest of biscuits in a jar = 112 – 32 = 80 So, 80 biscuits are put in the jar by the Trevor Question 5. Li feeds her dog 3 cups of food each day. About how many cups of food does her dog eat in 28 days? Options: a. 60 cups b. 70 cups c. 80 cups d. 90 cups Answer: c. 80 cups Explanation: As per the given data, Li feeds her dog per day = 3 cups of food Then, Li feeds her dog for 28 days = 3 x 28 = 84 cups of food So, Li feeds her dog with 84 cups of food in 28 days Question 6. Which symbol makes the number sentence true? 4 ■ 0 = 0 Options: a. + b. – c. × d. ÷ Answer: c. × Explanation: 4 x 0 = 0 ### Page No. 165 Question 1. Find 24 × 34. _____ Answer: 816 Explanation: Question 2. 1 2 × 1 2 ——– _____ Answer: 144 Explanation: Question 3. 3 1 × 2 4 ——- _____ Answer: 744 Explanation: Question 4. 2 5 × 4 3 ——- _____ Answer: 1,075 Explanation: Question 5. 3 7 × 2 4 ——- _____ Answer: 888 Explanation: Question 6. 5 4 × 1 5 ——- _____ Answer: 810 Explanation: Question 7. 8 7 × 1 6 ——- _____ Answer: 1,392 Explanation: Question 8. 6 2 × 5 6 ——- _____ Answer: 3,472 Explanation: Question 9. 4 9 × 6 3 ——- _____ Answer: 3,087 Explanation: Practice: Copy and Solve Record the product. Question 10. 38 × 47 _____ Answer: 1,786 Explanation: Question 11. 46 × 27 _____ Answer: 1,242 Explanation: Question 12. 72 × 53 _____ Answer: 3,816 Explanation: Question 13. 98 × 69 _____ Answer: 6,762 Explanation: Question 14. 53 × 68 _____ Answer: 3,604 Explanation: Question 15. 76 × 84 _____ Answer: 6,384 Explanation: Question 16. 92 × 48 _____ Answer: 4,416 Explanation: Question 17. 37 × 79 _____ Answer: 2,923 Explanation: Reason Abstractly Algebra Find the unknown digits. Complete the problem. Question 18. Type below: ___________ Answer: 1,824 Explanation: Question 19. Type below: ___________ Answer: 7,954 Explanation: Question 20. Type below: ___________ Answer: 1,908 Explanation: Question 21. Type below: ___________ Answer: 952 Explanation: ### Page No. 166 Use the picture graph for 22–24. Question 22. Use Graphs A fruit-packing warehouse is shipping 15 boxes of grapefruit to a store in Santa Rosa, California. What is the total weight of the shipment? ______ pounds Answer: 1275 pounds Explanation: From the given data, A fruit packing warehouse is shipping 15 boxes of grapefruit to store in Santa Rose, California Grapefruit weight per box = 85 pounds Total weight of the shipment = 85 x 15 So, the total weight of the shipment = 1275 pounds Question 23. How much less do 13 boxes of tangelos weigh than 18 boxes of tangerines? ______ pounds Answer: 450 pounds Explanation: As per the given data, Tangelos weight per box = 90 pounds Then, the weight of the 13 boxes of tangelos = 90 x 13 And, the weight of the 18 boxes of tangelos = 90 x 18 1620 – 1170 = 450 So, 13 boxes of tangelos weight are 450 pounds less than 18 boxes of tangelos weight Question 24. What is the weight of 12 boxes of oranges? ______ pounds Answer: 1,080 pounds Explanation: The weight of the oranges per box = 90 pounds then, weight of 12 boxes oranges = 90 x 12 So, weight of 12 boxes oranges = 1,080 pounds Question 25. Each person in the United States eats about 65 fresh apples each year. Based on this estimate, how many apples do 3 families of 4 eat each year? ______ apples Answer: 780 apples Explanation: From the given data, Each person in the united states eats fresh apples per year = 65 3 families of 4 persons = 3 x 4 = 12 persons Then, the number of apples eat by 12 persons = 65 x 12 So, the total number of apples eat by 12 persons per year = 780 Question 26. The product 26 × 93 is greater than 25 × 93. How much greater? Explain how you know without multiplying. ______ Answer: The difference is 93 26 x 93 is one more group of 93 than 25 x 93 Question 27. Margot wants to use partial products to find 22 × 17. Write the numbers in the boxes to show 22 × 17. Type below: __________ Answer: Explanation: 22 x 17 (20 + 2) x 17 20 x 17 + 2 x 17 20 x (10 + 7) + 2 x (10 + 7) (20 x 10) + (20 x 7) + (2 x 10) + (2 x 7) ### Common Core – Page No. 167 Multiply Using Partial Products Record the product. Question 1. 2 3 × 7 9 ——— 1, 4 0 0 2 1 0 1 8 0 + 2 7 ——– 1, 8 1 7 Answer: 1, 8 1 7 Explanation: 2 3 × 7 9 ——— 1, 4 0 0 2 1 0 1 8 0 + 2 7 ——– 1, 8 1 7 Question 2. 5 6 × 3 2 ——- _______ Answer: 1,792 Explanation: Question 3. 8 7 × 6 4 ——- _______ Answer: 5,568 Explanation: Question 4. 3 3 × 2 5 ——- _______ Answer: 825 Explanation: Question 5. 9 4 × 1 2 ——- _______ Answer: 1,128 Explanation: Question 6. 5 1 × 7 7 ——- _______ Answer: 3,927 Explanation: Question 7. 6 9 × 4 9 ——- _______ Answer: 3,381 Explanation: Question 8. 8 6 × 8 4 ——- _______ Answer: 7,224 Explanation: Question 9. 9 8 × 4 2 ——- _______ Answer: 4,116 Explanation: Question 10. 7 3 × 3 7 ——- _______ Answer: 2,701 Explanation: Question 11. 8 5 × 5 1 ——- _______ Answer: 4,335 Explanation: Problem Solving Question 12. Evelyn drinks 8 glasses of water a day, which is 56 glasses of water a week. How many glasses of water does she drink in a year? (1 year = 52 weeks) _______ glasses Answer: 2,912 glasses Explanation: As per the given data, Evelyn drinks 8 glasses of water a day Evelyn drinks water per week = 56 glasses Then, the number of glasses per 52 weeks = 52 x 56 Total number of glasses of water drink by Evelyn per year = 2912 glasses of water Question 13. Joe wants to use the Hiking Club’s funds to purchase new walking sticks for each of its 19 members. The sticks cost$26 each. The club has $480. Is this enough money to buy each member a new walking stick? If not, how much more money is needed? Is the money enough? _______ How much more is needed? _______ Answer: This amount is not enough to buy walking sticks Still,$14 amount is needed to buy walking sticks

Explanation:
From the given data,
Joe wants to use the Hiking club funds to purchase new walking sticks for each of its 19 members
Cost per each stick = $26 Total walking sticks cost per 19 members =$26 x 19

Total cost for walking sticks for 19 members = $494 The club has =$480
This amount is not enough to buy walking sticks
Still, $14 amount is needed to buy walking sticks ### Common Core – Page No. 168 Lesson Check Question 1. A carnival snack booth made$76 selling popcorn in one day. It made 22 times as much selling cotton candy. How much money did the snack booth make selling
cotton candy?
Options:
a. $284 b.$304
c. $1,562 d.$1,672

d. $1,672 Explanation: As per the given data, A carnival snack booth made popcorn in one day =$76
It made 22 times as much selling cotton candy
Then, total selling cotton candy made by snack booth = $76 x 22 So,$1672 money snack booth will get for selling cotton candy

Question 2.
What are the partial products of
42 × 28?
Options:
a. 800, 80, 40, 16
b. 800, 16
c. 800, 40, 320, 16
d. 80, 16

c. 800, 40, 320, 16

Explanation:

So, partial products of 42 x 28 are 800, 40, 320, 16

Spiral Review

Question 3.
Last year, the city library collected 117 used books for its shelves. This year, it collected 3 times as many books. How many books did it collect this year?
Options:
a. 832
b. 428
c. 351
d. 72

c. 351

Explanation:
From the given data,
Last year, the number of used books collected by city library by its shelves = 117 books
This year, it collected 3 times as many books = 3 x 117 =351 books
Total number of books collected by the city library for this year = 351 books

Question 4.
Washington Elementary has 232 students. Washington High has 6 times as many students. How many students does Washington High have?
Options:
a. 1,392
b. 1,382
c. 1,292
d. 1,281

a. 1,392

Explanation:
As per the given data,
The number of students in Washington elementary = 232 students
Washington High has 6 times as many students = 6 x 232 = 1392
Total number of students in Washington High = 1392 students

Question 5.
What are the partial products of 35 × 7?
Options:
a. 10, 12
b. 21, 35
c. 210, 35
d. 350, 21

c. 210, 35

Explanation:
Partial products of 35 x 7 are 210, 35

Question 6.
Shelby has ten $5 bills and thirteen$10 bills. How much money does Shelby have in all?
Options:
a. $15 b.$60
c. $63 d.$180

d. $180 Explanation: From the given data, Shelby has ten$5 bills and thirteen $10 bills = (10 x$5) + (13 x $10) = ($50) + ($130) =$180
Total money with Shelby = $180 ### Page No. 169 Question 1. Explain how to find 40 × 50 using mental math. Type below: __________ Answer: 200 Explanation: 40 x 50 By using mental math 4 x 5 = 20 40 x 50 = 200 Question 2. What is the first step in estimating 56 × 27? Type below: __________ Answer: 18 centimeters Explanation: Round to the nearest values. So, the first step of the estimated 56 x 27 is rounding to the nearest values that is 60 x 30 Choose a method. Then find the product. Question 3. 35 × 10 = _____ Answer: 350 Explanation: By using the place value method You can take 10 as 1 ten 35 x 10 = 35 x 1 ten = 35 ten 35 x 10 = 350 Question 4. 19 × 20 = _____ Answer: 380 Explanation: 19 x 20 By using the associative property You can think of 20 as (2 x 10) 19 x 20 = 19 x (2 x 10) = (19 x 2) x 10 = 38 x 10 19 x 20 = 380 Question 5. 12 × 80 = _____ Answer: 960 Explanation: Use the halving and doubling strategy half of the 80 to make the problem simpler 80/ 2 = 40 Multiply 40 with 12 40*12 = 480 Double the 480 2*480= 960 12*80 = 960 Question 6. 70 × 50 = _____ Answer: 3,500 Explanation: 70 x 50 By using the place value method You can take 50 as 5 tens 70 x 50 = 70 x 5 tens = 350 tens 70 x 50 = 3,500 Question 7. 58 × 40 = _____ Answer: 2,320 Explanation: By using the associative property You can think of 40 as (4 x 10) 58 x 40 = 58 x (4 x 10) = (58 x 4) x 10 = 232 x 10 58 x 40 = 2,320 Question 8. 30 × 40 = _____ Answer: 1,200 Explanation: Use the halving and doubling strategy half of the 40 to make the problem simpler 40/ 2 = 20 Multiply 20 with 30 20*30 = 600 Double the 600 2*600= 1200 30*40 = 1,200 Question 9. 14 × 60 = _____ Answer: 840 Explanation: By using the place value method You can take 60 as 6 tens 14 x 60 = 14 x 6 tens = 84 tens 14 x 60 = 840 Question 10. 20 × 30 = _____ Answer: 600 Explanation: By using the associative property You can think of 30 as (3 x 10) 20 x 30 = 20 x (3 x 10) = (20 x 3) x 10 = 60 x 10 20 x 30 = 600 Question 11. 16 × 90 = _____ Answer: 1,440 Explanation: Use the halving and doubling strategy half of the 90 to make the problem simpler 90/ 2 = 45 Multiply 45 with 16 16*45 = 720 Double the 720 2*720= 1440 16*90 = 1,440 Estimate the product. Choose a method. Question 12. 81 × 38 Estimate: _____ Answer: 3,200 Explanation: Round to the nearest tens. 81 is close to 80; 38 is close to 40; 80 x 40 = 3,200 Question 13. 16 ×$59
Estimate: $_____ Answer:$120

Explanation:
Round to the nearest tens.
16 is close to 20; $59 is close to$60;
Use the mental math to find the product of 20 x $60 2 x$6 = $12 20 x$60 = $120 Estimated product of 16 x$59 = $120 Question 14. 43 × 25 Estimate: _____ Answer: 1,000 Explanation: Round to the nearest tens. 43 is close to 40; 25 is close to 25; 40 x 25 = 1000 Estimated product of 43 x 25 = 1,000 Question 15. 76 × 45 Estimate: _____ Answer: 3,200 Explanation: Round to the nearest tens. 76 is close to 80; 45 is close to 40; Use the mental math 8 x 4 = 32 80 x 40 = 3200 So, the estimated product of 76 x 45 = 3,200 Question 16. 65 ×$79
Estimate: _____

$4,800 Explanation: Round to the nearest tens. 65 is close to 60;$79 is close to $80; Use the mental math 6 x$8 = $48 60 x$80 = $4800 So, estimated product of 65 x$79 = $4,800 Question 17. 92 × 38 Estimate: _____ Answer: 3,600 Explanation: Round to the nearest tens. 92 is close to 90; 38 is close to 40; Use the mental math, then 9 x 4 = 36 90 x 40 = 3,600 So, estimated product of 92 x 38 = 3,600 Question 18. 37 × 31 Estimate: _____ Answer: 1,200 Explanation: Round to the nearest tens. 37 is close to 40; 31 is close to 30; Use the mental math, then 4 x 3 = 12 40 x 30 = 1,200 So, estimated product of 37 x 31 = 1,200 Question 19. 26 ×$59
Estimate: _____

$1,800 Explanation: Round to the nearest tens. 26 is close to 30;$59 is close to $60; Use the mental math, then 3 x$6 = $18 30 x$60 = $1,800 So, estimated product of 26 x$59 = $1,800 Question 20. 54 × 26 Estimate: _____ Answer: 18 centimeters Explanation: Round to the nearest tens. 54 is close to 50; 26 is close to 30; Use the mental math 5 x 3 = 15 50 x 30 = 1,500 So, estimated product of 54 x 26 = 1,500 Question 21. 52 × 87 Estimate: _____ Answer: 4,500 Explanation: Round to the nearest tens. 52 is close to 50; 87 is close to 90; Use the mental math 5 x 9 = 45 50 x 90 = 4500 So, estimated product of 52 x 87 = 4,500 Question 22. 39 × 27 Estimate: _____ Answer: 18 centimeters Explanation: Round to the nearest tens. 39 is close to 40; 27 is close to 30; Use the mental math 4 x 3 = 12 40 x 30 = 1,200 So, estimated product of 39 x 27 = 1,200 Question 23. 63 × 58 Estimate: _____ Answer: 3,600 Explanation: Round to the nearest tens. 63 is close to 60; 58 is close to 60; Use the mental math 6 x 6 = 36 60 x 60 = 3,600 So, estimated product of 63 x 58 = 3,600 ### Page No. 170 Question 24. Ms. Traynor’s class is taking a field trip to the zoo. The trip will cost$26 for each student. There are 22 students in her class. What is a good estimate for the cost of the students’ field trip?
Type below:
__________

18 centimeters

Explanation:
As per the given data,
Ms. Traynor’s class is taking a field trip to the zoo
Cost of the trip for each student = $26 Total number of students in her class = 22 The total cost of the trip for students =$26 x 22
Round to the nearest tens.
26 is close to 30; 22 is close to 20;
Use the mental math
$3 x 2 =$6
$30 x 20 =$600
Then, the total estimated cost for the trip for students = $600 Question 25. Tito wrote the following on the board. What is the unknown number? ______ Answer: 400 Explanation: An unknown number is 50 x 8 = 400 Question 26. What are the partial products that result from multiplying 15 × 32? Type below: __________ Answer: Partial products are 300, 150, 20, 10 Explanation: Partial products are 300, 150, 20, 10 Question 27. A city bus company sold 39 one-way tickets and 20 round-trip tickets from West Elmwood to East Elmwood. One-way tickets cost$14. Round trip tickets cost $25. How much money did the bus company collect?$ ______

$1,046 Explanation: As per the given data, Number of one – way tickets sold by the city bus company = 39 Round trip tickets from west Elmwood to east Elmwood = 20 Cost of one – way tickets =$14
Then, cost of 39 one – way tickets = 39 x $14 =$546
Cost of round trip tickets = $25 Then, cost of 20 round trip tickets =$25 x 20 = $500 Total money collected by the city bus company =$546 + $500 =$1,046

### Page No. 173

Question 1.
Look at the problem. Complete the sentences.
Multiply ____ and ____ to get 0.
Multiply ____ and ____ to get 1,620.
0 + 1,620 = ____

_____

Multiply 27 and 0 to get 0.
Multiply 27 and 6 to get 1,620.
Add the partial products. 0 + 1,620 = 1,620.

Estimate. Then find the product.

Question 2.
6 8
× 5 3
——-
Estimate: _________
Product: __________

Estimate: 3,500
Product: 3,604

Explanation:
68 is closer to 70 and 53 is closer to 50
Estimate: 70 x 50 = 3,500
60 x 53 = 3180
8 x 53 = 424
3180 + 424 = 3604
Product 3,604

Question 3.
6 1
× 5 4
——-
Estimate: _________
Product: __________

Estimate: 3,000
Product: 3,294

Explanation:
61 is closer to 60 and 54 is closer to 50
Estimate: 60 x 50 = 3,000
60 x 54 = 3240
1 x 54 = 54
3240 + 54 = 3294
Product 3,294

Question 4.
9 0
× 2 7
——-
Estimate: _________
Product: __________

Estimate: 2,700
Product: 2,430

Explanation:
27 is closer to 30
Estimate: 90 x 30 = 2,700
90 x 27 = 2430
Product 2,430

Question 5.
3 0
× 4 7
——-
Estimate: _________
Product: __________

Estimate: 1,500
Product: 1,410

Explanation:
47 is closer to 50
Estimate: 30 x 50 = 1,500
30 x 47 = 1410
Product 1,410

Question 6.
7 8
× 5 6
——-
Estimate: _________
Product: __________

Estimate: 4,800
Product: 4,368

Explanation:
78 is closer to 80 and 56 is closer to 60
Estimate: 80 x 60 = 4,800
70 x 56 = 3920
8 x 56 = 448
3920 + 448 = 4368
Product 4,368

Question 7.
2 7
× 2 5
——-
Estimate: _________
Product: __________

Estimate: 600
Product: 675

Explanation:
27 is closer to 30 and 25 is closer to 20
Estimate: 30 x 20 = 600
20 x 25 = 500
7 x 25 = 175
500 + 175 = 675
Product 675

Practice: Copy and Solve Estimate. Then find the product.

Question 8.
34 × 65
Estimate: _________
Product: __________

Estimate: 1,800
Product: 2,210

Explanation:
34 is closer to 30 and 65 is closer to 60
Estimate: 30 x 60 = 1,800
30 x 65 = 1950
4 x 65 = 260
1950 + 260 = 2210
Product 2,210

Question 9.
42 × $13 Estimate:$ _________
Product: $_________ Answer: Estimate:$400
Product: $546 Explanation: 42 is closer to 40 and 13 is closer to 10 Estimate: 40 x 10 = 400 40 x$13 = $520 2 x$13= $26$520 + $26 =$546
Product $546 Question 10. 60 × 17 Estimate: _________ Product: __________ Answer: Estimate: 1,200 Product: 1,020 Explanation: 17 is closer to 20 Estimate: 60 x 20 = 1,200 60 x 17 = 1020 Product = 1,020 Question 11. 62 × 45 Estimate: _________ Product: __________ Answer: Estimate: 2,400 Product: 2,790 Explanation: 62 is closer to 60 and 45 is closer to 40 Estimate: 60 x 40 = 2,400 60 x 45 = 2700 2 x 45= 90 2700 + 90 = 2790 Product 2,790 Question 12. 57 ×$98
Estimate: $_________ Product:$ _________

Estimate: 6,000
Product: 5,586

Explanation:
57 is closer to 60 and 98 is closer to 100
Estimate: 60 x 100 = 6,000
50 x 98 = 4900
7 x 98= 686
4900 + 686 = 5586
Product 5,586

Look for a Pattern Algebra Write a rule for the pattern.
Use your rule to find the unknown numbers.

Question 13.

Rule _____________
Type below:
_________

Explanation:
1 hour = 60 min
Then, 5hr = 5 x 60 = 300 min
10hr = 10 x 60 = 600 min
15hr = 15 x 60 = 900 min
20hr = 20 x 60 = 1200 min
25hr = 25 x 60 = 1500 min

Question 14.
Owners of a summer camp are buying new cots for their cabins. There are 16 cabins. Each cabin needs 6 cots. Each cot costs $92. How much will the new cots cost?$ _______

$8,832 Explanation: As per the given data, Owners pf a summer camp are buying new cots for their cabins Number of cabins = 16 Each cabin needs 6 cots Then, total cots = 16 x 6 = 96 Each cot cost =$92
Then, cost for total cots = $92 x 96 92 is closer to 90 and 96 is closer to 100 Estimate = 90 x 100 = 9,000 90 x 96 = 8640 2 x 96 = 192 8640 + 192 = 8832 Product = 8,832 Question 15. A theater has 28 rows of 38 seats downstairs and 14 rows of 26 seats upstairs. How many seats does the theater have? ______ seats Answer: 1,428 seats Explanation: As per the given data, A theatre has 28 rows of 38 seats downstairs = 28 x 38 = 1064 14 rows of 26 seats upstairs = 14 x 26 = 364 Total number of seats = 1064 + 364 = 1,428 seats ### Page No. 174 Question 16. Machine A can label 11 bottles in 1 minute. Machine B can label 12 bottles in 1 minute. How many bottles can both machines label in 15 minutes? a. What do you need to know? Type below: __________ Answer: number of bottles labeled by Machine A and Machine B in 15 minutes Question 16. b. What numbers will you use? Type below: __________ Answer: 15x 11 and 15 x 12 Question 16. c. Tell why you might use more than one operation to solve the problem. Type below: __________ Answer: To find out the total number of bottle made by both machines A & B Question 16. d. Solve the problem. So, both machines can label ____ bottles in ____ minutes. Type below: __________ Answer: Machine A can label 11 bottles in 1 minute Then, the number of bottles labeled by machine A in 15 minutes = 15 x 11 = 165 Machine B can label 12 bottles in 1 minute Then, number of bottles labelled by Machine B in 15 minutes = 15 x 12 = 180 Total bottles labelled by both the machines in 15 minutes = 165 + 180 = 345 Question 17. Make Sense of Problems A toy company makes wooden blocks. A carton holds 85 blocks. How many blocks can 19 cartons hold? ______ blocks Answer: 1,615 blocks Explanation: From the given data, A toy company makes wooden blocks A carton holds 85 blocks Then, number of blocks hold by 19 cartons = 19 x 85 = 1615 Total number of blocks held by 19 cartons = 1,615 Question 18. A company is packing cartons of candles. Each carton can hold 75 candles. So far, 50 cartons have been packed, but only 30 cartons have been loaded on a truck. How many more candles are left to load on the truck? ______ candles Answer: 1,500 candles Explanation: As per the given data, A company is packing cartons of candles Each carton can hold 75 candles Then, number of candles hold by 50 cartons = 50 x 75 = 3750 Number of candles hold by 30 cartons = 30 x 75 = 2250 50 cartons have been packed, but only 30 cartons have been loaded on a truck Remaining candles are left to load on truck = 3750 – 2250 = 1,500 Question 19. Mr. Garcia’s class raised money for a field trip to the zoo. There are 23 students in his class. The cost of the trip will be$17 for each student. What is the cost for all the students? Explain how you found your answer.
$______ Answer:$391

Explanation:
As per the given data,
Mr. Garcia’s class raised money for a field trip to the zoo
Total number of students in his class = 23 students
Cost of the trip for each student = $17 Then, total cost for all the students =$17 x 23 = $391 ### Common Core – Page No. 175 Multiply with Regrouping Estimate. Then find the product. Question 1. Estimate: 2,700 Think: 87 is close to 90 and 32 is close to 30. 90 × 30 = 2,700 Answer: 2,784 Explanation: Think: 87 is close to 90 and 32 is close to 30. 90 × 30 = 2,700 Question 2. 7 3 × 2 8 ——– Estimate: ______ Product: _______ Answer: Estimate: 2,100 Product: 2,044 Explanation: Estimate: 73 is close to 70; 28 is close to 30. So, 70 x 30 = 2,100. Product: Write 73 as 7 tens and 3 ones. Multiply 28 by 3 ones. 2 28 x 73 ——– 84 <– 3 x 28 Multiply 28 by 7 tens 5 28 x 73 ——– 1960 <– 70 x 28 Add the partial products. 84 + 1960 = 2,044. So, 73 x 28 = 2,044. Question 3. 4 8 × 3 8 ——– Estimate: ______ Product: _______ Answer: Estimate: 2,000 Product: 1,824 Explanation: 48 is close to 50 and 38 is close to 40. Estimate: 50 × 40 = 2,000 40 x 38 = 1520 8 x 38 = 304 1520 + 304 = 1824. Product: 1,824 Question 4. 5 9 × 5 2 ——– Estimate: ______ Product: _______ Answer: Estimate: 3,000 Product: 3,068 Explanation: 59 is close to 60 and 52 is close to 50. Estimate: 60 × 50 = 3,000 50 x 52 = 2600 9 x 52 = 468 2600 + 468 = 3068. Product: 3,068. Question 5. 8 4 × 4 0 ——– Estimate: ______ Product: _______ Answer: Estimate: 3,200 Product: 3,360 Explanation: 84 is close to 80 and 40 is close to 40. Estimate: 80 × 40 = 3,200 80 x 40 = 3,200 4 x 40 = 160 3200 + 160 = 3,360. Product: 3,360. Question 6. 8 3 × 7 7 ——– Estimate: ______ Product: _______ Answer: Estimate: 6,400 Product: 6,391 Explanation: 83 is close to 80 and 77 is close to 80. Estimate: 80 × 80 = 6,400 80 x 77 = 6,160 3 x 77 = 231 6,160 + 231 = 6,391. Product: 6,391. Question 7. 9 1 × 1 9 ——– Estimate: ______ Product: _______ Answer: Estimate: 1,800 Product: 1,729 Explanation: 91 is close to 90 and 19 is close to 20. Estimate: 90 × 20 = 1,800 90 x 19 = 1,710 1 x 19 = 19 1,710+ 19 = 1,729. Product: 1,729. Problem Solving Question 8. Baseballs come in cartons of 84 baseballs. A team orders 18 cartons of baseballs. How many baseballs does the team order? _______ baseballs Answer: 1,512 baseballs Explanation: To find total baseballs, 84 x 18 80 x 18 = 1,440 4 x 18 = 72 84 x 18 = 1,512 Question 9. There are 16 tables in the school lunch room. Each table can seat 22 students. How many students can be seated at lunch at one time? _______ students Answer: 352 students Explanation: Total Students = 16 x 22 10 x 22 = 220 6 x 22 = 132 220 + 132 = 352. 352 students can be seated at lunch at one time ### Common Core – Page No. 176 Lesson Check Question 1. The art teacher has 48 boxes of crayons. There are 64 crayons in each box. Which is the best estimate of the number of crayons the art teacher has? Options: a. 2,400 b. 2,800 c. 3,000 d. 3,500 Answer: c. 3,000 Explanation: 1. Total number of crayons = 48 x 64 48 is close to 50; 64 is close to 60 50 x 60 = 3,000. The art teacher has about to 3, 000 crayons. Question 2. A basketball team scored an average of 52 points in each of 15 games. How many points did the team score in all? Options: a. 500 b. 312 c. 780 d. 1,000 Answer: c. 780 Explanation: Total Points = 52 x 15 50 x 15 = 750 2 x 15 = 30 750 + 30 = 780. The basketball team scored 780 points in total. Spiral Review Question 3. One Saturday, an orchard sold 83 bags of apples. There are 27 apples in each bag. Which expression represents the total number of apples sold? Options: a. 16 + 6 + 56 + 21 b. 160 + 60 + 56 + 21 c. 160 + 60 + 560 + 21 d. 1,600 + 60 + 560 + 21 Answer: d. 1,600 + 60 + 560 + 21 Explanation: Total number of apples sold = 83 x 27 80 x 27 = 2,160 3 x 27 = 81 2,160 + 81 = 2,241. The total number of apples sold = 2,241. 16 + 6 + 56 + 21 = 99 not equal to 2,241 160 + 60 + 56 + 21 = 297 not equal to 2,241 160 + 60 + 560 + 21 = 801 not equal to 2,241 1,600 + 60 + 560 + 21 = 2,241 equal to 2,241 1,600 + 60 + 560 + 21 = 2,241 is correct. Question 4. Hannah has a grid of squares that has 12 rows with 15 squares in each row. She colors 5 rows of 8 squares in the middle of the grid blue. She colors the rest of the squares red. How many squares does Hannah color red? Options: a. 40 b. 140 c. 180 d. 220 Answer: b. 140 Explanation: Hannah has a grid of squares that has 12 rows with 15 squares in each row = 12 x 15 = 180. The grid of squares in blue = 5 x 8 = 40. The grid of squares in red = 180 – 40 = 140. Question 5. Gabriella has 4 times as many erasers a Leona. Leona has 8 erasers. How many erasers does Gabriella have? Options: a. 32 b. 24 c. 12 d. 2 Answer: a. 32 Explanation: Gabriella have 4 x 8 = 32 erasers. Question 6. Phil has 3 times as many rocks as Peter. Together, they have 48 rocks. How many more rocks does Phil have than Peter? Options: a. 36 b. 24 c. 16 d. 12 Answer: b. 24 Explanation: Phil has 3 times as many rocks as Peter. Together, they have 48 rocks If Peter has x rocks, Phil has 3x rocks 3x + x = 48. 4x = 48. x = 48/4 = 12. Peter has 12 rocks. Phil has 3 x 12 = 36 rocks. Phil has 36 – 12 = 24 more rocks than Peter. ### Page No. 179 Question 1. Find the product. Estimate: ______ Product: _______ Answer: Estimate: 1,500 Product: 1,566 Explanation: 54 x 29 Estimate: Think 54 is close to 50; 29 is close to 30. 50 x 30 = 1,500 Product: 20 x 5 tens = 100 tens 20 x 4 ones = 80 ones 9 x 5 tens = 45 tens 9 x 4 ones = 36 ones. Add partial products. 1000 + 80 + 450 + 36 = 1,566. Estimate. Then choose a method to find the product. Question 2. 3 6 × 1 4 ——- Estimate: ______ Product: _______ Answer: Estimate: 400 Product: 504 Explanation: 36 x 14 Estimate: Think 36 is close to 40; 14 is close to 10. 40 x 10 = 400 Product: 10 x 3 tens = 30 tens 10 x 6 ones = 60 ones 4 x 3 tens = 12 tens 4 x 6 ones = 24 ones. Add partial products. 300 + 60 + 120 + 24 = 504. Question 3. 6 3 × 4 2 ——- Estimate: ______ Product: _______ Answer: Estimate: 2,400 Product: 2646 Explanation: 63 x 42 Estimate: Think 63 is close to 60; 42 is close to 40. 60 x 40 = 2400 Product: 40 x 6 tens = 240 tens 40 x 3 ones = 120 ones 2 x 6 tens = 12 tens 2 x 3 ones = 6 ones. Add partial products. 2400 + 120 + 120 + 6 = 2646. Question 4. 8 4 × 5 3 ——- Estimate: ______ Product: _______ Answer: Estimate: 4,000 Product: 4,452 Explanation: 84 x 53 Estimate: Think 84 is close to 80; 53 is close to 50. 80 x 50 = 4,000 Product: 50 x 8 tens = 400 tens 50 x 4 ones = 200 ones 3 x 8 tens = 24 tens 3 x 4 ones = 12 ones. Add partial products. 4000 + 200 + 240 + 12 = 4,452. Question 5. 7 1 × 1 3 ——- Estimate: ______ Product: _______ Answer: Estimate: 700 Product: 923 Explanation: 71 x 13 Estimate: Think 71 is close to 70; 13 is close to 10. 70 x 10 = 700 Product: 10 x 7 tens = 70 tens 10 x 1 ones = 10 ones 3 x 7 tens = 21 tens 3 x 1 ones = 3 ones. Add partial products. 700 + 10 + 210 + 3 = 923. Practice: Copy and Solve Estimate. Find the product. Question 6. 29 ×$82
Estimate: $_______ Product:$ _______

Estimate: $2,400 Product:$2,378

Explanation:
29 x $82 Estimate: Think 29 is close to 30;$82 is close to $80. 30 x$80 = $2,400 Product:$80 x 2 tens = $160 tens$80 x 9 ones = $720 ones$2 x 2 tens = $4 tens$2 x 9 ones = $18 ones. Add partial products.$1600 + $720 +$40 + $18 =$2,378.

Question 7.
57 × 79
Estimate: _______
Product: _______

Estimate: 4,800
Product: 4,503

Explanation:
57 x 79
Estimate: Think 57 is close to 60; 79 is close to 80.
60 x 80 = 4,800
Product:
70 x 5 tens = 350 tens
70 x 7 ones = 490 ones
9 x 5 tens = 45 tens
9 x 7 ones = 63 ones.
3500 + 490 + 450 + 63 = 4,503.

Question 8.
80 × 27
Estimate: _______
Product: _______

Estimate: 2,400
Product: 2,160

Explanation:
80 x 27
Estimate: Think 27 is close to 30.
30 x 80 = 2,400
Product:
20 x 8 tens = 160 tens
20 x 0 ones = 0 ones
7 x 8 tens = 56 tens
7 x 0 ones = 0 ones.
1600 + 0 + 560 + 0 = 2,160.

Question 9.
32 × $75 Estimate:$ _______
Product: $_______ Answer: Estimate:$2,100
Product: $2,400 Explanation: 32 ×$75
Estimate: Think 32 is close to 30; $75 is close to$70.
30 x $70 =$2,100
Product:
$70 x 3 tens =$210 tens
$70 x 2 ones =$140 ones
$5 x 3 tens =$15 tens
$5 x 2 ones =$10 ones.
$2100 +$140 + $150 +$10 = $2,400. Question 10. 55 × 48 Estimate: _______ Product: _______ Answer: Estimate: 2,750 Product: 2,640 Explanation: 55 × 48 Estimate: Think 48 is close to 50. 55 x 50 = 2,750 Product: 40 x 5 tens = 200 tens 40 x 5 ones = 200 ones 8 x 5 tens = 40 tens 8 x 5 ones = 40 ones. Add partial products. 2000 + 200 + 400 + 40 = 2,640. Question 11. 19 ×$82
Estimate: $_______ Product:$ _______

Estimate: $1,600 Product:$1,558

Explanation:
19 × $82 Estimate: Think 19 is close to 20;$82 is close to $80. 20 x$80 = $1,600 Product:$80 x 1 tens = $80 tens$80 x 9 ones = $720 ones$2 x 1 tens = $2 tens$2 x 9 ones = $18 ones. Add partial products.$800 + $720 +$20 + $18 =$1,558.

Question 12.
25 × $25 Estimate:$ _______
Product: $_______ Answer: Estimate:$625
Product: $625 Explanation: 25 ×$25
Estimate:
25 x $25 =$625
Product:
$20 x 2 tens =$40 tens
$20 x 5 ones =$100 ones
$5 x 2 tens =$10 tens
$5 x 5 ones =$25 ones.
$400 +$100 + $100 +$25 = $625. Question 13. 41 × 98 Estimate: _______ Product: _______ Answer: Estimate: 4,000 Product: 4,018 Explanation: 41 × 98 Estimate: Think 41 is close to 40; 98 is close to 100. 40 x 100 = 4,000 Product: 90 x 4 tens = 360 tens 90 x 1 ones = 90 ones 8 x 4 tens = 32 tens 8 x 1 ones = 8 ones. Add partial products. 3600 + 90 + 320 + 8 = 4,018. Identify Relationships Algebra Use mental math to find the number. Question 14. 30 × 14 = 420, so 30 × 15 = ______ Answer: 30 × 15 = 450 Explanation: 30 × 15 = 30 + 420 30 × 15 = 450 Question 15. 25 × 12 = 300, so 25 × ______ = 350 Answer: 25 x 14 = 350 Explanation: 25 × 12 = 300 For every next multiplication, the product value is increased by 25. 25 x 13 = 325. 25 x 14 =350. Question 16. The town conservation manager bought 16 maple trees for$26 each. She paid with five $100 bills. How much change will the manager receive? Explain.$ ______

$84 Explanation: The town conservation manager bought 16 maple trees for$26 each = 16 x $26 =$416.
She paid with five $100 bills = 5 x$100 = $500. The manager receive =$500 – $416 =$84.

Question 17.
Each of 25 students in Group A read for 45 minutes. Each of 21 students in Group B read for 48 minutes. Which group read for more minutes? Explain.
_________

Group A read for more minutes than Group B.

Explanation:
Group A read for 25 x 45 = 1125 minutes.
Group B read for 21 x 48 = 1008 minutes.
Group A read for more minutes than Group B.

### Page No. 180

Question 18.
Martin collects stamps. He counted 48 pages in his collector’s album. The first 20 pages each have 35 stamps in 5 rows. The rest of the pages each have 54 stamps. How many stamps does Martin have in his album?

a. What do you need to know?
Type below:
_________

The total stamps in the first 20 pages + The total stamps in the remaining pages.

Question 18.
b. How will you use multiplication to find the number of stamps?
Type below:
_________

The first 20 pages each have 35 stamps in 5 rows.
So, 35 x 5 = 175 stamps.

Question 18.
c. Tell why you might use addition and subtraction to help solve the problem.
Type below:
_________

As mentioned that the number of stamps available in the first 20 pages and the number of stamps available in the rest of the pages. We need to add all pages to get 48 pages stamps.

Question 18.
d. Show the steps to solve the problem.
Type below:
_________

Martin has 48 pages in his collector’s album.
The first 20 pages each have 35 stamps in 5 rows.
So, 35 x 5 = 175 stamps.
The first 20 pages have 175 stamps.
The rest of the pages each have 54 stamps.
So, total stamps = 175 + 54 = 229 stamps.

Question 18.
e. Complete the sentences.
Martin has a total of _____ stamps on the first 20 pages.
There are _____ more pages after the first 20 pages in Martin’s album.
There are _____ stamps on the rest of the pages.
There are _____ stamps in the album.
Type below:
_________

Martin has a total of __175___ stamps on the first 20 pages.
There are __24___ more pages after the first 20 pages in Martin’s album.
There are __54___ stamps on the rest of the pages.
There are ___229__ stamps in the album.

Question 19.
Select the expressions that have the same product as 35 × 17. Mark all that apply.
Options:
a. (30 × 10) + (30 × 7) + (5 × 10) + (5 × 7)
b. (30 × 17) + (5 × 17)
c. (35 × 30) + (35 × 5) + (35 × 10) + (35 × 7)
d. (35 × 10) + (35 × 7)
e. (35 × 10) + (30 × 10) + (5 × 10) + (5 × 7)
f. (35 × 30) + (35 × 5)

a. (30 × 10) + (30 × 7) + (5 × 10) + (5 × 7)
b. (30 × 17) + (5 × 17)
d. (35 × 10) + (35 × 7)

Explanation:
35 × 17
30 x 10 =300
30 x 7 = 210
5 x 10 = 50
5 x 7 = 35
300 + 210 + 50 + 35 = 595.
a. (30 × 10) + (30 × 7) + (5 × 10) + (5 × 7) = 300 + 210 + 50 + 35 = 595 equal to 595.
b. (30 × 17) + (5 × 17) = 510 + 85 = 595 equal to 595.
c. (35 × 30) + (35 × 5) + (35 × 10) + (35 × 7) = 1050 + 175 + 350 + 245 = 1820 not equal to 595.
d. (35 × 10) + (35 × 7) = 350 + 245 = 595 equal to 595
e. (35 × 10) + (30 × 10) + (5 × 10) + (5 × 7) = 350 + 300 + 50 + 35 = 735 not equal to 595.
f. (35 × 30) + (35 × 5) = 1050 + 175 = 1,225 not equal to 595.

### Common Core – Page No. 181

Choose a Multiplication Method

Estimate. Then choose a method to find the product.

Question 1.
Estimate: 1,200
3 1
× 4 3
——-
9 3
+ 1, 2 4 0
————
1, 3 3 3

Estimate: 1,200
Product: 1, 3 3 3

Explanation:
Estimate: 1,200
3 1
× 4 3
——-
9 3
+ 1, 2 4 0
————
1, 3 3 3

Question 2.
6 7
× 8 5
——-
Estimate: _____
Product: ______

Estimate: 6,300
Product: 5,695

Explanation:
Estimate: 67 is close to 70; 85 is close to 90.
70 x 90 = 6,300.
Product: 67 x 85
80 x 6 tens = 480 tens
80 x 7 ones = 560 ones
5 x 6 tens = 30 tens
5 x 7 ones = 35 ones.
4800 + 560 + 300 + 35 = 5,695.

Question 3.
6 8
× 3 8
——-
Estimate: _____
Product: ______

Estimate: 2,800
Product: 2,584

Explanation:
Estimate: 68 is close to 70; 38 is close to 40.
70 x 40 = 2,800.
Product: 68 x 38
30 x 6 tens = 180 tens
30 x 8 ones = 240 ones
8 x 6 tens = 48 tens
8 x 8 ones = 64 ones.
1800 + 240 + 480 + 64 = 2,584.

Question 4.
9 5
× 1 7
——-
Estimate: _____
Product: ______

Estimate: 1,700
Product: 1,615

Explanation:
Estimate: 95 is close to 100.
100 x 17 = 1,700.
Product: 95 x 17
10 x 9 tens = 90 tens
10 x 5 ones = 50 ones
7 x 9 tens = 63 tens
7 x 5 ones = 35 ones.
900 + 50 + 630 + 35 = 1,615.

Question 5.
4 9
× 5 4
——-
Estimate: _____
Product: ______

Estimate: 2,500
Product: 2,646

Explanation:
Estimate: 49 is close to 50; 54 is close to 50.
50 x 50 = 2,500.
Product: 49 x 54
50 x 4 tens = 200 tens
50 x 9 ones = 450 ones
4 x 4 tens = 16 tens
4 x 9 ones = 36 ones.
2000 + 450 + 160 + 36 = 2,646.

Question 6.
9 1
× 2 6
——-
Estimate: _____
Product: ______

Estimate: 2,700
Product: 2,366

Explanation:
Estimate: 91 is close to 90; 26 is close to 30.
90 x 30 = 2,700.
Product: 49 x 54
20 x 9 tens = 180 tens
20 x 1 ones = 20 ones
6 x 9 tens = 54 tens
6 x 1 ones = 6 ones.
1800 + 20 + 540 + 6 = 2,366.

Question 7.
8 2
× 1 9
——-
Estimate: _____
Product: ______

Estimate: 1,600
Product: 1,558

Explanation:
Estimate: 82 is close to 80; 19 is close to 20.
80 x 20 = 1,600.
Product: 82 x 19
10 x 8 tens = 80 tens
10 x 2 ones = 20 ones
9 x 8 tens = 72 tens
9 x 2 ones = 18 ones.
800 + 20 + 720 + 18 = 1,558.

Question 8.
4 6
× 2 7
——-
Estimate: _____
Product: ______

Estimate: 1,500
Product: 1,242

Explanation:
Estimate: 46 is close to 50; 27 is close to 30.
50 x 30 = 1,500.
Product: 46 x 27
20 x 4 tens = 80 tens
20 x 6 ones = 120 ones
7 x 4 tens = 28 tens
7 x 6 ones = 42 ones.
800 + 120 + 280 + 42 = 1,242.

Question 9.
4 1
× 3 3
——-
Estimate: _____
Product: ______

Estimate: 1,200
Product: 1,353

Explanation:
Estimate: 41 is close to 40; 33 is close to 30.
40 x 30 = 1,200.
Product: 41 x 33
30 x 4 tens = 120 tens
30 x 1 ones = 30 ones
3 x 4 tens = 12 tens
3 x 1 ones = 3 ones.
1200 + 30 + 120 + 3 = 1,353.

Question 10.
9 7
× 1 3
——-
Estimate: _____
Product: ______

Estimate: 1,300
Product: 1,261

Explanation:
Estimate: 97 is close to 100.
100 x 13 = 1,300.
Product: 97 x 13
10 x 9 tens = 90 tens
10 x 7 ones = 70 ones
3 x 9 tens = 27 tens
3 x 7 ones = 21 ones.
900 + 70 + 270 + 21 = 1,261.

Question 11.
7 5
× 6 9
——-
Estimate: _____
Product: ______

Estimate: 5,600
Product: 5,195

Explanation:
Estimate: 75 is close to 80; 69 is close to 70.
80 x 70 = 5,600.
Product: 75 x 69
60 x 7 tens = 420 tens
60 x 5 ones = 300 ones
9 x 7 tens = 63 tens
9 x 5 ones = 45 ones.
4200 + 300 + 630 + 45 = 5,195.

Problem Solving

Question 12.
A movie theatre has 26 rows of seats. There are 18 seats in each row. How many seats are there in all?
______ seats

468 seats

Explanation:
26 x 18 = 468 seats.
20 x 18 = 360
6 x 18 = 108
108+360 = 468.

Question 13.
Each class at Briarwood Elementary collected at least 54 cans of food during the food drive. If there are 29 classes in the school, what was the least number of
cans collected?
______ cans

1,566 cans

Explanation:
Each class at Briarwood Elementary collected at least 54 cans of food.
If there are 29 classes in the school,
the least number of cans collected = 54 x 29 = 1,566 cans.

### Common Core – Page No. 182

Lesson Check

Question 1.
A choir needs new robes for each of its 46 singers. Each robe costs $32. What will be the total cost for all 46 robes? Options: a.$1,472
b. $1,372 c.$1,362
d. $230 Answer: a.$1,472

Explanation:
46 x $32 40 x$32 = $1,280 6 x$32 = $192$1,280 + $192 =$1,472

Question 2.
A wall on the side of a building is made up of 52 rows of bricks with 44 bricks in each row. How many bricks make up the wall?
Options:
a. 3,080
b. 2,288
c. 488
d. 416

b. 2,288

Explanation:
52 x 44
50 x 44 = 2,200
2 x 44 = 88
2,200 + 88 = 2,288.
2,288 bricks make up the wall.

Spiral Review

Question 3.
Which expression shows how to multiply 4 × 362 by using place value and expanded form?
Options:
a. (4 × 3) + (4 × 6) + (4 × 2)
b. (4 × 300) + (4 × 600) +(4 × 200)
c. (4 × 300) + (4 × 60) + (4 × 20)
d. (4 × 300) + (4 × 60) + (4 × 2)

d. (4 × 300) + (4 × 60) + (4 × 2)

Explanation:
4 × 362 = 1,448
a. (4 × 3) + (4 × 6) + (4 × 2) = 12 + 24 + 8 = 44 not equal to 1,448.
b. (4 × 300) + (4 × 600) +(4 × 200) = 1200 + 2400 + 800 = 4,400 not equal to 1,448.
c. (4 × 300) + (4 × 60) + (4 × 20) = 1200 + 240 + 80 = 1,520 not equal to 1,448.
d. (4 × 300) + (4 × 60) + (4 × 2) = 1200 + 240 + 8 = 1,448 equal to 1,448.

Question 4.
Use the model below. What is the product 4 x 492?

Options:
a. 16 + 36 + 8 = 60
b. 160 + 36 + 8 = 204
c. 160 + 360 + 8 = 528
d. 1,600 + 360 + 8 = 1,968

d. 1,600 + 360 + 8 = 1,968

Explanation:

1,600 + 360 + 8 = 1,968

Question 5.
What is the sum 13,094 + 259,728?
Options:
a. 272,832
b. 272,822
c. 262,722
d. 262,712

c. 262,722

Explanation:
13,094 + 259,728 = 262,722

Question 6.
During the 2008–2009 season, there were 801,372 people who attended the home hockey games in Philadelphia. There were 609,907 people who attended the home hockey games in Phoenix. How much greater was the home attendance in Philadelphia than in Phoenix that season?
Options:
a. 101,475
b. 191,465
c. 201,465
d. 202,465

b. 191,465

Explanation:
801,372 – 609,907 = 191,465
Philadelphia attendance is 191,465 greater than in Phoenix that season.

### Page No. 185

Question 1.
An average of 74 reports with bird counts were turned in each day in June. An average of 89 were turned in each day in July. How many reports were turned in for both months? (Hint: There are 30 days in June and 31 days in July.)
First, write the problem for June.
Type below:
__________

Given that An average of 74 reports with bird counts was turned in each day in June.
For June Month, there are 30 days = 30 x 74 = 2,220.

Question 1.
Next, write the problem for July.
Type below:
__________

An average of 89 reports with bird counts was turned in each day in July.
For July Month, there are 31 days = 31 x 89 = 2,759.

Question 1.
Last, find and add the two products.
____________ reports were turned in for both months.
Type below:
__________

Given that An average of 74 reports with bird counts was turned in each day in June.
For June Month, there are 30 days = 30 x 74 = 2,220.
An average of 89 reports with bird counts was turned in each day in July.
For July Month, there are 31 days = 31 x 89 = 2,759.
Add two products to get the total number of reports that were turned in for both months.
2,220 + 2,759 = 4,979.

Question 2.
What if an average of 98 reports were turned in each day for the month of June? How many reports were turned in for June? Describe how your answer for June would be different.
______ reports

720 more reports

Explanation:
Given that an average of 98 reports was turned in each day for the month of June.
June has 30 days.
Total number of reports were turned in for June = 30 x 98 = 2, 940.
From the above answer, 98 − 74 = 24. So, there would be 30 × 24, or 720 more reports.

Question 3.
There are 48 crayons in a box. There are 12 boxes in a carton. Mr. Johnson ordered 6 cartons of crayons for the school. How many crayons did he get?
______ crayons

3,456 crayons

Explanation:
There are 48 crayons in a box.
There are 12 boxes in a carton.
So, 1 carton = 48 x 12 = 576 crayons.
If Mr. Johnson ordered 6 cartons of crayons for the school, 6 x 576 crayons = 3,456 crayons.
He gets 3,456 crayons.

Question 4.
Make Sense of Problems Each of 5 birdwatchers reported seeing 15 roseate spoonbills in a day. If they each reported seeing the same number of roseate spoonbills over 14 days, how many would be reported?
______ roseate spoonbills

1,050 roseate spoonbills

Explanation:
Given that, 1 day –>5 birdwatchers reported 15 roseate spoonbills = 5 x 15 = 75 roseate spoonbills.
So, in 14 days –> 5 birdwatchers reported 75 x 14 = 1,050 roseate spoonbills.

### Page No. 186

Question 5.
On each of Maggie’s bird-watching trips, she has seen at least 24 birds. If she has taken 4 of these trips each year over the past 16 years, at least how many birds has Maggie seen?
at least ______ birds

Maggie seen 1,536 birds

Explanation:
Given that, 1 trip –> Maggie seen 24 birds.
For 1 year she goes for 4 bird-watching trips.
So, she has seen 4 x 24 = 96 birds for 1 year.
For 16 years, 16 x 96 = 1,536 birds have Maggie seen.

Question 6.
Make Sense of Problems There are 12 inches in a foot. In September, Mrs. Harris orders 32 feet of ribbon for the Crafts Club. In January, she orders 9 feet less. How many inches of ribbon does Mrs. Harris order? Explain how you found your answer.
______ inches

660 inches

Explanation:
There are 12 inches in a foot.
In September, Mrs. Harris orders 32 feet of ribbon for the Crafts Club = 32 x 12 = 384.
In January, she orders 9 feet less = 32 – 9 = 23.
So, in January, she orders 23 x 12 = 276.
Mrs. Harris order 276 + 384 = 660 inches of ribbon in total.
(or)
9 less than 32 is 23, so I added 23 + 32 = 55.
Then, I multiplied 55 × 12 = 660.

Question 7.
Lydia is having a party on Saturday. She decides to write a riddle on her invitations to describe her house number on Cypress Street. Use the clues to find Lydia’s address.

______ Cypress Street

14827 Cypress Street

Explanation:
Given that tens digit is 5 less than 7 = 7 – 5 = 2. 2 is the tens digit.
The thousands digit is twice the digit in the tens place = 2 x 2 = 4.
The hundreds digit is the greatest even number that is less than 10 i.e, 8.
The ones digit is the product of 7 and 1 = 7 x 1 = 7.
The ten thousands digit is the difference between the hundreds digit and the ones digit. So, 8 – 7 = 1.
Lydia’s address ( house number ) is 14827 Cypress Street.

Question 8.
A school is adding 4 rows of seats to the auditorium. There are 7 seats in each row. Each new seat costs $99. What is the total cost for the new seats? Show your work.$ ______

$2,772 Explanation: Given that A school is adding 4 rows of seats to the auditorium. There are 7 seats in each row. So, 7 x 4 = 28 seats are available in an auditorium. Each new seat costs$99.
28 x $99 =$2,772 for total cost of the new seats.

### Common Core – Page No. 187

Problem Solving Multiply 2 – Digit numbers

Solve each problem. Use a bar model to help.

Question 1.
Mason counted an average of 18 birds at his bird feeder each day for 20 days. Gloria counted an average of 21 birds at her bird feeder each day for 16 days. How many more birds did Mason count at his feeder than Gloria counted at hers?

Birds counted by Mason: 18 × 20 = 360
Birds counted by Gloria: 21 × 16 = 336
Draw a bar model to compare.
Subtract. 360 – 336 = 24
So, Mason counted 24 more birds.

Birds counted by Mason: 18 × 20 = 360
Birds counted by Gloria: 21 × 16 = 336
Draw a bar model to compare.

Subtract. 360 – 336 = 24
So, Mason counted 24 more birds.

Question 2.
The 24 students in Ms. Lee’s class each collected an average of 18 cans for recycling. The 21 students in Mr. Galvez’s class each collected an average of 25 cans for recycling. How many more cans were collected by Mr. Galvez’s class than Ms. Lee’s class?
______ more cans

The number of cans collected by Ms. Lee’s class = 18 x 24 = 432.
The number of cans collected by Mr. Galvez’s class = 25 x 21 = 525.
Use Bar Model

Subtract. 525 – 432 = 93 more cans.
So, Mr. Galvez’s class collected 93 more cans than Ms. Lee’s class.

Question 3.
At East School, each of the 45 classrooms has an average of 22 students. At West School, each of the 42 classrooms has an average of 23 students. How many more students are at East School than at West School?
______ more students

Students in East school = 45 x 22 = 990.
Students in West School = 42 x 23 = 966.
Use Bar Model

Subtract. 990 – 966 = 24.
So, East School has 24 students more than West School.

Question 4.
A zoo gift shop orders 18 boxes of 75 key rings each and 15 boxes of 80 refrigerator magnets each. How many more key rings than refrigerator magnets does the gift shop order?
______ more key rings

Number of Key Rings = 75 x 18 = 1,350.
Number of Refrigerator Magnets= 80 x 15 = 1,200.
Use Bar Model

Subtract. 1,350 – 1,200 = 150.
So, key rings are 150 more than refrigerator magnets.

### Common Core – Page No. 188

Lesson Check

Question 1.
Ace Manufacturing ordered 17 boxes with 85 ball bearings each. They also ordered 15 boxes with 90 springs each. How many more ball bearings than springs did they order?
Options:
a. 5
b. 85
c. 90
d. 95

d. 95

Explanation:
Number of ball bearings = 85 x 17 = 1,445.
Number of springs = 90 x 15 = 1,350.
Use Bar Model

Subtract. 1,445 – 1,350 = 95.
So, ball bearings are 95 more than springs.

Question 2.
Elton hiked 16 miles each day on a 12-day hiking trip. Lola hiked 14 miles each day on her 16-day hiking trip. In all, how many more miles did Lola hike than Elton hiked?
Options:
a. 2 miles
b. 18 miles
c. 32 miles
d. 118 miles

c. 32 miles

Explanation:
Hiking trip by Elton = 12 x 16 = 192.
Hiking trip by Lola = 16 x 14 = 224.
Use Bar Model

Subtract. 224 – 192 = 32.
So, the Hiking trip by Lola is 32 times more than the Hiking trip by Elton.

Spiral Review

Question 3.
An orchard has 24 rows of apple trees. There are 35 apple trees in each row. How many apple trees are in the orchard?
Options:
a. 59
b. 192
c. 740
d. 840

d. 840

Explanation:
An orchard has 24 rows of apple trees. There are 35 apple trees in each row.
24 x 35 = 840 apple trees are in the orchard.

Question 4.
An amusement park reported 354,605 visitors last summer. What is this number rounded to the nearest thousand?
Options:
a. 354,600
b. 355,000
c. 360,000
d. 400,000

b. 355,000

Explanation:
An amusement park reported 354,605 visitors last summer. 4,605 is close to 5,000. So, the answer is 355,000.

Question 5.
Attendance at the football game was 102,653. What is the value of the digit 6?
Options:
a. 6
b. 60
c. 600
d. 6,000

c. 600

Explanation:
Digit 6 is at hundreds of positions. So, the answer is 6 x 100 = 600.

Question 6.
Jill’s fish weighs 8 times as much as her parakeet. Together, the pets weigh 63 ounces. How much does the fish weigh?
Options:
a. 7 ounces
b. 49 ounces
c. 55 ounces
d. 56 ounces

d. 56 ounces

Explanation:
Let Jill’s parakeet = X.
Jill’s fish weighs 8 times as much as her parakeet = 8X.
Together, the pets weigh 63 ounces.
X + 8X = 63.
9X = 63.
X = 63/9 = 7.
So, Jill’s parakeet =7.
Jill’s fish = 7 x 8 = 56 ounces.

### Review/Test – Page No. 189

Question 1.
Explain how to find 40 × 50 using mental math
Type below:
_________

200

Explanation:
40 x 50
By using mental math
4 x 5 = 20
40 x 50 = 200

Mrs. Traynor’s class is taking a field trip to the zoo. The trip will cost $26 for each student. There are 22 students in her class. Question 2. Part A Round each factor to estimate the total cost of the students’ field trip.$ ______

$600 Explanation: Total cost of the students’ field trip = 22 x$26.
22 x $26 20 x$30 = $600 The total cost would be about$600.

Question 2.
Part B
Use compatible numbers to estimate the total cost of the field trip.
$______ Answer:$500

Explanation:
If we use compatible numbers to estimate the total cost of the field trip.
22 x $26 20 × 25 = 500 The total cost would be about$500.

Question 2.
Part C
Which do you think is the better estimate? Explain.
Better estimate: _________

Using rounded numbers is a better estimate. When rounded numbers are used, one estimated factor was $4 more than the actual factor and the other estimated factor was$2 that is less than the actual factor. So, the estimate should be close to the actual one. When compatible numbers are used both estimated factors were less than the actual factors. So, the product will be an underestimate.

### Review/Test – Page No. 190

For numbers 3a–3e, select Yes or No to show if the answer is correct.

Question 3.
3a. 35 × 10 = 350
i. yes
ii. no

i. yes

Explanation:
35 x 10 = 350
30 x 10 = 300.
5 x 10 = 50.
300 + 50 = 350.

Question 3.
3b. 19 × 20 = 380
i. yes
ii. no

i. yes

Explanation:
19 × 20 = 380
19 x 20 = 19 x 2 tens.
19 x 20 = 38 tens = 380.

Question 3.
3c. 12 × 100 = 120
i. yes
ii. no

ii. no

Explanation:
12 x 100 = 120.
10 x 100 = 1000
2 x 100 = 200.
1000 + 200 = 1200.

Question 3.
3d. 70 × 100 = 7,000
i. yes
ii. no

i. yes

Explanation:
70 x 100 = 7,000
100 x 7 tens = 700 tens = 7,000

Question 3.
3e. 28 × 30 = 2,100
i. yes
ii. no

ii. no

Explanation:
28 × 30
20 x 30 = 600
8 x 30 = 240
600 + 240 = 840

Question 4.
There are 23 boxes of pencils in Mr. Shaw’s supply cabinet. Each box contains 100 pencils. How many pencils are in the supply cabinet?
_____ penciles

2,300 pencils

Explanation:
23 x 100 = 2,300 pencils are in the supply cabinet.

Question 5.
Which would provide a reasonable estimate for each product? Write the estimate beside the product. An estimate may be used more than once
23 × 38 __________
31 × 32 __________
46 × 18 __________
39 × 21 __________

23 × 38 –> 25 x 40
31 x 32 –> 30 × 30
46 × 18 –> 50 × 20
39 × 21 –> 25 × 40

Explanation:
23 × 38; 23 is close to 25; 38 is close to 40.
So, the estimated product is 25 x 40
31 x 32; 31 is close to 30; 32is close to 30.
So, the estimated product is 30 × 30
46 × 18; 46 is close to 50; 18 is close to 20.
So, the estimated product is 50 × 20
39 × 21; 39 is close to 40; 21 is close to 25.
So, the estimated product is 25 × 40

Question 6.
There are 26 baseball teams in the league. Each team has 18 players. Write a number sentence that will provide a reasonable estimate for the number of players in the league. Explain how you found your estimate.
Type below:
__________

There are 26 baseball teams in the league. Each team has 18 players.
26 x 18
25 x 20
We Rounded each factor to its close factor, then simplified the multiplication.

Question 7.
The model shows 48 × 37. Write the partial products.

Type below:
__________

Partial Products are 1200, 240, 280, 56

### Review/Test – Page No. 191

Question 8.
Jess made this model to find the product 32 × 17. Her modelis incorrect.

Part A
What did Jess do wrong?
Type below:
__________

Question 8.
Part B
Redraw the model so that it is correct.

Type below:
__________

Question 8.
Part C
What is the actual product 32 × 17?
______

544

Explanation:
32 × 17
10 x 32 = 320
7 x 32 = 224
320 + 224 = 544.

Question 9.
Tatum wants to use partial products to find 15 × 32. Write the numbers in the boxes to show 15 × 32.

Type below:
__________

### Review/Test – Page No. 192

Question 10.
Which product is shown by the model? Write the letter of the product on the line below the model.

Type below:
__________

C                                              A                                                  B
10 + 3 = 13
10 + 3 = 13
13 x 13
2. 10 + 7 = 17
30 + 6 = 36
17 x 36
3. 20 + 4 = 24
10 + 4 = 14
24 x 14

Question 11.
Mrs. Jones places 3 orders for school T-shirts. Each order has 16 boxes of shirts and each box holds 17 shirts. How many T-shirts does Mrs. Jones order? Use partial products to help you.
Type below:
__________

816 T-shirts

Explanation:
Mrs. Jones places 3 orders for school T-shirts. Each order has 16 boxes of shirts and each box holds 17 shirts.
Each box has 17 shirts.
16 boxes = 16 x 17 = 272.
Each order = 16 boxes = 272 shirts.
3 orders = 3 x 272 = 816 shirts.
Mrs. Jones order 816 T-shirts.

Question 12.
Write the unknown digits. Use each digit exactly once.

Type below:
__________

90 x 40 = 3,600
90 x 6 = 540
3 x 40 = 120
3 x 6 = 18.
3,600 + 540 + 120 + 8 = 4,278.

Question 13.
Mike has 16 baseball cards. Niko has 17 times as many baseball cards as Mike does. How many baseball cards does Niko have?
________ baseball cards

272 baseball cards

Explanation:
Mike has 16 baseball cards. Niko has 17 times as many baseball cards as Mike does.
Niko have 16 x 17 = 272 baseball cards.

Question 14.
Multiply.
36 × 28 = ________

1,008

Explanation:
36 x 28
20 x 30 = 600
20 x 6 = 120
8 x 30 = 240
8 x 6 = 48
600 + 120 + 240 + 48 = 1,008

### Review/Test – Page No. 193

Question 15.
A farmer planted 42 rows of tomatoes with 13 plants in each row. How many tomato plants did the farmer grow?
42 × 13 = ______ tomato plants

420 + 126 = 546 tomato plants

Explanation:
42 × 13
10 x 42 = 420
3 x 42 = 126
420 + 126 = 546 tomato plants

Question 16.
Select another way to show 25 × 18. Mark all that apply.
Options:
a. (20 × 10) + (20 × 8) + (5 × 10) + (5 × 8)
b. (25 × 20) + (25 × 5) + (25 × 10) + (25 × 8)
c. (20 × 18) + (5 × 10) + (5 × 8)
d. (25 × 10) + (25 × 8)
e. (25 × 20) + (25 × 5)

a. (20 × 10) + (20 × 8) + (5 × 10) + (5 × 8)
c. (20 × 18) + (5 × 10) + (5 × 8)
d. (25 × 10) + (25 × 8)

Explanation:
25 × 18
10 x 25 = 250
8 x 25 = 200
250 + 200 = 450.
a. (20 × 10) + (20 × 8) + (5 × 10) + (5 × 8) = 200 + 160 + 50 + 40 = 450
b. (25 × 20) + (25 × 5) + (25 × 10) + (25 × 8) = 500 + 125 + 250 + 200 = 1,075
c. (20 × 18) + (5 × 10) + (5 × 8) = 360 + 50 + 40 = 450
d. (25 × 10) + (25 × 8) = 250 + 200 = 450
e. (25 × 20) + (25 × 5) = 500 + 125 = 625

Question 17.
Terrell runs 15 sprints. Each sprint is 65 meters. How many meters does Terrell run? Show your work.
______ meters

975 meters

Explanation:
Terrell run 15 x 65 = 975 meters.

Question 18.
There are 3 new seats in each row in a school auditorium. There are 15 rows in the auditorium. Each new seat cost $74. What is the cost for the new seats? Explain how you found your answer.$ ______

$3,330 Explanation: Given that There are 3 new seats in each row in a school auditorium. There are 15 rows in the auditorium. Each new seat cost$74.
So, 3 x 15 = 45 seats are available in an auditorium.
Each new seat costs $74. 45 x$74 = $3,330 for total cost of the new seats. Question 19. Ray and Ella helped move their school library to a new building. Ray packed 27 boxes with 25 books in each box. Ella packed 23 boxes with 30 books in each box. How many more books did Ella pack? Show your work. ______ books Answer: 15 books Explanation: Ray packed 27 x 25 = 675 books. Ella packed 23 x 30 = 690 books Ella packed 690 – 675 = 15 books more than Ray. ### Review/Test – Page No. 194 Question 20. Julius and Walt are finding the product of 25 and 16. Part A Julius’ answer is incorrect. What did Julius do wrong? Type below: __________ Answer: Julius multiplied 25 by 10 and then multiplied 25 by 6 correctly. He added the two partial products incorrectly. Question 20. Part B What did Walt do wrong? Type below: __________ Answer: Walt multiplied 6 by 5 and got 300 instead of 30 Question 20. Part C What is the correct product? Type below: __________ Answer: 25 x 16 = 400 Question 21. A clothing store sells 26 shirts and 22 pairs of jeans. Each item of clothing costs$32.
Part A
What is a reasonable estimate for the total cost of the clothing?
$______ Answer:$1500

Explanation:
A clothing store sells 26 shirts and 22 pairs of jeans. 26 + 22 = 48 clothes.
Each item of clothing costs $32. 48 x$32
50 x $30 =$1500

Question 21.
Part B
What is the exact answer for the total cost of the clothing? Show or explain how you found your answer.
$______ Answer:$1,536

Explanation:
48 x $32 40 x$32 = $1,280 8 x$32 = $256$1,280 + $256 =$1,536

### Page No. 199

Question 1.
A restaurant has 68 chairs. There are six chairs at each table. About how many tables are in the restaurant?
Estimate. 68 ÷ 6
Think: What number times 6 is about 68?
10 × 6 = ___
11 × 6 = ___
12 × 6 = ___
68 is closest to ______, so the best estimate is about _______ tables are in the restaurant.
Type below:
__________

68 is close to 70 and 6 is close to 5.
So, 70/5 = 12.
10 × 6 = __60_
11 × 6 = _66__
12 × 6 = _72__
68 is closest to ___66___, so the best estimate is about 11 x 6 = 66 tables are in the restaurant.

Find two numbers the quotient is between. Then estimate the quotient.

Question 2.
41 ÷ 3
between _______ and _______

between 13 and 14

Explanation:
13 x 3 = 39; 14 x 3 = 42.
The quotient of 41 ÷ 3 is between 13 and 14.

Question 3.
192 ÷ 5
between _______ and _______

between 30 and 40

Explanation:
30 x 5 = 150; 40 x 5 = 200.
The quotient of 192 ÷ 5 is between 30 and 40.

Find two numbers the quotient is between. Then estimate the quotient.

Question 4.
90 ÷ 7
between _______ and _______

between 12 and 13

Explanation:
12 x 7 = 84; 13 x 7 = 91.
The quotient of 90 ÷ 7 is between 12 and 13.

Question 5.
67 ÷ 4
between _______ and _______

between 16 and 17

Explanation:
16 x 4 = 64; 17 x 4 = 68.
The quotient of 67 ÷ 4 is between 16 and 17.

Question 6.
281 ÷ 9
between _______ and _______

between 30 and 40

Explanation:
30 x 9 = 270; 40 x 9 = 360.
The quotient of 281 ÷ 9 is between 30 and 40.

Question 7.
102 ÷ 7
between _______ and _______

between 14 and 15

Explanation:
14 x 7 = 98; 15 x 7 = 105.
The quotient of 102 ÷ 7 is between 14 and 15.

Question 8.
85 ÷ 6
between _______ and _______

between 14 and 15

Explanation:
14 x 6 = 84; 15 x 6 = 90.
The quotient of 85 ÷ 6 is between 14 and 15.

Question 9.
220 ÷ 8
between _______ and _______

between 20 and 30

Explanation:
20 x 8 = 160; 30 x 8 = 240.
The quotient of 220 ÷ 8 is between 20 and 30.

Decide whether the actual quotient is greater than or less than the estimate given. Write < or >.

Question 10.
83 ÷ 8 _______ 10

>

Explanation:
83 ÷ 8 = 10.375 > 10

Question 11.
155 ÷ 4 _______ 40

<

Explanation:
155 ÷ 4 = 38.75 < 40

Question 12.
70 ÷ 6 _______ 11

>

Explanation:
70 ÷ 6 = 11.666 > 11

Question 13.
What’s the Question? A dolphin’s heart beats 688 times in 6 minutes. Answer: about 100 times.
Type below:
__________

About how many times does a dolphin’s heart beats in 1 minute?

Question 14.
Analyze A mother bottlenose ate about 278 pounds of food in one week. About how much food did she eat in a day?

Explanation:
278 ÷ 7
The quotient of 278 ÷ 7 is between 39 and 40.

Question 15.
Tanya has $42 to spend at the Dolphin Island store. T-shirts sell for$7 each and a pair of sunglasses sells for $6. Tanya buys 3 T-shirts. How many pairs of sunglasses can she buy with the amount of money she has left? _____ pairs of sunglasses Answer: 3 pairs of sunglasses Explanation: Given that Tanya has$42 to spend at the Dolphin Island store. T-shirts sell for $7 each and a pair of sunglasses sell for$6.
Tanya buys 3 T-shirts = 3 x $7 =$21.
pair of sunglasses = $42 –$21 = $21. 1 pair of sunglasses sells for$6.
So, $21 ÷$7 = 3.
3 pairs of sunglasses can Tanya buy with the amount of money she has left.

### Page No. 200

Question 16.
If a bottlenose dolphin can eat 175 pounds of fish, squid, and shrimp in a week, about how many pounds of food does it eat in a day? Milo says the answer is about 20 pounds. Leah says the answer is about 30 pounds. Who is correct? Explain.

________

The bottlenose dolphin can eat 25 pounds for 1 day.
Both answers are correct. Because the 25 pounds is in between 20 and 30 pounds.

Explanation:
1 week = 7 days.
The bottlenose dolphin can eat 175 pounds for 7 days.
For 1 day = 175 ÷ 7 = 25 pounds.
The bottlenose dolphin can eat 25 pounds for 1 day.
Both answers are correct. Because the 25 pounds is in between 20 and 30 pounds.

Question 17.
Four families went out for lunch. The total food bill came to $167. The families also left a$30 tip for the waitress. If each family spent the same amount, about how much did each family spend on dinner? Explain how you found your answer.
$______ Answer:$98.5

Explanation:
Four families went out for lunch. The total food bill came to $167. The families also left a$30 tip for the waitress.
So, total amount = $167 +$30 = $197. If each family spent the same amount =$197 ÷ 2 = $98.5 Each family spent$98.5.

Question 18.
There are 6 showings of a film about Van Gogh at the Art Museum. A total of 459 people saw the film. The same number of people were at each showing. About how many people were at each showing? Circle the numbers the quotient is between. Then explain how you found your answer.
40 50 60 70 80
Type below:
_________

40 50 60 70 80
I found multiples of 6 that 459 is between. 70 × 6 = 420 and 80 × 6 = 480. Since 459 is closer to 480, 459 ÷ 6 is about 80.

### Conclusion

Hope the data shared about Go Math Grade 4 Answer Key Chapter 3 Multiply 2- Digit Number has helped you in your preparation. If you feel any learning is missing do give us your suggestions and we will consider them if possible. Just keep on visiting our site to get the latest update on Grade 4 Go Math HMH Answer Keys for other chapters as well.

## Go Math Grade 5 Answer Key Chapter 1 Place Value, Multiplication, and Expressions

It’s really difficult to find Solutions for all the Problems in Go Math Grade 5 Chapter 1 all in one place. Now, you will no longer have such difficulties. We are providing Go Math Grade 5 Answer Key Chapter 1 Place Value, Multiplication, and Expressions. Check out Step by Step Solutions provided for various lessons and the topics in it. Practice using the 3rd Grade Go Math Answer Key Ch 1 Place Value, Multiplication, and Expressions and score better grades in the exam.

Lesson 1: Investigate • Place Value and Patterns

Lesson 2: Place Value of Whole Numbers

Lesson 3: Algebra • Properties

Lesson 4: Algebra • Powers of 10 and Exponents

Lesson 5: Algebra • Multiplication Patterns

Mid-Chapter Checkpoint

Lesson 6: Multiply by 1-Digit Numbers

Lesson 7: Multiply by Multi-Digit Numbers

Lesson 8: Relate Multiplication to Division

Lesson 9: Problem Solving • Multiplication and Division

Lesson 10: Algebra • Numerical Expressions

Lesson 11: Algebra • Evaluate Numerical Expressions

Lesson 12: Algebra • Grouping Symbols

Review/Test

### Place Value and Patterns – Share and Show – Page No. 7

Complete the sentence.

Question 1.
500 is 10 times as much as ______

50

Explanation:
Let the unknown number is S.
500 = 10S
S = 500/10 = 50.
500 is 10 times as much as 50.

Question 2.
20,000 is $$\frac{1}{10}$$ of ______

2,00,000

Explanation:
Let the unknown number is S.
20,000 = $$\frac{1}{10}$$ S
S = 20,000 X 10 = 2,00,000

Question 3.
900 is $$\frac{1}{10}$$ of ______

9,000

Explanation:
Let the unknown number is S.
900 = $$\frac{1}{10}$$ S
S = 900 X 10 = 9,000

Question 4.
600 is 10 times as much as ______

60

Explanation:
Let the unknown number is S.
600 = 10S
S = 600/10 = 60.

Use place-value patterns to complete the table

Question 5.

 Numbers 10 times as much as $$\frac{1}{10}$$ of 10 ______ ______ 3,000 ______ ______ 800 ______ ______ 50 ______ ______

 Numbers 10 times as much as $$\frac{1}{10}$$ of 10 ___1___ ___100___ 3,000 ___300___ ___30,000___ 800 ___80___ ___8,000___ 50 ___5___ ___500___

Explanation:
1. 10 is 10 times as much as ______
Let the unknown number is S.
10 = 10S
S = 10/10 = 1.
10 is 10 times as much as 1.
10 is $$\frac{1}{10}$$ of ______
Let the unknown number is S.
10 = $$\frac{1}{10}$$ S
S = 10 X 10 = 100.
2. 3,000 is 10 times as much as ______
Let the unknown number is S.
3,000 = 10S
S = 3,000/10 = 300.
3,000 is 10 times as much as 300.
3,000 is $$\frac{1}{10}$$ of ______
Let the unknown number is S.
3,000 = $$\frac{1}{10}$$ S
S = 3,000 X 10 = 30,000.
3. 800 is 10 times as much as ______
Let the unknown number is S.
800 = 10S
S = 800/10 = 80.
800 is 10 times as much as 80.
800 is $$\frac{1}{10}$$ of ______
Let the unknown number is S.
800 = $$\frac{1}{10}$$ S
S = 800 X 10 = 8,000.
4. 50 is 10 times as much as ______
Let the unknown number is S.
50 = 10S
S = 50/10 = 5.
50 is 10 times as much as 5.
50 is $$\frac{1}{10}$$ of ______
Let the unknown number is S.
50 = $$\frac{1}{10}$$ S
S = 50 X 10 = 500.

Question 6.

 Numbers 10 times as much as $$\frac{1}{10}$$ of 400 ______ ______ 90 ______ ______ 6,000 ______ ______ 200 ______ ______

 Numbers 10 times as much as $$\frac{1}{10}$$ of 400 __40____ ___4,000___ 90 ___9___ ___900___ 6,000 __600____ __60,000____ 200 ___20___ ___2,000___

Explanation:
1. 400 is 10 times as much as ______
Let the unknown number is S.
400 = 10S
S = 400/10 = 40.
400 is 10 times as much as 40.
400 is $$\frac{1}{10}$$ of ______
Let the unknown number is S.
400 = $$\frac{1}{10}$$ S
S = 400 X 10 = 4,000.
2. 90 is 10 times as much as ______
Let the unknown number is S.
90 = 10S
S = 90/10 = 9.
90 is 10 times as much as 9.
90 is $$\frac{1}{10}$$ of ______
Let the unknown number is S.
90 = $$\frac{1}{10}$$ S
S = 90 X 10 = 900.
3. 6,000 is 10 times as much as ______
Let the unknown number is S.
6,000 = 10S
S = 6,000/10 = 600.
6,000 is 10 times as much as 600.
6,000 is $$\frac{1}{10}$$ of ______
Let the unknown number is S.
6,000 = $$\frac{1}{10}$$ S
S = 6,000 X 10 = 60,000.
4. 200 is 10 times as much as ______
Let the unknown number is S.
200 = 10S
S = 200/10 = 20.
200 is 10 times as much as 20.
200 is $$\frac{1}{10}$$ of ______
Let the unknown number is S.
200 = $$\frac{1}{10}$$ S
S = 200 X 10 = 2,000.

Complete the sentence with 100 or 1,000.

Question 13.
200 is ______ times as much as 2

200 is 100 times as much as 2

Explanation:
Let the unknown number is S.
200 = 2S
S = 200/2 = 100

Question 14.
4,000 is ______ times as much as 4

4,000 is 1000 times as much as 4

Explanation:
Let the unknown number is S.
4,000 = 2S
S = 4,000/2 = 1,000

Question 15.
700,000 is ______ times as much as 700

700,000 is 1,000 times as much as 700

Explanation:
Let the unknown number is S.
700,000 = 700S
S = 700,000/700 = 1,000

Question 16.
600 is ______ times as much as 6

600 is 100 times as much as 6

Explanation:
Let the unknown number is S.
600 = 6S
S = 600/6= 100

Question 17.
50,000 is ______ times as much as 500

50,000 is 100_ times as much as 500

Explanation:
Let the unknown number is S.
50,000 = 500S
S = 50,000/500= 100

Question 18.
30,000 is ______ times as much as 30

30,000 is 1,000 times as much as 30

Explanation:
Let the unknown number is S.
30,000 = 30S
S = 30,000/30 = 1,000

Question 19.
Explain how you can use place-value patterns to describe how 50 and 5,000 compare.
Type below:
__________

5,000 is 100 times as much as 50

Explanation:
5,000/50 = 100

### Place Value and Patterns – Problem Solving – Page No. 8

Sense or Nonsense?

Question 20.
Mark and Robyn used base-ten blocks to show that 300 is 100 times as much as 3. Whose model makes sense? Whose model is nonsense? Explain your reasoning.

Type below:
__________

Robyn’s model makes sense. Because the given data 300 is 100 times as much as 3. It clearly states that there are 100 model-blocks and one model blocks should take to solve the problem.

Question 20.
Explain how you would help Mark understand why he should have used small cubes instead of longs.
Type below:
__________

Mark’s drew 100 model-blocks and 10 model-blocks which. To get 300 is 100 times as much as 3, he needs to do 300/3 = 100 model blocks.

### Place Value of Whole Numbers – Share and Show – Page No. 11

Complete the place-value chart to find the value of each digit.

Question 1.

Type below:
__________

7,333,820

Explanation:
7 x 1,000,000 = 7,000,000
3 x 100,000 = 300,000
3 x 1,000 = 3000
8 x 100 = 800
2 x 10 = 20

Write the value of the underlined digit.

Question 2.
1,574,833
__________

4,000

Explanation:
(1 x 1,000,000) + (5 x 1,00,000) + (7 x 10,000) + (4 x 1,000) + (8 x 100) + (3 x 10) + (3 x 1)
4 x 1,000 = 4 thousands = 4,000

Question 3.
598,102
__________

100

Explanation:
(5 x 1,00,000) + (9 x 10,000) + (8 x 1,000) + (1 x 100) + (0 x 10) + (2 x 1)
1 x 100 = 4 hundreds = 100

Question 4.
7,093,455
__________

90,000

Explanation:
(7 x 1,000,000) + (0 x 1,00,000) + (9 x 10,000) + (3 x 1,000) + (4 x 100) + (5 x 10) + (5 x 1)
9 x 10,000 = 9 ten-thousands = 90,000

Question 5.
301,256,878
__________

3,00,000,000

Explanation:
(3 x 1,00,000,000) + (0 x 10,000,000) + (1 x 1,000,000) + (2 x 1,00,000) + (5 x 10,000) + (6 x 1,000) + (8 x 100) + (7 x 10) + (8 x 1)
3 x 1,00,000,000 = 3 hundred- millions = 3,00,000,000

Write the number in two other forms.

Question 6.
(8 × 100,000) + (4 × 1,000) + (6 × 1) =
__________

80,4006
Eight Hundred Four Thousand Six

Explanation:
(8 × 100,000) + (4 × 1,000) + (6 × 1) = 800,000 + 4,000 + 6 = 80,4006

Question 7.
seven million, twenty thousand, thirty-two
__________

7,020,032
Seven Million Twenty Thousand Thirty-Two

Explanation:
seven million = 7,000,000
twenty thousand = 20,000
thirty-two = 32

Write the value of the underlined digit.

Question 8.
849,567,043
__________

40,000,000

Explanation:
(8 x 1,00,000,000) + (4 x 10,000,000) + (9 x 1,000,000) + (5 x 1,00,000) + (6 x 10,000) + (7 x 1,000) + (0 x 100) + (4 x 10) + (3 x 1)
4 x 10,000,000 = 4 ten- millions = 40,000,000

Question 9.
9,422,850
__________

4,00,000

Explanation:
(9 x 1,000,000) + (4 x 1,00,000) + (2 x 10,000) + (2 x 1,000) + (8 x 100) + (5 x 10) + (0 x 1)
4 x 1,00,000 = 4 Hundred Thousand = 4,00,000

Question 10.
96,283
__________

90,000

Explanation:
(9 x 10,000) + (6 x 1,000) + (2 x 100) + (8 x 10) + (3 x 1)
9 x 10,000 = 9 ten-thousands = 90,000

Question 11.
498,354,021
__________

4,00,000,000

Explanation:
(4 x 1,00,000,000) + (9 x 10,000,000) + (8 x 1,000,000) + (3 x 1,00,000) + (5 x 10,000) + (4 x 1,000) + (0 x 100) + (2 x 10) + (1 x 1)
4 x 1,00,000,000 = Four Hundred Million = 4,00,000,000

Question 12.
791,350
__________

300

Explanation:
(7 x 1,00,000) + (9 x 10,000) + (1 x 1,000) + (3 x 100) + (5 x 10) + (0 x 1)
3 x 100 = 3 hundred = 300

Question 13.
27,911,534
__________

7,000,000

Explanation:
(2 x 10,000,000) + (7 x 1,000,000) + (9 x 1,00,000) + (1 x 10,000) + (1 x 1,000) + (5 x 100) + (3 x 10) + (4 x 1)
7 x 1,000,000 = Seven Million = 7,000,000

Question 14.
105,980,774
__________

80,000

Explanation:
(1 x 1,00,000,000) + (0 x 10,000,000) + (5 x 1,000,000) + (9 x 1,00,000) + (8 x 10,000) + (0 x 1,000) + (7 x 100) + (7 x 10) + (4 x 1)
8 x 10,000 = 8 ten-thousand = 80,000

Question 15.
8,265,178
__________

5,000

Explanation:
(8 x 1,000,000) + (2 x 1,00,000) + (6 x 10,000) + (5 x 1,000) + (1 x 100) + (7 x 10) + (8 x 1)
5 x 1,000 = 5 one-thousand = 5,000

Write the number in two other forms.

Question 16.
345,000
Type below:
__________

Three Hundred Forty-Five Thousand
(3 x 1,00,000) + (4 x 10,000) + (5 x 1,000) + (0 x 100) + (0 x 10) + (0 x 1)

Question 17.
119,000,003
Type below:
__________

One Hundred Nineteen Million Three
(1 x 100,000,000) + (1 x 10,000,000) + (9 x 1,000,000) + (0 x 1,00,000) + (0 x 10,000) + (0 x 1,000) + (0 x 100) + (0 x 10) + (3 x 1)

### Place Value of Whole Numbers – Problem Solving – Page No. 12

Use the table for 18–19.

Question 18.
Which planet is about 10 times as far as Earth is from the Sun?

__________

Saturn

Explanation:
Saturn = 1,427,000/10 = 142,700 which is 10 times as far as Earth

Question 19.
Which planet is about $$\frac{1}{10}$$ of the distance Uranus is from the Sun?
__________

Mars

Explanation:
Mars = 227,900
$$\frac{1}{10}$$ x 2,871,000 = 287,100
Which planet is about $$\frac{1}{10}$$ of the distance Uranus is from the Sun

Question 20.
What’s the Error? Matt wrote the number four million, three hundred five thousand, seven hundred sixty-two as 4,350,762. Describe and correct his error.
Type below:
__________

Matt switched 2 digits in the thousands period: 4,305,762

Question 21.
Explain how you know that the values of the digit 5 in the numbers 150,000 and 100,500 are not the same.
Type below:
__________

In 150,000, the digit 5 is in the ten-thousands place, So, its value is 50,000; in 100,500, the digit 5 is in the hundreds place. So, its value is 500.

Question 22.
Test Prep In the number 869,653,214, which describes how the digit 6 in the ten-millions place compares to the digit 6 in the hundred-thousands place?
Options:
A. 10 times as much as
B. 100 times as much as
C. 1,000 times as much as
D.$$\frac{1}{10}$$ of

B. 100 times as much as

Explanation:
869,653,214
(8 x 100,000,000) + (6 x 10,000,000) + (9 x 1,000,000) + (6 x 1,00,000) + (5 x 10,000) + (3 x 1,000) + (2 x 100) + (1 x 10) + (4 x 1)
6 x 10,000,000 = 60,000,000
6 x 1,00,000 = 6,00,000
60,000,000/6,00,000 = 100

### Properties – Share and Show – Page No. 15

Use properties to find 4 × 23 × 25.

Question 1.
23 × × 25 ________ Property of Multiplication
23 × ( × ) ________ Property of Multiplication
23 ×
__________
____

23 x 4 x 25; Commutative Property of Multiplication
23 x (4 x 25); Associative Property of Multiplication
23 x 100
2,300

Use properties to find the sum or product.

Question 2.
89 + 27 + 11 = ____

89 + (27 + 11); Associative Property of Addition
89 + 38
127

Question 3.
9 × 52 = ____

468

Explanation:
9 x 52
Write 52 = (50 + 2)
9 x (50 + 2)
(9 x 50) + (9 x 2); Distributive Property of Multiplication
450 + 18
468

Question 4.
107 + 0 + 39 + 13 = ____

107 + 0 + 39 + 13
(107 + 0) + (39 + 13); Associative Property of Addition
107 + 0 = 107; Identity Property of Addition
107 + 52 = 159

Complete the equation, and tell which property you used.

Question 5.
9 × (30 + 7) = (9 × ____) + (9 × 7)

9 × (30 + 7) = (9 ×30) + (9 × 7)
Distributive Property of Multiplication

Explanation:
9 x (30 + 7)
(9 x 30) + (9 x 7); Distributive Property of Multiplication
270 + 63 = 333

Question 6.
0 + ____ = 47

Explanation:
0 + 47 = 47; Identity Property of Addition

Question 6.
Describe how you can use properties to solve problems more easily.
Type below:
__________

Using Properties of Addition and Properties of Multiplication, we can solve problems more easily. Simplifying problems is easy with the properties.

Practice: Copy and Solve Use properties to find the sum or product.

Question 7.
3 × 78 = ____

234, Associative Property of Multiplication

Explanation:
Write 78 as 6 x 13
3 x 6 x 13
(3 x 6) x 13; Associative Property of Multiplication
18 x 13 = 234

Question 8.
4 × 60 × 5 = ____

1,200; Associative Property of Multiplication

Explanation:
4 x 60 x 5
4 x (60 x 5); Associative Property of Multiplication
4 x 300 = 1,200

Question 9.
21 + 25 + 39 + 5 = ____

Explanation:
(21 + 25) + (39 + 5); Associative Property of Addition
46 + 44 = 90

Complete the equation, and tell which property you used.

Question 10.
11 + (19 + 6) = (11 + ____) + 6

11 + (19 + 6) = (11 + 19) + 6; Associative Property of Addition

Question 11.
25 + 14 = ____ + 25

25 + 14 = 14 + 25; Commutative Property of Addition

Question 12.
Show how you can use the Distributive Property to rewrite and find (32 × 6) + (32 × 4).
____

(32 × 6) + (32 × 4) = 32 x (6 + 4); Distributive Property

### Properties – Problem Solving – Page No. 16

Question 13.
Three friends’ meals at a restaurant cost $13,$14, and $11. Use parentheses to write two different expressions to show how much the friends spent in all. Which property does your pair of expressions demonstrate?$ ____

$38; Associative Law of Addition Explanation: Three friends’ meals at a restaurant cost$13, $14, and$11.
Friends spent in all = $13 +$14 + $11$13 + ($14 +$11) = ($13 +$14) + $11 Associative Law of Addition Question 14. Jacob is designing an aquarium for a doctor’s office. He plans to buy 6 red blond guppies, 1 blue neon guppy, and 1 yellow guppy. The table shows the price list for the guppies. How much will the guppies for the aquarium cost?$ ____

$162 Explanation: Jacob is designing an aquarium for a doctor’s office. He plans to buy 6 red blond guppies, 1 blue neon guppy, and 1 yellow guppy. (6 x$22) + (1 x $11) + (1 x$19) = $132 +$11 + $19 =$162

Question 15.
Sylvia bought 8 tickets to a concert. Each ticket costs $18. To find the total cost in dollars, she added the product 8 × 10 to the product 8 × 8, for a total of 144. Which property did Sylvia use? i. Distributive Property ii. Associative Property Answer: i. Distributive Property Explanation: Sylvia bought 8 tickets to a concert. Each ticket costs$18.
To find the total cost in dollars = 8 x $18 Using Distributive Property (8 × 10) + (8 × 8) = 8 x (10 + 8) = 144. Question 16. Sense or Nonsense? Julie wrote (15 – 6) – 3 = 15 – (6 – 3). Is Julie’s equation sense or nonsense? Do you think the Associative Property works for subtraction? Explain. __________ Answer: Nonsense; (15 – 6) – 3 = 9 – 3 = 6. 15 – (6 – 3) = 15 – 3 = 12 6 not equal to 12. So, Associative Property does not work for subtraction Question 17. Test Prep Canoes rent for$29 per day. Which expression can be used to find the cost in dollars of renting 6 canoes for a day?
Options:
A. (6 + 20) + (6 + 9)
B. (6 × 20) + (6 × 9)
C. (6 + 20) × (6 + 9)
D. (6 × 20) × (6 × 9)

B. (6 × 20) + (6 × 9)

Explanation:
Canoes rent for $29 per day. For renting 6 canoes for a day, 6 x$29
6 x $(20 + 9) = (6 x 20) + (6 x 9) ### Powers of 10 and Exponents – Share and Show – Page No. 18 Write in exponent form and word form. Question 1. 10 × 10 Exponent form: Word form: Type below: __________ Answer: Exponent form: 102 Word form: the second power of ten Explanation: 10 × 10 Base = 10; Exponent = 2; Exponent Form: 102 Word Form: the second power of ten Question 2. 10 × 10 × 10 × 10 Exponent form: Word form: Type below: __________ Answer: Exponent Form: 104 Word Form: the fourth power of ten Explanation: 10 × 10 × 10 × 10 Base = 10; Exponent = 4; Exponent Form: 104 Word Form: the fourth power of ten Find the value. Question 3. 102 = ____ Answer: 100 Explanation: 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; Question 4. 4 × 102 = ____ Answer: 400 Explanation: 4 × 102 = 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 4 x 100 = 400 Question 5. 7 × 102 = ____ Answer: 700 Explanation: 7 × 102 = 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 7 x 100 = 700 ### Powers of 10 and Exponents – On Your Own – Page No. 19 Write in exponent form and word form. Question 6. 10 × 10 × 10 exponent form: word form: Type below: __________ Answer: Exponent form: 103 Word form: the third power of ten Explanation: 10 × 10 × 10 Base = 10; Exponent = 3; Exponent Form: 103 Word Form: the third power of ten Question 7. 10 × 10 × 10 × 10 × 10 exponent form: word form: Type below: __________ Answer: Exponent form: 105 Word form: the fifth power of ten Explanation: 10 × 10 × 10 × 10 × 10 Base = 10; Exponent = 5; Exponent Form: 105 Word Form: the fifth power of ten Find the value. Question 8. 104 = ____ Answer: 10,000 Explanation: 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; Question 9. 2 × 103 = ____ Answer: 2,000 Explanation: 2 × 103 = 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 2 x 1,000 = 2,000 Question 10. 6 × 104 = ____ Answer: 60,000 Explanation: 6 × 104 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; 6 x 10,000 = 60,000 Complete the pattern. Question 11. 7 × 100 = 7 × 1 = _______ 7 × 101 = 7 × 10 = _______ 7 × 102 = 7 × 10 × 10 = _______ 7 × 103 = 7 × 10 × 10 × 10 = _______ 7 × 104 = 7 × 10 × 10 × 10 × 10 = _______ Answer: 7 × 100 = 7 × 1 = 7 7 × 101 = 7 × 10 = 70 7 × 102 = 7 × 10 × 10 = 7 x 100 = 700 7 × 103 = 7 × 10 × 10 × 10 = 7 x 1,000 = 7,000 7 × 104 = 7 × 10 × 10 × 10 × 10 = 7 x 10,000 = 70,000 Explanation: 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; Question 12. 9 × 100 = _______ = 9 9 × 101 = _______ = 90 9 × 102 = _______ = 900 9 × 103 = _______ = 9,000 9 × 104 = _______ = 90,000 Answer: 9 × 100 = 9 x 1 = 9 9 × 101 = 9 x 10 = 90 9 × 102 = 9 x 10 x 10 = 900 9 × 103 = 9 x 10 x 10 x 10= 9,000 9 × 104 = 9 x 10 x 10 x 10 x 10 = 90,000 Explanation: 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; Question 13. 12 × 100 = 12 × 1 = _______ 12 × 101 = 12 × 10 = _______ 12 × 102 = 12 × 10 × 10 = _______ 12 × 103 = 12 × 10 × 10 × 10 _______ 12 × 104 = 12 × 10 × 10 × 10 × 10 _______ Answer: 12 × 100 = 12 × 1 = 12 12 × 101 = 12 × 10 = 120 12 × 102 = 12 × 10 × 10 = 1,200 12 × 103 = 12 × 10 × 10 × 10 = 12,000 12 × 104 = 12 × 10 × 10 × 10 × 10 = 120,000 Explanation: 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; Question 14. 103 = 10 × 10n What is the value of n? Think: 103 = 10 × () × (), or 10 × () The value of n is …….. n = ______ Answer: 2 Explanation: 103 = 10 × 10n 103 = 10 x 10 x 10 = 10 x 102 The value of n is 2 Question 15. Explain how to write 50,000 using exponents. Type below: __________ Answer: 5 x 104 Explanation: 5 x 10,000 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; 10,000 = 10 x 10 x 10 x 10 = 104 5 x 104 ### Powers of 10 and Exponents – UNLOCK the Problem – Page No. 20 Question 16. Lake Superior is the largest of the Great Lakes. It covers a surface area of about 30,000 square miles. How can you show the estimated area of Lake Superior as a whole number multiplied by a power of ten? a. What are you asked to find? Options: A. 3 × 102 sq mi B. 3 × 103 sq mi C. 3 × 104 sq mi D. 3 × 105 sq mi Answer: C. 3 × 104 sq mi Explanation: Lake Superior is the largest of the Great Lakes. It covers a surface area of about 30,000 square miles. 3 x 10,000 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; 10,000 = 10 x 10 x 10 x 10 = 104 3 x 104 Question 16. b. How can you use a pattern to find the answer? Type below: __________ Answer: 3 x 10,000 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; 10,000 = 10 x 10 x 10 x 10 = 104 3 x 104 Question 16. c. Write a pattern using the whole number 3 and powers of ten. 3 × 101 = 3 × 10 = 3 × 102 = = 3 × 103 = = 3 × 104 = = Type below: __________ Answer: 3 × 101 = 3 × 10 = 3 × 102 = 3 x 10 x 10 = 300 3 × 103 = 3 x 10 x 10 x 10 = 3,000 3 × 104 = 3 x 10 x 10 x 10 x 10 = 30,000 Question 16. d. Fill in the correct answer choice above. Type below: __________ Answer: 3 × 104 = 3 x 10 x 10 x 10 x 10 = 30,000 Question 17. The Earth’s diameter through the equator is about 8,000 miles. What is the Earth’s estimated diameter written as a whole number multiplied by a power of ten? Options: A. 8 × 101 miles B. 8 × 102 miles C. 8 × 103 miles D. 8 × 104 miles Answer: C. 8 × 103 miles Explanation: The Earth’s diameter through the equator is about 8,000 miles. 8 x 1,000 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 1,000 = 10 x 10 x 10 8 x 1,000 = 8 x 103 Question 18. The Earth’s circumference around the equator is about 25 × 103 miles. What is the Earth’s estimated circumference written as a whole number? Options: A. 250,000 miles B. 25,000 miles C. 2,500 miles D. 250 miles Answer: B. 25,000 miles Explanation: The Earth’s circumference around the equator is about 25 × 103 miles. 25 × 103 miles; 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 25 x 1,000 = 25,000 miles ### Multiplication Patterns – Share and Show – Page No. 22 Use mental math and a pattern to find the product. Question 1. • What basic fact can you use to help you find 30×4,000? 30 × 4,000 = ____ Answer: 3 x 4 = 12 Explanation: 30 × 4,000 The basic fact is 3 x 4 = 12 Use mental math to complete the pattern. Question 2. 1 × 1 = 1 1 × 101 = _______ 1 × 102 = _______ 1 × 103 = _______ Answer: 1 × 1 = 1 1 × 101 = 10 1 × 102 = 100 1 × 103 = 1,000 Explanation: 1 × 1 = 1 1 × 101 = 1 x 10 = 10 1 × 102 = 1 x 10 x 10 = 100 1 × 103 = 1 x 10 x 10 x 10 = 1,000 Question 3. 7 × 8 = 56 (7 × 8) × 101 = _______ (7 × 8) × 102 = _______ (7 × 8) × 106 = _______ Answer: 7 × 8 = 56 (7 × 8) × 101 = 560 (7 × 8) × 102 = 5,600 (7 × 8) × 106 = 56,000,000 Explanation: 7 × 8 = 56 (7 × 8) × 101 = 56 x 10 = 560 (7 × 8) × 102 = 56 x 10 x 10 = 5,600 (7 × 8) × 106 = 56 x 10 x 10 x 10 x 10 x 10 x 10 = 56,000,000 Question 4. 6 × 5 = _______ 6 × 5 × _______ = 300 6 × 5 × _______ = 3000 6 × 5 × _______ = 30,000 Answer: 6 × 5 = 30 6 × 5 × 101 = 300 6 × 5 × 103 = 3000 6 × 5 × 104 = 30,000 Explanation: 6 × 5 = 30 6 × 5 × 10 = 300 6 × 5 × 10 x 10 x 10 = 3000 6 × 5 × 10 x 10 x 10 x 10 = 30,000 On Your Own Use mental math to complete the pattern. Question 5. 9 × 5 = 45 (9 × 5) × 101 = _______ (9 × 5) × 102 = _______ (9 × 5) × 103 = _______ Answer: 9 × 5 = 45 (9 × 5) × 101 = 450 (9 × 5) × 102 = 4,500 (9 × 5) × 103 = 45,000 Explanation: 9 × 5 = 45 (9 × 5) × 101 = 45 x 10 = 450 (9 × 5) × 102 = 45 x 10 x 10 = 4,500 (9 × 5) × 103 = 45 x 10 x 10 x 10 = 45,000 Question 6. 3 × 7 = 21 (3 × 7) × 101 = _______ (3 × 7) × 102 = _______ (3 × 7) × 103 = _______ Answer: 3 × 7 = 21 (3 × 7) × 101 = 210 (3 × 7) × 102 = 2,100 (3 × 7) × 103 = 21,000 Explanation: 3 × 7 = 21 (3 × 7) × 101 = 21 x 10 = 210 (3 × 7) × 102 = 21 x 10 x 10 = 2,100 (3 × 7) × 103 = 21 x 10 x 10 x 10 = 21,000 Question 7. 5 × 4 = _______ (5 × 4) × _______ = 200 (5 × 4) × _______ = 2,000 (5 × 4) × _______ = 20,000 Answer: 5 × 4 = 20 (5 × 4) × 101 = 200 (5 × 4) × 102 = 2,000 (5 × 4) × 103 = 20,000 Explanation: 5 × 4 = 20 (5 × 4) × 10 = 200 (5 × 4) × 10 x 10 = 2,000 (5 × 4) × 10 x 10 x 10 = 20,000 Question 8. 5 × 7 = _______ (5 × 7) × _______ = 350 (5 × 7) × _______ = 3,500 (5 × 7) × _______ = 35,000 Answer: 5 × 7 = 35 (5 × 7) × 101 = 350 (5 × 7) × 102 = 3,500 (5 × 7) × 103 = 35,000 Explanation: 5 × 7 = 35 (5 × 7) × 10 = 350 (5 × 7) × 10 x 10 = 3,500 (5 × 7) × 10 x 10 x 10 = 35,000 Question 9. 4 × 2 = 8 (4 × 2) × 101 = _______ (4 × 2) × 102 = _______ (4 × 2) × 103 = _______ Answer: 4 × 2 = 8 (4 × 2) × 101 = 80 (4 × 2) × 102 = 800 (4 × 2) × 103 = 8,000 Explanation: 4 × 2 = 8 (4 × 2) × 101 = 8 x 10 = 80 (4 × 2) × 102 = 8 x 10 x 10 = 800 (4 × 2) × 103 = 8 x 10 x 10 x 10 = 8,000 Question 10. 6 × 7 = 42 (6 × 7) × 101 = _______ (6 × 7) × 102 = _______ (6 × 7) × 103 = _______ Answer: 6 × 7 = 42 (6 × 7) × 101 = 420 (6 × 7) × 102 = 4,200 (6 × 7) × 103 = 42,000 Explanation: 6 × 7 = 42 (6 × 7) × 101 = 42 x 10 = 420 (6 × 7) × 102 = 42 x 10 x 10 = 4,200 (6 × 7) × 103 = 42 x 10 x 10 x 10 = 42,000 Use mental math and a pattern to find the product. Question 11. (6 × 6) × 101 = ____ Answer: (6 × 6) × 101 = 360 Explanation: 6 x 6 =36 (6 × 6) × 101 = 36 x 10 = 360 Question 12. (7 × 4) × 103 = ____ Answer: 28,000 Explanation: 7 x 4 = 28 (7 × 4) × 101 = 28 x 10 = 280 (7 × 4) × 102 = 28 x 10 x 10 = 2,800 (7 × 4) × 103 = 28 x 10 x 10 x 10 = 28,000 Question 13. (9 × 8) × 102 = ____ Answer: 7,200 Explanation: (9 × 8) = 72 (9 × 8) × 101 = 72 x 10 = 720 (9 × 8) × 102 = 72 x 10 x 10 = 7,200 Question 14. (4 × 3) × 102 = ____ Answer: 1,200 Explanation: (4 × 3) = 12 (4 × 3) × 101 = 12 x 10 = 120 (4 × 3) × 102 = 12 x 10 x 10 = 1,200 Question 15. (2 × 5) × 103 = ____ Answer: 10,000 Explanation: (2 × 5) = 10 (2 × 5) × 101 = 10 x 10 = 100 (2 × 5) × 102 = 10 x 10 x 10 = 1,000 (2 × 5) × 103 = 10 x 10 x 10 x 10 = 10,000 Question 16. (2 × 8) × 102 = ____ Answer: 1,600 Explanation: (2 × 8) = 16 (2 × 8) × 101 = 16 x 10 = 160 (2 × 8) × 102 = 16 x 10 x 10 = 1,600 Question 17. (6 × 5) × 103 = ____ Answer: 30,000 Explanation: (6 × 5) = 30 (6 × 5) × 101 = 30 x 10 = 300 (6 × 5) × 102 = 30 x 10 x 10 = 3,000 (6 × 5) × 103 = 30 x 10 x 10 x 10 = 30,000 Question 18. (8 × 8) × 104 = ____ Answer: 640,000 Explanation: (8 × 8) = 64 (8 × 8) × 101 = 64 x 10 = 640 (8 × 8) × 102 = 64 x 10 x 10 = 6,400 (8 × 8) × 103 = 64 x 10 x 10 x 10 = 64,000 (8 × 8) × 104 = 64 x 10 x 10 x 10 x 10 = 640,000 Question 19. (7 × 8) × 104 = ____ Answer: 560,000 Explanation: (7 × 8) = 56 (7 × 8) × 101 = 56 x 10 = 560 (7 × 8) × 102 = 56 x 10 x 10 = 5,600 (7 × 8) × 103 = 56 x 10 x 10 x 10 = 56,000 (7 × 8) × 104 = 56 x 10 x 10 x 10 x 10 = 560,000 ### Multiplication Patterns – Share and Show – Page No. 23 Use mental math to complete the table. Question 20. 1 roll = 50 dimes ; Think:50 dimes per roll × 20 rolls =(5 × 2) × (10 × 10) Type below: __________ Answer: Explanation: 1 roll = 50 dimes ; Think:50 dimes per roll × 20 rolls =(5 × 2) × (10 × 10) = 10 x 102 50 dimes per roll × 30 rolls = (5 x 3) x (10 × 10) = 15 x 102 50 dimes per roll × 40 rolls = (5 x 4) x (20 × 10) = 20 x 102 50 dimes per roll × 50 rolls = (5 x 5) x (10 × 10) = 25 x 102 50 dimes per roll × 60 rolls = (5 x 6) x (10 × 10) = 30 x 102 50 dimes per roll × 70 rolls = (5 x 7) x (10 × 10) = 35 x 102 50 dimes per roll × 80 rolls = (5 x 8) x (10 × 10) = 40 x 102 50 dimes per roll × 90 rolls = (5 x 9) x (10 × 10) = 45 x 102 50 dimes per roll × 100 rolls = (5 x 10) x (10 × 10) = 50 x 102 Question 21. 1 roll = 40 quarters ; Think:40 quarters per roll × 20 rolls =(4 × 2) × (10 × 10) Type below: __________ Answer: Explanation: 1 roll = 40 quarters ; Think:40 quarters per roll × 20 rolls =(4 × 2) × (10 × 10) = 8 x 102 40 quarters per roll × 30 rolls =(4 × 3) × (10 × 10) = 12 x 102 40 quarters per roll × 40 rolls =(4 × 4) × (10 × 10) = 16 x 102 40 quarters per roll × 50 rolls =(4 × 5) × (10 × 10) = 20 x 102 40 quarters per roll × 60 rolls =(4 × 6) × (10 × 10) = 24 x 102 40 quarters per roll × 70 rolls =(4 × 7) × (10 × 10) = 28 x 102 40 quarters per roll × 80 rolls =(4 × 8) × (10 × 10) = 32 x 102 40 quarters per roll × 90 rolls =(4 × 9) × (10 × 10) = 36 x 102 40 quarters per roll × 100 rolls =(4 × 10) × (10 × 10) = 40 x 102 Question 22. Type below: __________ Answer: Explanation: 80 x 800 = 64 x 103 80 x 6 = (8 x 6) x 10 = 48 x 101 80 x 70 = (8 x 7) x (10 x 10) = 56 x 102 80 x 9,000 = (8 x 9) x (10 x 10 x 10 x 10) = 64 x 104 Question 23. Type below: __________ Answer: Explanation: Given that 90 x 9,000 = (9 x 9) x 10 x 10 x 10 x 10 = 81 x 104 90 x 6 = (9 x 6) x 10 = 54 x 101 90 x 70 = (9 x 7) x (10 x 10) = 63 x 102 90 x 800 = (9 x 8) x (10 x 10 x 10) = 72 x 103 Problem Solving Use the table for 24–26. Question 24. What if you magnified the image of a cluster fly by 9 × 103 ? What would the length appear to be? ____ mm Answer: 9,000 mm Explanation: 9 × 103 = 9 x 10 x 10 x 10 = 9,000 Question 25. If you magnified the image of a fire ant by 4 × 103 and a tree hopper by 3 × 103 , which insect would appear longer? How much longer? ____ mm Answer: 103 mm Explanation: fire ant: 4 × 103 = 4 x 10 x 10 x 10 = 4,000 mm tree hopper: 3 × 103 = 3 x 10 x 10 x 10 = 3,000 mm 4,000 > 3,000. So, fire ant appears to be longer. 4,000 – 3,000 = 1,000 = 103 Question 26. John wants to magnify the image of a fire ant and a crab spider so they appear to be the same length. How many times their actual sizes would he need to magnify each image? Fire ant by _______ times Crab spider by ______ times Answer: Fire ant by 5 times Crab spider by 4 times Explanation: Given that Fire ant = 4 mm crab spider = 5 mm So, to make them have the same lengths, multiply fire ant by 5 mm and multiply Crab spider by 4 mm ### Multiplication Patterns – Share and Show – Page No. 24 Question 27. What does the product of any whole-number factor multiplied by 100 always have? Explain. Type below: __________ Answer: The product of any whole number factor multiplied by 100 has two digits which are 0 in ones and tens place. Example: 2 x 100 = 200 Question 28. Test Prep How many zeros are in the product (5 × 4) × 104? Options: A. 3 B. 4 C. 5 D. 6 Answer: C. 5 Explanation: (5 × 4) × 104 = 20 x 104 = 2 x 105 5 zeroes Use patterns and mental math to solve. Question 29. A human body has about 30 times as many platelets as white blood cells. A small sample of blood has 8×103 white blood cells. About how many platelets are in the sample? ______ platelets Answer: 24 x 104 platelets Explanation: Let the number of platelets = s. s = 30 x 8×103 s = 30 x 8 x 10 x 10 x 10 = (3 x 8) x (10 x 10 x 10 x 10) = 24 x 104 Question 30. Basophils and monocytes are types of white blood cells. A blood sample has about 5 times as many monocytes as basophils. If there are 60 basophils in the sample, about how many monocytes are there? ______ monocytes Answer: 3 x 102 monocytes Explanation: Let the number of monocytes = S S = 5 x 60 = 300 = 3 x 100 S = 3 x 102 Question 31. Lymphocytes and eosinophils are types of white blood cells. A blood sample has about 10 times as many lymphocytes as eosinophils. If there are 2 × 102 eosinophils in the sample, about how many lymphocytes are there? ______ lymphocytes Answer: 2 × 103 lymphocytes Explanation: Lymphocytes and eosinophils are types of white blood cells. A blood sample has about 10 times as many lymphocytes as eosinophils. There are 2 × 102 eosinophils in the sample Then, Lymphocytes = 10 x 2 × 102 eosinophils = 2 × 103 Question 32. An average person has 6 × 102 times as many red bloods cells as white blood cells. A small sample of blood has 7 × 103 white blood cells. About how many red blood cells are in the sample? ______ red blood cells Answer: 42 x 10 red blood cells Explanation: Let the red blood cells = S S = 7 × 103 x 6 × 102 S = 42 x 10 ### Mid-Chapter Checkpoint – Vocabulary – Page No. 25 Choose the best term for the box. Question 1. A group of three digits separated by commas in a multidigit number is a __ ________ Answer: Period Question 2. An __ is the number that tells how many times a base is used as a factor ________ Answer: exponent Concepts and Skills Complete the sentence. Question 3. 7 is $$\frac{1}{10}$$ of ______ Answer: 70 Explanation: Let the unknown number is S. 7 = $$\frac{1}{10}$$ S S = 7 X 10 = 70 Question 4. 800 is 10 times as much as ______ Answer: 80 Explanation: Let the unknown number is S. 800 = 10S S = 800/10 = 80. Write the value of the underlined digit. Question 5. 6,581,678 ________ Answer: 80,000 Explanation: (6 x 1,000,000) + (5 x 1,00,000) + (8 x 10,000) + (1 x 1,000) + (6 x 100) + (7 x 10) + (8 x 1) 8 x 10,000 = 80,000 Question 6. 25,634 ________ Answer: 600 Explanation: (2 x 10,000) + (5 x 1,000) + (6 x 100) + (3 x 10) + (4 x 1) 6 x 100 = 600 Question 7. 34,634,803 ________ Answer: 4,000,000 Explanation: (3 x 10,000,000) + (4 x 1,000,000) + (6 x 1,00,000) + (3 x 10,000) + (4 x 1,000) + (8 x 100) + (0 x 10) + (3 x 1) 4 x 1,000,000 = 4,000,000 Question 8. 2,764,835 ________ Answer: 700,000 Explanation: (2 x 1,000,000) + (7 x 1,00,000) + (6 x 10,000) + (4 x 1,000) + (8 x 100) + (3 x 10) + (5 x 1) 7 x 1,00,000 = 700,000 Complete the equation, and tell which property you used. Question 9. 8 × (14 + 7) = ________ + (8 × 7) Answer: 8 × (14 + 7) = (8 x 14) + (8 × 7); Distributive Property of Multiplication Explanation: 8 × (14 + 7) (8 x 14) + (8 × 7); Distributive Property of Multiplication Question 10. 7 + (8 + 12) = ________ + 12 Answer: 7 + (8 + 12) = (7 + 8) + 12 Associative Property of Addition Find the value. Question 11. 103 = ______ Answer: 1,000 Explanation: 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; Question 12. 6 × 102 = ______ Answer: 600 Explanation: 6 × 102 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 6 x 100 = 600 Question 13. 4 × 104 = ______ Answer: 40,000 Explanation: 4 × 104 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 104 = 10 x 10 x 10 x 10 = 10,000; 4 x 10,000 = 40,000 Use mental math and a pattern to find the product. Question 14. 70 × 300 = ______ Answer: 21,000 Explanation: 70 × 300 = (7 x 3) x (10 x 10 x 10) = 21 x 1,000 = 21,000 Question 15. (3 × 4) × 103 = ______ Answer: 12,000 Explanation: (3 × 4) × 103 100 = 1; 101 = 1 x 10 = 10; 102 = 10 x 10 = 100; 103 = 10 x 10 x 10 = 1,000; 12 x 1,000 = 12,000 ### Mid-Chapter Checkpoint – Page No. 26 Fill in the bubble completely to show your answer. Question 16. DVDs are on sale for$24 each. Which expression can be used to find the cost in dollars of buying 4 DVDs?
Options:
A. (4 + 20) + (4 + 4)
B. (4 × 20) + (4 × 4)
C (4 + 20) × (4 + 4)
D. (4 × 20) × (4 × 4)

B. (4 × 20) + (4 × 4)

Explanation:
24 can be written as 25 – 1
4 x 24 = 4 x (20 + 4) = (4 x 20) + (4 x 4)

Question 17.
The Muffin Shop chain of bakeries sold 745,305 muffins last year. Which choice shows that number in expanded form?
Options:
A. (7 × 100,000) + (45 × 10,000) + (3 × 100) + (5 × 10)
B. (7 × 100,000) + (4 × 10,000) + (5 × 1,000) + (5 × 10)
C. (7 × 100,000) + (4 × 10,000) + (5 × 1,000) + (3 × 100) + (5 × 1)
D. (7 × 100,000) + (4 × 10,000) + (3 × 100) + (5 × 1)

C. (7 × 100,000) + (4 × 10,000) + (5 × 1,000) + (3 × 100) + (5 × 1)

Explanation:
First, we can write 745,305 as:
700,000 + 40, 000 + 5,000 + 300 + 5
(7 x 100,000) + (4 x 10,000) + (5 x 1,000) + (3 x 100) + 5

Question 18.
The soccer field at Mario’s school has an area of 6,000 square meters. How can Mario show the area as a whole number multiplied by a power of ten?
Options:
A. 6 × 104 sq m
B. 6 × 103 sq m
C. 6 × 102 sq m
D. 6 × 101 sq m

B. 6 × 103 sq m

Explanation:
6,000 square meters = 6 x 1,000 = 6 x 10 x 10 x 10 = 6 × 103 sq m

Question 19.
Ms. Alonzo ordered 4,000 markers for her store. Only $$\frac{1}{10}$$ of them arrived. How many markers did she receive?
Options:
A. 4
B. 40
C. 400
D. 1,400

C. 400

Explanation:
Ms. Alonzo ordered 4,000 markers for her store. Only $$\frac{1}{10}$$ of them arrived.
4,000 x $$\frac{1}{10}$$ = 400

Question 20.
Mark wrote the highest score he made on his new video game as the product of 70 × 6,000. What was his score?
Options:
A. 420
B. 4,200
C. 42,000
D. 420,000

D. 420,000

Explanation:
Mark wrote the highest score he made on his new video game as the product of 70 × 6,000.
(7 x 6) x (10 x 10 x 10 x 10) = 42 x 10,000 = 420,000

### Multiply by 1-digit numbers – Share and Show – Page No. 29

Complete to find the product.

Question 1.
6 × 796           Estimate: 6 × ___ = ___

______

4,776

Explanation:

Estimate. Then find the product.

Question 2.
Estimate: ___
6 0 8
×   8
———-
Estimate: ________
Product: 608 × 8 = ________

Estimate: 6,000
Product: 608 × 8 = 4,864

Explanation:
Estimate: 608 is close to 600; 8 is close to 10
600 x 10 = 6,000
608 x 8
Multiply the ones; 8 x 8 = 64. 4 ones and 6 tens. Write the ones and the
regrouped tens.
Multiply the tens; 0 x 8 = 0 + 6 = 6
Multiply the hundreds; 6 x 8 = 48.
So, 4,864 is the product of 608 × 8
Product: 4,864

Question 3.
Estimate: __
5 5 6
×   4
———–
Estimate: ________
Product: 556 × 4 = ________

Estimate: 2,780
Product: 556 × 4 = 2,224

Explanation:
Estimate: 556 is close to 550; 4 is close to 5
556 x 5 = 2,780
556 × 4
Multiply the ones; 6 x 4 = 24. 4 ones and 2 tens. Write the ones and the
regrouped tens.
Multiply the tens; 5 x 4 = 20 + 2 = 22; 2 tens and 2 hundreds. Write the tens and regrouped hundreds.
Multiply the hundreds; 5 x 4 = 20; 20 + 2 = 22.
So, 2,224 is the product of 556 × 4
Product: 2,224

Question 4.
Estimate:
1,925
×    7
———–
Estimate: ________
Product: 1,925 × 7 = ________

Estimate: 10,000
Product: 1,925 × 7 = 13,475

Explanation:
Estimate: 1,925 is close to 2000; 7 is close to 5
2,000 x 5 = 10,000
1,925 × 7
Multiply the ones; 7 x 5 = 35. 5 ones and 3 tens. Write the ones and the
regrouped tens.
Multiply the tens; 7 x 2 = 14; 14 + 3 = 17; 7 tens and 1 hundreds. Write the tens and regrouped hundreds.
Multiply the hundreds; 7 x 9 = 63; 63 + 1 = 64. 4 hundred and 6 thousand Write the hundreds and regrouped thousands.
Multiply the thousands; 7 x 1 = 7; 7 + 6 = 13
So, 13,475 is the product of 1,925 × 7
Product: 13,475

Estimate. Then find the product.

Question 5.
Estimate:__
7 9 4
×   3
———-
Estimate: ________
Product:794 × 3 = ________

Estimate: 800
Product:794 × 3 = 2,382

Explanation:
Estimate: 794 is close to 800 and 3 is close to 1
800 x 1 = 800
794 x 3 = (700 + 90 + 4) x 3 = (700 x 3) + (90 x 3) + (4 x 3) = 2100 + 270 + 12 = 2,382

Question 6.
Estimate:___
8 2 2
×   6
———-
Estimate: ________
822 × 6 = ________

Estimate: 4,000
822 × 6 = 4,932

Explanation:
Estimate: 822 is close to 800 and 6 is close to 5
800 x 5 = 4,000
822 × 6 = (800 + 20 + 2) x 6 = (800 x 6) + (20 x 6) + (2 x 6) = 4800 + 120 + 12 = 4,932

Question 7.
Estimate:
3,102
×    5
———–
Estimate: ________
Product: 3,102 × 5 = ________

Estimate: 15,500
Product: 3,102 × 5 = 15,510

Explanation:
Estimate: 3,102 is close to 3,100 and 5 is close to 5
3,100 x 5 = 15,500
3,102 x 5 = (3,000 + 100 + 0 + 2) x 5 = (3000 x 5) + (100 x 5) + 0 + (2 x 5) = 15,000 + 500 + 0 + 10 = 15,510

Algebra Solve for the unknown number.

Question 8.
3 9 6
×   6
———
2, 3 6
396 × 6 = 23 ______ 6

7

Explanation:
396 x 6 = (300 + 90 + 6) x 6 = (300 x 6) + (90 x 6) + (6 x 6) = 1800 + 540 + 36 = 2376. So, the unknown number is 7

Question 9.
5,1 2
×   8
——–
16
Type below:
__________

5127 x 8 = 41,016.
Unknown numbers = 7 and 410

Explanation:
5,127 x 8 = (5000 + 100 + 20 + 7) x 8 = (5000 x 8) + (100 x 8) + (20 x 8) + (7 x 8) = 40000 + 800 + 160 + 56 = 41,016

Question 10.
8, 5 6
×    7
———
60,03
Type below:
__________

8,576 x 7 = 60,032

Explanation:
8,576 x 7 = (8000 + 500 + 70 + 6) x 7 = (8000 x 7) + (500 x 7) + (70 x 7) + (6 x 7) = 56000 + 3500 + 490 + 42 = 60,032

Practice: Copy and Solve Estimate. Then find the product.

Question 11.
116 × 3 = _______
Estimate: _______

Estimate: 300
116 × 3 = 348

Explanation:
Estimate: 116 is close to 100;
100 x 3 = 300
116 x 3
6 x 3 =18; add ones and regroup tens
3 x 1 = 3; 3 + 1 = 4
3 x 1 = 3
So, 348 is the product

Question 12.
338 × 4 = _______
Estimate: _______

338 × 4 = 1,352
Estimate: 1,200

Explanation:
Estimate: 338 is close to 300;
300 x 4 = 1,200
338 × 4
8 x 4 =32; add ones and regroup tens
3 x 4 = 12; 12 + 3 = 15; add tens and regroup hundreds
3 x 4 = 12; 12 + 1 = 13
So, 1352 is the product

Question 13.
6 × 219 = _______
Estimate: _______

6 × 219 = 1,314
Estimate: 1200

Explanation:
Estimate: 219 is close to 200
200 x 6 = 1200
6 × 219
6 x 9 = 54; add ones and regroup tens
6 x 1 = 6; 6 + 5 = 11; add tens and regroup hundreds
6 x 2 = 12; 12 + 1 = 13
So, 1,314

Question 14.
7 × 456 = _______
Estimate: _______

7 × 456 = 3192
Estimate: 3500

Explanation:
Estimate: 456 is close to 500
500 x 7 = 3500
7 x 456
7 x 6 = 42; add ones and regroup tens
7 x 5 = 35; 35 + 4 = 39; add tens and regroup hundreds
7 x 4 = 28; 28 + 3 = 31
So, 3192

Question 15.
5 × 1,012 = _______
Estimate: _______

5 × 1,012 = 5,060
Estimate: 5,000

Explanation:
Estimate: 1,012 is close to 1,000
1,000 x 5 = 5,000
5 × 1,012
5 x 2 = 10; add ones and regroup tens
5 x 1 = 5; 5 + 1 = 6; add tens and regroup hundreds
5 x 0 = 0
5 x 1 = 5
So, 5,060

Question 16.
2,921 × 3 = _______
Estimate: _______

2,921 × 3 = 8,763
Estimate: 9,000

Explanation:
Estimate: 2,921 is close to 3,000
3,000 x 3 = 9,000
2,921 × 3
3 x 1 = 3;
3 x 2 = 6;
3 x 9 = 27; add hundreds and regroup thousands
3 x 2 = 6; 6 + 2 = 8
So, 8,763

Question 17.
8,813 × 4 = _______
Estimate: _______

8,813 × 4 = 35,252
Estimate: 3,600

Explanation:
Estimate: 8,813 is close to 9,000
9,000 x 4 = 3,600
8,813 × 4
4 x 3 = 12; add ones and regroup tens
4 x 1 = 4; 4 + 1 = 5;
4 x 8 = 32; add hundreds and regroup thousands
4 x 8 = 32; 32 + 3 = 35
So, 35,252

Question 18.
9 × 3,033 = _______
Estimate: _______

Explanation:
Estimate: 3,033 is close to 3,000
3,000 x 9 = 27,000
9 × 3,033
9 x 3 = 27; add ones and regroup tens
9 x 3 = 27; 27 +  = 11; add tens and regroup hundreds
6 x 2 = 12; 12 + 1 = 13
So, 1,314

### Multiply by 1-digit numbers – Problem Solving – Page No. 30

What’s the Error?

Question 19.
The Plattsville Glee Club is sending 8 of its members to a singing contest in Cincinnati, Ohio. The cost will be $588 per person. How much will it cost for the entire group of 8 students to attend? Both Brian and Jermaine solve the problem. Brian says the answer is$40,074. Jermaine’s answer is $4,604. Estimate the cost. A reasonable estimate is _$ ______

$4,800 Explanation: The Plattsville Glee Club is sending 8 of its members to a singing contest in Cincinnati, Ohio. The cost will be$588 per person.
So, for entire group 8 x $588 =$4,704
Jermaine’s answer is correct. Because the $4,604 is close to$4,704
588 is close to 600. So, 600 x 8 = $4,800 Question 19. Although Jermaine’s answer seems reasonable, neither Brian nor Jermaine solved the problem correctly. Find the errors in Brian’s and Jermaine’s work. Then, solve the problem correctly. What error did Brian make? Explain. Type below: __________ Answer: When Brian multiplied the tens, he wrote the total number of tens in the product instead of regrouping, so the place values of his product are incorrect. Question 19. What error did Jermaine make? Explain. Type below: __________ Answer: Jermaine regrouped the wrong amount of hundreds. He regrouped the tens as 6 hundred instead of 7 hundred.$588 x 8 = \$4,704

Question 19.
How could you predict that Jermaine’s answer might be incorrect using your estimate?
Type below:
__________

I used 600 × 8 to estimate the product; 588 is 12 less than 600. Since 12 × 8 = 96, and 4,604 is almost 200 less than the estimate of 4,800, the answer is probably too low.

### Multiply by 2-digit numbers – Share and Show – Page No. 33

Complete to find the product

Question 1.

Type below:
__________

2,752

Explanation:
64 x 3 = 192
64 x 40 = 2,560
2,560 + 192 = 2,752

Question 2.

Type below:
__________

21,698

Explanation:
571 x 8 = 4,568
571 x 30 = 17,130
17,130 + 4,568 = 21,698

Estimate. Then find the product.

Question 3.
Estimate:____
2 4
× 1 5
———-
Estimate: ________
Product: ________

Estimate: 300
Product: 360

Explanation:
2 4 x 15
Estimate: 20 x 15 = 300
24 x 5 = 120
24 x 10 = 240
Product:: 240 + 120 = 360

Question 4.
Estimate:____
3 7
× 6 3
———-
Estimate: ________
Product: ________

Estimate: 2,400
Product: 2,331

Explanation:
37 x 63
Estimate: 40 x 60 = 2,400
37 x 3 = 111
37 x 60 = 2220
Product:: 2220 + 111 = 2,331

Question 5.
Estimate:____
3 8 4
× 4 5
———-
Estimate: ________
Product: ________

Estimate: 20,000
Product: 17,280

Explanation:
384 x 45
Estimate: 400 x 50 = 20,000
384 x 5 = 1920
384 x 40 = 15,360
Product:: 15,360 + 1920 = 17,280

Estimate. Then find the product.

Question 6.
Estimate:____
2 8
× 2 2
———-
Estimate: ________
Product: ________

Estimate: 600
Product: 616

Explanation:
28 x 22
Estimate: 30 x 20 = 600
28 x 2 = 56
28 x 20 = 560
Product:: 56 + 560 = 616

Question 7.
Estimate:____
9 3
× 7 6
———-
Estimate: ________
Product: ________

Estimate: 7200
Product: 7,068

Explanation:
93 x 76
Estimate: 90 x 80 = 7200
93 x 6 = 558
93 x 70 = 6,510
Product:: 558 + 6,510 = 7,068

Question 8.
Estimate:____
2 9 5
× 5 1
———-
Estimate: ________
Product: ________

Estimate: 15,000
Product: 15,045

Explanation:
295 x 51
Estimate: 300 x 50 = 15,000
295 x 1 = 295
295 x 50 = 14,750
Product:: 295 + 14,750 = 15,045

Practice: Copy and Solve Estimate. Then find the product.

Question 9.
Estimate: ________
54 × 31 = ________

Estimate: 1,500
Product: 1,674

Explanation:
54 x 31
Estimate: 50 x 30 = 1,500
54 x 1 = 54
54 x 30 = 1,620
Product:: 54 + 1,620 = 1,674

Question 10.
Estimate: ________
42 × 26 = ________

Estimate: 1,200
Product: 1,092

Explanation:
42 x 26
Estimate: 40 x 30 = 1,200
42 x 6 = 252
42 x 20 = 840
Product:: 252 + 840 = 1,092

Question 11.
Estimate: ________
38 × 64 = ________

Estimate: 2,400
Product: 2,432

Explanation:
38 × 64
Estimate: 40 x 60 = 2,400
38 x 4 = 152
38 x 60 = 2,280
Product:: 152 + 2,280 = 2,432

Question 12.
Estimate: ________
63 × 16 = ________

Estimate: 1,200
Product: 1,008

Explanation:
63 x 16
Estimate: 60 x 20 = 1,200
63 x 6 = 378
63 x 10 = 630
Product:: 378 + 630 = 1,008

Question 13.
Estimate: ________
204 × 41 = ________

Estimate: 8,000
Product: 8,364

Explanation:
204 × 41
Estimate: 200 x 40 = 8,000
204 x 1 = 204
204 x 40 = 8,160
Product:: 204 + 8,160 = 8,364

Question 14.
Estimate: ________
534 × 25 = ________

Estimate: 15,000
Product: 13,350

Explanation:
534 x 25
Estimate: 500 x 30 = 15,000
534 x 5 = 2,670
534 x 20 = 10,680
Product:: 2,670 + 10,680 = 13,350

Question 15.
Estimate: ________
722 × 39 = ________

Estimate: 28,000
Product: 28,158

Explanation:
722 × 39
Estimate: 700 x 40 = 28,000
722 x 9 = 6,498
722 x 30 = 21,660
Product:: 6,498 + 21,660 = 28,158

Question 16.
Estimate: ________
957 × 43 = ________