## Big Ideas Math Answers Grade K Chapter 12 Identify Three-Dimensional Shapes

Big Ideas Math Book Answers has created a sequence of lessons in all the chapters. Get Big Ideas Math Answers Grade K Chapter 12 Identify Three-Dimensional Shapes on this. This pdf link will make understanding concepts of 3-Dimensional shapes so easy. The 3-D shapes are cone, cylinder, cube, cuboid, etc. So following the Big Ideas Math Answers Grade K Chapter 12 Identify Three-Dimensional Shapes is necessary to get notified of the topics. So it will be easy for you to understand the concepts behind each and every lesson.

## Big Ideas Math Book Grade K Answer Key Chapter 12 Identify Three-Dimensional Shapes

The topics covered in this chapter are Vocabulary, Two- and Three-Dimensional Shapes, Cubes and Spheres etc. So, Download Big Ideas Math Answers Grade K Chapter 12 Identify Three-Dimensional Shapes PDF for free. For more practice questions simply go to the performance task and cumulative practice which is given at the end of the chapter. Just click on the below-attached links and start your preparation from now.

Vocabulary

Lesson: 1 Two- and Three-Dimensional Shapes

Lesson: 2 Describe Three-Dimensional Shapes

Lesson: 3 Cubes and Spheres

Lesson: 4 Cones and Cylinders

Lesson: 5 Build Three-Dimensional Shapes

Lesson: 6 Positions of Solid Shapes

Chapter 12: Identify Three-Dimensional Shapes

### Identify Three-Dimensional Shapes Vocabulary

Directions:
Circle each can. Draw a square around each box. Count and write how many of each two-dimensional shape you draw.

### Lesson 12.1 Two- and Three-Dimensional Shapes

Explore and Grow

Directions:
Circle any triangles, rectangles, squares, hexagons, and circles you see in the picture. Use another color to circle any objects in the picture that match the blue shapes shown. Tell what you notice about each shape.

two-dimensional
rectangle
circle
triangle
hexagon
Total11 shapes
three-dimensional
cylinder
sphere
cube
cone
Total 8 shapes
Explanation:
A two-dimensional shape is a shape that has length and width but no depth. … A circle is one example of a two-dimensional shape. Example Two. A rectangle is another example of a two-dimensional shape.

Triangles, Rectangles, Squares, Hexagons, and Circles all these shapes are all 2-D shapes.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width, and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
All the objects in the picture represent the 3-D shapes so, they are circled with a different color.

Think and Grow

Directions:
Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct.

two-dimensional
rectangle
circle
three-dimensional
cylinder
sphere
cuboid
cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Apply and Grow: Practice

Directions:
1 – 4 Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct.

Question 1.

two-dimensional
rectangle
circle
three-dimensional
cylinder
cube
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangles.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 2.

two-dimensional
circle
square
hexagon
three-dimensional
sphere
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 3.

two-dimensional
Triangle
three-dimensional
Cuboid
Triangle prism
Cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 4.

two-dimensional
0
three-dimensional
Cylinder
Sphere
Cone

Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Think and Grow: Modeling Real Life

Directions:
Circle any shapes in the picture that are solids. Draw rectangles around any shapes in the picture that are flats. Count and write how many solids and flats you find.

three-Dimensional
________
– – – – – – – –
________

two-dimensional
________
– – – – – – – –
________

two-dimensional
Rectangles
Total 7 flat surfaces
three-dimensional
Cubes
Sphere
Cone
Cylinder
Total 8 solids
Explanation:
Solid figures are three-dimensional. A face is a flat surface of a solid.

### Two- and Three-Dimensional Shapes Homework & Practice 12.1

Directions:
1 – 3 Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct.

Question 1.

two-dimensional
Triangle
Square
three-dimensional
Cylinder
Sphere
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 2.

two-dimensional
Hexagon
Circle
three-dimensional
Cube
Cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 3.

two-dimensional
Rectangle
Square
three-dimensional
Cube
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Directions:
4 and 5 Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct. 6 Circle any three-dimensional shapes in the picture. Count and write the number. Draw rectangles around any two-dimensional shapes in the picture. Count and write the number.

Question 4.

two-dimensional
Circle
three-dimensional
Cylinder
Sphere
Cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 5.

two-dimensional
Rectangle
Circle
Triangle
three-dimensional
Cube
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 6.

Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

### Lesson 12.2 Describe Three-Dimensional Shapes

Explore and Grow

Directions:
Cut out the Roll, Stack, Slide Sort Cards. Sort the cards into the categories shown.

rolls

stacks

slides

Think and Grow

Directions:

• Look at the solid shape on the left that rolls. Circle the other solid shapes that roll.
• Look at the solid shapes on the left that stack. Circle the other solid shapes that stack.
• Look at the solid shape on the left that slides. Circle the other solid shapes that slide.

Explanation:
Solid shapes that can roll are circled with Brown.
Solid shapes that can slide are circled with Yellow.
Solid shapes that can stack are circled with Blue.

The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere , cylinder , cone shape.

Apply and Grow: Practice

Directions:
1 Look at the solid shape on the left that rolls. Circle the other solid shapes that roll. 2 Circle the solid shapes that roll and slide. 3 Circle the solid shapes that stack and slide. 4 Circle the solid shape that does not stack or slide.

Question 1.

Given:
The cylinder can roll,roll and slide, stack and slide
The cube can slide and stack.
The sphere can only roll.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 2.

Given:
The cylinder can roll, stack and slide, roll and slide
Cone can roll, roll and slide.
The cube can slide and stack.
Explanation:
Solid shapes that can roll are circled with Brown.
Solid shapes that can slide are circled with Yellow.
Solid shapes that can stack are circled with Blue.

The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 3.

Given:
A ball that represents a Sphere. The ball can only roll.
A wooden log which represent cylinder. Log can roll, roll and slide, stack and slide.
The wooden box which represents cube. The box can slide and stack.
Hat represent cone. hat can roll, roll and slide.

Explanation:
Solid shapes that can roll are circled with Brown.
Solid shapes that can slide are circled with Yellow.
Solid shapes that can stack are circled with Blue.
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 4.

Given:
The ball which represents Sphere. The ball can only roll.
Glue stick which represents cylinder. Glue stick can roll,roll and slide, stack and slide.
The box which represents cube. Box can slide and stack.
Birthday Hat represents cone. Birthday hat can roll, roll and slide.

Explanation:
Solid shapes that can roll are circled with Brown.
Solid shapes that can slide are circled with Yellow.
Solid shapes that can stack are circled with Blue.
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Think and Grow: Modeling Real Life

Directions:
You stack the 3 objects shown. Write 1 below the object you place at the bottom of the stack, write 2 below the object you stack next, and write 3 below the object you stack last. Tell why you chose this order.

Given:
cylinder shaped Oats box and piggy bank. Oats box and piggy bank can roll, roll and slide, stack and slide.
cube shaped Cardboard box and a Wooden box . A cardboard box and a Wooden Box can slide and stack.
cone shaped Party hats. Party hats can roll, roll and slide.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere , cylinder , cone shape.

### Describe Three-Dimensional Shapes Homework & Practice 12.2

Directions:
1 Look at the solid shapes on the left that stack. Circle the other solid shapes that stack. 2 Look at the solid shape on the left that rolls. Circle the other solid shapes that roll. 3 Look at the solid shape on the left that slides. Circle the other solid shapes that slide.

Question 1.

Explanation:
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.

Question 2.

Explanation:
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 3.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.

Directions:
4 Circle the solid shapes that roll and stack. 5 Circle the solid shapes that stack and slide. 6 Circle the solid shape that does not roll. 7 You stack the 3 objects shown. Write 1 below the object you place at the bottom of the stack, write 2 below the object you stack next, and write 3 below the object you stack last. Tell why you chose this order.

Question 4.

Cylinder shaped objects can roll and slide.

Explanation:
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.
Shapes with a curved face can roll. Example sphere, cylinder, cone shape.

Question 5.

Given:
Cube shaped objects can slide and stake.
Cylinder shaped objects can roll and slide.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.

Question 6.

Given:
Cube shaped objects can slide and stake. Cube shaped solids that does not roll.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.

Question 7.

Explanation:
In the figure given we have 2 cylinder shaped objects and 1 cone shaped object.
Cylinders can stack, slide and roll. So, I used both the cylinder shaped objects at the bottom.
Cone can roll and slide. As cones shaped figures can not be stacked I used at the top.

### Lesson 12.3 Cubes and Spheres

Explore and Grow

Directions:
Cut out the Cube and Sphere Sort Cards. Sort the cards into the categories shown.

Think and Grow

Directions:
Circle the cube. Draw a rectangle around the sphere. Tell why your answers are correct.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Apply and Grow: Practice

Directions:
1 Circle the cube. Draw a rectangle around the sphere. Tell why your answers are correct. 2 – 4 Circle any object that looks like a cube. Draw a rectangle around any object that looks like a sphere. Tell why your answers are correct.

Question 1.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 2.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 3.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 4.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Think and Grow: Modeling Real Life

Directions:
Use Make a Cube to build your own number cube. Draw the shape of the flat surfaces of your cube. Count and write the number of flat surfaces.

________
– – – – – – – –
________ flat surfaces

### Cubes and Spheres Homework & Practice 12.3

Directions:
1 – 3 Circle the cube. Draw a rectangle around the sphere. Tell why your answers are correct.

Question 1.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Question 2.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Question 3.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Directions:
4 – 6 Circle any object that looks like a cube. Draw a rectangle around any object that looks like a sphere. Tell why your answers are correct. 7 Draw the shape of the flat surfaces of a die. Count and write the number of flat surfaces.

Question 4.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Question 5.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges

Question 6.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 7.

Explanation:
Dice is similar to a cube.
A cube is a region of space formed by six identical square faces joined along their edges.

### Lesson 12.4 Cones and Cylinders

Explore and Grow

Directions:
Cut out the Cone and Cylinder Sort Cards. Sort the cards into the categories shown.

Think and Grow

Directions:
Circle the cone. Draw a rectangle around the cylinder. Tell why your answers are correct.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Apply and Grow: Practice

Directions:
1 Circle the cone. Draw a rectangle around the cylinder. Tell why your answers are correct. 2 – 4 Circle any object that looks like a cone. Draw a rectangle around any object that looks like a cylinder. Tell why your answers are correct.

Question 1.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 2.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 3.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 4.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Think and Grow: Modeling Real Life

Directions:
Use Make a Cylinder to build a can of vegetables. Draw the shape of the flat surfaces of your can. Count and write the number of flat surfaces.

__________
– – – – – – – – – –
__________ flat surfaces

### Cones and Cylinders Homework & Practice 12.4

Directions:
1 – 3 Circle the cone. Draw a rectangle around the cylinder. Tell why your answers are correct.

Question 1.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 2.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 3.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Directions:
4 – 6 Circle any object that looks like a cone. Draw a rectangle around any object that looks like a cylinder. Tell why your answers are correct. 7 Draw the shape of the flat surface of a cone. Count and write the number of ﬂat surfaces.

Question 4.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 5.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 6.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 7.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

### Lesson 12.5 Build Three-Dimensional Shapes

Explore and Grow

Directions:
Use your materials to build one of the three-dimensional shapes shown. Circle the three-dimensional shape that you build.

Think and Grow

Directions:

• Use your materials to build the 2 shapes shown.
• Connect the 2 shapes that you build, as shown.
• Tell what solid shape you build.

Cube

Apply and Grow: Practice

Directions:
1 – 3 Use your materials to build the solid shape shown. 4 Use your materials to build a solid shape that has 6 square, ﬂat surfaces. Circle the shape you build.

Question 1.

Question 2.

Question 3.

Question 4.

Think and Grow: Modeling Real Life

Directions:

• Use your materials to build the castle tower in the picture.
• Circle the solid shapes that you use to build the tower.

Explanation
The above figure represent a cylinder base with cone on the top.

### Build Three-Dimensional Shapes Homework & Practice 12.5

Directions:
1 and 2 Use your materials to build the solid shape shown. 3 Use your materials to build the solid shape that has a curved surface and only 1 flat surface. Circle the shape you build. 4 Use your materials to build a solid shape that has no flat surfaces. Circle the shape you build.

Question 1.

Question 2.

Question 3.

Question 4.

Directions:
5 Use your materials to build the totem pole in the picture. Circle the solid shapes that you use to make the totem pole.

Question 5.

Explanation:
The above totem is a stack of three shapes. The bottom is in the shape of a Cube. The middle is in the shape of a Cylinder. The top is in the shape of a Cone.

### Lesson 12.6 Positions of Solid Shapes

Explore and Grow

Directions:
Place a counter beside the bench. Place a counter in front of the tree. Place a counter next to below the stairs. Place a counter the baby swing.

Think and Grow

Directions:

• Circle the object that looks like a cylinder that is next to the table. Draw a line through the object that looks like a cone that is below the shelf. Draw a rectangle around the object that looks like a sphere that is above the table.
• Circle the object that looks like a cube that is behind the shovel. Draw a line through the object that looks like a cylinder that is beside the tree. Draw a rectangle around the object that looks like a sphere that is in front of the tree.

Apply and Grow: Practice

Directions:
1 Circle the object that looks like a cylinder that is behind a paper cup. Draw a line through the object that looks like a sphere that is above the napkin dispenser. Draw a rectangle around the object that looks like a cone that is below a glass cup. 2 Circle the object that looks like a cone that is beside the log. Draw a line through the object that looks like a sphere that is above the log. Draw a rectangle around the object that looks like a cone that is in front of the log.

Question 1.

Question 2.

Think and Grow: Modeling Real Life

Directions: Use the City Scene Cards to place the objects on the picture.

• Place a dog in front of the boy crossing the street.
• Place a tree beside the building that looks like a cube.
• Place an object that looks like a sphere above the buildings. Place that object behind a cloud.
• Place an object that looks like a cone below the traffic light.
• Place a streetlight next to the girl on the sidewalk.

### Positions of Solid Shapes Homework & Practice 12.6

Directions:
1 Circle the object that looks like a sphere that is beside the pool. Draw a line through the object that looks like a cone that is next to the ball. Draw a rectangle around the object that looks like a cylinder that is behind the block.

Question 1.

Directions:
2 Circle the object that looks like a cone that is above the stuffed animal. Draw a line through the object that looks like a cylinder that is in front of the stuffed animal. Draw a rectangle around the object that looks like a cube that is below the stuffed animal. 3 Use the Construction Scene Cards to place the objects on the picture. Place a building below the object that is shaped like a cube. Place a tree beside that building. Place a blimp the traffic cone. Place a truck in front of the traffic cone.

Question 2.

Question 3.

### Identify Three-Dimensional Shapes Performance Task

Directions: 1 You pick up trash in the park. Draw lines to match each item with its correct recycling bin.

• The object that rolls but does not stack that is in front of the lamppost goes in the yellow bin.
• The object below the bench that does not roll goes in the blue bin.
• The object that has 1 flat surface that is behind an object that looks like a cylinder goes in the green bin.
• The object that stacks, slides, and rolls that are above an object that looks like a cube goes in the orange bin.
• The object in front of the tree that rolls and has 2 flat surfaces goes in the green bin.
• The object next to the tree that stacks and slides and has only flat surfaces goes in the green bin.
• The object that has a curved surface that does not stack that is beside the tree goes in the blue bin.
• The object that slides and rolls that is next to an object that has 6 flat surfaces goes in the blue bin.

Question 1.

### Identify Three-Dimensional Shapes Activity

Solid Shapes: Spin and Cover
Directions:
Take turns using the spinner to find which type of three-dimensional shape to cover. Use a counter to cover an object on the page. Repeat this process until you have covered all of the objects.

### Identify Three-Dimensional Shapes Chapter Practice

Directions:
1 and 2 Circle any three-dimensional shapes. Draw rectangles around any two-dimensional shapes. Tell why your answers are correct. 3 Look at the solid shape on the left that rolls. Circle the other solid shapes that roll. 4 Circle the solid shapes that stack and slide.

12.1 Two- and Three-Dimensional Shapes

Question 1.

two-dimensional
rectangle
circle
three-dimensional
Sphere
cube
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

Question 2.

two-dimensional
Triangle
three-dimensional
Cylinder
Cone
Explanation:
A two-dimensional shape is a shape that has length and width but no depth.
Examples: Circle, Triangle, Rectangle, Squares, Hexagons.
2-D shapes have been shaped with rectangle.
A three-dimensional shape can be defined as a solid figure or an object or shape that has three dimensions – length, width and height. Unlike two-dimensional shapes, three-dimensional shapes have thickness or depth.
Examples: Sphere, Torus, Cylinder, Cone, Cube, Cuboid, Triangular Pyramid, Square Pyramid.
3-D shapes have been shaped with circle.

12.2 Describe Three-Dimensional Shapes

Question 3.

Explanation:
Shapes with a curved face can roll. Example sphere , cylinder , cone shape

Question 4.

Explanation:
The object which has a flat surface can slide. Example Rectangle, cube, cuboid, cylinder shapes.
Shapes with a flat face can stack. Example Cube, Rectangle, Cylinder shape.

Directions:
5 Circle the cube. Draw a rectangle around the sphere. Tell why your answers are correct. 6 Circle any object that looks like a cube. Draw a rectangle around any object that looks like a sphere. Tell why your answers are correct. 7 and 8 Circle any object that looks like a cone. Draw a rectangle around any object that looks like a cylinder. Tell why your answers are correct.

12.3 Cubes and Spheres

Question 5.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

Question 6.

Explanation:
A sphere is a round, ball-shaped solid. It has one continuous surface with no edges or vertices.
A cube is a region of space formed by six identical square faces joined along their edges.

12.4 Cones and Cylinders

Question 7.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Question 8.

Explanation:
A Cone is a distinctive three-dimensional geometric figure that has a flat surface and a curved surface, pointed towards the top. The pointed end of the cone is called the apex, whereas the flat surface is called the base.

A Cylinder is a three-dimensional solid that holds two parallel bases joined by a curved surface, at a fixed distance.

Directions:
9 Use your materials to build the solid shape shown. 10 Use your materials to build a shape that has a curved surface and 2 flat surfaces. Circle the shape you build. 11 Use your materials to build the elf in the picture. Circle the solid shapes that you use to make the elf.

12.5 Build Three-Dimensional Shapes

Question 9.

Question 10.

Question 11.

Directions:
12 Circle the object that looks like a cylinder that is below the hat. Draw a line through the object that looks like a cone that is beside the cooler. Draw a rectangle around the object that looks like a cylinder that is in front of the hat. 13 Circle the object that looks like a sphere that is above the cone. Draw a line through the object that looks like a cylinder that is next to the cone. Draw a rectangle around the object that looks like a sphere that is behind the cone.

Question 12.

Question 13.

Final Words:

You can learn the difference between 2-D and 3-D shapes from here. Write your new ideas on your book and solve the problems in own way. Also create questions on your own and try to understand the concepts in depth. We hope the given info is helpful for all the students of Grade K. If you have any doubts regarding the concept you can post the comments in the below-mentioned comment box. Hope this Big Ideas Math Grade K Solution Key helps you to score good marks in the exams.

## Big Ideas Math Book 2nd Grade Answer Key Chapter 12 Solve Length Problems

Students can check out the topic-wise questions and solutions of Big Ideas Math Book Grade 2 Chapter 12 Solve Length Problems. The different lessons of BIM Grade 2 Chapter 12 Solve Length Problems are Solve Length Problems Vocabulary, Use a Number Line to Add and Subtract Lengths, Problem Solving: Length, Problem Solving: Missing Measurement, and Practice Measurement Problems.

Make use of Big Ideas Math 2nd Grade 12th Chapter Solve Length Problems Answer Key during your practice sessions. Make the most out of them and score better grades in your exams. Students can access whichever topic they feel like preparing by click on the quick links listed below. You will be directed to the chosen link.

Vocabulary

Lesson: 1 Use a Number Line to Add and Subtract Lengths

Lesson: 2 Problem Solving: Length

Lesson: 3 Problem Solving: Missing Measurement

Lesson: 4 Practice Measurement Problems

Chapter: 12 – Solve Length Problems

### Solve Length Problems Vocabulary

Organize It
Use the review words to complete the graphic organizer.

Explanation :
Bar model can be defined as a pictorial representation of a number in the form of bars or boxes used to solve number problems.

Define It
Match the review word to its model.

### Lesson 12.1 Use a Number Line to Add and Subtract Lengths

Explore and Grow

Your goldfish is 4 centimetres long. It grows 6 more centimetres. Use the number line and your ruler to show how long the goldfish is now.

Length of the gold fish = 4 centimetres
Increased length of gold fish = 6 centimetres
Length of gold fish now = 4 + 6 = 10 centimetres  .

Explanation:
Draw an arrow from 0 to 4 to represent 4. Then draw an arrow 6 units to the right representing adding +6.
So, 4 + 6 =0

What is the same about your ruler and the number line? What is different?
_________________________
_________________________
_________________________
_________________________
A number line is just that – a straight, horizontal line with numbers placed at even increments along the length. It’s not a ruler, so the space between each number doesn’t matter, but the numbers included on the line determine how it’s meant to be used.
Ruler a straight strip, typically marked at regular intervals and used to draw straight lines or measure distances.

Show and Grow

Question 1.
You swim 15 meters and take a break. Then you swim 10 meters. How many meters do you swim?

Distance covered while swimming = 15 meters
Distance covered while swimming after break = 10 metresters .
Therefore, 25 meters traveled in swimming

Explanation :
Draw an arrow from 0 to 10 to represent 10. Then draw an arrow 15 units to the right representing adding +15.
So, 10 + 15 = 25

Question 2.
A ribbon is 16 yards long. You cut off 7 yards. How long is the ribbon now?

Length of the ribbon = 16 yards
Decreased in the length of the ribbon = 7 yards
Length of the ribbon now = 16 – 7 = 9 yards .

Explanation :
Draw an arrow from 0 to 16 to represent 16. Then draw an arrow 7 units to the left representing Subtracting -7.
So, 16  – 7 = 9

Apply and Grow: Practice

Question 3.
A snake is 24 inches long. It sheds 14 inches of its skin. How much skin does it not shed?

Length of the snake = 24
Length of snake sheds = 14
Length of the snake didn’t shed = Total Length – shed length = 24 – 14 = 10 inches

Explanation :
Draw an arrow from 0 to 24 to represent 24. Then draw an arrow 14 units to the left representing Subtracting 10
So, 24 – 14 = 10 inches .

Question 4.
A photo is 15 centimetres long. You cut off 3 centimetres from the left and 3 centimetres from the right. How long is the photo now?

Length of the photo = 15 centimetres
Length of the photo cut from left = 3 centimetres
Length of the photo cut from right = 3 centimetres
Length of the photo now = 15 – 3 – 3 = 15 – 6 = 9 centimetres

Question 5.
Structure
Write an equation that matches the number line.

Explanation :
Draw an arrow from 0 to 11 to represent 11. Then draw an arrow 6 units to the right representing adding 6 and draw an arrow 3 units to the left representing subtracting 3 .
So, 11 + 6 – 3 = 14 .

Think and Grow: Modeling Real Life

You want to make a bracelet that is 6 inches long. You make 4 inches before lunch. You make 2 inches after lunch. Did you finish the bracelet?

Model:

Did you finish? Yes No
Length of the bracelet = 6 inches
Length of the bracelet made before lunch = 4 inches.
Length of the bracelet made after lunch = 2 inches.
Total Length of the bracelet made = 4 + 2 = 6 inches .
Yes , it is finished

Explanation :
Draw an arrow from 0 to 4 to represent 4. Then draw an arrow 2 units to the right representing adding 2
So, 4 + 2 = 6 inches .

Show and Grow

Question 6.
You are painting a fence that is 24 feet long. You paint 10 feet on Saturday. You paint 13 feet on Sunday. Did you finish painting the fence?

Length of the total fencing = 24 feet
Length of the fence painted on Saturday = 10 feet
Length of the fence painted on Sunday = 13 feet
Total length of the fencing painted = 10 + 13 = 23 feet.
No painting of fencing is not finished as it is painted 23 feet . still 1 feet left to paint ( 24 – 23 = 1 feet )

Explanation :
Draw an arrow from 0 to 10 to represent 10. Then draw an arrow 13 units to the right representing adding 13
So, 10 + 13  = 23 feet .

Question 7.
DIG DEEPER!
You throw a disc 9 meters. On your second throw, the disc travels 3 meters more than your ﬁrst throw. How many meters did the disc travel in all?

______ meters
Length of the first disc thrown = 9 metres
Length of the disc travels in second throw = 3 meters more than your ﬁrst throw = 9 + 3 = 12
Total length the disc travels in first and second throw = 9 + 12 = 21 meters .

Explanation :
Draw an arrow from 0 to 9 to represent 9. Then draw an arrow 12 units to the right representing adding 12
So, 9 + 12  = 21 meters .

### Use a Number Line to Add and Subtract Lengths Homework & Practice 12.1

Question 1.
You kick a ball 13 yards. Your friend kicks it back 9 yards. How far is the ball from you now?

Distance traveled by ball when i kicked the ball  = 13 yards
Distance traveled by ball when my friend kicked the ball = 9 yards back ward = -9 yards.
Distance of ball from me now = 13 yards – 9 yards = 4 yards.

Explanation :
Draw an arrow from 0 to 13 to represent 13. Then draw an arrow 9 units to the left representing Subtracting 9
So, 13- 9 = 4 yards .

Question 2.
Your shoelace is 20 inches long. Your friend’s is 4 inches longer than yours. How long is your friend’s shoelace?

Length of my shoelace = 20 inches
Length of my friend’s shoelace = 4 inches longer than me = 4 + 20 = 24 inches.

Explanation :
Draw an arrow from 0 to 20 to represent 20. Then draw an arrow 4 units to the right representing adding 4
So, 20 + 4 = 24 inches .

Question 3.
Structure
One power cord is 7 feet long. Another power cord is 5 feet long. Use the number line to find the combined length of the power cords.

Length of power cord = 7 feet
Length of another power cord = 5 feet
Total length of both cords = 7 + 5 = 12 feet

Explanation :
Draw an arrow from 0 to 7 to represent 7. Then draw an arrow 5 units to the right representing adding 5
So, 7 + 5  =12 feets .

Question 4.
Modeling Real Life
A worker needs to pave a bike path that is 25 feet long. He completes 13 feet on Monday and 11 feet on Tuesday. Did he complete the paving?

Length of the bike path = 25 feet
length paved on Monday = 13 feet
Length paved on Tuesday = 11 feet
Total length paved on Monday and Tuesday = 13 + 11 = 24 feet
1 feet is less to complete the length of bike path . So, Paving is not completed .

Explanation :
Draw an arrow from 0 to 13 to represent 13. Then draw an arrow 11 units to the right representing adding 11
So, 13+ 11  = 24 feets .

Question 5.
DIG DEEPER!
You throw a baseball 5 yards. On your second throw, the baseball travels 2 yards more than your ﬁrst throw. How many yards did the baseball travel in all?

_______ yards
Distance traveled by a base ball = 5 yards
Distance traveled by a base ball in second throw = 2 yards more than your ﬁrst throw = 5 + 2 = 7 yards .

Explanation :
Draw an arrow from 0 to 5 to represent 5. Then draw an arrow +2 units to the right representing adding 2
So, 5 + 2  = 7 yards .

Review & Refresh

Compare
Question 6.

210 = 200 + 10

Question 7.

532 = 500 + 20 + 3

### Lesson 12.2 Problem Solving: Length

Explore and Grow

How much longer is the red ribbon than the blue ribbon?

______ inches

Show and Grow

Question 1.
An orange fish is 10 centimeters long. A yellow fish is 35 centimeters long. A red fish is 19 centimeters long. How much longer is the yellow fish than the red fish?

Length of Orange fish =  10 centimeters
Length of Yellow fish = 35 centimeters
Length of red fish  = 19 centimeters
length of yellow fish is how much longer than the red fish = 35 – 19 = 16 centimeters

Apply and Grow: Practice

Question 2.
A green scarf is 60 inches long. An orange scarf is 45 inches long. A red scarf is 36 inches long. How much longer is the green scarf than the red scarf?

______ inches
Length of green scarf = 60 inches
Length of Orange scarf = 45 inches .
Length of Red scarf = 36 inches .
Length of green scarf is how much longer than red scarf = 60 – 36 = 24 inches .

Question 3.
DIG DEEPER!
A pink ribbon is 90 centimeters long. A purple ribbon is 35 centimeters long. A blue ribbon is46 centimeters long. How much longer is the pink ribbon than the total length of the purple and blue ribbons?

_______ centimeters
Length of pink ribbon = 90 centimeters
Length of purple ribbon = 35 centimeters
Length of blue ribbon = 46 centimeters
The total length of the purple and blue ribbons = 35 + 46 = 81  centimeters.
Length of pink ribbon is how much longer than the total length of the purple and blue ribbons = 90 – 81 = 9 centimeters

Question 4.
Structure
How much taller is Student 3 than the shortest student?

________ inches
Taller student is student 3 = 53 inches .
Shorter student is student 2 = 48 inches .
Taller Student is how much taller than shorter student = 53 – 48 = 5 inches .

Think and Grow: Modeling Real Life

You hop 27 inches and then 24 inches. Your friend hops 3 inches less than you. How far does your friend hop?
Think: What do you know? What do you need to find?
Model:
_______ inches
Length hoped by me is 27 and 24 inches = 27 + 24 = 51 inches
Length hoped by my friend = 3 inches less than me = 51 – 3 = 48 inches .

Show and Grow

Question 5.
You throw a ball 36 feet and then 41 feet. Your friend throws a ball 5 feet farther than you. How far does your friend throw the ball?

______ feet
Distance traveled by ball in first throw = 36 feet
Distance traveled by ball in Second throw = 41 feet
Total Distance traveled by balls = 36 + 41 = 77 feet
Distance traveled by ball when my friend throws = 5 feet farther than me = 77 + 5 = 82 feet .

Question 6.
DIG DEEPER!
A black horse runs 53 meters and then 45 meters. A brown horse runs 62 meters and then 31 meters. Which horse ran the longer distance in all? How many more meters did the horse run?

Black horse Brown horse
_________ more meters
Distance traveled by black horse is 53 meters and then 45 meters. = 53 + 45 = 98 meters
Distance traveled by Brown horse is 62 meters and then 31 meters = 62 + 31 = 93 meters
Black Horse runs longer distance than brown horse
Black horse travels 98 – 93 = 5 meters more than brown horse .

### Problem Solving: Length Homework & Practice 12.2

Question 1.
The distance to the principal’s office is 24 yards. The distance to the bathroom is 15 yards. The distance to your teacher’s desk is 2 yards. How much farther away is the principal’s office than the bathroom?

_______ yards
The distance to the principal’s office = 24 yards
The distance to the bathroom = 15 yards
The distance to your teacher’s desk = 2 yards

The principal’s office is 9 yards farther away than the bathroom .

Question 2.
YOU BE THE TEACHER
You launch a rocket 63 meters. Your friend launches it 28 meters, and your cousin launches it 86meters. Your cousin says that he launches the rocket 58 meters farther than you. Is he correct? Explain.

Distance traveled by my rocket = 63 meters
Distance traveled by my friend rocket = 28 meters
Distance traveled by my cousin’s rocket = 86 meters
Distance differences between my rocket and my cousins rocket = 86 – 63 = 23 meters .
My Rockets is 23 meters farther than my cousin rocket not 58 meters.
So above statement is wrong .
Explanation :
Distance differences between my rocket and my cousins rocket = 86 – 63 = 23 meters .
My Rockets is 23 meters farther than my cousin rocket not 58 meters.

Question 3.
Modeling Real Life
You create a drawing that is 15 centimeters long and then add on 7 more centimeters. Your friend creates a drawing that is 3 centimeters longer than yours. How long is your friend’s drawing?

_______ centimeters
Length of my Drawing = 15 centimeters
Length of my drawing after adding 7 centimeters = 15 + 7 = 22 centimeters
Length of my Friends drawing = 3 centimeters longer than my drawing = 22 + 3 = 25 centimeters
Therefore length of my friend’s drawing = 25 centimeters .

Question 4.
DIG DEEPER!
A frog hops 36 inches and then 22 inches. A toad hops 14 inches and then 43 inches. Which animal hopped the longer distance in all? How many more inches did the animal hop?
________ more inches
Distance hopped by frog is 36 inches and then 22 inches. = 36 + 22 = 58 inches .
Distance hopped by toad is 14 inches and then 43 inches = 14 + 43 = 57 inches .
Frog hopped more distance than toad
Difference of distance hopped by frog and toad = 58 – 57 = 1 inch .

Review & Refresh

Question 5.
635 + 10 = ______
635 + 100 = _____
635 + 10 = 645
635 + 100 = 735
Explanation :
The ones digit remains the same when you add ten. The tens digit increases by 1 every time you add ten
The ones digit remains the same and the tens digit remains the same when you add hundred. The hundred digit increases by 1 every time you add hundred

Question 6.
824 + _____ = 924
824 + _____ = 834
824 + 100 = 924
824 + 10 = 834
Explanation :
The ones digit remains the same when you add ten. The tens digit increases by 1 every time you add ten
The ones digit remains the same and the tens digit remains the same when you add hundred. The hundred digit increases by 1 every time you add hundred

Question 7.
309 + _____ = 409
309 + _____ = 319
309 + 100 = 409
309 + 10 = 319
Explanation :
The ones digit remains the same when you add ten. The tens digit increases by 1 every time you add ten
The ones digit remains the same and the tens digit remains the same when you add hundred. The hundred digit increases by 1 every time you add hundred

### Lesson 12.3 Problem Solving: Missing Measurement

Explore and Grow

You and your friend each have a piece of yarn. The total length of both pieces is 16 centimeters. Use a ruler to measure your yarn. Then draw your friend’s yarn.

Explain how you found the length of your friend’s yarn.
_______________________
_______________________
_______________________
Total Length of the yarn  = 16 centimeters
Length of my yarn = 7.8 centimeters .
Length of my friend’s yarn = 16 – 7.8 =  8.2 centimeters

Show and Grow

Question 1.
A rope is 31 meters long. You cut a piece off. Now the rope is 14 meters long. How much rope did you cut off?

So, ? = ______.
______ inches
Length of the rope = 31 meters
Length of the rope after cut off = 14 meters.
Length of the cut off rope = 31 – 14 = 17 meters .

Apply and Grow: Practice

Question 2.
A celery stalk is 20 centimeters long. You cut off the leaves. Now it is 13 centimeters long. How much did you cut off?

______ centimeters
Length of celery stalk = 20 centimeters
Length of celery stacking after chopping the leaves = 13 centimeters .
Length of the chopped leaves = 20 – 13 = 7 centimeters .

Question 3.
Descartes walked some and then ran 39 yards. He went a total of 75 yards. How far did he walk?
______ yards
Total Distance traveled by Descartes = 75 yards
Distance traveled by running = 39 yards
Distance Traveled by walking = 75 – 39 = 36 yards

Question 4.
Your coat zipper is 18 inches long. The zipper gets stuck at 11 inches. How much of the zipper will not zip?

_____ inches
Length of coat zipper = 18 inches
Length of zipper got stuck = 11 inches
Length of zipper will not zip = 18 – 11 = 7 inches .

Question 5.
Number Sense
The path to school is 181 meters long in all. How long is the missing part of the path?

_______ meters
Total path to school = 181 meters
The path from the house to first turn = 74 meters
The path from the first turn to the second turn = 86 meters.
Path from second turn to school is missing = 181 – 74 – 86 = 181 – 160 = 21 meters

Think and Grow: Modeling Real Life

You make a paper chain that is 8 feet long. You add 7 feet of chain to the end. Then 6 feet of the chain breaks off. How long is the chain now?
Think: What do you know? What do you need to find?

_____ feet
Length of paper chain = 8 feet
Length of chain added in the end = 7 feet .
Length of chain broken = 6 feet
Total length of the chain now = 8 + 7 – 6 = 15 – 6 = 9 feet .
Explanation :

Show and Grow

Question 6.
You build a tower that is 48 centimeters tall. You add 34 centimeters to the tower height. Your tower breaks and 29 centimeters fall off. How tall is your tower now?
______ centimeters
Height of the tower = 48 centimeters
Height added to the tower = 34 centimeters
Total Height of the tower now = 48 + 34 = 82 centimeters .
Height at which tower broken = 29 centimeters
Height of the tower after broken = 82 – 29 = 53 centimeters

Question 7.
A football team is 78 yards away from scoring. They gain 15 yards on the first play and 21 yards on the second play. How far is the team from scoring now?

_______ yards
The distance of foot ball team from scoring = 75 yards
Distance gain in first play = 15 yards
Distance gain in second play = 21 yards.
Distance of foot ball team from scoring now = 75 – 15 – 21 = 39 yards .

### Problem Solving: Missing Measurement Homework & Practice 12.3

Question 1.
A piece of fabric is 36 inches long. Another piece is 18 inches long. What is the total length of both pieces of fabric?

_____ inches
Length of first fabric = 36 inches
Length of second fabric = 18 inches
Total length of both fabrics = 36 + 18 = 54 inches.

Question 2.
A rose is 61 centimeters long. You cut off some of the stem. Now it is 48 centimeters long. How much did you cut off?

______ Centimeters
Length of rose = 61 centimeters
Length of rose after chopping stem = 48 centimeters .
Length of chopped stem = 61 – 48 = 13 centimeters .

Question 3.
Number Sense
Newton’s balloon is 18 inches long. Descartes’s balloon is 23 inches long. Your friend’s balloon is 12 inches long. Which sentences are true?

Newton’s balloon is 6 inches longer than your friend’s.
Your friend’s balloon is 11 inches longer than Descartes’s.
Descartes’s balloon is 5 inches longer than Newton’s.
Statement 1 is true and Statement 3 is true
Length of Newton’s balloon =18 inches
Length of Descartes’s balloon = 23 inches .
Length of  my Friend’s balloon = 12 inches .
Explanation :
Statement 1 :
Newton balloon and my friends balloon difference in length = 18 – 12 = 6 inches.
It means Newton balloon is 6 inches longer than my friend balloon .
Statement 2 :
Descartes balloon and my friends balloon difference in length= 23 – 12 = 11 inches .
Descartes balloon is 11 inches longer than my friends balloon .
so statement is wrong .
Statement 3 :
Descartes balloon and newtons balloon difference in length = 23 – 18 = 5 inches.
Descartes balloon is 5 inches longer than newton balloon .
Statement is true .

Question 4.
DIG DEEPER!

Equation 3 is true

Question 5.
Modeling Real Life
A piece of wood is 16 feet long. You cut off 6 feet, but it is still too long. You cut off 2 more feet. How long is the piece of wood now?

______ feet
Length of wood = 16 feet
Length of wood chopped = 6 feet
Length of wood chopped again = 2 feet
Length of wood after chopping = 16 – 6 – 2 = 16 – 8 = 8 feet .

Review & Refresh

Question 6.

Question 7.

Question 8.

### Lesson 12.4 Practice Measurement Problems

Explore and Grow

Newton’s piece of string is 24 centimeters long. He gives Descartes 12 centimeters of the string. How long is the string that Newton has left? Draw a picture and write an equation to solve.

______ cm
Length of Newton’s peice of string = 24 centimeters
Length of string given to Descartes = 12 centimeters
Length of string left over with Newton = 24 – 12 = 12 centimeters .
24 – 12 = 12 is the equation .

Compare the lengths of string. Is one longer, or are they the same length? Explain.

__________________________
__________________________

Both the lengths are same
Newton’s length = Descarte’s length = 12 centimeters .

Show and Grow

Question 1.
Your blanket is 66 inches long. Your friend’s blanket is 9 inches longer than yours. How long is your friend’s blanket?

So, ? = ______
_____ inches
Length of my blanket = 66 inches
Length of my friend’s blanket = 9 inches longer than yours. = 66 + 9 = 75 inches
My friends blanket is 9 inches longer than my blanket .

Apply and Grow: Practice

Question 2.
A blue whale is 31 meters long. A humpback whale is 16 meters long. How much longer is the blue whale than the humpback whale?

_____ meters
Length of blue whale = 31 meters
Length of hump back whale = 16 meters
Differences in the length of blue whale and hump back whale = 31 – 16 = 15 meters.
Length of blue whale is 15 meters longer than hump back whale .

Question 3.
Newton runs 450 meters. Descartes runs 25 meters less than Newton. How far do they run in all?

______ meters
Distance traveled by Newtons in running= 450 meters
Distance traveled by Descartes in running= 25 meters less than Newton = 450 – 25 = 425 meters .
Total Distance traveled by Newton and Descartes = 450 + 425 = 875 meters .

Question 4.
Reasoning
Solve the problem below two different ways.
You want to read 100 books during the school year. You read 25 books in the fall and 54 books in the winter. How many books do you still need to read?
______ books
Total Number of books to read = 100 books
Number of books read in fall = 25 books
Number of books read in winter = 54 books
Number of books still needed to read = 100 – 25 – 54 = 100 – 79 = 21 books

Think and Grow: Modeling Real Life

A yellow subway train is 18 meters longer than a blue subway train. The yellow subway train is 92 meters long. How long is the blue subway train?

Equation:
______ meters
Length of yellow subway tarin = 92 meters .
Length of yellow subway train = 18 meters longer than a blue subway train
Length of blue subway train = 92 – 18 = 74

Show and Grow

Question 5.
A brown rabbit hops 24 inches less than a white rabbit. The brown rabbit hops 48 inches. How many inches does the white rabbit hop?
_____ inches
Distance traveled by brown rabbit by hopping = 24 inches less than a white rabbit
Distance traveled by brown rabbit = 48 inches .
Distance traveled by white rabbit = 48 + 24 = 72 inches

Question 6.
Your kite string is 47 yards long. You tie 6 yards of string to the end. Now your kite string is 21 yards longer than your friend’s kite string. How long is your friend’s kite string?

_______ yards
Length of the kite  string = 47 yards
Length attached to kite in the end = 6 yards
Length of my kite string now =47 + 6 = 53 yards
Length of my kite string= 21 yards longer than your friend’s kite string
Length of my friend’s kite string = 53 – 21 = 32 yards .

### Practice Measurement Problems Homework & Practice 12.4

Question 1.
A swimming pool is 28 feet long. The pool cover is 32 feet long. How much longer is the pool cover?

So, ? = ______
_____ feet
Length of swimming pool = 28 feet
Length of pool cover = 32 feet
Difference of length in pool and pool cover = 32 – 28 = 4 feet
So, the pool cover is 4 feet longer than pool

Question 2.
Writing
Write and solve a word problem about the colored pencils.

Which color pencil is longer and how much centimetres it is longer from shorter pencil ?
Explanation :
The length of green pencil = 8 cm
The length of red pencil = 11 cm
The length of blue pencil = 15 cm
green pencil is the shorter pencil
Blue pencil is the longer pencil
Differences in the lengths of blue and green pencils = 15 – 8 = 7 cms
The blue pencil is 7 cms longer than green pencil .

Question 3.
Modeling Real Life
You cast out your fishing line 14 yards less than your friend. Your friend casts out her line 33 yards. How many yards do you cast out your fishing line?

______ yards
length of my friend’s fishing line = 33 yards
length of my fishing line= 14 yards less than my friend’s
Length of my fishing line= 33 – 14 = 19 yards

Question 4.
Modeling Real Life
Your nightstand is 24 inches tall. You put a 20-inch lamp on it. Now your nightstand and lamp are 19 inches taller than your bed. How tall is your bed?

______ inches
Height of my nightstand = 24 inches
Height of the lamp = 20 inch
Height of nightstand and lamp = 24 + 20 = 42 inches.
Height of bed = nightstand and lamp are 19 inches taller than your bed.
Height of bed = 42 – 19 = 23 inches .

Review & Refresh

Question 5.
Write the number in expanded form and word form.
645
______ + ____ + ______ _________
645 = 600 + 40 + 5
Explanation :
Six hundred and forty five represents six hundreds plus four tens and five ones
When numbers are separated into individual place values and decimal places is called expanded form

Question 6.
Write the number in standard form and word form.
800 + 60 + 2
_____ _______________________
862
Explanation:
eight hundred and sixty two = 8 hundreds plus 6 tens and 2ones .
When numbers are separated into individual place values and decimal places is called expanded form

### Solve Length Problems Performance Task

Question 1.
A recorder is 1 foot long. A clarinet is 24 inches long. Which instrument is longer? How much longer is the instrument?

Recorder Clarinet
_____ inches
Length of the Recorder = 1 foot = 12 inches
Length of the clarinet = 24 inches
Clarinet is longer
Difference in the length of clarinet and Recorder = 24 – 12 = 12 inches .
Clarinet is 12 inches longer than Recorder .

Question 2.
A piano has 27 more keys than a keyboard. There are 52 white keys and 36 black keys on a piano.
a. How many keys are on the keyboard?

_____ keys
Number of white keys on piano = 52
Number of black keys on piano = 36
Total Number of keys on piano = 52 + 36 = 88 keys
piano has 27 more keys than a keyboard
Number of keys on key board = 88 – 27 = 61 keys .

b. The number of black keys on the piano is equal to the number of white keys on the keyboard. How many black keys are on the keyboard?
______ black keys
Number of black keys on piano = 36 = number of white keys on keyboard
Number of black keys on keyboard = 61 – 36 = 25 keys .

Question 3.
A drum set has drums and cymbals on stands.
a. The cymbals are 77 centimeters from the ground. You raise the stand 18 centimeters. The cymbals are now 23 centimeters higher than one of the drums. What is the height of the drum?

_______ centimeters
Length of cymbals from ground = 77 centimeters .
Length of the stand = 18 centimeters .
Length of cymbals now = 77 + 18 = 95 centimeters
cymbals are now 23 centimeters higher than one of the drums.
Height of the drums = 95 – 23 =72 centimeters .

b. Another drum is 60 centimeters from the ground. You raise it 12 centimeters. Are both drums the same height?
Yes No
Height of the drum = 60 centimeters
Height of the drum raised = 12 centimeters.
Height of the drum now = 60 + 12 = 72 centimeters .
Both the drums are at equal heights of 72 centimeters
Yes, both drums have the same height .

### Solve Length Problems Activity

Draw and Cover
To Play: Players take turns. On your turn, pick a Draw and Cover Card and solve. Then cover the sea turtle that has the answer. Continue playing until all sea turtles are covered.

### Solve Length Problems Chapter Practice

12.1 Use a Number Line to Add and Subtract Lengths

Question 1.
You throw a ball 12 yards. Your friend throws it back 8 yards. How far is the ball from you now?

Distance traveled by ball when thrown by me= 12 yards
Distance traveled by my friend after throwing = – 8 yards. ( – represents back direction )
Distance of ball far from me  = 12 – 8 = 4 yards.

Explanation:
Draw an arrow from 0 to 12 to represent 12. Then draw an arrow 4 units to the left representing subtracting 4.
So, 12 – 8 = 4 yards .

12.2 Problem Solving: Length

Question 2.
Your cat’s first collar was 6 inches long. Now your cat has a collar that is 13 inches long. Your puppy’s collar is 11 inches long. How much longer is your cat’s collar now?

Length of cat first collar = 6 inches
Length of cat collar now  = 13 inches .
Difference in cat collar now and first collar = 13 – 6 = 7 inches
Cat collar is 7 inches longer than first collar

12.3 Problem Solving: Missing Measurement

Question 3.
You must be 54 inches tall to ride a roller coaster. At 8 years old, you were 48 inches tall. You grow 3 inches the next year. How much more do you still need to grow to be able to ride the roller coaster?

______ inches
Required Height to ride a aroller coaster = 54
My Height at the age of 8 yaers = 48 inches.
Next year my height = 48 + 3 = 51 inches.
Height required more for me to ride a roller coaster = 51 – 48 = 3 inches

Question 4.
Number Sense
A car tire is 61 centimeters tall. A truck tire is 84 centimeters tall. A monster truck tire is167 centimeters tall. Which sentences are true?
The car tire is 23 centimeters taller than the truck tire.
The truck tire is 83 centimeters shorter than the monster truck tire.
The monster truck tire is 106 centimeters taller than the car tire.

Statements 2 and 3 are true .
Length of car tire = 61 centimeters
Length of truck tire = 84 centimeters
Length of monster truck = 167 centimeters
Explanation:
Statement 1:
Car tire and truck tire difference in lengths = 84 – 61 = 23 centimeters
Truck tire is 23 centimeters  longer than car tire
Statement is false
Statement 2 :
Truck tire and monster tire lengths differences = 167 – 84 = 83 centimeters
The truck tire is 83 centimeters shorter than the monster truck tire.
Statement is true .
Statement 3 :
Monster truck and car tire lengths differences = 167 – 61 = 106 centimeters
The monster truck tire is 106 centimeters taller than the car tire.
Statement is true .

12.4 Practice Measurement Problems

Question 5.
A kangaroo jumps 24 feet. A frog jumps 19 feet less than the kangaroo. How far does the frog jump?

______ feet
Height of kangaroo jump = 24 feet
Height of frog jump = 19 feet less than the kangaroo. = 24 – 19 = 5 feet

Question 6.
A store owner wants to add on to the parking lot to make it 38 meters long. It is currently 21 meters long. How many meters does the store owner want to add?
_____ meters
Length of parking after adding parking = 38 meters
Current length = 21 meters
Increase in the length = 38 – 21 = 17 meters .

### Solve Length Problems Cumulative Practice

Question 1.
Which expressions have a sum less than 12?

Expressions : 5 + 3 , 1 +0 and 4 +6
Explanation :
5 + 3 = 8
4 + 6 = 10
1 + 0 = 1
7 + 8 = 15

Question 2.
Find each difference.

Question 3.
A blue sailboat is 44 feet long. A white sailboat is 36 feet long. A green sailboat is 22 feet long. Which sentences are true?

Statement 2 and 3 are true .
Length of blue sail boat = 44 feet
Length of white sail boat = 36 feet
Length of green sail boat = 22 feet
Explanation:
Statement 1 :
Difference of blue and green sail boat = 44 – 22 = 22 feet
The blue sail boat is 22 feet longer than green boat .
Statement 1 is wrong .
Statement 2 :
Difference of white and green sail boat = 36 – 22 = 14 feet.
The green sail boat is 14 feet shorter than white sail boat .
Statement 2 is true .
Statement 3 :
Difference of blue and green sail boat = 44 – 22 = 22 feet
The green sail boat is 22 feet shorter than blue boat .
Statement 3 is true .
Statement 4 :
Difference of blue and white sail boat = 44 – 36 = 8 feet
The blue sail boat is 8 feet longer than white sail boat .
Statement 4 is wrong .

Question 4.
What is the value of the underlined digit?
739

3 tens

Question 5.
Use mental math to solve.
403 – 10 = ______
898 – 100 = _____
640 – 10 = ______
204 – 10 = ______
843 – _____ = 833
_______ – 100 = 731
403 – 10 = 393
898 – 100 = 798
640 – 10 = 630
204 – 10 = 194
843 – 10 = 833
831- 100 = 731

Question 6.
The cracker is about 2 inches long. What is the best estimate of the length of the cracker box?

Length of cracker = 2 inches
The best estimate of the length of the cracker box = 3 inches

Question 7.
You take 14 pictures on Friday. You take 20 more on Saturday. Your friend takes 34 pictures in all on Friday and Saturday. How many pictures did you and your friend take in all?

Number of pictures taken on Friday by me= 14 pictures
Number of pictures taken on Saturday by me= 20 more on Saturday = 20 pictures
Number of pictures taken by friend on Friday and Saturday = 34 pictures
Total Number of pictures taken by me and my friend = 14 + 20 + 34 = 68 pictures

Question 8.
Which expressions are equal to 245 + 386?

245 + 386 = 631
Expression  are 631 and 200 + 300 + 40 + 80+ 5 + 6
Explanation :
631
500 + 130 + 11 = 630 + 11 = 641
200 + 300 + 40 + 80+ 5 + 6 = 500+120+11 = 620 +11 = 631
500 + 120 + 5 = 625

Question 9.
Newton runs 7 yards, takes a break, and runs 3 more yards. Which number line shows how many yards Newton runs?

Distance traveled by Newton =7 yards
Distance traveled by newton after break = 3 yards
Total Distance Traveled by Newton = 7 + 3 = 10 yards
Picture 4 represents correct .

Explanation :
in picture 4 the distance traveled by newton will be shown where the arrow ends at 10 .

Question 10
Find the sum.

Question 11.
Find each difference.
80 – 53 = ?
79 – 13 = ?
90 – 32 = ?
64 – 40 = ?
80 – 53 = 27
79 – 13 = 66
90 – 32 = 58
64 – 40 = 24

Question 12.
Complete the sentences using centimeters or meters.
A teacher’s desk is about 2 ________ long.
A paper clip is about 8 ________ long.
A carrot is about 12 _________ long.
A boat is about 20 _______ long.
A teacher’s desk is about 2 meters long.
A paper clip is about 8 centimeters long.
A carrot is about 12 centimeters long.
A boat is about 20 meters long.

Final Words:

## Big Ideas Math Algebra 2 Answers Chapter 3 Quadratic Equations and Complex Numbers

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### Quadratic Equations and Complex Numbers Maintaining Mathematical Proficiency

Simplify the expression.
Question 1.
$$\sqrt{27}$$

Question 2.
–$$\sqrt{112}$$

Question 3.
$$\sqrt{\frac{11}{64}}$$

Question 4.
$$\sqrt{\frac{147}{100}}$$

Question 5.
$$\sqrt{\frac{18}{49}}$$

Question 6.
–$$\sqrt{\frac{65}{121}}$$

Question 7.
–$$\sqrt{80}$$

Question 8.
$$\sqrt{32}$$

Factor the polynomial.
Question 9.
x2 − 36

Question 10.
x2 − 9

Question 11.
4x2 − 25

Question 12.
x2 − 22x + 121

Question 13.
x2 + 28x + 196

Question 14.
49x2 + 210x + 225

Question 15.
ABSTRACT REASONING
Determine the possible integer values of a and c for which the trinomial ax2+ 8x+c is factorable using the Perfect Square Trinomial Pattern. Explain your reasoning.

### Quadratic Equations and Complex Numbers Mathematical Practices

Mathematically proficient students recognize the limitations of technology

Monitoring Progress

Question 1.
Explain why the second viewing window in Example 1 shows gaps between the upper and lower semicircles, but the third viewing window does not show gaps.

Use a graphing calculator to draw an accurate graph of the equation. Explain your choice of viewing window.
Question 2.
y = $$\sqrt{x^{2}-1.5}$$

Question 3.
y = $$\sqrt{x-2.5}$$

Question 4.
x2 + y2= 12.25

Question 5.
x2 + y2 = 20.25

Question 6.
x2 + 4y2 = 12.25

Question 7.
4x2 + y2 = 20.25

### Lesson 3.1 Solving Quadratic Equations

Essential Question How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

EXPLORATION 1

Matching a Quadratic Function with Its Graph
Work with a partner. Match each quadratic function with its graph. Explain your reasoning. Determine the number of x-intercepts of the graph.
a. f(x) = x2 − 2x
b. f(x) = x2 − 2x + 1
c. f(x) = x2 − 2x + 2
d. f(x) = −x2 + 2x
e. f(x) = −x2 + 2x − 1
f. f(x) = −x2 + 2x − 2

EXPLORATION 2

Work with a partner. Use the results of Exploration 1 to find the real solutions (if any) of each quadratic equation.
a. x2 − 2x = 0
b. x2 − 2x + 1 = 0
c. x2 − 2x + 2 = 0
d. −x2 + 2x = 0
e. −x2 + 2x − 1 = 0
f. −x2 + 2x − 2 = 0

Question 3.
How can you use the graph of a quadratic equation to determine the number of real solutions of the equation?

Question 4.
How many real solutions does the quadratic equation x2 + 3x + 2 = 0 have? How do you know? What are the solutions?

Monitoring Progress

Solve the equation by graphing.
Question 1.
x2 − 8x + 12 = 0

Question 2.
4x2 − 12x + 9 = 0

Question 3.
$$\frac{1}{2}$$x2 = 6x − 20

Solve the equation using square roots.
Question 4.
$$\frac{2}{3}$$x2 + 14 = 20

Question 5.
−2x2 + 1 = −6

Question 6.
2(x − 4)2 = −5

Solve the equation by factoring.
Question 7.
x2 + 12x + 35 = 0

Question 8.
3x2 − 5x = 2

Find the zero(s) of the function.
Question 9.
f(x) = x2 − 8x

Question 10.
f(x) = 4x2 + 28x + 49

Question 11.
WHAT IF?
The magazine initially charges $21 per annual subscription. How much should the magazine charge to maximize annual revenue? What is the maximum annual revenue? Answer: Question 12. WHAT IF? The egg container is dropped from a height of 80 feet. How does this change your answers in parts (a) and (b)? Answer: ### Solving Quadratic Equations 3.1 Exercises Vocabulary and Core Concept Check Question 1. WRITING Explain how to use graphing to find the roots of the equation ax2 + bx + c = 0. Answer: Question 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. Monitoring Progress and Modeling with Mathematics In Exercises 3–12, solve the equation by graphing. Question 3. x2 + 3x + 2 = 0 Answer: Question 4. −x2 + 2x + 3 = 0 Answer: Question 5. y = x2 − 9 Answer: Question 6. −8 = −x2 − 4 Answer: Question 7. 8x = −4 − 4x2 Answer: Question 8. 3x2 = 6x − 3 Answer: Question 9. 7 = −x2 − 4x Answer: Question 10. 2x = x2 + 2 Answer: Question 11. $$\frac{1}{5}$$x2 + 6 = 2x Answer: Question 12. 3x = $$\frac{1}{4}$$x2 + 5 Answer: In Exercises 13–20, solve the equation using square roots. Question 13. s2 = 144 Answer: Question 14. a2 = 81 Answer: Question 15. (z − 6)2 = 25 Answer: Question 16. (p − 4)2 = 49 Answer: Question 17. 4(x − 1)2 + 2 = 10 Answer: Question 18. 2(x + 2)2 − 5 = 8 Answer: Question 19. $$\frac{1}{2}$$r2 − 10 = $$\frac{3}{2}$$r2 Answer: Question 20. $$\frac{1}{5}$$x2 + 2 = $$\frac{3}{5}$$x2 Answer: Question 21. ANALYZING RELATIONSHIPS Which equations have roots that are equivalent to the x-intercepts of the graph shown? A. −x2 − 6x − 8 = 0 B. 0 = (x + 2)(x + 4) C. 0 = −(x + 2)2 + 4 D. 2x2 − 4x − 6 = 0 E. 4(x + 3)2 − 4 = 0 Answer: Question 22. ANALYZING RELATIONSHIPS Which graph has x-intercepts that are equivalent to the roots of the equation (x − $$\frac{3}{2}$$)2 = $$\frac{25}{4}$$? Explain your reasoning. Answer: ERROR ANALYSIS In Exercises 23 and 24, describe and correct the error in solving the equation. Question 23. Answer: Question 24. Answer: Question 25. OPEN-ENDED Write an equation of the form x2 = d that has (a) two real solutions, (b) one real solution, and (c) no real solution. Answer: Question 26. ANALYZING EQUATIONS Which equation has one real solution? Explain. A. 3x2 + 4 = −2(x2 + 8) B. 5x2 − 4 = x2 − 4 C. 2(x + 3)2 = 18 D. $$\frac{3}{2}$$x2 − 5 = 19 Answer: In Exercises 27–34, solve the equation by factoring. Question 27. 0 = x2 + 6x + 9 Answer: Question 28. 0 = z2 − 10z + 25 Answer: Question 29. x2 − 8x = −12 Answer: Question 30. x2 − 11x = −30 Answer: Question 31. n2 − 6n = 0 Answer: Question 32. a2 − 49 = 0 Answer: Question 33. 2w2 − 16w = 12w − 48 Answer: Question 34. −y + 28 + y2 = 2y + 2y2 Answer: MATHEMATICAL CONNECTIONS In Exercises 35–38, find the value of x. Question 35. Area of rectangle = 36 Answer: Question 36. Area of circle = 25π Answer: Question 37. Area of triangle = 42 Answer: Question 38. Area of trapezoid = 32 Answer: In Exercises 39–46, solve the equation using any method. Explain your reasoning. Question 39. u2 = −9u Answer: Question 40. $$\frac{t^{2}}{20}$$ + 8 = 15 Answer: Question 41. −(x + 9)2 = 64 Answer: Question 42. −2(x + 2)2 = 5 Answer: Question 43. 7(x − 4)2 − 18 = 10 Answer: Question 44. t2 + 8t + 16 = 0 Answer: Question 45. x2 + 3x + $$\frac{5}{4}$$ = 0 Answer: Question 46. x2 − 1.75 = 0.5 Answer: In Exercises 47–54, find the zero(s) of the function. Question 47. g(x) = x2 + 6x + 8 Answer: Question 48. f(x) = x2 − 8x + 16 Answer: Question 49. h(x) = x2 + 7x − 30 Answer: Question 50. g(x) = x2 + 11x Answer: Question 51. f(x) = 2x2 − 2x − 12 Answer: Question 52. f(x) = 4x2 − 12x + 9 Answer: Question 53. g(x) = x2 + 22x + 121 Answer: Question 54. h(x) = x2 + 19x + 84 Answer: Question 55. REASONING Write a quadratic function in the form f(x) = x2 + bx + c that has zeros 8 and 11. Answer: Question 56. NUMBER SENSE Write a quadratic equation in standard form that has roots equidistant from 10 on the number line. Answer: Question 57. PROBLEM SOLVING A restaurant sells 330 sandwiches each day. For each$0.25 decrease in price, the restaurant sells about 15 more sandwiches. How much should the restaurant charge to maximize daily revenue? What is the maximum daily revenue?

Question 58.
PROBLEM SOLVING
An athletic store sells about 200 pairs of basketball shoes per month when it charges $120 per pair. For each$2 increase in price, the store sells two fewer pairs of shoes. How much should the store charge to maximize monthly revenue? What is the maximum monthly revenue?

Question 59.
MODELING WITH MATHEMATICS
Niagara Falls is made up of three waterfalls. The height of the Canadian Horseshoe Falls is about 188 feet above the lower Niagara River. A log falls from the top of Horseshoe Falls.
a. Write a function that gives the height h (in feet) of the log after t seconds. How long does the log take to reach the river?
b. Find and interpret h(2) − h(3).

Question 60.
MODELING WITH MATHEMATICS
According to legend, in 1589, the Italian scientist Galileo Galilei dropped rocks of different weights from the top of the Leaning Tower of Pisa to prove his conjecture that the rocks would hit the ground at the same time. The height h (in feet) of a rock after t seconds can be modeled by h(t) = 196 − 16t2.

a. Find and interpret the zeros of the function. Then use the zeros to sketch the graph.
b. What do the domain and range of the function represent in this situation?

Question 61.
PROBLEM SOLVING
You make a rectangular quilt that is 5 feet by 4 feet. You use the remaining 10 square feet of fabric to add a border of uniform width to the quilt. What is the width of the border?

Question 62.
MODELING WITH MATHEMATICS
You drop a seashell into the ocean from a height of 40 feet. Write an equation that models the height h (in feet) of the seashell above the water after t seconds. How long is the seashell in the air?

Question 63.
WRITING
The equation h = 0.019s2 models the height h (in feet) of the largest ocean waves when the wind speed is s knots. Compare the wind speeds required to generate 5-foot waves and 20-foot waves.

Question 64.
CRITICAL THINKING
Write and solve an equation to find two consecutive odd integers whose product is 143.

Question 65.
MATHEMATICAL CONNECTIONS
A quadrilateral is divided into two right triangles as shown in the figure. What is the length of each side of the quadrilateral?

Question 66.
ABSTRACT REASONING
Suppose the equation ax2 + bx + c = 0 has no real solution and a graph of the related function has a vertex that lies in the second quadrant.
a. Is the value of a positive or negative? Explain your reasoning.
b. Suppose the graph is translated so the vertex is in the fourth quadrant. Does the graph have any x-intercepts? Explain.

Question 67.
REASONING
When an object is dropped on any planet, its height h (in feet) after t seconds can be modeled by the function h = −$$\frac{g}{2}$$t2 + h0, where h0 is the object’s initial height and g is the planet’s acceleration due to gravity. Suppose a rock is dropped from the same initial height on the three planets shown. Make a conjecture about which rock will hit the ground first. Justify your answer.

Question 68.
PROBLEM SOLVING
A café has an outdoor, rectangular patio. The owner wants to add 329 square feet to the area of the patio by expanding the existing patio as shown. Write and solve an equation to find the value of x. By what distance should the patio be extended?

Question 69.
PROBLEM SOLVING
A flea can jump very long distances. The path of the jump of a flea can be modeled by the graph of the function y = −0.189x2 + 2.462x, where x is the horizontal distance (in inches) and y is the vertical distance (in inches). Graph the function. Identify the vertex and zeros and interpret their meanings in this situation.

Question 70.
HOW DO YOU SEE IT?
An artist is painting a mural and drops a paintbrush. The graph represents the height h (in feet) of the paintbrush after t seconds.

a. What is the initial height of the paintbrush?
b. How long does it take the paintbrush to reach the ground? Explain.

Question 71.
MAKING AN ARGUMENT
Your friend claims the equation x2 + 7x =−49 can be solved by factoring and has a solution of x = 7. You solve the equation by graphing the related function and claim there is no solution. Who is correct? Explain.

Question 72.
ABSTRACT REASONING
Factor the expressions x2 − 4 and x2 − 9. Recall that an expression in this form is called a difference of two squares. Use your answers to factor the expression x2 − a2. Graph the related function y = x2 − a2. Label the vertex, x-intercepts, and axis of symmetry.

Question 73.
DRAWING CONCLUSIONS
Consider the expression x2 + a2, where a > 0.
a. You want to rewrite the expression as (x + m)(x + n). Write two equations that m and n must satisfy.
b. Use the equations you wrote in part (a) to solve for m and n. What can you conclude?

Question 74.
THOUGHT PROVOKING
You are redesigning a rectangular raft. The raft is 6 feet long and 4 feet wide. You want to double the area of the raft by adding to the existing design. Draw a diagram of the new raft. Write and solve an equation you can use to find the dimensions of the new raft.

Question 75.
MODELING WITH MATHEMATICS
A high school wants to double the size of its parking lot by expanding the existing lot as shown. By what distance x should the lot be expanded?

Maintaining Mathematical Proficiency

Find the sum or difference.
Question 76.
(x2 + 2) + (2x2 − x)

Question 77.
(x3 + x2 − 4) + (3x2 + 10)

Question 78.
(−2x + 1) − (−3x2 + x)

Question 79.
(−3x3 + x2 − 12x) − (−6x2 + 3x − 9)

Find the product.
Question 80.
(x + 2)(x − 2)

Question 81.
2x(3 − x + 5x2)

Question 82.
(7 − x)(x − 1)

Question 83.
11x(−4x2 + 3x + 8)

### Lesson 3.2 Complex Numbers

Essential Question What are the subsets of the set of complex numbers?
In your study of mathematics, you have probably worked with only real numbers, which can be represented graphically on the real number line. In this lesson, the system of numbers is expanded to include imaginary numbers. The real numbers and imaginary numbers compose the set of complex numbers.

EXPLORATION 1

Classifying Numbers
Work with a partner. Determine which subsets of the set of complex numbers contain each number.
a. $$\sqrt{9}$$
b. $$\sqrt{0}$$
c. −$$\sqrt{4}$$
d. $$\sqrt{\frac{4}{9}}$$
e. $$\sqrt{2}$$
f. $$\sqrt{-1}$$

EXPLORATION 2

Work with a partner. Use the definition of the imaginary unit i to match each quadratic equation with its complex solution. Justify your answers.
a. x2 − 4 = 0
b. x2 + 1 = 0
c. x2 − 1 = 0
d. x2 + 4 = 0
e. x2 − 9 = 0
f. x2 + 9 = 0
A. i
B. 3i
C. 3
D. 2i
E. 1
F. 2

Question 3.
What are the subsets of the set of complex numbers? Give an example of a number in each subset.

Question 4.
Is it possible for a number to be both whole and natural? natural and rational? rational and irrational? real and imaginary? Explain your reasoning.

Monitoring Progress

Find the square root of the number.
Question 1.
$$\sqrt{-4}$$

Question 2.
$$\sqrt{-12}$$

Question 3.
−$$\sqrt{-36}$$

Question 4.
2$$\sqrt{-54}$$

Find the values of x and y that satisfy the equation.
Question 5.
x + 3i = 9 − yi

Question 6.
9 + 4yi = −2x + 3i

Question 7.
WHAT IF?
In Example 4, what is the impedance of the circuit when the capacitor is replaced with one having a reactance of 7 ohms?

Perform the operation. Write the answer in standard form.
Question 8.
(9 − i ) + (−6 + 7i )

Question 9.
(3 + 7i ) − (8 − 2i )

Question 10.
−4 − (1 + i) − (5 + 9i)

Question 11.
(−3i)(10i)

Question 12.
i(8 − i)

Question 13.
(3 + i)(5 −i)

Solve the equation.
Question 14.
x2 = −13

Question 15.
x2= −38

Question 16.
x2 + 11 = 3

Question 17.
x2 − 8 = −36

Question 18.
3x2 − 7 = −31

Question 19.
5x2 + 33 = 3

Find the zeros of the function.
Question 20.
f(x) = x2 + 7

Question 21.
f(x) = −x2 − 4

Question 22.
f(x) = 9x2 + 1

### Complex Numbers 3.2 Exercises

Vocabulary and Core Concept Check
Question 1.
VOCABULARY
What is the imaginary unit i defined as and how can you use i?

Question 2.
COMPLETE THE SENTENCE
For the complex number 5 + 2i, the imaginary part is ____ and the real part is ____.

Question 3.
WRITING
Describe how to add complex numbers.

Question 4.
WHICH ONE DOESN’T BELONG?
Which number does not belong with the other three? Explain your reasoning.

Monitoring Progress and Modeling with Mathematics

In Exercises 5–12, find the square root of the number.
Question 5.
$$\sqrt{-36}$$

Question 6.
$$\sqrt{-64}$$

Question 7.
$$\sqrt{-18}$$

Question 8.
$$\sqrt{-24}$$

Question 9.
2$$\sqrt{-16}$$

Question 10.
−3$$\sqrt{-49}$$

Question 11.
−4$$\sqrt{-32}$$

Question 12.
6$$\sqrt{-63}$$

In Exercises 13–20, find the values of x and y that satisfy the equation.
Question 13.
4x + 2i = 8 + yi

Question 14.
3x + 6i = 27 + yi

Question 15.
−10x + 12i = 20 + 3yi

Question 16.
9x − 18i = −36 + 6yi

Question 17.
2x − yi = 14 + 12i

Question 18.
−12x + yi = 60 − 13i

Question 19.
54 − $$\frac{1}{7}$$yi = 9x− 4i

Question 20.
15 − 3yi = $$\frac{1}{2}$$x + 2i

In Exercises 21–30, add or subtract. Write the answer in standard form.
Question 21.
(6 − i) + (7 + 3i)

Question 22.
(9 + 5i) + (11 + 2i )

Question 23.
(12 + 4i) − (3 − 7i)

Question 24.
(2 − 15i) − (4 + 5i)

Question 25.
(12 − 3i) + (7 + 3i)

Question 26.
(16 − 9i) − (2 − 9i)

Question 27.
7 − (3 + 4i) + 6i

Question 28.
16 − (2 − 3i) − i

Question 29.
−10 + (6 − 5i) − 9i

Question 30.
−3 + (8 + 2i) + 7i

Question 31.
USING STRUCTURE
Write each expression as a complex number in standard form.
a. $$\sqrt{-9}+\sqrt{-4}-\sqrt{16}$$
b. $$\sqrt{-16}+\sqrt{8}+\sqrt{-36}$$

Question 32.
REASONING
The additive inverse of a complex number z is a complex number za such that z + za = 0. Find the additive inverse of each complex number.
a. z = 1 + i
b. z = 3 − i
c. z = −2 + 8i

In Exercises 33–36, find the impedance of the series circuit.
Question 33.

Question 35.

In Exercises 37–44, multiply. Write the answer in standard form.
Question 37.
3i(−5 + i)

Question 38.
2i(7 − i)

Question 39.
(3 − 2i)(4 + i)

Question 40.
(7 + 5i)(8 − 6i)

Question 41.
(4 − 2i)(4 + 2i)

Question 42.
(9 + 5i)(9 − 5i)

Question 43.
(3 − 6i)2

Question 44.
(8 + 3i)2

JUSTIFYING STEPS In Exercises 45 and 46, justify each step in performing the operation.
Question 45.
11 − (4 + 3i) + 5i

Question 46.
(3 + 2i)(7 − 4i)

REASONING In Exercises 47 and 48, place the tiles in the expression to make a true statement.
Question 47.
(____ − ____i) – (____ − ____i ) = 2 − 4i

Question 48.
____i(____ + ____i ) = −18 − 10i

In Exercises 49–54, solve the equation. Check your solution(s).
Question 49.
x2 + 9 = 0

Question 50.
x2 + 49 = 0

Question 51.
x2 − 4 = −11

Question 52.
x2 − 9 = −15

Question 53.
2x2 + 6 = −34

Question 54.
x2 + 7 = −47

In Exercises 55–62, find the zeros of the function.
Question 55.
f(x) = 3x2 + 6

Question 56.
g(x) = 7x2 + 21

Question 57.
h(x) = 2x2 + 72

Question 58.
k(x) = −5x2 − 125

Question 59.
m(x) = −x2 − 27

Question 60.
p(x) = x2 + 98

Question 61.
r(x) = − $$\frac{1}{2}$$x2 − 24

Question 62.
f(x) = −$$\frac{1}{5}$$x2 − 10

ERROR ANALYSIS In Exercises 63 and 64, describe and correct the error in performing the operation and writing the answer in standard form.
Question 63.

Question 64.

Question 65.
NUMBER SENSE
Simplify each expression. Then classify your results in the table below.
a. (−4 + 7i) + (−4 − 7i)
b. (2 − 6i) − (−10 + 4i)
c. (25 + 15i) − (25 − 6i)
d. (5 + i)(8 − i)
e. (17 − 3i) + (−17 − 6i)
f. (−1 + 2i)(11 − i)
g. (7 + 5i) + (7 − 5i)
h. (−3 + 6i) − (−3 − 8i)

Question 66.
MAKING AN ARGUMENT
The Product Property ofSquare Roots states $$\sqrt{a}$$ • $$\sqrt{b}$$ = $$\sqrt{ab}$$ . Your friend concludes $$\sqrt{-4}$$ • $$\sqrt{-9}$$ = $$\sqrt{36}$$ = 6. Is your friend correct? Explain.

Question 67.
FINDING A PATTERN
Make a table that shows the powers of i from i1 to i8 in the first row and the simplified forms of these powers in the second row. Describe the pattern you observe in the table. Verify the pattern continues by evaluating the next four powers of i.

Question 68.
HOW DO YOU SEE IT?
The graphs of three functions are shown. Which function(s) has real zeros? imaginary zeros? Explain your reasoning.

In Exercises 69–74, write the expression as a complex number in standard form.
Question 69.
(3 + 4i) − (7 − 5i) + 2i(9 + 12i)

Question 70.
3i(2 + 5i) + (6 − 7i) − (9 + i)

Question 71.
(3 + 5i)(2 − 7i4)

Question 72.
2i3(5 − 12i )

Question 73.
(2 + 4i5) + (1 − 9i6) − (3 +i7)

Question 74.
(8 − 2i4) + (3 − 7i8) − (4 + i9)

Question 75.
OPEN-ENDED
Find two imaginary numbers whose sum and product are real numbers. How are the imaginary numbers related?

Question 76.
COMPARING METHODS
Describe the two different methods shown for writing the complex expression in standard form. Which method do you prefer? Explain.

Question 77.
CRITICAL THINKING
Determine whether each statement is true or false. If it is true, give an example. If it is false, give a counterexample.
a. The sum of two imaginary numbers is an imaginary number.
b. The product of two pure imaginary numbers is a real number.
c. A pure imaginary number is an imaginary number.
d. A complex number is a real number.

Question 78.
THOUGHT PROVOKING
Create a circuit that has an impedance of 14 − 3i.

Maintaining Mathematical Proficiency

Determine whether the given value of x is a solution to the equation.
Question 79.
3(x − 2) + 4x − 1 = x − 1; x = 1

Question 80.
x3 − 6 = 2x2 + 9 − 3x; x = −5

Question 81.
−x2 + 4x = 19 — 3x2; x = −$$\frac{3}{4}$$

Write a quadratic function in vertex form whose graph is shown.
Question 82.

Question 83.

Question 84.

### Lesson 3.3 Completing the Square

Essential Question How can you complete the square for a quadratic expression?

EXPLORATION 1

Using Algebra Tiles to Complete the Square
Work with a partner. Use algebra tiles to complete the square for the expression x2 + 6x.

a. You can model x2 + 6x using one x2-tile and six x-tiles. Arrange the tiles in a square. Your arrangement will be incomplete in one of the corners.
b. How many 1-tiles do you need to complete the square?
c. Find the value of c so that the expression x2 + 6x + c is a perfect square trinomial.
d. Write the expression in part (c) as the square of a binomial.

EXPLORATION 2

Drawing Conclusions
Work with a partner.
a. Use the method outlined in Exploration 1 to complete the table.

b. Look for patterns in the last column of the table. Consider the general statement x2 + bx + c = (x + d)2. How are d and b related in each case? How are c and d related in each case?
c. How can you obtain the values in the second column directly from the coefficients of x in the first column?

Question 3.
How can you complete the square for a quadratic expression?

Question 4.
Describe how you can solve the quadratic equation x2 + 6x = 1 by completing the square.

Monitoring Progress

Solve the equation using square roots. Check your solution(s).
Question 1.
x2 + 4x + 4 = 36

Question 2.
x2 − 6x + 9 = 1

Question 3.
x2 − 22x + 121 = 81

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
Question 4.
x2 + 8x + c

Question 5.
x2 − 2x + c

Question 6.
x2 − 9x + c

Solve the equation by completing the square.
Question 7.
x2 − 4x + 8 = 0

Question 8.
x2 + 8x − 5 = 0

Question 9.
−3x2 − 18x − 6 = 0

Question 10.
4x2 + 32x = −68

Question 11.
6x(x + 2) = −42

Question 12.
2x(x − 2) = 200

Write the quadratic function in vertex form. Then identify the vertex.
Question 13.
y = x2 − 8x + 18

Question 14.
y = x2 + 6x + 4

Question 15.
y = x2 − 2x − 6

Question 16.
WHAT IF?
The height of the baseball can be modeled by y = −16t2 + 80t + 2. Find the maximum height of the baseball. How long does the ball take to hit the ground?

### Completing the Square 3.3 Exercises

Vocabulary and Core Concept Check
Question 1.
VOCABULARY
What must you add to the expression x2 + bx to complete the square?

Question 2.
COMPLETE THE SENTENCE
The trinomial x2 − 6x + 9 is a ____ because it equals ____.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–10, solve the equation using square roots. Check your solution(s).
Question 3.
x2 − 8x + 16 = 25

Question 4.
r2 − 10r + 25 = 1

Question 5.
x2 − 18x + 81 = 5

Question 6.
m2 + 8m + 16 = 45

Question 7.
y2 − 24y + 144 = −100

Question 8.
x2 − 26x + 169 = −13

Question 9.
4w2 + 4w + 1 = 75

Question 10.
4x2 − 8x + 4 = 1

In Exercises 11–20, find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.
Question 11.
x2 + 10x + c

Question 12.
x2 + 20x + c

Question 13.
y2 − 12y + c

Question 14.
t2 − 22t + c

Question 15.
x2 − 6x + c

Question 16.
x2 + 24x + c

Question 17.
z2 − 5z + c

Question 18.
x2 + 9x + c

Question 19.
w2 + 13w + c

Question 20.
s2 − 26s + c

In Exercises 21–24, find the value of c. Then write an expression represented by the diagram.
Question 21.

Question 22.

Question 23.

Question 24.

In Exercises 25–36, solve the equation by completing the square.
Question 25.
x2 + 6x + 3 = 0

Question 26.
s2 + 2s − 6 = 0

Question 27.
x2 + 4x − 2 = 0

Question 28.
t2 − 8t − 5 = 0

Question 29.
z(z + 9) = 1

Question 30.
x(x + 8) = −20

Question 31.
7t2 + 28t + 56 = 0

Question 32.
6r2 + 6r + 12 = 0

Question 33.
5x(x + 6) = −50

Question 34.
4w(w − 3) = 24

Question 35.
4x2 − 30x = 12 + 10x

Question 36.
3s2 + 8s = 2s − 9

Question 37.
ERROR ANALYSIS
Describe and correct the error in solving the equation.

Question 38.
ERROR ANALYSIS
Describe and correct the error in finding the value of c that makes the expression a perfect square trinomial.

Question 39.
WRITING
Can you solve an equation by completing the square when the equation has two imaginary solutions? Explain.

Question 40.
ABSTRACT REASONING
Which of the following are solutions of the equation x2 − 2ax + a2 = b2? Justify your answers.
A. ab
B. −a − b
C. b
D. a
E. a − b
F. a + b

USING STRUCTURE In Exercises 41–50, determine whether you would use factoring, square roots, or completing the square to solve the equation. Explain your reasoning. Then solve the equation.
Question 41.
x2 − 4x − 21 = 0

Question 42.
x2 + 13x + 22 = 0

Question 43.
(x + 4)2 = 16

Question 44.
(x − 7)2 = 9

Question 45.
x2 + 12x + 36 = 0

Question 46.
x2 − 16x + 64 = 0

Question 47.
2x2 + 4x − 3 = 0

Question 48.
3x2 + 12x + 1 = 0

Question 49.
x2 − 100 = 0

Question 50.
4x2 − 20 = 0

MATHEMATICAL CONNECTIONS In Exercises 51–54, find the value of x.
Question 51.
Area of rectangle = 50

Question 52.
Area of parallelogram = 48

Question 53.
Area of triangle = 40

Question 54.
Area of trapezoid = 20

In Exercises 55–62, write the quadratic function in vertex form. Then identify the vertex.
Question 55.
f(x) = x2 − 8x + 19

Question 56.
g(x) = x2 − 4x − 1

Question 57.
g(x) = x2 + 12x + 37

Question 58.
h(x) = x2 + 20x + 90

Question 59.
h(x) = x2 + 2x − 48

Question 60.
f(x) = x2 + 6x − 16

Question 61.
f(x) = x2 − 3x + 4

Question 62.
g(x) = x2 + 7x + 2

Question 63.
MODELING WITH MATHEMATICS
While marching, a drum major tosses a baton into the air and catches it. The height h (in feet) of the baton t seconds after it is thrown can be modeled by the function h = −16t2 + 32t + 6.
a. Find the maximum height of the baton.
b. The drum major catches the baton when it is 4 feet above the ground. How long is the baton in the air?

Question 64.
MODELING WITH MATHEMATICS
A firework explodes when it reaches its maximum height. The height h (in feet) of the firework t seconds after it is launched can be modeled by h = $$-\frac{500}{9} t^{2}+\frac{1000}{3} t$$ + 10. What is the maximum height of the firework? How long is the firework in the air before it explodes?

Question 65.
COMPARING METHODS
A skateboard shop sells about 50 skateboards per week when the advertised price is charged. For each \$1 decrease in price, one additional skateboard per week is sold. The shop’s revenue can be modeled by y = (70 − x)(50 + x).

a. Use the intercept form of the function to find the maximum weekly revenue.
b. Write the function in vertex form to find the maximum weekly revenue.
c. Which way do you prefer? Explain your reasoning.

Question 66.
HOW DO YOU SEE IT?
The graph of the function f(x) = (x − h)2 is shown. What is the x-intercept? Explain your reasoning.

Question 67.
WRITING
At Buckingham Fountain in Chicago, the height h (in feet) of the water above the main nozzle can be modeled by h = −162 + 89.6t, where t is the time (in seconds) since the water has left the nozzle. Describe three different ways you could find the maximum height the water reaches. Then choose a method and find the maximum height of the water.

Question 68.
PROBLEM SOLVING
A farmer is building a rectangular pen along the side of a barn for animals. The barn will serve as one side of the pen. The farmer has 120 feet of fence to enclose an area of 1512 square feet and wants each side of the pen to be at least 20 feet long.
a. Write an equation that represents the area of the pen.
b. Solve the equation in part (a) to find the dimensions of the pen.

Question 69.
MAKING AN ARGUMENT
Your friend says the equation x2 + 10x = −20 can be solved by either completing the square or factoring. Is your friend correct? Explain.

Question 70.
THOUGHT PROVOKING
Write a function g in standard form whose graph has the same x-intercepts as the graph of f(x) = 2x2 + 8x + 2. Find the zeros of each function by completing the square. Graph each function.

Question 71.
CRITICAL THINKING
Solve x2 + bx + c = 0 by completing the square. Your answer will be an expression for x in terms of b and c.

Question 72.
DRAWING CONCLUSIONS
In this exercise, you will investigate the graphical effect of completing the square.
a. Graph each pair of functions in the same coordinate plane.
y = x2 + 2x y = x2 − 6x
y = (x + 1)2 y = (x − 3)2
b. Compare the graphs of y = x2 + bx and y = (x + $$\frac{b}{2}$$)2. Describe what happens to the graph of y = x2 + bx when you complete the square.

Question 73.
MODELING WITH MATHEMATICS
In your pottery class, you are given a lump of clay with a volume of 200 cubic centimeters and are asked to make a cylindrical pencil holder. The pencil holder should be 9 centimeters high and have an inner radius of 3 centimeters. What thickness x should your pencil holder have if you want to use all of the clay?

Maintaining Mathematical Proficiency

Solve the inequality. Graph the solution.
Question 74.
2x − 3 < 5

Question 75.
4 − 8y ≥ 12

Question 76.
$$\frac{n}{3}$$ + 6 > 1

Question 77.
−$$\frac{2s}{5}$$ ≤ 8

Graph the function. Label the vertex, axis of symmetry, and x-intercepts.
Question 78.
g(x) = 6(x − 4)2

Question 79.
h(x) = 2x(x − 3)

Question 80.
f(x) = x2 + 2x + 5

Question 81.
f(x) = 2(x + 10)(x − 12)

### Quadratic Equations and Complex Numbers Study Skills: Creating a Positive Study Environment

3.1–3.3 What Did You Learn?

Core Vocabulary

Core Concepts

Mathematical Practices
Question 1.
Analyze the givens, constraints, relationships, and goals in Exercise 61 on page 101.

Question 2.
Determine whether it would be easier to find the zeros of the function in Exercise 63 on page 117 or Exercise 67 on page 118.

Study Skills: Creating a Positive Study Environment

• Set aside an appropriate amount of time for reviewing your notes and the textbook, reworking your notes, and completing homework.
• Set up a place for studying at home that is comfortable, but not too comfortable. The place needs to be away from all potential distractions.
• Form a study group. Choose students who study well together, help out when someone misses school, and encourage positive attitudes.

### Quadratic Equations and Complex Numbers 3.1–3.3 Quiz

Solve the equation by using the graph. Check your solution(s).
Question 1.
x2 − 10x + 25 = 0

Question 2.
2x2 + 16 = 12x

Question 3.
x2 = −2x + 8

Solve the equation using square roots or by factoring. Explain the reason for your choice.
Question 4.
2x2 − 15 = 0

Question 5.
3x2 − x − 2 = 0

Question 6.
(x + 3)2 = 8
ans;

Question 7.
Find the values of x and y that satisfy the equation 7x − 6i = 14 + yi.

Question 8.
(2 + 5i) + (−4 + 3i)

Question 9.
(3 + 9i) − (1 − 7i)

Question 10.
(2 + 4i)(−3 − 5i)

Question 11.
Find the zeros of the function f(x) = 9x2 + 2. Does the graph of the function intersect the x-axis? Explain your reasoning.

Solve the equation by completing the square.
Question 12.
x2 − 6x + 10 = 0

Question 13.
x2 + 12x + 4 = 0

Question 14.
4x(x + 6) = −40

Question 15.
Write y = x2 − 10x + 4 in vertex form. Then identify the vertex.

Question 16.
A museum has a café with a rectangular patio. The museum wants to add 464 square feet to the area of the patio by expanding the existing patio as shown.

a. Find the area of the existing patio.
b. Write an equation to model the area of the new patio.
c. By what distance x should the length of the patio be expanded?

Question 17.
Find the impedance of the series circuit.

Question 18.
The height h (in feet) of a badminton birdie t seconds after it is hit can be modeled by the function h = −16t2 + 32t + 4.
a. Find the maximum height of the birdie.
b. How long is the birdie in the air?

### Lesson 3.4 Using the Quadratic Formula

Essential Question How can you derive a general formula for solving a quadratic equation?

EXPLORATION 1

Work with a partner. Analyze and describe what is done in each step in the development of the Quadratic Formula.

EXPLORATION 2

Work with a partner. Use the Quadratic Formula to solve each equation.
a. x2 − 4x + 3 = 0
b. x2 − 2x + 2 = 0
c. x2 + 2x − 3 = 0
d. x2 + 4x + 4 = 0
e. x2 − 6x + 10 = 0
f. x2 + 4x + 6 = 0

Question 3.
How can you derive a general formula for solving a quadratic equation?

Question 4.
Summarize the following methods you have learned for solving quadratic equations: graphing, using square roots, factoring, completing the square, and using the Quadratic Formula.

Monitoring Progress

Solve the equation using the Quadratic Formula.
Question 1.
x2 − 6x + 4 = 0

Question 2.
2x2 + 4 = −7x

Question 3.
5x2 = x + 8

Solve the equation using the Quadratic Formula.
Question 4.
x2 + 41 = −8x

Question 5.
−9x2 = 30x + 25

Question 6.
5x − 7x2 = 3x + 4

Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation.
Question 7.
4x2 + 8x + 4 = 0

qm 8.
$$\frac{1}{2}$$x2 + x − 1 = 0

Question 9.
5x2 = 8x − 13

Question 10.
7x2 − 3x = 6

Question 11.
4x2 + 6x = −9

Question 12.
−5x2 + 1 = 6 − 10x

Question 13.
Find a possible pair of integer values for a and c so that the equation ax2 + 3x + c = 0 has two real solutions. Then write the equation.

Question 14.
WHAT IF?
The ball leaves the juggler’s hand with an initial vertical velocity of 40 feet per second. How long is the ball in the air?

### Using the Quadratic Formula 3.4 Exercises

Vocabulary and Core Concept Check
Question 1.
COMPLETE THE SENTENCE
When a, b, and c are real numbers such that a ≠ 0, the solutions of the quadratic equation ax2 + bx + c = 0 are x= ____________.

Question 2.
COMPLETE THE SENTENCE
You can use the ____________ of a quadratic equation to determine the number and type of solutions of the equation.

Question 3.
WRITING
Describe the number and type of solutions when the value of the discriminant is negative.

Question 4.
WRITING
Which two methods can you use to solve any quadratic equation? Explain when you might prefer to use one method over the other.

Monitoring Progress and Modeling with Mathematics

In Exercises 5–18, solve the equation using the Quadratic Formula. Use a graphing calculator to check your solution(s).
Question 5.
x2 − 4x + 3 = 0

Question 6.
3x2 + 6x + 3 = 0

Question 7.
x2 + 6x + 15 = 0

Question 8.
6x2 − 2x + 1 = 0

Question 9.
x2 − 14x = −49

Question 10.
2x2 + 4x = 30

Question 11.
3x2 + 5 = −2x

Question 12.
−3x = 2x2 − 4

Question 13.
−10x = −25 − x2

Question 14.
−5x2 − 6 = −4x

Question 15.
−4x2 + 3x = −5

Question 16.
x2 + 121 = −22x

Question 17.
−z2 = −12z + 6

Question 18.
−7w + 6 = −4w2

In Exercises 19–26, find the discriminant of the quadratic equation and describe the number and type of solutions of the equation.
Question 19.
x2 + 12x + 36 = 0

Question 20.
x2 − x + 6 = 0

Question 21.
4n2 − 4n − 24 = 0

Question 22.
−x2 + 2x + 12 = 0

Question 23.
4x2 = 5x − 10

Question 24.
−18p = p2 + 81

Question 25.
24x = −48 − 3x2

Question 26.
−2x2 − 6 = x2

Question 27.
USING EQUATIONS
What are the complex solutions of the equation 2x2− 16x+ 50 = 0?
A. 4 + 3i, 4 − 3i
B. 4 + 12i, 4 − 12i
C. 16 + 3i, 16 − 3i
D. 16 + 12i, 16 − 12i

Question 28.
USING EQUATIONS
Determine the number and type of solutions to the equation x2 + 7x = −11.
A. two real solutions
B. one real solution
C. two imaginary solutions
D. one imaginary solution

ANALYZING EQUATIONS In Exercises 29–32, use the discriminant to match each quadratic equation with the correct graph of the related function. Explain your reasoning.
Question 29.
x2 − 6x + 25 = 0

Question 30.
2x2 − 20x + 50 = 0

Question 31.
3x2 + 6x − 9 = 0

Question 32.
5x2 − 10x − 35 = 0

ERROR ANALYSIS In Exercises 33 and 34, describe and correct the error in solving the equation.
Question 33.

Question 34.

OPEN-ENDED In Exercises 35–40, find a possible pair of integer values for a and c so that the quadratic equation has the given solution(s). Then write the equation.
Question 35.
ax2 + 4x + c = 0; two imaginary solutions

Question 36.
ax2 + 6x + c = 0; two real solutions

Question 37.
ax2 − 8x + c = 0; two real solutions

Question 38.
ax2 − 6x + c = 0; one real solution

Question 39.
ax2 + 10x = c; one real solution

Question 40.
−4x + c = −ax2; two imaginary solutions

USING STRUCTURE In Exercises 41–46, use the Quadratic Formula to write a quadratic equation that has the given solutions.
Question 41.
x = $$\frac{-8 \pm \sqrt{-176}}{-10}$$

Question 42.
x = $$\frac{15 \pm \sqrt{-215}}{22}$$

Question 43.
x = $$\frac{-4 \pm \sqrt{-124}}{-14}$$

Question 44.
x = $$\frac{-9 \pm \sqrt{137}}{4}$$

Question 45.
x = $$\frac{-4 \pm 2}{6}$$

Question 46.
x = $$\frac{2 \pm 4}{-2}$$

COMPARING METHODS In Exercises 47–58, solve the quadratic equation using the Quadratic Formula. Then solve the equation using another method. Which method do you prefer? Explain.
Question 47.
3x2 − 21 = 3

Question 48.
5x2 + 38 = 3

Question 49.
2x2 − 54 = 12x

Question 50.
x2 = 3x + 15

Question 51.
x2 − 7x + 12 = 0

Question 52.
x2 + 8x − 13 = 0

Question 53.
5x2 − 50x = −135

Question 54.
8x2 + 4x + 5 = 0

Question 55.
−3 = 4x2 + 9x

Question 56.
−31x + 56 = −x2

Question 57.
x2 = 1 − x

Question 58.
9x2 + 36x + 72 = 0

MATHEMATICAL CONNECTIONS In Exercises 59 and 60, find the value for x.
Question 59.
Area of the rectangle = 24 m2

Question 6.
Area of the triangle = 8ft2

Question 61.
MODELING WITH MATHEMATICS
A lacrosse player throws a ball in the air from an initial height of 7 feet. The ball has an initial vertical velocity of 90 feet per second. Another player catches the ball when it is 3 feet above the ground. How long is the ball in the air?

Question 62.
NUMBER SENSE
Suppose the quadratic equation ax2 + 5x + c = 0 has one real solution. Is it possible for a and c to be integers? rational numbers? Explain your reasoning. Then describe the possible values of a and c.

Question 63.
MODELING WITH MATHEMATICS
In a volleyball game, a player on one team spikes the ball over the net when the ball is 10 feet above the court. The spike drives the ball downward with an initial vertical velocity of 55 feet per second. How much time does the opposing team have to return the ball before it touches the court?

Question 64.
MODELING WITH MATHEMATICS
An archer is shooting at targets. The height of the arrow is 5 feet above the ground. Due to safety rules, the archer must aim the arrow parallel to the ground.

a. How long does it take for the arrow to hit a target that is 3 feet above the ground?
b. What method did you use to solve the quadratic equation? Explain.

Question 65.
PROBLEM SOLVING
A rocketry club is launching model rockets. The launching pad is 30 feet above the ground. Your model rocket has an initial vertical velocity of 105 feet per second. Your friend’s model rocket has an initial vertical velocity of 100 feet per second.
a. Use a graphing calculator to graph the equations of both model rockets. Compare the paths.
b. After how many seconds is your rocket 119 feet above the ground? Explain the reasonableness of your answer(s).

Question 66.
PROBLEM SOLVING
The number A of tablet computers sold (in millions) can be modeled by the function A = 4.5t2 + 43.5t + 17, where t represents the year after 2010.

a. In what year did the tablet computer sales reach 65 million?
b. Find the average rate of change from 2010 to 2012 and interpret the meaning in the context of the situation.
c. Do you think this model will be accurate after a new, innovative computer is developed? Explain.

Question 67.
MODELING WITH MATHEMATICS
A gannet is a bird that feeds on fish by diving into the water. A gannet spots a fish on the surface of the water and dives 100 feet to catch it. The bird plunges toward the water with an initial vertical velocity of −88 feet per second.

a. How much time does the fish have to swim away?
b. Another gannet spots the same fish, and it is only 84 feet above the water and has an initial vertical velocity of −70 feet per second. Which bird will reach the fish first? Justify your answer.

Question 68.
USING TOOLS
You are asked to find a possible pair of integer values for a and c so that the equation ax2 − 3x + c = 0 has two real solutions. When you solve the inequality for the discriminant, you obtain ac < 2.25. So, you choose the values a = 2 and c = 1. Your graphing calculator displays the graph of your equation in a standard viewing window. Is your solution correct? Explain.

Question 69.
PROBLEM SOLVING
Your family has a rectangular pool that measures 18 feet by 9 feet. Your family wants to put a deck around the pool but is not sure how wide to make the deck. Determine how wide the deck should be when the total area of the pool and deck is 400 square feet. What is the width of the deck?

Question 70.
HOW DO YOU SEE IT?
The graph of a quadratic function y = ax2 + bx + c is shown. Determine whether each discriminant of ax2 + bx + c = 0 is positive, negative, or zero. Then state the number and type of solutions for each graph. Explain your reasoning.

Question 71.
CRITICAL THINKING
Solve each absolute value equation.
a. |x2 – 3x – 14| = 4
b. x2 = |x| + 6

Question 72.
MAKING AN ARGUMENT
The class is asked to solve the equation 4x2 + 14x + 11 = 0. You decide to solve the equation by completing the square. Your friend decides to use the Quadratic Formula. Whose method is more efficient? Explain your reasoning.

Question 73.
ABSTRACT REASONING
For a quadratic equation ax2 + bx + c = 0 with two real solutions, show that the mean of the solutions is $$\frac{b}{2a}$$. How is this fact related to the symmetry of the graph of y = ax2 + bx + c?

Question 74.
THOUGHT PROVOKING
Describe a real-life story that could be modeled by h = −16t2 + v0t + h0 . Write the height model for your story and determine how long your object is in the air.

Question 75.
REASONING
Show there is no quadratic equation ax2+bx+c= 0 such that a, b, and c are real numbers and 3i and −2i are solutions.

Question 76.
MODELING WITH MATHEMATICS
The Stratosphere Tower in Las Vegas is 921 feet tall and has a “needle” at its top that extends even higher into the air. A thrill ride called Big Shot catapults riders 160 feet up the needle and then lets them fall back to the launching pad.

a. The height h (in feet) of a rider on the Big Shot can be modeled by h = −16t2 + v0 t + 921, where t is the elapsed time (in seconds) after launch and v0 is the initial vertical velocity (in feet per second). Find v0 using the fact that the maximum value of h is 921 + 160 = 1081 feet.
b. A brochure for the Big Shot states that the ride up the needle takes 2 seconds. Compare this time to the time given by the model h = −16t2 + v0 t + 921, where v0 is the value you found in part (a). Discuss the accuracy of the model.

Maintaining Mathematical Proficiency

Solve the system of linear equations by graphing.
Question 77.
−x + 2y = 6
x + 4y = 24

Question 78.
y = 2x − 1
y = x + 1

Question 79.
3x + y = 4
6x + 2y = −4

Question 80.
y = −x + 2
−5x + 5y = 10

Graph the quadratic equation. Label the vertex and axis of symmetry.
Question 81.
y = −x2 + 2x + 1

Question 82.
y = 2x2 − x + 3

Question 83.
y = 0.5x2 + 2x + 5

Question 84.
y = −3x2 − 2

### Lesson 3.5 Solving Nonlinear Systems

Essential Question How can you solve a nonlinear system of equations?

EXPLORATION 1

Solving Nonlinear Systems of Equations
Work with a partner. Match each system with its graph. Explain your reasoning. Then solve each system using the graph.

EXPLORATION 2

Solving Nonlinear Systems of Equations
Work with a partner. Look back at the nonlinear system in Exploration 1(f). Suppose you want a more accurate way to solve the system than using a graphical approach.
a. Show how you could use a numerical approach by creating a table. For instance, you might use a spreadsheet to solve the system.

b. Show how you could use an analytical approach. For instance, you might try solving the system by substitution or elimination.

Question 3.
How can you solve a nonlinear system of equations?

Question 4.
Would you prefer to use a graphical, numerical, or analytical approach to solve the given nonlinear system of equations? Explain your reasoning.

Solve the system using any method. Explain your choice of method.
Question 1.
y = −x2 + 4
y = −4x + 8

Question 2.
x2 + 3x + y = 0
2x + y = 5

Question 3.
2x2 + 4x − y =−2
x2 + y = 2

Solve the system.
Question 4.
x2 + y2 = 16
y = −x + 4

Question 5.
x2 + y2 = 4
y = x + 4

Question 6.
x2 + y2 = 1
y = $$\frac{1}{2}$$x + $$\frac{1}{2}$$

Solve the equation by graphing.
Question 7.
x2 − 6x + 15 = −(x − 3)2 + 6

Question 8.
(x + 4)(x − 1) = −x2 + 3x + 4

### Solving Nonlinear Systems 3.5 Exercises

Vocabulary and Core Concept Check
Question 1.
WRITING
Describe the possible solutions of a system consisting of two quadratic equations.

Question 2.
WHICH ONE DOESN’T BELONG?
Which system does not belong with the other three? Explain your reasoning.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–10, solve the system by graphing. Check your solution(s).
Question 3.
y = x + 2
y = 0.5(x + 2)2

Question 4.
y = (x − 3)2 + 5
y = 5

Question 5.
y = $$\frac{1}{3}$$x + 2
y = −3x2 − 5x − 4

Question 6.
y = −3x2 − 30x − 71
y = −3x − 17

Question 7.
y = x2 + 8x + 18
y = −2x2 − 16x − 30

Question 8.
y = −2x2 − 9
y = −4x − 1

Question 9.
y = (x − 2)2
y = −x2 + 4x − 2

Question 10.
y = $$\frac{1}{2}$$(x + 2)2
y = −$$\frac{1}{2}$$x2 + 2

In Exercises 11–14, solve the system of nonlinear equations using the graph.
Question 11.

Question 12.

Question 13.

Question 14.

In Exercises 15–24, solve the system by substitution.
Question 15.
y = x + 5
y = x2 − x + 2

Question 16.
x2 + y2 = 49
y = 7 − x

Question 17.
x2 + y2 = 64
y = −8

Question 18.
x = 3
−3x2 + 4x − y = 8

Question 19.
2x2 + 4x − y = −3
−2x + y = −4

Question 20.
2x − 3 = y + 5x2
y = −3x − 3

Question 21.
y = x2 − 1
−7 = −x2 − y

Question 22.
y + 16x − 22 = 4x2
4x2 − 24x + 26 + y = 0

Question 23.
x2 + y2 = 7
x + 3y = 21

Question 24.
x2 + y2 = 5
−x + y = −1

Question 25.
USING EQUATIONS
Which ordered pairs are solutions of the nonlinear system?
y = $$\frac{1}{2}$$x2 − 5x + $$\frac{21}{2}$$
y = −$$\frac{1}{2}$$x + $$\frac{13}{2}$$
A. (1, 6)
B. (3, 0)
C. (8, 2.5)
D. (7, 0)

Question 26.
USING EQUATIONS
How many solutions does the system have? Explain your reasoning.
y = 7x2 − 11x + 9
y = −7x2 + 5x − 3
A. 0
B. 1
C. 2
D. 4

In Exercises 27–34, solve the system by elimination.
Question 27.
2x2 − 3x −y =−5
−x + y = 5

Question 28.
−3x2 + 2x − 5 = y
−x + 2 = −y

Question 29.
−3x2 + y = −18x + 29
−3x2 − y = 18x − 25

Question 30.
y = −x2 − 6x 10
y = 3x2 + 18x + 22

Question 31.
y + 2x = −14
−x2 − y − 6x = 11

Question 32.
y = x2 + 4x + 7
−y = 4x + 7

Question 33.
y = −3x2 − 30x − 76
y = 2x2 + 20x + 44

Question 34.
−10x2 + y = −80x + 155
5x2 + y = 40x − 85

Question 35.
ERROR ANALYSIS
Describe and correct the error in using elimination to solve a system.

Question 36.
NUMBER SENSE
The table shows the inputs and outputs of two quadratic equations. Identify the solution(s) of the system. Explain your reasoning.

In Exercises 37–42, solve the system using any method. Explain your choice of method.
Question 37.
y = x2 − 1
−y = 2x2 + 1

Question 38.
y = −4x2 − 16x − 13
−3x2 + y + 12x = 17

Question 39.
−2x + 10 + y = $$\frac{1}{3}$$x2
y = 10

Question 40.
y = 0.5x2 − 10
y = −x2 + 14

Question 41.
y = −3(x − 4)2 + 6
(x − 4)2 + 2 − y = 0

Question 42.
−x2 + y2 = 100
y = −x + 14

USING TOOLS In Exercises 43–48, solve the equation by graphing.
Question 43.
x2 + 2x = −$$\frac{1}{2}$$x2 + 2x

Question 44.
2x2 − 12x − 16 = −6x2 + 60x − 144

Question 45.
(x + 2)(x − 2) = −x2 + 6x − 7

Question 46.
−2x2 − 16x − 25 = 6x2 + 48x + 95

Question 47.
(x − 2)2 − 3 = (x + 3)(−x + 9) − 38

Question 48.
(−x + 4)(x + 8) − 42 = (x + 3)(x + 1) − 1

Question 49.
REASONING
A nonlinear system contains the equations of a constant function and a quadratic function. The system has one solution. Describe the relationship between the graphs.

Question 50.
PROBLEM SOLVING
The range (in miles) of a broadcast signal from a radio tower is bounded by a circle given by the equation x2 + y2= 1620.

A straight highway can be modeled by the equation y = −$$\frac{1}{3}$$x + 30.
For what lengths of the highway are cars able to receive the broadcast signal?

Question 51.
PROBLEM SOLVING
A car passes a parked police car and continues at a constant speed r. The police car begins accelerating at a constant rate when it is passed. The diagram indicates the distance d (in miles) the police car travels as a function of time t (in minutes) after being passed. Write and solve a system of equations to find how long it takes the police car to catch up to the other car.

Question 52.
THOUGHT PROVOKING
Write a nonlinear system that has two different solutions with the same y-coordinate. Sketch a graph of your system. Then solve the system.

Question 53.
OPEN-ENDED
Find three values for m so the system has no solution, one solution, and two solutions. Justify your answer using a graph.
3y = −x2 + 8x − 7
y = mx + 3

Question 54.
MAKING AN ARGUMENT
You and a friend solve the system shown and determine that x = 3 and x = −3. You use Equation 1 to obtain the solutions (3, -3), (3, −3), (−3, 3), and (−3, −3). Your friend uses Equation 2 to obtain the solutions (3, 3) and (−3, −3). Who is correct? Explain your reasoning.
x2 + y2 = 18 Equation 1
x − y = 0 Equation 2

Question 55.
COMPARING METHODS
Describe two different ways you could solve the quadratic equation. Which way do you prefer? Explain your reasoning.
−2x2 + 12x − 17 = 2x2 − 16x + 31

Question 56.
ANALYZING RELATIONSHIPS
Suppose the graph of a line that passes through the origin intersects the graph of a circle with its center at the origin. When you know one of the points of intersection, explain how you can find the other point of intersection without performing any calculations.

Question 57.
WRITING
Describe the possible solutions of a system that contains (a) one quadratic equation and one equation of a circle, and (b) two equations of circles. Sketch graphs to justify your answers.

Question 58.
HOW DO YOU SEE IT?
The graph of a nonlinear system is shown. Estimate the solution(s). Then describe the transformation of the graph of the linear function that results in a system with no solution.

Question 59.
MODELING WITH MATHEMATICS
To be eligible for a parking pass on a college campus, a student must live at least 1 mile from the campus center.

a. Write equations that represent the circle and Oak Lane.
b. Solve the system that consists of the equations in part (a).
c. For what length of Oak Lane are students not eligible for a parking pass?

Question 60.
CRITICAL THINKING
Solve the system of three equations shown.
x2 + y2 = 4
2y = x2 − 2x + 4
y = −x + 2

Maintaining Mathematical Proficiency

Solve the inequality. Graph the solution on a number line.
Question 61.
4x − 4 > 8

Question 62.
−x + 7 ≤ 4 − 2x

Question 63.
−3(x − 4) ≥ 24

Write an inequality that represents the graph.
Question 64.

Question 65.

Question 66.

Essential Question How can you solve a quadratic inequality?

EXPLORATION 1

Work with a partner. The graphing calculator screen shows the graph of f(x) = x2 + 2x − 3.

Explain how you can use the graph to solve the inequality x2 + 2x − 3 ≤ 0.
Then solve the inequality.

EXPLORATION 2

Work with a partner. Match each inequality with the graph of its related quadratic function. Then use the graph to solve the inequality.
a. x2 − 3x + 2 > 0
b. x2 − 4x + 3 ≤ 0
c. x2 − 2x − 3 < 0
d. x2 + x − 2 ≥ 0
e. x2 − x − 2 < 0
f. x2 − 4 > 0

Question 3.
How can you solve a quadratic inequality?

Question 4.
Explain how you can use the graph in Exploration 1 to solve each inequality. Then solve each inequality.

Monitoring Progress

Graph the inequality.
Question 1.
y ≥ x2 + 2x − 8

Question 2.
y ≤ 2x2 −x − 1

Question 3.
y > −x2 + 2x + 4

Question 4.
Graph the system of inequalities consisting of y ≤ −x2 and y > x2 − 3.

Solve the inequality.
Question 5.
2x2 + 3x ≤ 2

Question 6.
−3x2 − 4x + 1 < 0

Question 7.
2x2 + 2 > −5x

Question 8.
WHAT IF?
In Example 6, the area must be at least 8500 square feet. Describe the possible lengths of the parking lot.

Vocabulary and Core Concept Check
Question 1.
WRITING
Compare the graph of a quadratic inequality in one variable to the graph of a quadratic inequality in two variables.

Question 2.
WRITING
Explain how to solve x2 + 6x − 8 < 0 using algebraic methods and using graphs.

Monitoring Progress and Modeling with Mathematics

In Exercises 3–6, match the inequality with its graph. Explain your reasoning.
Question 3.
y ≤ x2 + 4x + 3

Question 4.
y > −x2 + 4x − 3

Question 5.
y < x2 − 4x + 3

Question 6.
y ≥ x2 + 4x + 3

In Exercises 7–14, graph the inequality.
Question 7.
y < −x2

Question 8.
y ≥ 4x2

Question 9.
y > x2 − 9

Question 10.
y < x2 + 5

Question 11.
y ≤ x2 + 5x

Question 12.
y ≥ −2x2 + 9x − 4

Question 13.
y > 2(x + 3)2 − 1

Question 14.
y ≤ (x − $$\frac{1}{2}$$)2 + $$\frac{5}{2}$$

ANALYZING RELATIONSHIPS In Exercises 15 and 16, use the graph to write an inequality in terms of f (x) so point P is a solution.
Question 15.

Question 16.

ERROR ANALYSIS In Exercises 17 and 18, describe and correct the error in graphing y ≥ x2 + 2.
Question 17.

Question 18.

Question 19.
MODELING WITH MATHEMATICS
A hardwood shelf in a wooden bookcase can safely support a weight W (in pounds) provided W ≤ 115x2, where x is the thickness (in inches) of the shelf. Graph the inequality and interpret the solution.

Question 20.
MODELING WITH MATHEMATICS
A wire rope can safely support a weight W (in pounds) provided W ≤ 8000d2, where d is the diameter (in inches) of the rope. Graph the inequality and interpret the solution.

In Exercises 21–26, graph the system of quadratic inequalities.
Question 21.
y ≥ 2x2
y < −x2 + 1

Question 22.
y > −5x2
y > 3x2 − 2

Question 23.
y ≤ −x2 + 4x − 4
y < x2 + 2x − 8

Question 24.
y ≥ x2 − 4
y ≤ −2x2 + 7x + 4

Question 25.
y ≥ 2x2 + x − 5
y < −x2 + 5x + 10

Question 26.
y ≥ x2 − 3x − 6
y ≥ x2 + 7x + 6

In Exercises 27–34, solve the inequality algebraically.
Question 27.
4x2 < 25

Question 28.
x2 + 10x + 9 < 0

Question 29.
x2 − 11x ≥ −28

Question 30.
3x2 − 13x > −10

Question 31.
2x2 − 5x − 3 ≤ 0

Question 32.
4x2 + 8x − 21 ≥ 0

Question 33.
$$\frac{1}{2}$$x2 − x > 4

Question 34.
−$$\frac{1}{2}$$x2 + 4x ≤ 1

In Exercises 35–42, solve the inequality by graphing.
Question 35.
x2 − 3x + 1 < 0

Question 36.
x2 − 4x + 2 > 0

Question 37.
x2 + 8x > −7

Question 38.
x2 + 6x < −3

Question 39.
3x2 − 8 ≤ − 2x

Question 40.
3x2 + 5x − 3 < 1

Question 41.
$$\frac{1}{3}$$x2 + 2x ≥ 2

Question 42.
$$\frac{3}{4}$$x2 + 4x ≥ 3

Question 43.
DRAWING CONCLUSIONS
Consider the graph of the function f(x) = ax2 + bx + c.

a. What are the solutions of ax2 + bx + c < 0?
b. What are the solutions of ax2 + bx + c > 0?
c. The graph of g represents a reflection in the x-axis of the graph of f. For which values of x is g(x) positive?

Question 44.
MODELING WITH MATHEMATICS
A rectangular fountain display has a perimeter of 400 feet and an area of at least 9100 feet. Describe the possible widths of the fountain.

Question 45.
MODELING WITH MATHEMATICS
The arch of the Sydney Harbor Bridge in Sydney, Australia, can be modeled by y = −0.00211x2 + 1.06x, where x is the distance (in meters) from the left pylons and y is the height (in meters) of the arch above the water. For what distances x is the arch above the road?

Question 46.
PROBLEM SOLVING
The number T of teams that have participated in a robot-building competition for high-school students over a recent period of time x(in years) can be modeled by T(x) = 17.155x2 + 193.68x + 235.81, 0 ≤ x ≤ 6.
After how many years is the number of teams greater than 1000? Justify your answer.

Question 47.
PROBLEM SOLVING
A study found that a driver’s reaction time A(x) to audio stimuli and his or her reaction time V(x) to visual stimuli (both in milliseconds) can be modeled by
A(x) = 0.0051x2 − 0.319x + 15, 16 ≤ x ≤ 70
V(x) = 0.005x2 − 0.23x + 22, 16 ≤ x ≤ 70
where x is the age (in years) of the driver.
a. Write an inequality that you can use to find the x-values for which A(x) is less than V(x).
b. Use a graphing calculator to solve the inequality A(x) < V(x). Describe how you used the domain 16 ≤ x ≤ 70 to determine a reasonable solution. c. Based on your results from parts (a) and (b), do you think a driver would react more quickly to a traffic light changing from green to yellow or to the siren of an approaching ambulance? Explain.

Question 48.
HOW DO YOU SEE IT?
The graph shows a system of quadratic inequalities.

a. Identify two solutions of the system.
b. Are the points (1, −2) and (5, 6) solutions of the system? Explain.
c. Is it possible to change the inequality symbol(s) so that one, but not both of the points in part (b), is a solution of the system? Explain.

Question 49.
MODELING WITH MATHEMATICS
The length L (in millimeters) of the larvae of the black porgy fish can be modeled by L(x) = 0.00170x2 + 0.145x + 2.35, 0 ≤ x ≤ 40 where x is the age (in days) of the larvae. Write and solve an inequality to find at what ages a larva’s length tends to be greater than 10 millimeters. Explain how the given domain affects the solution.

Question 50.
MAKING AN ARGUMENT
You claim the system of inequalities below, where a and b are real numbers, has no solution. Your friend claims the system will always have at least one solution. Who is correct? Explain.
y < (x + a)2
y < (x + b)2

Question 51.
MATHEMATICAL CONNECTIONS
The area A of the region bounded by a parabola and a horizontal line can be modeled by A= $$\frac{2}{3}$$bh, where b and h are as defined in the diagram. Find the area of the region determined by each pair of inequalities.

Question 52.
THOUGHT PROVOKING
Draw a company logo that is created by the intersection of two quadratic inequalities. Justify your answer.

Question 53.
REASONING
A truck that is 11 feet tall and 7 feet wide is traveling under an arch. The arch can be modeled by y = −0.0625x2 + 1.25x + 5.75, where x and y are measured in feet.

a. Will the truck fit under the arch? Explain.
b. What is the maximum width that a truck 11 feet tall can have and still make it under the arch?
c. What is the maximum height that a truck 7 feet wide can have and still make it under the arch?

Maintaining Mathematical Proficiency

Graph the function. Label the x-intercept(s) and the y-intercept.
Question 54.
f(x) = (x + 7)(x − 9)

Question 55.
g(x) = (x − 2)2 − 4

Question 56.
h(x) = −x2 + 5x − 6

Find the minimum value or maximum value of the function. Then describe where the function is increasing and decreasing.
Question 57.
f(x) = −x2 − 6x − 10

Question 58.
h(x) = $$\frac{1}{2}$$(x + 2)2 − 1

Question 59.
f(x) = −(x − 3)(x + 7)

Question 60.
h(x) = x2 + 3x − 18

### Quadratic Equations and Complex Numbers Performance Task: Algebra in Genetics: The Hardy-Weinberg Law

3.4–3.6 What Did You Learn?

Core Vocabulary

Core Concepts
Section 3.4
Solving Equations Using the Quadratic Formula, p. 122
Analyzing the Discriminant of ax2+bx+c= 0, p. 124
Methods for Solving Quadratic Equations, p. 125
Modeling Launched Objects, p. 126

Section 3.5
Solving Systems of Nonlinear Equations, p. 132
Solving Equations by Graphing, p. 135

Section 3.6
Graphing a Quadratic Inequality in Two Variables, p. 140
Solving Quadratic Inequalities in One Variable, p. 142

Mathematical Practices
Question 1.
How can you use technology to determine whose rocket lands first in part (b) of Exercise 65 on page 129?

Question 2.
What question can you ask to help the person avoid making the error in Exercise 54 on page 138?

Question 3.
Explain your plan to find the possible widths of the fountain in Exercise 44 on page 145.

Performance Task: Algebra in Genetics: The Hardy-Weinberg Law
Some people have attached earlobes, the recessive trait. Some people have free earlobes, the dominant trait. What percent of people carry both traits?
To explore the answers to this question and more, go to BigIdeasMath.com.

### Quadratic Equations and Complex Numbers Chapter Review

3.1 Solving Quadratic Equations (pp. 93–102)

Question 1.
Solve x2 − 2x − 8 = 0 by graphing.

Solve the equation using square roots or by factoring.
Question 2.
3x2 − 4 = 8

Question 3.
x2 + 6x − 16 = 0

Question 4.
2x2 − 17x = −30

Question 5.
A rectangular enclosure at the zoo is 35 feet long by 18 feet wide. The zoo wants to double the area of the enclosure by adding the same distance x to the length and width. Write and solve an equation to find the value of x. What are the dimensions of the enclosure?

3.2 Complex Numbers (pp. 103–110)

Question 6.
Find the values x and y that satisfy the equation 36 − yi = 4x + 3i.

Perform the operation. Write the answer in standard form.
Question 7.
(−2 + 3i ) + (7 − 6i )

Question 8.
(9 + 3i ) − (−2 − 7i )

Question 9.
(5 + 6i )(−4 + 7i )

Question 10.
Solve 7x2 + 21 = 0.

Question 11.
Find the zeros of f(x) = 2x2 + 32.

3.3 Completing the Square (pp. 111–118)

Question 12.
An employee at a local stadium is launching T-shirts from a T-shirt cannon into the crowd during an intermission of a football game. The height h (in feet) of the T-shirt after t seconds can be modeled by h = −16t2 + 96t + 4. Find the maximum height of the T-shirt.

Solve the equation by completing the square.
Question 13.
x2 + 16x + 17 = 0

Question 14.
4x2 + 16x + 25 = 0

Question 15.
9x(x − 6) = 81

Question 16.
Write y = x2 − 2x + 20 in vertex form. Then identify the vertex.

3.4 Using the Quadratic Formula (pp. 121–130)

Solve the equation using the Quadratic Formula.
Question 17.
−x2 + 5x = 2

Question 18.
2x2 + 5x = 3

Question 19.
3x2 − 12x + 13 = 0

Find the discriminant of the quadratic equation and describe the number and type of solutions of the equation.
Question 20.
−x2 − 6x − 9 = 0

Question 21.
x2 − 2x − 9 = 0

Question 22.
x2 + 6x + 5 = 0

3.5 Solving Nonlinear Systems (pp. 131–138)

Solve the system by any method. Explain your choice of method.
Question 23.
2x2 − 2 = y
−2x + 2 = y

Question 24.
x2 − 6x + 13 = y
−y = −2x + 3

Question 25.
x2 + y2 = 4
−15x + 5 = 5y

Question 26.
Solve −3x2 + 5x − 1 = 5x2 − 8x − 3 by graphing.

Graph the inequality.
Question 27.
y > x2 + 8x + 16

Question 28.
y ≥ x2 + 6x + 8

Question 29.
x2 + y ≤ 7x − 12

Graph the system of quadratic inequalities.
Question 30.
x2 − 4x + 8 > y
−x2 + 4x + 2 ≤ y

Question 31.
2x2 − x ≥ y − 5
0.5x2> y − 2x− 1

Question 32.
−3x2 − 2x ≤ y + 1
−2x2 + x − 5 > −y

Solve the inequality.
Question 33.
3x2 + 3x − 60 ≥0

Question 34.
−x2 − 10x < 21

Question 35.
3x2 + 2 ≤ 5x

### Quadratic Equations and Complex Numbers Chapter Test

Solve the equation using any method. Provide a reason for your choice.
Question 1.
0 = x2 + 2x + 3

Question 2.
6x = x2 + 7

Question 3.
x2 + 49 = 85

Question 4.
(x + 4)(x − 1) = −x2 + 3x + 4

Explain how to use the graph to find the number and type of solutions of the quadratic equation. Justify your answer by using the discriminant.
Question 5.
$$\frac{1}{2}$$x2 + 3x + $$\frac{9}{2}$$ = 0

Question 6.
4x2 + 16x + 18 = 0

Question 7.
−x2 + $$\frac{1}{2}$$x + $$\frac{3}{2}$$ = 0

Solve the system of equations or inequalities.
Question 8.
x2 + 66 = 16x − y
2x − y = 18

Question 9.
y ≥ $$\frac{1}{4}$$x2 − 2
y < −(x + 3)2x − y = 18 + 4

Question 10.
0 = x2 + y2 − 40
y = x + 4

Question 11.
Write (3 + 4i )(4 − 6i ) as a complex number in standard form.

Question 12.
The aspect ratio of a widescreen TV is the ratio of the screen’s width to its height, or 16 : 9. What are the width and the height of a 32-inch widescreen TV? Justify your answer. (Hint: Use the Pythagorean Theorem and the fact that TV sizes refer to the diagonal length of the screen.)

Question 13.
The shape of the Gateway Arch in St. Louis, Missouri, can be modeled by y = −0.0063x2 + 4x, where x is the distance (in feet) from the left foot of the arch and y is the height (in feet) of the arch above the ground. For what distances x is the arch more than 200 feet above the ground? Justify your answer.

Question 14.
You are playing a game of horseshoes. One of your tosses is modeled in the diagram, where x is the horseshoe’s horizontal position (in feet) and y is the corresponding height (in feet). Find the maximum height of the horseshoe. Then find the distance the horseshoe travels. Justify your answer.

### Quadratic Equations and Complex Numbers Cumulative Assessment

Question 1.
The graph of which inequality is shown?

A. y2 > x2 + x – 6
B. y ≥ x2 + x – 6
C. y > x2 – x – 6
D. y ≥ x2 – x – 6

Question 2.
Classify each function by its function family. Then describe the transformation of the parent function.
a. g(x) = x + 4
b. h(x) = 5
c. h(x) = x2 − 7
d. g(x) = −∣x + 3∣− 9
e. g(x) = $$\frac{1}{4}$$(x − 2)2 − 1
f. h(x) = 6x+ 11

Question 3.
Two baseball players hit back-to-back home runs. The path of each home run is modeled by the parabolas below, where x is the horizontal distance (in feet) from home plate and y is the height (in feet) above the ground. Choose the correct symbol for each inequality to model the possible locations of the top of the outfield wall.(HSA-CED.A.3)

Question 4.
You claim it is possible to make a function from the given values that has an axis of symmetry at x = 2. Your friend claims it is possible to make a function that has an axis of symmetry at x = −2. What values can you use to support your claim? What values support your friend’s claim?(HSF-IF.B.4)

Question 5.
Which of the following values are x-coordinates of the solutions of the system?
y = x2 – 6x + 14
y = 2x + 7

Question 6.
The table shows the altitudes of a hang glider that descends at a constant rate. How long will it take for the hang glider to descend to an altitude of 100 feet? Justify your answer.

A. 25 seconds
B. 35 seconds
C. 45 seconds
D. 55 seconds

Question 7.
Use the numbers and symbols to write the expression x2 + 16 as the product of two binomials. Some may be used more than once. Justify your answer.

Question 8.
Choose values for the constants h and k in the equation x = $$\frac{1}{4}$$( y − k)2 + h so that each statement is true.(HSA-CED.A.2)

Question 9.
Which of the following graphs represents a perfect square trinomial? Write each function in vertex form by completing the square.

As per the student’s understanding level, only these Big Ideas Math Answers Grade 1 Chapter 3 More Addition and Subtraction Situations are designed and helping them to learn all primary mathematical concepts in a simple way. Learn the basic concepts of addition and subtraction from our Big Ideas Math Book 1st Grade Answers Chapter 3 More Addition and Subtraction Situations. Check out the topics covered in this chapter from the below section.

Lesson 1: Solve Add To Problems with Start Unknown

Lesson 2: Solve Take From Problems with Change Unknown

Lesson 3: Solve Take From Problems with Start Unknown

Lesson 4: Compare Problems: Bigger Unknown

Lesson 5: Compare Problems: Smaller Unknown

Lesson 6: True or False Equations

Lesson 7: Find Numbers That Make 10

Lesson 8: Fact Families

### More Addition and Subtraction Situations Vocabulary

Organize It

Review Words:
count on
number line

Use the review words to complete the graphic organizer.

Define It

Use your vocabulary cards to identify the words.

### Lesson 3.1 Solve Add To Problems with Start Unknown

Explore and Grow

Use counters to model each problem.

Explaination:
Given Sum is 5 and a addend of 2. So, to get the sum 5 we need to count 3 more from 2.
2 + 3 = 5.

Explaination:
Given 5 as sum and an addend 2. To get the sum 5 we need to count 3 more from 2.
3 + 2 = 5

Show and Grow

Question 1.

_______ + 5 = 8
3 + 5 = 8

Explaination:
Given: 8 as sum and an addend 5. To get the sum 8 we need to add 3 more to 5.
3 + 5 = 8

Question 2.

_______ + 2 = 8
1 + 2 = 3

Explaination:
Given: 3 as sum and an addend 2. To get the sum 3 we need to add 1 more to 2.
1 + 2 = 3

Apply and Grow: practice

Question 3.

_________ + 3 = 7

Explaination:
Given: 7 as sum and an addend 3. To get the sum 7 we need to add 4 more to 3.
4 + 3 = 7

Question 4.

_______ + 8 = 10

Explaination:
Given: 10 as sum and an addend 8.  We need to add 2 more to 8 to get 10 as sum.
2 + 8 = 10.

Question 5.
_______ + 4 = 9
5 + 4 = 9
Explaination:
Given: 9 as sum and an addend 4. To get the sum 9 we need to add 5 more to 4.
5 + 4 = 9

Question 6.
_________ + 0 = 5
5 + 0 = 5
Explaination:

Adding 0 to any number gives the sum as the number itself.
Given: 5 as sum and an addend 0.
so we add 5 to get the sum 5.

Question 7.
MP Structure
Circle the equation that matches the model.

8 + 2 = 10            2 + 6 = 8
2 + 2 = 4             4 + 6 = 10

Explaination :
There are 2 counters in 1st  box and 6 counters in 2nd box.
Total number of boxes = 2
2 + 6 = 8.

Think and Grow: Modeling Real Life

There are some ladybugs. 2 more join them. Now there are 9. How many ladybugs were there to start?

Model:

Addition equation: 7 + 2 = 9

There were 7 ladybugs at the start.
Explaination:

given

Show and Grow

Question 8.
You have some books. You get 4 more books. Now you have 10. How many books did you have to start?

Model:

_________ books

Addition equation: 6 + 4 = 10

### Solve Add To Problems with Start Unknown Practice 3.1

Question 1.

________ + 5 = 10

5 + 5 = 10

Question 2.

_________ + 1 = 4

3 + 1 = 4

Question 3.
_________ + 2 = 7
5 + 2 = 7

Question 4.
_________ + 3 = 9
6 + 3 = 9

Question 5.
MP Structure
Circle the equations that match the model.

6 + 1 = 7                      4 + 2 = 6
3 + 4 = 7                      1 + 5 = 6

Question 6.
Modeling Real Life
There are some hippos. 6 more join them. Now there are 9. How many hippos were there to start?

___________ hippos
Total number of hippos = 9
Number of hippos joined = 6
Number of hippos at the start are 3

Review & Refresh

Question 7.
There are 8 on a free.
5 fall off.
How many are left?

________ – ________ = _________
Number of leaf on the trees = 8
Number of fallen leafs = 5
Total number of leafs left on the tree=
8 – 5 = 3 leafs left.

Question 8.
You have 3 .
You lose 1 .
How many im – 19 do you have left?
________ – ________ = _________
Number of balls i have = 3
Number of balls i lose = 1
Total number of left with me =
3 – 1 = 2 balls left.

### Lesson 3.2 Solve Take From Problems with Change Unknown

Explore and Grow

Use counters to model each problem.

Show and Grow

Question 1.

7 – _____ = 2

7 – 5 = 2

Question 2.

5 – ______ = .3

5 – 2 = 3

Apply and Grow: Practice

Question 3.

8 – ______ = 2

8 – 6 = 2

Question 4.

9 – ______ = 6

9 – 3 = 6

Question 5.
3 – _______ = 0
3 – 3 = 0

Question 6.
10 – ________ = 5
10 – 5 = 5

Question 7.
MP Repeated Reasoning
Match each model with its correct equation.

Think and Grow: Modeling Real Life

You have 9 coins. You toss some of them into a fountain. You have 5 left. How many coins did you toss?

Model:

Subtraction equation

________ coins.
Total number of coins i have = 9 coins
Number of coins left with me = 5 coins
Number of coins i tossed in fountain = 9 – 5 = 4 coins

Show and Grow

Question 8.
You have 8 crayons. You lose some of them. You have 2 left. How many crayons did you lose?

Model:

Subtraction equation:

__________ crayons
Total number of crayons = 8
Number of crayons left after loosing = 2
Number of crayons left = 8 – 2 = 6

### Solve Take From Problems with Change Unknown Practice 3.2

Question 1.
8 – ? = 4
8 – ______ = 4

Question 2.
7 – ? = 3

7 – ______ = 3

Question 3.
5 – _______ = 4
5 – 1 = 4

Question 4.
6 – ______ = 6
6 – 0 = 6

Question 5.
MP Repeated Reasoning
Match each model with its correct equation.

Question 6.
Modeling Real Life
You have 10 toys. Your friend borrows some of them. You have 7 left. How many toys did your friend borrow?

________ toys

Total number of toys = 10
Number of toys left after borrowing = 7
Number of toys my friend borrowed = 10 – 7 = 3 toys.

Review & Refresh

Question 7.
You have 6
How many do you have now?
_______ + _______ = ______
Number of banana’s i have = 6
Number of banana’s bought = 3
Total number of banana’s= 6 + 3 = 9 banana’s.

Question 8.
You have .
How many do you have now?
_______ + _______ = ______
number of bat i have =1
number of more bat i bought = 1
total number of bat i have = 1 + 1 = 2 bats.

### Lesson 3.3 Solve Take From Problems with Start Unknown

Explore and Grow

Use counters to model each problem.

3 + 4 = _______

_____ – 3 = 4

3 + 4 = 7
7 – 3 = 4

Show and Grow

Question 1.
? – 2 = 4

Think 2 + 4 = _______.
So, ______ – 2 = 4

Think 2 + 4 = 6.
So, 6 – 2 = 4

Question 2.
? – 1 = 1

Think 1 + 1 = _______.
So, ______ – 1 = 1

Think 1 + 1 = 2.
So, 2 – 1 = 1.

Apply and Grow: Practice

Question 3.
? – 2 = 5

Think 2 + 5 = _______.
So, ______ – 2 = 5

Think 2 + 5 = 7.
So, 7 – 2 = 5

Question 4.
? – 3 = 6

Think 3 + 6 = _______.
So, ______ – 3 = 6.

Think 3 + 6 = 9.
So, 9 – 3 = 6.

Question 5.
? – 4 = 1
Think 4 + 1 = _______.
So, ______ – 4 = 1.
Think 4 + 1 = 5.
So, 5 – 4 = 1.

Question 6.
? – 6 = 4
Think 6 + 4 = _______.
So, ______ – 6 = 4.
Think 6 + 4 = 10.
So, 10 – 6 = 4.

Question 7.
MP Structure
Circle the equations that match the model.

8 – 2 = 6                  8 + 2 = 10
6 – 2 = 4                  2 + 6 = 8

Think and Grow: Modeling Real Life

A group of students are at a playground. 2 of them leave. There are 8 left. How many students were there to start?

Model:

Subtraction equation:

__________ students

Number of students leave = 2
Number of students remaining = 8
Total number of students = 8 + 2 = 10 students.
subtraction equation: 10 – 2 = 8.

Show and Grow

Question 8.
You have some strawberries. You eat 9 of them. You have 0 left. How many strawberries did you have to start?

Model:

Subtraction equation:

_________ strawberries

Number of strawberries i ate = 9
Number of strawberries left after eating = 0
Total number of strawberries i had at the start = 9 + 0 = 9 strawberries.
Subtraction equation: 9 – 9 = 0.

### Solve Take From Problems with Start Unknown Practice 3.3

Question 1.
? – 7 = 2

Think 7 + 2 = _______.
So, ______ – 7 = 2.

Think 7 + 2 = 9.
So, 9 – 7 = 2.

Question 2.
? – 2 = 8

Think 2 + 8 = _______.
So, ______ – 2 = 8.

Think 2 + 8 = 10.
So, 10 – 2 = 8.

Question 3.
? – 3 = 0
Think 3 + 0 = _______.
So, ______ – 3 = 0.
Think 3 + 0 = 3.
So, 3 – 3 = 0.

Question 4.
? – 2 = 0
Think 2 + 6 = _______.
So, ______ – 2 = 6.
Think 2 + 6 = 8.
So, 8 – 2 = 6.

Question 5.
MP Structure
Circle the equations that match the model.

8 – 1 = 7 9 + 1 = 10
9 – 8 = 1 8 + 1 = 9

Question 6.
Modeling Real Life
There are some people on a trolley. 4 of them exit. There are 4 people left. How many people were on the trolley to start?

__________ people

Number of people who exit the trolley = 4
Number of people left on trolley = 4
Total number of people on the trolley = 4 + 4 = 8

Review & Refresh

Question 7.
There are 5 blue balloons and 3 red balloons. How many more blue balloons are there?

________ – _________ = ________ more blue balloons
Given
Blue Balloons = 5
Red Balloons = 3
5 – 3 = 2 more blue balloons are there.

### Lesson 3.4 Compare Problems: Bigger Unknown

Explore and Grow

Use counters to model the story.

Newton has 5 balls. Descartes has 2 more balls than Newton. How many balls does Descartes have?

Number of balls with newton = 5
Number of balls with Descartes = 5 + 2 = 7.

Show and Grow

Question 1.

7 + _______ = _______ trading cards

7 + 3 = 10 trading cards.

Apply and Grow: Practice

Question 2.
Your friend has I soccer ball. You have 2 more than your friend. How many soccer balls do you have?

1 + _______ = _______ soccer balls

1 + 2 = 3 soccer balls

Question 3.
Your friend swims 4 more laps than you. You swim 3 laps. How many laps does your friend swim?

_______ + _______ = _______ laps

Question 4.
MP Precision
Your friend catches 5 more fish than you. You catch 2 fish. How many fish does your friend catch? Circle the bar model that matches the problem.

Think and Grow: Modeling Real Life

Your friend has 1 yellow flower and 2 red flowers. You have 3 more flowers than your friend. How many flowers do you have?

Model:

________ flowers
Model:

Addition equation: 3 + 3 = 6 flowers.

Show and Grow

Question 5.
Your friend has 6 gray shirts and 2 blue shirts. You have 2 more shirts than your friend. How many shirts do you have?

Model:

_________ shirts

Addition equation: 8 + 2 = 10 shirts

### Compare Problems: Bigger Unknown Practice 3.4

Question 1.
You have 3 key chains. Your friend has 5 more than you. How many key chains does your friend have?

3 + ______ = ______ key chains

3 + 5 = 8 key chains

Question 2.
You have 8 more bracelets than your friend. Your friend has 2 bracelets. How many bracelets do you have?

_______ + ______ = ______ bracelets

2 + 8 = 10 bracelets.

Question 3.
MP Precision
You have 1 seashell. Your friend has 8 more than you. How many seashells does your friend have? Circle the bar model that matches the problem.

Question 4.
Modeling Real Life
Your friend has 2 comic books and 2 mystery books. You have 3 more books than your friend. How many books do you have?

_________ books

4 + 3 = 7 books.

Review & Refresh

Question 5.
There are 2 blue crayons and 6 red crayons. How many fewer blue crayons are there?

_______ – ______ = _______ fewer blue crayons
Number of red crayons = 6
Number of blue crayons = 2
Number of fewer blue crayons = 6 – 2 = 4 blue crayons.

### Lesson 3.5 Compare Problems: Smaller Unknown

Explore and Grow

Use counters to model the story.

Newton has 5 treats. Descartes has 2 fewer treats than Newton. How many treats does Descartes have?

Number of treats with newton = 5
Number of treats with Descartes = 5 – 2 = 3.

Show and Grow

Question 1.
Your friend has 8 stones. You have I fewer than your friend. How many stones do you have?

_______ – _______ = ________ stones
_______ + _______ = ________ stones

8 – 1 = 7 stones
7 + 1 = 8 stones

Apply and Grow Practice

Question 2.
You blow 5 bubbles. Your friend blows 2 fewer than you. How many bubbles does your friend blow?

_______ – _______ = ________ bubbles
_______ + _______ = ________ bubbles

5 – 2 = 3 bubbles
3 + 2 = 5 bubbles.

Question 3.
You have 3 fewer oranges than your friend. Your friend has 9 oranges. How many oranges do you have?

_______ ○ ______ = _______ oranges

9 – 3 = 6 oranges

Question 4.
DIG DEEPER!
Complete the bar model. Do both equations match the bar model?

Both the equations matches.

Think and Grow: Modeling Real Life

Your friend has 2 blue markers and 7 yellow I markers. You have 5 fewer markers than your friend. How many markers do you have?

Model:

Equation:

_________ markers

Equation: 9 – 5 = 4
I have 4 markers

Show and Grow

Question 5.
Your friend has 6 tennis balls and I baseball. You have 2 fewer balls than your friend. How many balls do you have?

Model:

Equation:

_________ balls

Equation: 7 – 2 = 5

I have 5 balls

### Compare Problems: Smaller Unknown Practice 3.5

Question 1.
Your friend has 9 awards. You have 5 fewer than your friend. How many awards do you have?

______ – _____ = _____ awards
______ + _____ = _____ awards

9 – 5 = 4 awards
4 + 5 = 9 awards

Question 2.
Your friend finds 2 fewer bugs than you. You find 4 bugs. How many bugs does your friend find?

_______ ○ ______ = _______ bugs

4 – 2 = 2 bugs
My friend found 2 bugs.

Question 3.
MP Reasoning
Complete the bar model. Circle the equation that matches the bar model.

Question 4.
Modeling Real Life
Your friend has 8 black cats and 2 orange cats. You have 7 fewer cats than your friend. How many cats do you have?

_______ cats

Number of cats my friend have= 8 + 2 = 10 cats
Number of cats i have = 10 – 7 = 3

Review & Refresh

Question 5.
Write the numbers of shirts and shorts. Are the numbers equal? Circle the thumbs up for yes or the thumbs down for no.

### Lesson 3.6 True or False Equations

Explore and Grow

Color the flowers that have a sum or difference of 6.

Show and Grow

Is the equation true or false?

Question 1.

Question 2.

Apply and Grow: Practice

Is the equation true or false?

Question 3.

Question 4.

Question 5.

10 – 4 = 6
6 – 0 = 6

Question 6.

5- 2 = 3
7 – 4 = 3

Question 7.

Question 8.

Question 9.
MP Number Sense
Circle all of the equations that are true.

Think and Grow: Modeling Real Life

You have 7 marbles. You lose 2 of them. Your friend has 4 marbles and finds 3 more. Do you and your friend have the same number of marbles?

Equation:

Number of marbles I have = 7 – 2 = 5
Number of marbles my friend have = 4 + 3 = 7
Equation:

Show and Grow

Question 10.
You have I balloon. You blow up 3 more. Your friend has 5 balloons. I of your friend’s balloons pops. Do you and your friend have the same number of balloons?

Equation:

Number of balloons i have= 1 + 3 = 4
Number of balloons my friend have = 5 – 1 = 4
Equation:

### True or False Equations Practice 3.6

Is the equation true or false?

Question 1.

Question 2.

Question 3.

Question 4.

Question 5.
MP Number Sense
Circle all of the equations that are false.

Question 6.
Modeling Real Life
You have 5 crayons. You find 3 more. Your friend has 7 crayons and finds I more. Do you and your friend have the same number of crayons?

Number of crayons with me = 5 + 3 = 8
Number of crayons with my friend = 7 + 1 = 8

Review & Refresh

Question 7.
Circle the triangles.

### Lesson 3.7 Find Numbers That Make 10

Explore and Grow

Place some red counters on the ten frame. Add yellow counters to fill the frame. Write an equation to match.

Show and Grow

Question 1.

Question 2.

Question 3.

Question 4.

Apply and Grow Practice

Question 5.

4 + _____ = 10

Question 6.

7 + _____ = 10

Question 7.
1 + _____ = 10
1 + 9 = 10

Question 8.
8 + _____ = 10
8 + 2 = 10

Question 9.
_____ + 2 = 10
8 + 2 = 10

Question 10.
_____ + 5 = 10
5 + 5 = 10

Question 11.
_____ + 3 = 10
7 + 3 = 10

Question 12.
_____ + 0 = 10
10 + 0 = 10

Question 13.
DIG DEEPER!
Match the numbers that have a sum of 10.

Think and Grow: Modeling Real Life

Model:

________ jump ropes
Number of jump ropes= 7
Teacher bought some ropes and total = 10
Number of ropes teacher brought = 10 – 7 = 3

Addition equation: 7 + 3 = 10

3 jump ropes.

Show and Grow

Question 14.
There are 2 penguins. Some more join them. Now there are 10. How many more penguins joined them?

Model:

________ penguins
Number of penguins at present= 2
Total number of penguins = 10
Number of penguins joined = 10 – 2 = 8

Addition equation: 2 + 8 = 10

8 penguins joined.

### Find Numbers That Make 10 Practice 3.7

Question 1.

8 + _______ = 10

Question 2.

1 + _______ = 10

Question 3.
10 + _______ = 10
10 + 0 = 10

Question 4.
5 + _______ = 10
5 + 5 = 10

Question 5.
_______ + 4 = 10
6 + 4 = 10

Question 6.
_______ + 0 = 10
10 + 0 = 10

Question 7.
DIG DEEPER!
Match the numbers that have a sum of 10.

Question 8.
Modeling Real Life
You have 3 baseball cards. Your friend gives you some more. Now you have 10. How many baseball cards did your friend give you?

________ baseball cards
Number of baseball cards with me= 3
Total number of baseball cards i have = 10
Number of baseball cards given by my friend = 10 -3 = 7 cards.

Review & Refresh

Find the sum. Then change the order of the addends. Write the new addition problem.

Question 9.

Question 10.

### Lesson 3.8 Fact Families

Explore and Grow

Use linking cubes to model the equations.

Show and Grow

Question 1.
Complete the fact family.

1 + 8 = ______                          9 – ______ = ______

______ + ______ = ______            9 – ______ = ______

1 + 8 = 9                         9 – 8 = 1

8 + 1 = 9                         9 – 1 = 8

Apply and Grow: Practice

Complete the fact family.

Question 2.

4 + 2 = ______                              6 – ______ = ______

______ + ______ = ______                6 – ______ = ______

4 + 2 = 6                             6 – 2 = 4

2 + 4 = 6                             6 – 4 = 2

Question 3.

3 + 6 = ______                                 ______ – 6 = 3

______ + ______ = ______                   ______ – ______ = ______

3 + 6 = 9                                9 – 6 = 3

6 + 3 = 9                                9 – 3 = 6

Question 4.
7 + 1 = ______                                   ______ – 7 = _______

______ + ______ = ______                    ______ – 1 = ______

7 + 1 = 8                                   8 – 7 = 1

1 + 7 = 8                                   8 – 1 = 7

Question 5.
DIG DEEPER!
Cross out the equation that does not belong in the fact family.
5 + 3 = 8             5 – 3 = 2
3 + 5 = 8             8 – 5 = 3
5 + 3 = 8
3 + 5 = 8             8 – 5 = 3

Think and Grow: Modeling Real Life

You have 3 puppets. Your friend has 7 puppets. How many fewer puppets do you have?

Model:

Equation:

________ fewer puppets
Number of puppets i have = 3
Number of puppets my friend have = 7
Difference between puppets between me and my friend:

Equation: 7 – 3 = 4

I have 4 fewer puppets.

Show and Grow

Question 6.
There are 2 spoons and 8 forks. How many more forks are there?

Model:

Equation:

________ more forks
Number of spoons = 2
Number of forks = 8
difference between spoons and folks:
Equation: 8 – 2 = 6

There are 6 more forks.

### Fact Families Practice 3.8

Question 1.

1 + 5 = ______                          6 – ______ = _______

______ + ______ = ______            6 – ______ = ______

1 + 5 = 6                         6 – 5 = 1

5 + 1 = 6                         6 – 1 = 5

Question 2.
3 + 7 = ______                          ______ – 7 = 3

______ + ______ = ______            ______ – _______ = ______

3 + 7 = 10                          10 – 7 = 3

7 + 3 = 10                          10 – 3 = 7

Question 3.
Complete the fact family.
3 + 0 = ______                           ______ – 3 = _______

______ + ______ = ______             ______ – 0 = ______

3 + 0 = 3                           3 – 3 = 0

0 + 3 = 3                           3 – 0 = 3

Question 4.
DIG DEEPER!
Cross out the equation that does not belong in the fact family.
6 + 4 = 10               10 – 6 = 4
4 + 6 = 10                6 – 4 = 2
6 + 4 = 10               10 – 6 = 4
4 + 6 = 10

Question 5.
Modeling Real Life
There are 7 fish and 2 frogs. How many fewer frogs are there?

________ fewer frogs
Number of fish = 7
Number of frogs = 2
Difference between frogs and fish = 7 – 2 = 5.
There are 5 fewer frogs.

Review & Refresh

Circle the objects that holds more.

Question 6.

Question 7.

Question 1.

_______ loaf
Total number of bread loaves i have = 8.
Number of raisin breads i have = 3.
Number of banana breads i have = 8 – 3 = 5.
Total number of bread loaves my friend have = 10.
Number of banana bread my friend have = 6.
Difference between the banana breads is : 6 – 5 = 1.
My friend have 1 more loaves of banana bread than me.

Question 2.
You give away 3 loaves of banana bread and 3 loaves of raisin bread. Your friend gives away 1 more loaf of bread than you. How many loaves of bread does your friend give away?

_______ loaves
Number of banana bread loaves i gave away = 3
Number of raisin bread loaves i gave away = 3
Total number of bread loaves i gave away = 3 + 3 = 6.
The number of bread loaf my friend gave away is 1 more then = 6 + 1 = 7.
My friend gave away 7 loaves of bread.

Question 3.
You and your friend make boxes of muffins. Does each box have the same number of muffins?

Yes              No
Number of muffins in the boxes i make = 3 + 7 = 10.
Number of muffins in the boxes my friend make = 5 + 4 = 9.
Difference between the muffins = 10 – 9 = 1
NO  the boxes does not have same number of muffins.

Question 1.
? + 4 = 6

_______ + 4 = 6

2 + 4 = 6

Question 2.
? + 2 = 8

_______ + 2 = 8

6 + 2 = 8

Question 3.
5 – ? = 4

5 – ______ = 4

5 – 1 = 4

Question 4.
7 – ? = 7

7 – ______ = 7

7 – 0 = 7

### Solve Take From Problems with Start Unknown Homework & Practice 3.3

Question 5.
? – 6 = 3

Think 6 + 3 = ______.
So, ______ – 6 = 3.

Think 6 + 3 = 9.
So, 9 – 6 = 3.

Question 6.
? – 3 = 1

Think 3 + 1 = ______.
So, ______ – 3 = 1.

Think 3 + 1 = 4.
So, 4 – 3 = 1.

Question 7.
MP Structure
Circle the equation that matches the model.

6 – 6 = 0 4 – 2 = 2 6 – 2 = 4

### Compare Problems: Bigger Unknown Homework & Practice 3.4

Question 8.
Your friend has 3 stickers. You have 4 more than your friend. How many stickers do you have?

### Compare Problems: Smaller Unknown Homework & Practice 3.5

Question 9.
Your friend has 5 stuffed animals. You have 2 fewer than your friend. How many stuffed animals do you have?

Question 10.
Modeling Real Life Your friend has 4 dogs and 2 cots. You have I fewer pet than your friend. How many pets do you have?

### True or False Homework & Practice 3.6

Is the equation true or false?

Question 11.

Question 12.

Question 13.
MP Number Sense
Circle all of the equations that are true.

Question 14.

3 + _______ = 10

Question 15.

6 + _______ = 10

### Fact Families Homework & Practice 3.8

Question 16.
Complete the fact family.
8 + 1 = _______                         ________ – 8 = 1
________ + _______ = _______      _______ – _______ = ______
8 + 1 = 9                         9 – 8 = 1
1 + 8 = 9                         9 – 1 = 8

Question 17.
Modeling Real Life
There are 2 slides and 6 swings on a playground. How many more swings are there?

__________ more swings
Number of slides = 2
Number of swings = 6
Number of more swings = 6 – 2 = 4 more swings.

### More Addition and Subtraction Situations Cumulative Practice 1 – 3

Question 1.
4 + 4 = ______
○ 4
○ 6
○ 9
○ 8

Question 2.
Shade the circle next to the addition equation that you can use to solve 8 – 3.
○ 8 + 3 = 11
○ 3 + 5 = 8
○ 1 + 8 = 9
○ 5 + 2 = 7

Question 3.
Circle the equation that matches the bar model.

Question 4.
You take 10 pictures. Your friend takes 3 pictures. Shade the circle next to the equation that shows how many more pictures you take.

○ 3 + 3 = 6
○ 10 – 3 = 7
○ 10 + 3 = 13
○ 3 – 1 = 2

Question 5.
Shade the circle next to the number that completes the addition equation.
________ + 7 = 9

○ 1
○ 2
○ 3
○ 4

Question 6.
There are 6 .
3 more join them.
How many are there now?
_________ + __________ = _________
Number of = 6
Total number of = 6 + 3 = 9

Question 7.
Is each equation true or false?

Question 8.
Use the picture to write a subtraction equation.

________ – 0 = _______

5 – 0 = 5

Question 9.
You have 8 beads. 5 are orange. The rest are blue. Shade the circles next to the equations that describe the beads.

○ 3 + 5 = 8
○ 8 – 2 = 6
○ 4 + 4 = 8
○ 8 – 5 = 3

Question 10.
Use the numbers shown to write two equations.

Given   3     8     5
3 + 5 = 8      5 + 3 = 8

Question 11.
Shade the circle next to the equation that does not belong in the fact family.

○ 3 + 1 = 4
○ 4 – 3 = 1
○ 1 + 3 = 4
○ 3 – 1= 2

Question 12.
There are 3 rabbits. 3 more join them. Shade the circle next to the equation that shows how many rabbits there are in all.

○ 3 + 2 = 5
○ 3 + 4 = 7
○ 4 + 1 = 5
○ 3 + 3 = 6

Conclusion:

Hope you are all satisfied with the answers provided in the Bigideas Math Grade 1 Chapter 3 More Addition and Subtraction Situations. Get different methods to solve the problems in our Big Ideas Math Solution Key 1st Grade Chapter 3 More Addition & Subtraction Situations. Thus, the students who are interested to learn the basics quickly can follow the methods given here.

## Big Ideas Math Answers Grade 1 Chapter 6 Count and Write Numbers to 120

Download Big Ideas Math Answers Grade 1 Chapter 6 Count and Write Numbers to 120 pdf for free from here. The solutions for all the questions are given in a simple manner by subject experts. The Chapter Count and Write Numbers to 120 includes Count to 120 by Ones, Count to 120 by Tens, Count and Write Numbers to 120, etc. This chapter discusses the methods to count the numbers to 120. Go through the below sections to find topic-wise links.

## Big Ideas Math Book 1st Grade Answer Key Chapter 6 Count and Write Numbers to 120

To make your preparation perfect you have to follow the best guide. Big Ideas Math Book 1st Grade Answer Key Chapter 6 Count and Write Numbers to 120 is the best study material which consists of clear-cut explanations for all the problems. So, access the links and start your preparation for the exams.

Vocabulary

Lesson: 1 Count to 120 by Ones

Lesson: 2 Count to 120 by Tens

Lesson: 3 Compose Numbers 11 to 19

Lesson: 4 Tens

Lesson: 5 Tens and Ones

Lesson: 6 Make Quick Sketches

Lesson: 7 Understand Place Value

Lesson: 8 Write Numbers in Different Ways

Lesson: 9 Count and Write Numbers to 120

Chapter: 6 – Count and Write Numbers to 120

### Count and Write Numbers to 120 Vocabulary

Organize It

Review Words:
column
hundred chart
row

Use the review words to complete the graphic organizer.

Explanation:
The given graphic organizer title is called as hundred chart,
and numbers 41 to 50 are called as row, numbers 4,14,24
to 84, 94 are called as column and numbers 10,20,30 to 90,100

Define It

Use your vocabulary cards to identify the words.

Explanation:
Given in the picture to identify words 2 – Two and 3 – Three.

### Lesson 6.1 Count to 120 by Ones

Explore and Grow

Point to each number as you count to 120. Color the first two rows and the last two rows. How are the rows the same? How are they different?

Explanation:
As we move from 1,2,3 count to 120  have pointed with an arrow for each number. Colored with green first two rows from 1 to 10, 11 to 20 and the last two rows from 101 to 110, 111 to 120.
The rows are same – as we move from number 1 to 120 from first number to second number the number is increased by 1 as 1, 1 + 1 = 2, 2 + 1 = 3 and so on 119 + 1 = 120 means the next number is addition of 1 plus the previous number.
The rows are different – as we see the numbers are not the same they differ its 1,2 ,3 to 119, 120 from 1 to 120.

Show and Grow

Count by ones to write the missing numbers.

Question 1.
82, _____83_____, ____84____, _____85_____, ____86______, ____87______
The missing numbers after 82 are  83, 84, 85, 86 ,87

Explanation:
Given the starting number as 82 we count by ones and
write the next missing numbers as  82 + 1 = 83,
83 + 1 = 84, 84 + 1 = 85 , 85 +1 = 86, 86 + 1 = 87,
now the missing numbers after 82  are 83, 84, 85, 86 ,87.

Question 2.
103, ____104______, ____105______, ___106_______, ____107______, ____108______
The missing numbers after 103 are 104,105,106,107,108.

Explanation:
Given the starting number as 103 we count by ones and
write the next missing numbers as 103 + 1  = 104,
104 + 1 =105, 105 + 1 = 106, 106 + 1 = 107,107 + 1 = 108,
now the missing numbers after 103 are 104,105,106,107,108.

Apply and Grow: Practice

Count by ones to write the missing numbers.

Question 3.
56, ____57______, ____58____, ____59______, ____60______, ____61______
The missing numbers after 56 are 57,58,59,60,61

Explanation:
Given the starting number as 56 we count by ones and
write the next missing numbers as 56 + 1 = 57, 57 + 1 = 58,
58 + 1 = 59, 59 + 1 = 60, 60 + 1 = 61,
now the missing numbers after 56 are 57,58,59,60,61.

Question 4.
98, _____99_____, ____100______, ____101______, ____102______, ____103______
The missing numbers after 98 are 99,100,101,102,103

Explanation:
Given the starting number as 98 we count by ones and
write the next missing numbers as 98 + 1 = 99,99 + 1 = 100,
100 + 1 = 101, 101 + 1 = 102, 102 + 1 = 103,
So the missing numbers after 98 are 98,99,100,101,102,103.

Question 5.
115, ____116______, ____117______, ____118______, ____119______, ___120_______
The missing numbers after 115 are 116,117,118,119,120

Explanation:
Given the starting number as 115 we count by ones and
write the next missing numbers as 115 + 1 = 116,
116 + 1 = 117, 117 + 1 = 118, 118 + 1 = 119, 119 + 1 = 120,
So the missing numbers after 115 are 116,117,118,119,120.

Question 6.
___40_______, ____41______, 42, ____43______, ____44______, ____45______
The missing numbers before and after 42 are 40,41,43,44,45

Explanation:
Given number is 42 and 2 numbers before 42 are
count 1 before 42 is 42- 1= 41 and count 1 before 41 is
41 – 1 = 40 and count 1 after 42 is 42 + 1 = 43, 43 + 1 = 44,
44 + 1 = 45, So the missing numbers before and after 42 are 40,41,43,44,45.

Write the missing numbers in the chart.

Question 7.

The missing numbers in the chart are 33,35,42,44.

Explanation:
The missing number after 32 is 32 + 1 = 33,
the number after 34 is 34 + 1 = 35,
the number before 43 is 43 – 1 = 42 and
number after 43 is 43 + 1 = 44, So the missing
numbers in the chart are 33,35,42,44.

Question 8.

The missing numbers in the chart are 81,83,84,92,94.

Explanation:
The missing number before 82 is 82 – 1 = 81,
the number after 82 is 82 + 1 = 83,now number
after 83 is 83 + 1 = 84.
The number after 91 is 91 + 1 = 92,
the number after 93 is 93 + 1 = 94, therefore the
missing numbers in the chart are 81,83,84,92,94.

Question 9.
MP Structure
Structure Write a number between 95 and 105. Then count by ones to write the next 7 numbers.
____96___, _____97____, _____98_____, ____99____, ___ 100___, ____101___, ___102__, ___103___
Let the number be 96 which is between 95 and 105 and
the next 7 numbers after 96 are 97,98,99,100,101,102,103.

Explanation:
Given to write a number between 95 and 105,
Let us take the number as 96, now we will write the
next 7 numbers are 96 + 1 = 97,97+ 1 = 98, 98 + 1 =99,
99 + 1 = 100,100 + 1 = 101, 101 + 1 = 102, 102 + 1 = 103,
therefore the next 7 numbers after 96 are 97,98,99,100,101,102,103.

Think and Grow: Modeling Real Life

You hove 108 bouncy balls. You want 112. How many more bouncy bails do you need?
Draw more bails to show 112:

______4______ more bouncy balls
4 more bouncy  balls are needed,

Explanation:
Given I have 108 bouncy balls and I want 112,
So I need 108 + 1 = 109, 109 + 1 = 110, 110 + 1 = 111,
111+ 1 = 112 or 112 – 108 = 4, so we need 4 more bouncy  balls.

Show and Grow

Question 10.
You have 66 rocks. You want 75. How many more rocks do you need?
Draw more rocks to show 75:

______9_______ more rocks

I need 9 more rocks.

Explanation:
Given I have 66 rocks and I want 75,
So more rocks I needed are 75 – 66 = 9.

### Count to 120 by Ones Practice 6.1

Count by ones to write the missing numbers.

Question 1.
57, _____58_____, _____59_____, ____60______, ____61______, ____62______
The missing numbers after 57 are 58,59,60,61,62.

Explanation:
Given the starting number as 57 we count by ones and
write the next missing numbers as 57 + 1 = 58,
58 + 1 = 59, 59 + 1 = 60, 60 + 1 = 61, 61 + 1 = 62,
So the missing numbers after 57 are 58,59,60,61,62.

Question 2.
109, ____110___, ____111___, ____112___, ____113___, __114_____
The missing numbers after 109 are 110,111,112,113,114

Explanation:
Given the starting number as 109 we count by ones and
write the next missing numbers as 109 + 1 = 110,
110 + 1 =111,111 + 1 = 112, 112 + 1 = 113, 113 + 1 = 114,
So the missing numbers after 109 are 110,111,112,113,114.

Question 3.
40, ____41___, ____42___, ___43___, ___44___, ___45___
The missing numbers after 40 are 41,42,43,44,45

Explanation:
Given the starting number as 40 we count by ones and
write the next missing numbers as 40 + 1 = 41,
41 + 1 = 42,42 + 1 = 43, 43 + 1 = 44, 44 + 1 = 45,
So the missing numbers after 40 are 40,41,42,43,44,45.

Question 4.
_____97_____, ____98______, ____99______, 100, ____101______, ____102______
The missing numbers before and after 100 are 97,98,99,101,102

Explanation:
Given number is 100 and 3 numbers before 100 are
count 1 before 100 is 100- 1= 99, count 1
before 99 is 99 – 1 = 98 and count 1 before 98 is
98 – 1 = 97, numbers after 100 are 100 + 1 = 101 and
101 + 1 = 102,therefore the missing numbers
before and after 100 are 97,98,99,101,102.

Write the missing numbers in the chart.

Question 5.

The missing numbers in the chart are 21,23,30,32.

Explanation:
The missing number after 20 is 20 + 1 = 21,
the number after 22 is 22 + 1 = 23,
the number before 31 is 31 – 1 = 30 and
number before 33 is 33 – 1 = 32, So the missing
numbers in the chart are 21,23,30,32.

Question 6.

The missing numbers in the chart are 102,104,105,113,115.

Explanation:
The missing number before 103 is 103 – 1 = 102,
the number after 103 is 103 + 1 = 104,
the number after 104 is 104 + 1 = 105,
the number after 112 is 112 + 1 = 113 and
the number after 114 is 114 + 1 = 115
So the missing numbers in the chart are 102,104,105,113,115.

Question 7.
MP Structure
Write a number between 85 and 95. Then count by ones to write the next 7 numbers.

__87__, _88___, _89___, __90___, _91__, _92_, _93_, _94__
Let the number be 87 which is between 85 and 95 and
the next 7 numbers after 87 are 88,89,90,91,92,93,94.

Explanation:
Given to write a number between 85 and 95,
Let us take the number as 87, now we will write the
next 7 numbers are 87 + 1 = 88,88+ 1 = 89, 89 + 1 =90,
90 + 1 = 91,91 + 1 = 92, 92 + 1 = 93, 93 + 1 = 94,
therefore the next 7 numbers after 87 are 88,89,90,91,92,93,94.

Question 8.
Modeling Real Life
There are 110 tokens. You want 119. How many more tokens do you need?

____9_____ more tokens
I need 9 more tokens.

Explanation :
Given there are 110 tokens. and I want 119,
so I needed 119 – 110 = 9 more tokens.

Review & Refresh

Question 9.
? – 6 = 4

Think 6 + 4 = ___10_____ .
So, ______10__ – 6 = 4.

6 + 4 = 10
so, 10 – 6 = 4
Explanation:
Given ___ – 6 = 4, Let us take the missing number as X,
therefore X – 6 = 4, X = 4 + 6 =10,the equation is 10 – 6 = 4.

### Lesson 6.2 Count to 120 by Tens

Explore and Grow

Count to 10. Circle the number. Count 10 more. Circle the number. Continue until you reach 120.

Explanation:

Start counting from 1 to 10 circled, Next from 11 to 20 counted
circled, counted till 120 and circled.

Show and Grow

Count by tens to write the missing numbers.

Question 1.
70, __80___, __90__, __100__, _110__, __120__
The missing numbers after 70 are 80,90,100,110,120

Explanation:
Started from 70 counting by 10’s so 70 + 10 = 80,
80 + 10 = 90, 90 + 10 = 100, 100 + 10 = 110, 110 + 10 = 120,
So the missing numbers after 70 are 80,90,100,110,120

Question 2.
31, __41___, __51__, _61__, __71__, _81__
The missing numbers after 31 are 41,51,61,71,81

Explanation:
Started from 31 counting by 10’s so 31 + 10 = 41,
41 + 10 = 51, 51 + 10 =61, 61 + 10 = 71 ,71 +10 = 81,
So the missing numbers after 31 are 41,51,61,71,81.

Apply and Grow: Practice

Count by tens to write the missing numbers

Question 3.
62, __72__, __82__, _92__, _102__, _112___
The missing numbers after 62 are 72,82,92,102,112

Explanation:
Started from 62 counting by 10’s are 62 + 10 = 72,
72 + 10 = 82, 82 + 10 =92, 92 + 10 = 102 ,102 +10 = 112,
So the missing numbers after 62 are 72,82,92,102,112

Question 4.
43, __53__, _63__, __73__, __83__, _93__
The missing numbers after 43 are 53,63,73,83,93

Explanation:
Started from 43 counting by 10’s are 43 + 10 = 53,
53 + 10 = 63, 63 + 10 =73, 73 + 10 = 83 ,83 +10 = 93,
So the missing numbers after 43 are 53,63,73,83,93.

Question 5.
___10__, __20___, 30, __40__, __50___, __60__
The missing numbers before and after 30 are 10,20,40,50,60

Explanation:
Started from 30 counting by 10’s numbers before and after 30 are
before 30 – 10 = 20, 20 – 10 = 10 and after 30 are 30 + 10 = 40,
40 + 10 = 50, 50 + 10 = 60, therefore numbers before and after 30
are 10,20,40,50,60.

Question 6.
Write the missing numbers from the chart. Then count on by tens to write the next three numbers.

The missing numbers in the chart are 71,81 and
count on 10’s the next three numbers after 81 are 91,101,111.

Explanation:
Given numbers in the chart, the number missing before 72
is 72 – 1 = 71 and number missing before 82 in the chart is
82 – 1 = 81,now counting by 10’s next three numbers after 81 are
81 + 10 = 91,91 + 10 = 101,101 + 10 = 111.
Therefore the missing numbers in the chart are 71,81 and
count on 10’s the next three numbers after 81 are 91,101,111.

Question 7.
YOU BE THE TEACHER
Your friend counts by tens starting with 27. Is your friend correct? Show how you know.
27, 37, 47, 67, 77, 87
No, Friend is incorrect. As 57 is missing.

Explanation:
Given friend counts by tens starting with 27 and
gets 27, 37, 47, 67, 77, 87 but after 47 we should
get 47 + 10 =  57 as 57 is missing friend is incorrect.

Think and Grow: Modeling Real Life

You have 50 points. On your next turn, you knock over 6 cans. How many points do you have now?

Write the numbers:

____110______ points
I have 110 points now.

Explanation:
Given I have 50 points on my next turn I knock
over 6 cans and 10 points for each knocked over can
means I got 6 X 10 = 60 points, So in total I have
50 + 60 = 110 points now.

Show and Grow

Question 8.
You have 21 points. On your next turn, 3 beanbags land in the circle. How many points do you have now?

Write the numbers:

_____51_____ points
I have 51 points now.

Explanation:
Given I have 21 points on my next turn 3 beanbags
land in the circle 10 points for each beanbag
means I got 3 X 10 = 30 points, So in total I have
21 + 30 = 51 points now.

### Count to 120 by Tens Practice 6.2

Count by tens to write the missing numbers.

Question 1.
69, __79_, _89__, _99_, _109_, _119__
The missing numbers after 69 are 79,89,99,109,119

Explanation:
Started from 69 counting by 10’s numbers after 69 are
69 + 10 = 79, 79 + 10 = 89, 89 + 10 = 99, 99 + 10 = 109,
109 + 10 = 119, therefore the missing numbers after 69
are 79,89,99,109,119.

Question 2.
41, __51__, _61___, _71__, _81__, _91__
The missing numbers after 41 are 51,61,71,81,91

Explanation:
Started from 41 counting by 10’s numbers after 41 are
41 + 10 = 51,51 + 10 = 61, 61 + 10 = 71, 71 + 10 = 81,
81 + 10 =91, So the missing numbers after 41 are 51,61,71,81,91.

Question 3.
16, __26__, _36__, _46__, _56__, __66__
The missing numbers after 16 are 26,36,46,56,66

Explanation:
Started from 16 counting by 10’s numbers after 16 are
16 + 10 = 26, 26 + 10 = 36,36 + 10 = 46, 46 + 10 = 56,
56 + 10 = 66, So the missing numbers after 16 are 26,36,46,56,66.

Question 4.
_64__, __74__, __84__, 94, __104_, __114__
The missing numbers before 94 are 84,74 ,64 and
after 94 are 104, 114.

Explanation:
Started from 94 counting by 10’s numbers before and after 94 are
before 94 – 10 = 84, 84 – 10 = 74 , 74 – 10 = 64 and after are
94 + 10 = 104,104 + 10 = 114, So the missing numbers before
94 are 84,74 ,64 and after 94 are 104, 114.

Question 5.
Write the missing numbers from the chart. Then count on by tens to write the next three numbers.

The missing numbers in the chart are 6,16 and
count on 10’s the next three numbers after 16 are 26,36,46.

Explanation:
Given numbers in the chart, the number missing after 5
is 5 + 1  = 6 and number missing after 15 in the chart is
15 + 1 = 16,now counting by 10’s next three numbers after 16 are
16 + 10 = 26,26 + 10 = 36,36 + 10 = 46.
Therefore the missing numbers in the chart are 6,16 and
count on 10’s the next three numbers after 16 are 26,36,46.

Question 6.
DIG DEEPER!
You count to 50. You only count 5 numbers. Did you count by ones or by tens? Show how you know.

I count by tens.

Explanation:
Given I count to 50 and count only 5 numbers , If I count by ones
I will get 50 numbers like 1,1+1=2, 2+ 1=3,…till I reach 49+ 1 = 50
as shown in the above picture, If I count by tens I get 5 numbers,
Like 10,10 + 10 = 20,20 + 30 = 40, 40 + 10 = 50, as I count 5 numbers
So I count by tens.

Question 7.
Modeling Real Life
You have 30 points. On your next turn, 4 balls stick to the wall. How many points do you have now?

____70_____ points
I have 70 points now.

Explanation:
Given I have 30 points. On my next turn,
4 balls stick to the wall and each ball we have 10 points,
So 4 X 10 = 40 points means in total I have 30 + 40 = 70 points now.

Review & Refresh

Question 8.
3 + 1 = ___4_____
3 + 1 = 4

Explanation:
We are going to add 1 to 3 we get 4.

Question 9.
5 – 1 = ____4_____
5 – 1 = 4

Explanation:
We will subtract 1 from 5 we get 4.

### Lesson 6.3 Compose Numbers 11 to 19

Explore and Grow

Color to show 13 and 17. What is the same about the numbers? What is different?

Both are same- 13, 17 are prime numbers.
Both 13 and 17 are different as
both are not same numbers, they both are not equal.

Explanation:
Taken numbers from 1 to 20 in the chart,
colored numbers 13 and 17 with green.
Both are the same- 13 and 17 are prime numbers-
these are the numbers, which are only divisible by 1 or
the number itself as 13 and 17 are divisible by 1 and only
by them they both are prime.
Both are different-As 13 and 17 are equal they differ.

Show and Grow

Question 1.
Circle 10 feathers. Complete the sentence.

____One____ ten and ___six____ ones is __sixteen____ .

One ten and six ones is sixteen.
1 ten and 6 ones is 16.

Explanation:
Circled 10 feathers, after counting we have total
16 feathers we write the sentence as
one ten and six ones is sixteen means 10 + 6 = 16.
1 ten and 6 ones is 16.

Apply and Grow: Practice

Circle 10 feathers. Complete the sentence.

Question 2.

___One_____ ten and __five___ ones is ____fifteen_____ .

One ten and five ones is fifteen.
1 ten and 5 ones is 15.

Explanation:
Circled 10 fishes, after counting we have total
15 fishes we write the sentence as
one ten and five ones is fifteen means 10 + 5 = 15.
1 ten and 5 ones is 15.

Question 3.

____One____ ten and ___three__ ones is ____thirteen_____ .

One ten and three ones is thirteen.
1 ten and 3 ones is 13.

Explanation:
Circled 10 sun flowers, after counting we have total
13 sunflowers we write the sentence as
one ten and three ones is thirteen means 10 + 3 = 13.
1 ten and 3 ones is 13.

Question 4.

___One_ ten and __seven__ ones is ___seventeen___ .

One ten and seven ones is seventeen.
1 ten and 7 ones is 17.

Explanation:
Circled 10 bees, after counting we have total
17 bees we write the sentence as
one ten and seven ones is seventeen means 10 + 7 = 17.
1 ten and 7 ones is 17.

Question 5.
MP Number Sense
Color to show the number. Complete the sentence.

___one___ ten and __six___ ones is __sixteen____ .

One ten and six ones is sixteen.
1 ten and 6 ones is 16.

Explanation:
Colored the shown number 16 with blue,
we write the sentence as
one ten and six ones is sixteen means 10 + 6 = 16.
1 ten and 6 ones is 16.

Think and Grow: Modeling Real Life

You have 15 footballs. A bag can hold 10. You fill a bag. How many footballs are not in the bag?

Draw a picture:

Write the missing numbers: __one_ ten and __five_ ones is __fifteen__ footballs

One ten and five ones is fifteen footballs.
1 ten and 5 ones is 15.
5 footballs are not there in the bag,
Zero ten and five ones or 0 ten and 5 ones is five or 5 footballs are missing.

Explanation:
Given I have 15 footballs means one ten and five ones is
fifteen or 1 ten and 5 ones is 15 and a bag can hold 10 footballs,
In the picture there are total 15 footballs in that I
have taken 10 in the bag so footballs which are not there
in the bag are 15 – 10 = 5, So missing number of footballs
in the bag are zero ten and five ones is five or 0 ten and 5 ones is 5.

Show and Grow

Question 6.
Your teacher has 18 calculators. A case can hold 10. Your teacher fills a case. How many calculators are not in the case?

Draw a picture:

Write the missing numbers: __one__ ten and __eights ones _ is eighteen_ calculators

One ten and eight ones is eighteen calculators.
1 ten and 8 ones is 18.
8 calculators are not there in the case,
Zero ten and eight ones is eight or
0 ten and 8 ones is 8 calculators are missing,

Explanation:
Given teacher ha 18 calculators means one ten and eight ones is
eighteen or 1 ten and 8 ones is 18 and a case can hold 10,
In the picture there are total 18 calculators  in that I
have taken 10 in the case so number of calculators which
are not there in the case are 18 – 10 = 8, So missing calculators
in the case are zero ten and eight ones is eight or 0 ten and 8 ones is 8.

### Compose Numbers 11 to 19 Practice 6.3

Circle 10 objects. Complete the sentence.

Question 1.

__one___ ten and ___one___ ones is eleven_ .

One ten and one ones is eleven.
1 ten and 1 ones is 11.

Explanation:
Circled 10 balls, after counting we have total
11 balls we write the sentence as
one ten and one ones is eleven means 10 + 1 = 11.
1 ten and 1 ones is 11.

Question 2.

__one____ ten and __nine____ ones is _nineteen_____ .

One ten and nine ones is nineteen.
1 ten and 9 ones is 19.

Explanation:
Circled 10 fishes, after counting we have total
19 fishes we write the sentence as
one ten and nine ones is nineteen means 10 + 9 = 19.
1 ten and 9 ones is 19.

Question 3.

___one___ ten and __three____ ones __thirteen____ .

One ten and three ones is thirteen.
1 ten and 3 ones is 13.

Explanation:
Circled 10 honeybees, after counting we have total
13 we write the sentence as
one ten and three ones is thirteen means 10 + 3 = 13.
1 ten and 3 ones is 13.

Question 4.
MP Number Sense
Color to show the number. Complete the sentence.

__one____ ten and _eight___ ones is eighteen__ .

One ten and eight ones is eighteen.
1 ten and 8 ones is 18.

Explanation:
Colored the shown number 18 with yellow,
we write the sentence as
one ten and eight ones is eighteen means 10 + 8 = 18.
1 ten and 8 ones is 18.

Question 5.
MP Number Sense Match.

Explanation:
Matched 1 ten and 3 ones with 13, 1 ten and 8 ones with 12
and 12 ones with 12 in the picture above.

Question 6.
Modeling Real Life
You have 16 books. A backpack can hold 10. You fill a backpack. How many books are not in the backpack?

_____6_____ books
There are 6 books which are not in the backpack.

Explanation:
Given I have 16 books means and a backpack
can hold 10 books, out of 16 books I
have taken 10 in the backpack so books which are not there
in the backpack are 16 – 10 = 6 books.

Review & Refresh

Question 7.
10 + 10 = ___20_____
10 + 10 = 20

Explanation:
We are going to add 10 with 10 we get 20.

Question 8.
10 + 10 = ___20______
10 + 10 = 20

Explanation:
We are going to add 10 with 10 we get 20.

### Lesson 6.4 Tens

Explore and Grow

Circle groups of 10. Write the number of groups.

How many counters are there in all?
____20______ counters

There are 20 counters in all.

Explanation:
In the given picture circled groups of 10, There are 2 groups,
and counted there are total 20 counters in all.

Show and Grow

Circle groups of 10. Complete the sentence.

Question 1.

____6__ ten and _0__ ones is 60______ .

6 groups, 6 ten and 0 ones is 60.

Explanation:
In the given picture circled groups of 10, There are 6 groups,
and counted there are total 60 in all. 6 ten and 0 ones is 60.

Question 2.

__5____ ten and ___0___ ones is 50_ .

5 groups, 5 ten and 0 ones is 50.

Explanation:
In the given picture circled groups of 10, There are 5 groups,
and counted there are total 50 in all. 5 ten and 0 ones is 50.

Apply and Grow: Practice

Circle groups of 10. Complete the sentence.

Question 3.

___7___ ten and ___0___ ones is 70______ .

7 groups, 7 ten and 0 ones is 70.

Explanation:
In the given picture circled groups of 10, There are 7 groups,
and counted there are total 70 in all. 7 ten and 0 ones is 70.

Question 4.

___9___ ten and ___0___ ones is _90__ .

9 groups, 9 ten and 0 ones is 90.

Explanation:
In the given picture circled groups of 10, There are 9 groups,
and counted there are total 90 in all. 9 ten and 0 ones is 90.

Question 5.

___1___ ten and ___0___ ones is __10____ .

1 group, 1 ten and 0 ones is 10.

Explanation:
In the given picture circled groups of 10, There are 1 group,
and counted there are total 10 in all. 1 ten and 0 ones is 10.

Question 6.
MP Number Sense
You have 4 groups of 10 linking cubes. How many linking cubes do you have?

Explanation:
Given I have 4 groups of 10 linking cubes, So I have
4 X 10 = 40 linking cubes.

Think and Grow: Modeling Real Life

You read 10 books every month. You 40 books. How many months does it take?

Draw a picture:
Write the missing numbers: ___4___ tens and ___0___ ones __4____ months

4 tens and 0 ones is 40 books.
It will take 4 months.

Explanation:
4 tens and 0 ones is 40 books.
So we will divide 40 by 10 we get 4, therefore it will take 4 months,
Show and Grow

Question 7.
There are 10 dog bones in each box. You need 20 bones. How many boxes do you need?

Draw a picture:

Write the missing numbers: ___2___ tens and ___0___ ones __2____ boxes

2 tens and 0 ones is 20 boxes.
We need 2 boxes.

Explanation:
Given there are 10 dog bones in each box and need 20 bones.
2 tens and 0 ones is 20 bones.
So we will divide 20 by 10 we get 2, therefore we need 2 boxes.

### Tens Practice 6.4

Circle groups of 10. Complete the sentence.

Question 1.

___8___ tens and __0____ ones is 80__ .

8 groups, 1 ten and 0 ones is 10.

Explanation:
In the given picture circled groups of 10, There are 8 groups,
and counted there are total 80 in all. 8 ten and 0 ones is 80.

Question 2.

___6___ tens and __0____ ones is __60____ .

6 groups, 6 ten and 0 ones is 60.

Explanation:
In the given picture circled groups of 10, There are 6 groups,
and counted there are total 60 in all. 6 ten and 0 ones is 60.

Question 3.

___3___ tens and __0____ ones _is 30_____ .

3 groups, 3 ten and 0 ones is 30.

Explanation:
In the given picture circled groups of 10, There are 3 groups,
and counted there are total 30 in all. 3 ten and 0 ones is 30.

Circle groups of 10. Complete the sentence.

Question 4.

___2___ tens and __0____ ones is __20____ .

2 groups, 2 ten and 0 ones is 20.

Explanation:
In the given picture circled groups of 10, There are 2 groups,
and counted there are total 20 in all. 2 ten and 0 ones is 20.

Question 5.
MP Number Sense
You have 7 groups of 10 linking cubes. How many linking cubes do you have?

Explanation:
Given I have 7 groups of 10 linking cubes, So I have
7 X 10 = 70 linking cubes.

Question 6.
Modeling Real Life
You swim 10 laps at every practice. You want to swim 50 laps. How many practices will it take?

_____5______ Practices

5 tens and 0 ones is 50 laps
It will take 5 practices.

Explanation:
Given I swim 10 laps at every practice, I want to swim 50 laps
5 tens and 0 ones is 50 laps.
So I will divide 50 by 10 I get 5, therefore I will take 5 practices.

Review & Refresh

Question 7.
4 + 3 + 4 = ____11________
4 + 3 + 4 = 11

Explanation:
First we add 4 and 3 then add 4 we get 11.

Question 8.
1 + 5 + 9 = _____15______
1 + 5 + 9 = 15

Explanation:
First we add 1 and 5 then add 9 we get 15.

Question 9.
2 + 2 + 1 = _____5______
2 + 2 + 1 = 5

Explanation:
First we add 2 and 2 then add 1 we get 5.

Question 10.
7 + 3 + 6 = _____16______
7 + 3 + 6 = 16

Explanation:
First we add 7 and 3 then add 6 we get 16.

### Lesson 6.5 Tens and Ones

Explore and Grow

Model 2 tens and 3 ones. Write the number.

2 tens and 3 ones is 23.

Explanation:
Model shown in the picture 2 tens and 3 ones,
So placed 2 ten blocks in tens place and 3 blocks in ones place,
making 2 tens and 3 ones ,therefore the number is 23.

Show and Grow

Question 1.

__2___ tens and ___1___ ones is ___21___ .

2 tens and 1 ones is 21

Explanation:
Given in the picture there are 2 tens block in tens place
and 1 block in ones make 2 tens and 1 ones is 21.

Apply and Grow: Practice

Question 2.

___3___ tens and __5____ ones is ___35___ .

3 tens and 5 ones is 35

Explanation:
Given in the picture there are 3 ten blocks in tens place
and 5 blocks in ones make 3 tens and 5 ones is 35.

Question 3.

___6___ tens and ___6___ ones is ___6___ .

6 tens and 6 ones is 66

Explanation:
Given in the picture there are 6 ten blocks in tens place
and 6 blocks in ones make 6 tens and 6 ones is 66.

Question 4.

__8____ tens and __9____ ones is __89____ .

8 tens and 9 ones is 89

Explanation:
Given in the picture there are 8 ten blocks in tens place
and 9 blocks in ones make 8 tens and 9 ones is 89.

Question 5.
YOU BE THE TEACHER
You have 92 linking cubes. Your friend says that there are 2 tens and 9 ones.
Is your friend correct? Show how you know.

No, friend is incorrect.

Explanation:
Given I have 92 linking cubes, friend says that there are 2 tens and 9 ones,
but 92 means as shown in picture we get 9 tens and 2 ones,
therefore friend is in correct.

Think and Grow: Modeling Real Life

Your teacher has 2 packages of dice and 3 extra dice. Each package has 10 dice. How many dice are there in all?

Draw a picture:

Write the missing numbers: ___2___ tens and __3____ ones _is__23___ dice

2 tens and 3 ones is 23 dice are there in all.

Explanation:
Given my teacher has 2 packages of dice and 3 extra dice.
Each package has 10 dice, 2 packages means 2 X 10 = 20 dice
and 3 extra dice making in all 2 tens and 3 ones is 23 dice as shown in picture.

Show and Grow

Question 6.
You have 3 boxes of colored pencils and 4 extra colored pencils. Each box has 10 pencils. How many colored pencils are there in all?

Draw a picture:

Write the missing numbers: ___3___ tens and __4____ ones _is_34__ colored pencils

3 tens and 4 ones is 34 colored pencils.

Explanation:
Given I have 3 boxes of colored pencils and 4 extra colored pencils.
Each box has 10 pencils, So 3 boxes have 3 X 10 = 30 pencils and 4 extra
makes 30 + 4 = 34, 3 tens and 4 ones is 34 colored pencils are there in all.

### Tens and Ones Practice 6.5

Question 1.

__7____ tens and ___9___ ones is __79____ .

7 tens and 9 ones is 79.

Explanation:
Given in the picture there are 7 ten blocks in tens place
and 9 blocks in ones make 7 tens and 9 ones is 79.

Question 2.

__8____ tens and ___1___ ones _is 81_____ .

8 tens and 1 ones is 81.

Explanation:
Given in the picture there are 8 ten blocks in tens place
and 1 blocks in ones make 8 tens and 1 ones is 81.

Question 3.

___2___ tens and __4___ ones is 24_ .

2 tens and 4 ones is 24.

Explanation:
Given in the picture there are 2 ten blocks in tens place
and 4 blocks in ones make 2 tens and 4 ones is 24.

Question 4.
YOU BE THE TEACHER
You have 17 linking cubes. Your friend says that there is 1 ten and 7 ones. Is your friend correct? Show how you know.

Yes, friend is correct.

Explanation:
Given I have 17 linking cubes, friend says that there is 1 ten and 7 ones,
means as shown in picture we get 1 ten and 7 ones, similar to what friend says,
therefore friend is  correct.

Question 5.
Modeling Real Life
You have 5 bags of apples and 1 extra apple. Each bag has 10 apples. How many apples are there in all?

_____51______ apples

5 tens and 1 ones is 51 apples.

Explanation:
Given I have 5 bags of apples and 1 extra apple.
Each bag has 10 apples, So 5 bags have 5 X 10 = 50 apples and 1 extra apple
makes 50 + 1 = 51, 5 tens and 1 ones is 51 apples are there in all.

Review & Refresh

Question 6.
_____4_____ + 6 = 10
4 + 6 = 10

Explanation:
Given ____ + 6 = 10, let us take the missing
number as X , X + 6 = 10, X = 10 – 6 = 4 making
the equation as 4 + 6 = 10.

Question 7.
_____6______ + 2 = 8
6 + 2 = 8

Explanation:
Given ____ + 2 = 8, let us take the missing
number as X , X + 2 = 8, X = 8 – 2 = 6 making
the equation as 6 + 2 = 8.

### Lesson 6.6 Make Quick Sketches

Explore and Grow

Model the number 2.

2 ones is 2.

Explanation:
Model shown in the picture are 2 ones,
So 2 blocks in ones place, as shown in picture
making 2 ones ,therefore the number is 2.

Show and Grow

Make a quick sketch. Complete the sentence.

Question 1.
72

__72____ is ____7____ tens and __2____ ones.

72 is 7 tens and 2 ones.

Explanation:
The picture has 7 ten blocks in tens place
and 2 blocks in ones makes 72 as 7 tens and 2 ones.

Question 2.
36

__36____ is ___3__ tens and __6_ ones.

36 is 3 tens and 6 ones.

Explanation:
The picture has 3 ten blocks in tens place
and 6 blocks in ones makes 36 as 3 tens and 6 ones.

Apply and Grow: Practice

Make a quick sketch. Complete the sentence.

Question 3.
45

___45___ is ___4_____ tens and ___5___ ones.

45 is 4 tens and 5 ones.

Explanation:
The picture has 4 ten blocks in tens place
and 5 blocks in ones makes 45 as 4 tens and 5 ones.

Question 4.
87

___87___ is ___8___ tens and ___7___ ones.

87 is 8 tens and 7 ones.

Explanation:
The picture has 8 ten blocks in tens place
and 7 blocks in ones makes 87 as 8 tens and 7 ones.

Question 5.
64

__64____ is ___6____ tens and ___4___ ones.

64 is 6 tens and 4 ones.

Explanation:
The picture has 6 ten blocks in tens place
and 4 blocks in ones makes 64 as 6 tens and 4 ones.

Question 6.
DIG DEEPER!
Which sketch shows 54?

First sketch shows 54.

Explanation:
Given 2 sketches in that first counts 54 and second counts
45, So first sketch shows 54, So selected first one for 54.

Think and Grow: Modeling Real Life

You need 58 plates for a party. You have 51. How many more plates do you need?

Complete the model:

____7_______ more plates

I need 7 more plates.

Explanation:
Given I need 58 plates for a party and I have 51,
therefore more plates needed are 58 – 51 = 7 plates
as shown in the picture above.

Show and Grow

Question 7.
You need 80 tickets for a prize. You have 73. How many more tickets do you need?

Complete the model:

_____7______ more tickets

I need 7 more tickets.

Explanation:
Given I need 80 tickets for a prize and I have 73,
therefore more tickets needed are 80 – 73 = 7 plates
as shown in the picture above.

### Make Quick Sketches Practice 6.6

Make a quick sketch. Complete the sentence.

Question 1.
27

__27____ is ____2____ tens and ___7___ ones.

27 is 2 tens and 7 ones.

Explanation:
The picture has 2 ten blocks in tens place
and 7 blocks in ones makes 27 as 2 tens and 7 ones.

Question 2.
61

__61__ is ___6__ tens and ___1___ ones.

61 is 6 tens and 1 one.

Explanation:
The picture has 6 ten blocks in tens place
and 1 blocks in one makes 61 as 6 tens and 1 one.

Question 3.
92

___92___ is ___9_____ tens and ___2___ ones.

92 is 9 tens and 2 ones.

Explanation:
The picture has 9 ten blocks in tens place
and 2 blocks in ones makes 92 as 9 tens and 2 ones.

Question 4.
DIG DEEPER
Which sketch shows 87?

First sketch shows 87.

Explanation:
Given 2 sketches in that first counts 87 and second counts
78, So first sketch shows 87, So selected first one for 87.

Question 5.
Modeling Real Life
You need 55 beads to make a necklace. You have 48. How many more beads do you need?

Explanation:
Given I need 55 beads to make necklace and I have 48,

Review & Refresh

Question 6.
Color the shapes that have only 4 sides.

Explanation:
Given some shapes and to color that have 4 sides so selected shapes
that has 4 sides and  have colored as shown above in the picture.

### Lesson 6.7 Understand Place Value

Explore and Grow

Newton has 2 rods. Make a quick sketch.
Write the number.

Newton has 2 rods or 20 cubes

Explanation:
Given Newton has 2 rods as shown in the picture means 20 cubes.

Descartes has 2 cubes. Make a quick sketch.
Write the number.

Descartes has 2 cubes

Explanation:
Given Descartes has 2 cubes as shown in picture 2 cubes.

How are the models alike? How are they different?
Both use cubes so alike,
Both differ in number so different.

Explanation:
Models are alike means both use cubes.
They are different Newton has 20 cubes and Descartes have 2 cubes,
20 and 2 are not the same and are not equal so they are different.

Show and Grow

Question 1.
Make a quick sketch. Complete the sentences.
64

_____6_____ tens is ____60______ .
____4______ ones is ____4______ .
_____6_____ tens and ____4______ ones is ___64_______.

6 tens is 60,
4 ones is 4, 6 tens and 4 ones is 64.

Explanation:
As shown in the picture above first we take 6 tens cubes as 60
and 4 ones cubes as 4, making 6 tens and 4 ones is 64.

Apply and Grow: Practice

Make a quick sketch. Complete the sentences.

Question 2.
72

_____7_____ tens is ____70______ .
_____2_____ ones is ____2______ .
______7____ tens and ____2______ ones is ___72__.

7 tens is 70,
2 ones is 2, 7 tens and 2 ones is 72.

Explanation:
As shown in the picture above first we take 7 tens cubes as 70
and 2 ones cubes as 2, making 7 tens and 2 ones is 72.

Question 3.
98

__9____ tens is ___90____ .
___8____ ones is ___8_____ .
___9____ tens and ____8____ ones is ___98___.

9 tens is 90,
8 ones is 8, 9 tens and 8 ones is 98.

Explanation:
As shown in the picture above first we take 9 tens cubes as 90
and 8 ones cubes as 8, making 9 tens and 8 ones is 98.

Question 4.
57
____5______ tens is ___50_______ .
_____7_____ ones is ____7______ .
______5____ tens and _____7_____ ones is ____57______.
5 tens is 50,
7 ones is 7, 5 tens and 7 ones is 57.

Explanation:
First we take 5 tens cubes as 50
and 7 ones cubes as 7, making 5 tens and 7 ones is 57.

Think and Grow: Modeling Real Life

You have 94 charms to make bracelets. There are lo charms on each bracelet. How many bracelets can you make?

Model:

Write the missing numbers: ___9___ tens and ___4___ ones ___9___ bracelets
9 tens and 4 ones is 94,
I make 9 bracelets.

Explanation:
Given I have 94 charms to make bracelets. There are lo charms
on each bracelet. So I divide 94 with 10 I get 9 as whole
and 4 as remainder so I cannot make with 4 charms,
I consider only whole so I take 9,therefore I can make 9 bracelets
or as I have 9 tens and 4 ones I take only tens so 9 bracelets I can make.

Show and Grow

Question 5.
You have 67 seeds. You plant 10 seeds in a row. How many rows can you plant?

Model:

Write the missing numbers: ___6___ tens and __7____ ones __6____ rows
6 tens and 7 ones is 67,
I can plant 6 rows.

Explanation:
Given I have 67 seeds. I can plant 10 seeds in a row
So I divide 67 with 10 I get 6 as whole
and 7 as remainder so I cannot make with 7 seeds a row ,
I consider only whole so I take 6,therefore I can make 6 rows only,
or as I have 6 tens and 7 ones I take only tens so 6 rows I can make.

### Understand Place Value Practice 6.7

Make a quick sketch. Complete the sentences.

Question 1.
81

_____8_____ tens is ___80_______ .
_____1_____ ones is ____1______ .
_____8_____ tens and ____1______ ones is ____81______.

8 tens is 80,
1 one is 1, 8 tens and 1 one is 81.

Explanation:
As shown in the picture above first we take 8 tens cubes as 80
and 1 one cubes as 1, making 8 tens and 1 one is 81.

Question 2.
53

_____5_____ tens is _____50_____ .
______3____ ones is ____3______ .
______5____ tens and _____3____ ones is ___53_____.

5 tens is 50,
3 ones is 3, 5 tens and 3 ones is 53.

Explanation:
As shown in the picture above first we take 5 tens cubes as 50
and 3 ones cubes as 3, making 5 tens and 3 ones is 53.

Question 3.
49
___4_______ tens is ____40______ .
_____9_____ ones is ____9______ .
______4____ tens and ____9__ ones is __49__.

4 tens is 40,
9 ones is 9, 4 tens and 9 ones is 49.

Explanation:
As shown in the picture above first we take 4 tens cubes as 40
and 9 ones cubes as 9, making 4 tens and 9 ones is 49.

Question 4.
76
____7______ tens is ____70______ .
______6____ ones is ____6______ .
____7______ tens and ____6______ ones is ____76______.

7 tens is 70,
6 ones is 6, 7 tens and 6 ones is 76.

Explanation:
As shown in the picture above first we take 7 tens cubes as 70
and 6 ones cubes as 6, making 7 tens and 6 ones is 76.

Question 5.
Modeling Real Life
You have 77 crayons. A box can hold 10 crayons. How many boxes can you fill?

_____7____ boxes
7 tens and 7 ones is 77,
I can fill 7 boxes.

Explanation:
Given I have 77 crayons, A box can hold 10 crayons,
So I divide 77 with 10 I get 7 as whole
and 7 as remainder I consider only whole so I take 7,
therefore I need 7 boxes to fill or as I have 7 tens and 7 ones to
fill I take only tens so 7 boxes I can fill.

Review & Refresh

Is the equation true or false?

Question 6.

4 + 9 ≠  2 + 3 + 5
Not equal, So false

Equation:
First we see 4 and 9 as 4 + 9 is 13 and
if we add 2,3 and 5 we get 10 ,
therefore 4 + 9 ≠ 2 + 3 +5, So false.

Question 7.

5 + 3 = 4 + 4
Is equal, So true

Explanation:
First we see 5 + 3 as we add 5 and 3 we get 8 and
4 + 4 is 8 both sides are equal, So equations are true.

### Lesson 6.8 Write Numbers in Different Ways

Explore and Grow

Model 27 two ways.

_____2___ tens and __7_____ ones is ___27_____ .
_____1___ ten and ___17____ ones is ___27_____ .
one way is

2 tens and 7 ones is 27.
Other way is

1 ten and 17 ones is 27.

Explanation:
One way is we add 2 tens and 7 ones equal to 27,
20 + 7 = 27 and other way is 1 ten and 17 ones is 27,
10 + 17 = 27.

Show and Grow

Question 1.
Model 25 two ways.

____2____ tens and ____5___ ones is 25.

2 tens and 5 ones is 25.

Explanation:
we add 20 and 5 we get 25, So 2 tens and 5 ones is 25.

____1____ tens and __15_____ ones is 25.

1 ten and 15 ones is 25

Explanation:
we add 10 and 25 we get 25, So 1 ten and 15 ones is 25.

Apply and Grow: Practice

Question 2.
Model 52 two ways.

____5____ tens and ___2____ ones is 52.

5 tens and 2 ones is 52.

Explanation:
We add 50 and 2 we get 52, So 5 tens and 2 ones is 52.

______4__ tens and ____12___ ones is 52.

4 tens and 12 ones is 52.

Explanation:
First we take 40 and add to 12 we get 52,
40 + 12 = 52, therefore 4 tens and 12 ones is 52.

Question 3.
Model 14 two ways.

___1_____ tens and ___4____ ones is 14.

1 ten and 4 ones is 14

Explanation:
we add 10 + 4 = 14,
So 1 ten and 4 ones is 14.

___0_____ tens and ___14____ ones is 14.

14 ones is 14

Explanation:
we add 0 + 14 = 14,
0 tens and 14 ones is 14.

Question 4.
DIG DEEPER!
Circle all of the ways that show 39.
3 + 9          2 tens and 9 ones      9 tens and 3 ones
10 + 29      3 tens and 19 ones    39 ones

Circled the ways that show 39 are 10 + 29 and 39 ones

Explanation:
As shown in picture circled all of the ways that show 39,
3 + 9 = 12 ≠ 39 so not circled,
2 tens and 9 ones is 29 ≠ 39 so not circled,
9 tens and 3 ones is 93 ≠ 39 so not circled,
10 + 29 = 39 = 39 so circled it,
3 tens and 19 ones is 30 + 19 is 49 ≠ 39 so not circled,
39 ones is 39 = 39 so circled it.

Think and Grow: Modeling Real Life

The models show how many seashells you and your friend have. Does your friend have the same number of seashells as you?

Circle: Yes         No

Show how you know:

No friend does not have the same number of
seashells as much as I have.

Explanation:
After counting I have total 25 seashells and friend have 30,
Both do not have same or equal number of seashells ,
So circled No as shown in the picture above.

Show and Grow

Question 5.
The models show how many erasers you and your friend have. Does your friend have the same number of erasers as you?

Circle: Yes          No

Show how you know:

Yes, I and my friend have the same number of erasers.

Explanation:
After counting I have total 40 erasers and friend also have 40,
Both have the same or equal number of erasers,
So circled Yes as shown in the picture above.

### Write Numbers in Different Ways Practice 6.8

Question 1.
Model 49 two ways.

__4______ tens and
___9____ ones
is 49.

4 tens and 9 ones is 49.

Explanation:
Adding 40 to 9 is 40 + 9 = 49,
so 4 tens and 9 ones is 49.

___30_____ tens and
____19____ ones
is 49.

30 tens and 19 ones is 49.

Explanation:
We add 30 and 19 we get 49,
So 30 tens and 19 ones make 49.

Question 2.
DIG DEEPER!
Circle all of the ways that show 45.
40 + 5              45 tens and 0 ones       4 tens and 5 ones
20 + 15            2 tens and 25 ones       54 ones

Circled the ways that show 45 are 40 + 5,
4 tens and 5 ones and 2 tens and 25 ones.

Explanation:
As shown in picture circled all of the ways that show 45,
40 + 5 = 45 = 45 so circled it,
45 tens and 0 ones is 450 ≠ 45 so not circled,
4 tens and 5 ones is 45 = 45 so circled it,
20 + 15 = 35 ≠ 45 so not circled,
2 tens and 25 ones is 20 + 25 is 45 = 45  so circled it,
54 ones is 54 ≠ 45 so not circled.

Question 3.
Modeling Real Life
The models show the number of toy cars you and your friend have. Does your friend have the same number of toy cars as you?

Circle: Yes        No

Show how you know:

Yes, friend have the same number of toy cars as I have.

Explanation:
After counting I have total 34 toy cars and friend also have 34 ,
Both have same or equal number of toy cars,
So circled Yes as shown in the picture above.

Review & Refresh

Circle the heavier object.

Question 4.

Circled ball.

Explanation:
Given ball and balloon as ball is
heavier object than balloon I circled it.

Question 5.

Circled 4 bananas

Explanation:
Given 4 bananas and 1 banana as 4 bananas are
heavier object than 1 banana I circled it.

### Lesson 6.9 Count and Write Numbers to 120

Explore and Grow

How many balls are there? How did you count?

Number of balls are 106. Counted one after another.

Explanation:
There are total 106 balls, we start pointing and counting
from first ball as one add next ball and so on counting
till we reach the last ball ,we get total 106 balls.

Show and Grow

Question 1.

10 tens and 7 ones is 107.

Explanation:
Given in the picture if we count there are
100 + 7 = 107 , 10 tens and 7 ones is 107.

Question 2.

11 tens and 5 ones is 115.

Explanation:
Given in the picture if we count there are
110 + 5 = 107 , 11 tens and 5 ones is 115.

Apply and Grow: Practice

Question 3.

6 tens and 4 ones is 64.

Explanation:
Given in the picture if we count there are
60 + 4 = 64 , 6 tens and 4 ones is 64.

Question 4.

9 tens and 8 ones is 98.

Explanation:
Given in the picture if we count there are
90 + 8 = 98 , 9 tens and 8 ones is 98.

Question 5.

10 tens and 1 one is 101.

Explanation:
Given in the picture if we count there are
100 + 1 = 101 , 10 tens and 1 one is 101.

Question 6.

10 tens and 3 ones is 103.

Explanation:
Given in the picture if we count there are
100 + 3 = 103 , 10 tens and 3 ones is 103.

Question 7.

11 tens and 0 ones is 110.

Explanation:
Given in the picture if we count there are
110 + 0 = 110 , 11 tens and 0 ones is 110.

Question 8
DIG DEEPER!
What number is equal to lo tens and 8 ones? Show how you know.

The number equal to 10 tens and 8 ones is 108

Explanation:
Given to find number equal to 10 tens and 8 ones,
as shown in figure taken 10 tens and 8 ones ,
the number is equal to 100 + 8 = 108.

Think and Grow: Modeling Real Life

Your teacher has 12 bags of balloons. Each bag has 10 balloons. How many balloons are there in all?

Model:

____120_______ balloons
There are 120 balloons are there in all.

Explanation:
Given my teacher has 12 bags of balloons and
each bag has 10 balloons so in all there are
12 X 10 = 120 balloons.

Show and Grow

Question 9.
A dentist has 10 boxes of toothbrushes and 9 extra toothbrushes. Each box has 10 toothbrushes. How many toothbrushes are there in all?

Model:

___109________ toothbrushes

10 tens and 9 ones is 109.
There are 109 tooth brushes are there in all.

Explanation:
Given a dentist has 10 boxes of toothbrushes and
9 extra toothbrushes, each box has 10 toothbrushes
means 10 x 10 = 100 and 9 extra makes 100 + 9 = 109 or
10 tens and 9 ones is 109 tooth brushes are there in all.

### Count and Write Numbers to 120 Practice 6.9

Question 1.

7 tens and 9 ones is 79.

Explanation:
Given in the picture if we count there are
70 + 9 = 79 , 7 tens and 9 ones is 79.

Question 2.

11 tens and 5 ones is 115.

Explanation:
Given in the picture if we count there are
110 + 5 = 115 , 11 tens and 5 ones is 115.

Question 3.

12 tens and 0 ones is 120.

Explanation:
Given in the picture if we count there are
120 + 0 = 120 , 12 tens and 0 ones is 120.

Question 4.