Eureka Math

Engage NY Eureka Math 8th Grade Module 6 Lesson 11 Answer Key

Eureka Math Grade 8 Module 6 Lesson 11 Exercise Answer Key

Exercise 1.
Old Faithful is a geyser in Yellowstone National Park. The following table offers some rough estimates of the length of an eruption (in minutes) and the amount of water (in gallons) in that eruption.

This data is consistent with actual eruption and summary statistics that can be found at the following links:
http://geysertimes.org/geyser.php?id = OldFaithful and http://www.yellowstonepark.com/2011/07/about – old – faithful/
a. Chang wants to predict the amount of water in an eruption based on the length of the eruption. What should he use as the dependent variable? Why?
Answer:
Since Chang wants to predict the amount of water in an eruption, the time length (in minutes) is the predictor, and the amount of water is the dependent variable.

b. Which of the following two scatter plots should Chang use to build his prediction model? Explain.

Answer:
The predicted variable goes on the vertical axis with the predictor on the horizontal axis. So, the amount of water goes on the y – axis. The plot on the graph on the right should be used.

c. Suppose that Chang believes the variables to be linearly related. Use the first and last data points in the table to create a linear prediction model.
Answer:
m = \(\frac{84000 – 3700}{4.5 – 1.5}\)≈1,566.7
So, y = a + (1,566.7)x.
Using either (1.5,3700) or (4.5,8400) allows students to solve for the intercept. For example, solving 3,700 = a + (1,566.7)(1.5) for a yields a = 1,349.95, or rounded to 1,350.0 gallons. Be sure students talk through the units in each step of the calculations.
The (informal) linear prediction model is y = 1,350.0 + 1,566.7x. The amount of water (y) is in gallons, and the length of the eruption (x) is in minutes.

d. A friend of Chang’s told him that Old Faithful produces about 3,000 gallons of water for every minute that it erupts. Does the linear model from part (c) support what Chang’s friend said? Explain.
Answer:
This question requires students to interpret slope. An additional minute in eruption length results in a prediction of an additional 1,566.7 gallons of water produced. So, Chang’s friend who claims Old Faithful produces 3,000 gallons of water a minute must be thinking of a different geyser.

e. Using the linear model from part (c), does it make sense to interpret the y – intercept in the context of this problem? Explain.
Answer:
No, it doesn’t make sense because if the length of an eruption is 0, then it cannot produce 1,350 gallons of water. (Convey to students that some linear models have y – intercepts that do not make sense within the context of a problem.)

Exercise 2.
The following table gives the times of the gold, silver, and bronze medal winners for the men’s 100 – meter race (in seconds) for the past 10 Olympic Games.

Data Source: https://en.wikipedia.org/wiki/100_metres_at_the_Olympics#Men

a. If you wanted to describe how mean times change over the years, which variable would you use as the independent variable, and which would you use as the dependent variable?
Answer:
Mean medal time (dependent variable) is being predicted based on year (independent variable).

b. Draw a scatter plot to determine if the relationship between mean time and year appears to be linear. Comment on any trend or pattern that you see in the scatter plot.
Answer:
The scatter plot indicates a negative trend, meaning that, in general, the mean race times have been decreasing over the years even though there is not a perfect linear pattern.

c. One reasonable line goes through the 1992 and 2004 data. Find the equation of that line.
Answer:
The slope of the line through (1992,10.01) and (2004,9.86) is \(\frac{10.01 – 9.86}{1992 – 2004}\) = – 0.0125.
To find the intercept using (1992,10.01), solve 10.01 = a + ( – 0.0125)(1992) for a, which yields
a = 34.91.
The equation that predicts the mean medal race time for an Olympic year is y = 34.91 + ( – 0.0125)x.
The mean medal race time (y) is in seconds, and the time (x) is in years.

Note to Teacher: In Algebra I, students learn a formal method called least squares for determining a “best – fitting” line. For comparison, the least squares prediction line is y = 34.3562 + ( – 0.0122)x.

d. Before he saw these data, Chang guessed that the mean time of the three Olympic medal winners decreased by about 0.05 second from one Olympic Game to the next. Does the prediction model you found in part (c) support his guess? Explain.
Answer:
The slope – 0.0125 means that from one calendar year to the next, the predicted mean race time for the top three medals decreases by 0.0125 second. So, between successive Olympic Games, which occur every four years, the predicted mean race time is reduced by 0.05 second because 4(0.0125) = 0.05.

e. If the trend continues, what mean race time would you predict for the gold, silver, and bronze medal winners in the 2016 Olympic Games? Explain how you got this prediction.
Answer:
If the linear pattern were to continue, the predicted mean time for the 2016 Olympics is 9.71 seconds because 34.91 – (0.0125)(2016) = 9.71.

f. The data point (1980,10.3) appears to have an unusually high value for the mean race time (10.3). Using your library or the Internet, see if you can find a possible explanation for why that might have happened.
Answer:
The mean race time in 1980 was an unusually high 10.3 seconds. In their research of the 1980 Olympic Games, students find that the United States and several other countries boycotted the games, which were held in Moscow. Perhaps the field of runners was not the typical Olympic quality as a result. Atypical points in a set of data are called outliers. They may influence the analysis of the data.

Eureka Math Grade 8 Module 6 Lesson 11 Problem Set Answer Key

Question 1.
From the United States Bureau of Census website, the population sizes (in millions of people) in the United States for census years 1790–2010 are as follows.

a. If you wanted to be able to predict population size in a given year, which variable would be the independent variable, and which would be the dependent variable?
Answer:
Population size (dependent variable) is being predicted based on year (independent variable).

b. Draw a scatter plot. Does the relationship between year and population size appear to be linear?
Answer:
The relationship between population size and year of birth is definitely nonlinear. Note that investigating nonlinear relationships is the topic of the next two lessons.

c. Consider the data only from 1950 to 2010. Does the relationship between year and population size for these years appear to be linear?
Answer:
Drawing a scatter plot using the 1950–2010 data indicates that the relationship between population size and year of birth is approximately linear, although some students may say that there is a very slight curvature to the data.

d. One line that could be used to model the relationship between year and population size for the data from 1950 to 2010 is y = – 4875.021 + 2.578x. Suppose that a sociologist believes that there will be negative consequences if population size in the United States increases by more than 2 \(\frac{3}{4}\) million people annually. Should she be concerned? Explain your reasoning.
Answer:
This problem is asking students to interpret the slope. Some students will no doubt say that the sociologist need not be concerned, since the slope of 2.578 million births per year is smaller than her threshold value of 2.75 million births per year. Other students may say that the sociologist should be concerned, since the difference between 2.578 and 2.75 is only 172,000 births per year.

e. Assuming that the linear pattern continues, use the line given in part (d) to predict the size of the population in the United States in the next census.
Answer:
The next census year is 2020.
– 4875.021 + (2.578)(2020) = 332.539
The given line predicts that the population then will be 332.539 million people.

Question 2.
In search of a topic for his science class project, Bill saw an interesting YouTube video in which dropping mint candies into bottles of a soda pop caused the soda pop to spurt immediately from the bottle. He wondered if the height of the spurt was linearly related to the number of mint candies that were used. He collected data using 1, 3, 5, and 10 mint candies. Then, he used two – liter bottles of a diet soda and measured the height of the spurt in centimeters. He tried each quantity of mint candies three times. His data are in the following table.

a. Identify which variable is the independent variable and which is the dependent variable.
Answer:
Height of spurt is the dependent variable, and number of mint candies is the independent variable because height of spurt is being predicted based on number of mint candies used.

b. Draw a scatter plot that could be used to determine whether the relationship between height of spurt and number of mint candies appears to be linear.
Answer:

c. Bill sees a slight curvature in the scatter plot, but he thinks that the relationship between the number of mint candies and the height of the spurt appears close enough to being linear, and he proceeds to draw a line.
Answer:
His eyeballed line goes through the mean of the three heights for three mint candies and the mean of the three heights for 10 candies. Bill calculates the equation of his eyeballed line to be
y = – 27.617 + (43.095)x,
where the height of the spurt (y) in centimeters is based on the number of mint candies (x). Do you agree with this calculation? He rounded all of his calculations to three decimal places. Show your work.
Yes, Bill’s equation is correct.
The slope of the line through (3,101.667) and (10,403.333) is \(\frac{403.333 – 101.667}{10 – 3}\) = 43.095.
The intercept could be found by solving 403.333 = a + (43.095)(10) for a, which yields
a = – 27.617.
So, a possible prediction line is y = – 27.617 + (43.095)x.

d. In the context of this problem, interpret in words the slope and intercept for Bill’s line. Does interpreting the intercept make sense in this context? Explain.
Answer:
The slope is 43.095, which means that for every mint candy dropped into the bottle of soda pop, the height of the spurt increases by 43.095 cm.
The y – intercept is (0, – 27.617). This means that if no mint candies are dropped into the bottle of soda pop, the height of the spurt is – 27.617 ft. This does not make sense within the context of the problem.

e. If the linear trend continues for greater numbers of mint candies, what do you predict the height of the spurt to be if 15 mint candies are used?
Answer:
– 27.617 + (43.095)(15) = 618.808
The predicted height would be 618.808 cm, which is slightly over 20 ft.

Eureka Math Grade 8 Module 6 Lesson 11 Exit Ticket Answer Key

Question 1.
According to the Bureau of Vital Statistics for the New York City Department of Health and Mental Hygiene, the life expectancy at birth (in years) for New York City babies is as follows.

Data Source: http://www.nyc.gov/html/om/pdf/2012/pr465 – 12_charts.pdf
a. If you are interested in predicting life expectancy for babies born in a given year, which variable is the independent variable, and which is the dependent variable?
Answer:
Year of birth is the independent variable, and life expectancy in years is the dependent variable.

b. Draw a scatter plot to determine if there appears to be a linear relationship between the year of birth and life expectancy.
Answer:
Life expectancy and year of birth appear to be linearly related.

c. Fit a line to the data. Show your work.
Answer:
Answers will vary. For example, the line through (2001,77.9) and (2009,80.6) is y = – 597.438 + (0.3375)x, where life expectancy (y) is in years, and the time (x) is in years.
Note to Teacher: The formal least squares line (Algebra I) is y = – 612.458 + (0.345)x.

d. Based on the context of the problem, interpret in words the intercept and slope of the line you found in part (c).
Answer:
Answers will vary based on part (c). The intercept says that babies born in New York City in Year 0 should expect to live around – 597 years! Be sure students actually say that this is an unrealistic result and that interpreting the intercept is meaningless in this problem. Regarding the slope, for an increase of 1 in the year of birth, predicted life expectancy increases by 0.3375 year, which is a little over four months.

e. Use your line to predict life expectancy for babies born in New York City in 2010.
Answer:
Answers will vary based on part (c).
– 597.438 + (0.3375)(2010) = 80.9
Using the line calculated in part (c), the predicted life expectancy for babies born in New York City in 2010 is 80.9 years, which is also the value given on the website.

Engage NY Eureka Math 8th Grade Module 6 Lesson 9 Answer Key

Eureka Math Grade 8 Module 6 Lesson 9 Exercise Answer Key

Example 1: Crocodiles and Alligators
Scientists are interested in finding out how different species adapt to finding food sources. One group studied crocodilians to find out how their bite force was related to body mass and diet. The table below displays the information they collected on body mass (in pounds) and bite force (in pounds).

Data Source: http://journals.plos.org/plosone/article?id = 10.1371/journal.pone.0031781#pone – 0031781 – t001
(Note: Body mass and bite force have been converted to pounds from kilograms and newtons, respectively.)
As you learned in the previous lesson, it is a good idea to begin by looking at what a scatter plot tells you about the data. The scatter plot below displays the data on body mass and bite force for the crocodilians in the study.

Exercises 1–6

Exercise 1.
Describe the relationship between body mass and bite force for the crocodilians shown in the scatter plot.
Answer:
As the body mass increases, the bite force tends to also increase.

Exercise 2.
Draw a line to represent the trend in the data. Comment on what you considered in drawing your line.
Answer:
The line should be as close as possible to the points in the scatter plot. Students explored this idea in Lesson 8.

Exercise 3.
Based on your line, predict the bite force for a crocodilian that weighs 220 pounds. How does this prediction compare to the actual bite force of the 220 – pound crocodilian in the data set?
Answer:
Answers will vary. A reasonable prediction is around 650 to 700 pounds. The actual bite force was 1,000 pounds, so the prediction based on the line was not very close for this crocodilian.

Exercise 4.
Several students decided to draw lines to represent the trend in the data. Consider the lines drawn by Sol, Patti, Marrisa, and Taylor, which are shown below.

For each student, indicate whether or not you think the line would be a good line to use to make predictions.
Explain your thinking.
a. Sol’s line
Answer:
In general, it looks like Sol’s line overestimates the bite force for heavier crocodilians and underestimates the bite force for crocodilians that do not weigh as much.

b. Patti’s line
Answer:
Patti’s line looks like it fits the data well, so it would probably produce good predictions. The line goes through the middle of the points in the scatter plot, and the points are fairly close to the line.

c. Marrisa’s line
Answer:
It looks like Marrisa’s line overestimates the bite force because almost all of the points are below the line.

d. Taylor’s line
Answer:
It looks like Taylor’s line tends to underestimate the bite force. There are many points above the line.

Exercise 5.
What is the equation of your line? Show the steps you used to determine your line. Based on your equation, what is your prediction for the bite force of a crocodilian weighing 200 pounds?
Answer:
Answers will vary. Students have learned from previous modules how to find the equation of a line. Anticipate students to first determine the slope based on two points on their lines. Students then use a point on the line to obtain an equation in the form y = mx + b (or y = a + bx). Students use their lines to predict a bite force for a crocodilian that weighs 200 pounds. A reasonable answer would be around 800 pounds.

Exercise 6.
Patti drew vertical line segments from two points to the line in her scatter plot. The first point she selected was for a dwarf crocodile. The second point she selected was for an Indian gharial crocodile.

a. Would Patti’s line have resulted in a predicted bite force that was closer to the actual bite force for the dwarf crocodile or for the Indian gharial crocodile? What aspect of the scatter plot supports your answer?
Answer:
The prediction would be closer to the actual bite force for the dwarf crocodile. That point is closer to the line (the vertical line segment connecting it to the line is shorter) than the point for the Indian gharial crocodile.

b. Would it be preferable to describe the trend in a scatter plot using a line that makes the differences in the actual and predicted values large or small? Explain your answer.
Answer:
It would be better for the differences to be as small as possible. Small differences are closer to the line.

Exercise 7: Used Cars

Exercise 7.
Suppose the plot below shows the age (in years) and price (in dollars) of used compact cars that were advertised in a local newspaper.

a. Based on the scatter plot above, describe the relationship between the age and price of the used cars.
Answer:
The older the car, the lower the price tends to be.

b. Nora drew a line she thought was close to many of the points and found the equation of the line. She used the points (13,6000) and (7,12000) on her line to find the equation. Explain why those points made finding the equation easy.

Answer:
The points are at the intersection of the grid lines in the graph, so it is easy to determine the coordinates of these points.

c. Find the equation of Nora’s line for predicting the price of a used car given its age. Summarize the trend described by this equation.
Answer:
Using the points, the equation is y = – 1000x + 19000, or Price = – 1000(age) + 19000. The slope of the line is negative, so the line indicates that the price of used cars decreases as cars get older.

d. Based on the line, for which car in the data set would the predicted value be farthest from the actual value? How can you tell?
Answer:
It would be farthest for the car that is 10 years old. It is the point in the scatter plot that is farthest from the line.

e. What does the equation predict for the cost of a 10 – year – old car? How close was the prediction using the line to the actual cost of the 10 – year – old car in the data set? Given the context of the data set, do you think the difference between the predicted price and the actual price is large or small?
Answer:
The line predicts a 10 – year – old car would cost about $9,000. – 1000(10) + 19000 = 9000. Compared to $4,040 for the 10 – year – old car in the data set, the difference would be $4,960. The prediction is off by about $5,000, which seems like a lot of money, given the prices of the cars in the data set.

f. Is $5,000 typical of the differences between predicted prices and actual prices for the cars in this data set? Justify your answer.
Answer:
No, most of the differences would be much smaller than $5,000. Most of the points are much closer to the line, and most predictions would be within about $1,000 of the actual value.

Eureka Math Grade 8 Module 6 Lesson 9 Problem Set Answer Key

Question 1.
The Monopoly board game is popular in many countries. The scatter plot below shows the distance from “Go” to a property (in number of spaces moving from “Go” in a clockwise direction) and the price of the properties on the Monopoly board. The equation of the line is P = 8x + 40, where P represents the price (in Monopoly dollars) and x represents the distance (in number of spaces).

Price of Property Versus Distance from “Go” in Monopoly

a. Use the equation to find the difference (observed value – predicted value) for the most expensive property and for the property that is 35 spaces from “Go.”
Answer:
The most expensive property is 39 spaces from “Go” and costs $400. The price predicted by the line would be 8(39) + 40, or $352. Observed price – predicted price would be $400 – $352 = $48. The price predicted for 35 spaces from “Go” would be 8(35) + 40, or $320. Observed price – predicted price would be $200 – $320 = – $120.

b. Five of the points seem to lie in a horizontal line. What do these points have in common? What is the equation of the line containing those five points?
Answer:
These points all have the same price. The equation of the horizontal line through those points would be
P = 200.

c. Four of the five points described in part (b) are the railroads. If you were fitting a line to predict price with distance from “Go,” would you use those four points? Why or why not?
Answer:
Answers will vary. Because the four points are not part of the overall trend in the price of the properties, I would not use them to determine a line that describes the relationship. I can show this by finding the total error to measure the fit of the line.

Question 2.
The table below gives the coordinates of the five points shown in the scatter plots that follow. The scatter plots show two different lines.

a. Find the predicted response values for each of the two lines.

Answer:

b. For which data points is the prediction based on Line 1 closer to the actual value than the prediction based on Line 2?
Answer:
It is only for data point A. For data point C, both lines are off by the same amount.

c. Which line (Line 1 or Line 2) would you select as a better fit? Explain.
Answer:
Line 2 is a better fit because it is closer to more of the data points.

Question 3.
The scatter plots below show different lines that students used to model the relationship between body mass (in pounds) and bite force (in pounds) for crocodilians.
Match each graph to one of the equations below, and explain your reasoning. Let B represent bite force (in pounds) and W represent body mass (in pounds).
Equation 1
B = 3.28W + 126

Equation 2
B = 3.04W + 351

Equation 3
B = 2.16W + 267

Equation:

Answer:
Equation: 3
The intercept of 267 appears to match the graph, which has the second largest intercept.

Equation:

Answer:
Equation: 2
The intercept of Equation 2 is larger, so it matches Line 2, which has a y – intercept closer to 400.

Equation:

Answer:
Equation: 1
The intercept of Equation 1 is the smallest, which seems to match the graph.

b. Which of the lines would best fit the trend in the data? Explain your thinking.
Answer:
Answers will vary. Line 3 would be better than the other two lines. Line 1 is not a good fit for larger weights, and Line 2 is above nearly all of the points and pretty far away from most of them. It looks like Line 3 would be closer to most of the points.

Question 4.
Comment on the following statements:
a. A line modeling a trend in a scatter plot always goes through the origin.
Answer:
Some trend lines go through the origin, but others may not. Often, the value (0, 0) does not make sense for the data.

b. If the response variable increases as the independent variable decreases, the slope of a line modeling the trend is negative.
Answer:
If the trend is from the upper left to the lower right, the slope for the line is negative because for each unit increase in the independent variable, the response decreases.

Eureka Math Grade 8 Module 6 Lesson 9 Exit Ticket Answer Key

Question 1.
The scatter plot below shows the height and speed of some of the world’s fastest roller coasters. Draw a line that you think is a good fit for the data.

Answer:
Students would draw a line based on the goal of a best fit for the given scatter plot. A possible line is drawn below.

Question 2.
Find the equation of your line. Show your steps.
Answer:
Answers will vary based on the line drawn. Let S equal the speed of the roller coaster and H equal the maximum height of the roller coaster.
m = \(\frac{115 – 85}{500 – 225}\) ≈ 0.11
S = 0.11H + b
85 = 0.11(225) + b
b ≈ 60
Therefore, the equation of the line drawn in Problem 1 is S = 0.11H + 60.

Question 3.
For the two roller coasters identified in the scatter plot, use the line to find the approximate difference between the observed speeds and the predicted speeds.
Answer:
Answers will vary depending on the line drawn by a student or the equation of the line. For the Top Thrill, the maximum height is about 415 feet and the speed is about 100 miles per hour. The line indicated in Problem 2 predicts a speed of 106 miles per hour, so the difference is about 6 miles per hour over the actual speed. For the Kinga Ka, the maximum height is about 424 feet with a speed of 120 miles per hour. The line predicts a speed of about 107 miles per hour, for a difference of 13 miles per hour under the actual speed. (Students can use the graph or the equation to find the predicted speed.)

Engage NY Eureka Math 8th Grade Module 6 Lesson 8 Answer Key

Eureka Math Grade 8 Module 6 Lesson 8 Exercise Answer Key

Example 1: Housing Costs
Let’s look at some data from one midwestern city that indicate the sizes and sale prices of various houses sold in this city.

Data Source: http://www.trulia.com/for_sale/Milwaukee,WI/5_p, accessed in 2013
A scatter plot of the data is given below.

Exercises 1–6

Exercise 1.
What can you tell about the price of large homes compared to the price of small homes from the table?
Answer:
Answers will vary. Students should make the observation that, overall, the larger homes cost more, and the smaller homes cost less. However, it is hard to generalize because one of the smaller homes costs nearly $150,000.

Exercise 2.
Use the scatter plot to answer the following questions.
a. Does the scatter plot seem to support the statement that larger houses tend to cost more? Explain your thinking.
Answer:
Yes, because the trend is positive, the larger the size of the house, the more the house tends to cost.

b. What is the cost of the most expensive house, and where is that point on the scatter plot?
Answer:
The house with a size of 5,232 square feet costs $1,050,000, which is the most expensive. It is in the upper right corner of the scatter plot.

c. Some people might consider a given amount of money and then predict what size house they could buy. Others might consider what size house they want and then predict how much it would cost. How would you use the scatter plot in Example 1?
Answer:
Answers will vary. Since the size of the house is on the horizontal axis and the price is on the vertical axis, the scatter plot is set up with price as the dependent variable and size as the independent variable. This is the way you would set it up if you wanted to predict price based on size. Although various answers are appropriate, move the discussion along using size to predict price.

d. Estimate the cost of a 3,000-square-foot house.
Answer:
Answers will vary. Reasonable answers range between $300,000 and $600,000.

e. Do you think a line would provide a reasonable way to describe how price and size are related? How could you use a line to predict the price of a house if you are given its size?
Answer:
Answers will vary; however, use this question to develop the idea that a line would provide a way to estimate the cost given the size of a house. The challenge is how to make that line. Note: Students are encouraged in the next exercise to first make a line and then evaluate whether or not it fits the data. This will provide a reasonable estimate of the cost of a house in relation to its size.

Exercise 3.
Draw a line in the plot that you think would fit the trend in the data.
Answer:
Answers will vary. Discuss several of the lines students have drawn by encouraging students to share their lines with the class. At this point, do not evaluate the lines as good or bad. Students may want to know a precise procedure or process to draw their lines. If that question comes up, indicate to students that a procedure will be developed in their future work (Algebra I) with statistics. For now, the goal is to simply draw a line that can be used to describe the relationship between the size of a home and its cost. Indicate that strategies for drawing a line will be explored in Exercise 5. Use the lines provided by students to evaluate the predictions in the following exercise. These predictions are used to develop a strategy for drawing a line. Use the line drawn by students to highlight their understanding of the data.

Exercise 4.
Use your line to answer the following questions:
a. What is your prediction of the price of a 3,000-square-foot house?
Answer:
Answers will vary. A reasonable prediction is around $500,000.

b. What is the prediction of the price of a 1,500-square-foot house?
Answer:
Answers will vary. A reasonable prediction is around $200,000.

Exercise 5.
Consider the following general strategies students use for drawing a line. Do you think they represent a good strategy for drawing a line that fits the data? Explain why or why not, or draw a line for the scatter plot using the strategy that would indicate why it is or why it is not a good strategy.
a. Laure thought she might draw her line using the very first point (farthest to the left) and the very last point (farthest to the right) in the scatter plot.
Answer:
Answers will vary. This may work in some cases, but those points might not capture the trend in the data. For example, the first point in the lower left might not be in line with the other points.

b. Phil wants to be sure that he has the same number of points above and below the line.
Answer:
Answers will vary. You could draw a nearly horizontal line that has half of the points above and half below, but that might not represent the trend in the data at all. Note: For many students just starting out, this seems like a reasonable strategy, but it often can result in lines that clearly do not fit the data. As indicated, drawing a nearly horizontal line is a good way to indicate that this is not a good strategy.

c. Sandie thought she might try to get a line that had the most points right on it.
Answer:
Answers will vary. That might result in, perhaps, three points on the line (knowing it only takes two to make a line), but the others could be anywhere. The line might even go in the wrong direction. Note: For students just beginning to think of how to draw a line, this seems like a reasonable goal; however, point out that this strategy may result in lines that are not good for predicting price.

d. Maree decided to get her line as close to as many of the points as possible.
Answer:
Answers will vary. If you can figure out how to do this, Maree’s approach seems like a reasonable way to find a line that takes all of the points into account.

Exercise 6.
Based on the strategies discussed in Exercise 5, would you change how you draw a line through the points? Explain your answer.
Answer:
Answers will vary based on how a student drew his original line. Summarize that the goal is to draw a line that is as close as possible to the points in the scatter plot. More precise methods are developed in Algebra I.

Example 2: Deep Water
Does the current in the water go faster or slower when the water is shallow? The data on the depth and velocity of the Columbia River at various locations in Washington State listed below can help you think about the answer.

Data Source: www.seattlecentral.edu/qelp/sets/011/011.html
a. What can you tell about the relationship between the depth and velocity by looking at the numbers in the table?
Answer:
Answers will vary. According to the table, as the depth increases, the velocity appears to decrease.

b. If you were to make a scatter plot of the data, which variable would you put on the horizontal axis, and why?
Answer:
Answers will vary. It might be easier to measure the depth and use that information to predict the velocity of the water, so the depth should go on the horizontal axis.

Exercises 7–9

Exercise 7.
A scatter plot of the Columbia River data is shown below.

a. Choose a data point in the scatter plot, and describe what it means in terms of the context.
Answer:
Answers will vary. For example, (4.6,1.39) would represent a place in the river that was 4.6 feet deep and had a velocity of 1.39 ft/sec.

b. Based on the scatter plot, describe the relationship between velocity and depth.
Answer:
The deeper the water, the slower the current velocity tends to be.

c. How would you explain the relationship between the velocity and depth of the water?
Answer:
Answers will vary. Sample response: Velocity may be a result of the volume of water. Shallow water has less volume, and as a result, the water runs faster. Note: Students may have several explanations. For example, they may say that depth is a result of less water runoff; therefore, water depth increases.

d. If the river is two feet deep at a certain spot, how fast do you think the current would be? Explain your reasoning.
Answer:
Answers will vary. Based on the data, it could be around 1.11 ft/sec, or it could be closer to 1.42 ft/sec, which is more in line with the pattern for the other points.

Exercise 8.
Consider the following questions:
a. If you draw a line to represent the trend in the plot, would it make it easier to predict the velocity of the water if you know the depth? Why or why not?
Answer:
Answers will vary. A line will help you determine a better prediction for 1.5 ft. or 5 ft., where the points are a bit scattered.

b. Draw a line that you think does a reasonable job of modeling the trend on the scatter plot in Exercise 7. Use the line to predict the velocity when the water is 8 feet deep.
Answer:
Answers will vary. A line is drawn in the following graph. Using this line, when the water is 8 ft. deep, the velocity is predicted to be 0.76 ft/sec.

Exercise 9.
Use the line to predict the velocity for a depth of 8.6 feet. How far off was your prediction from the actual observed velocity for the location that had a depth of 8.6 feet?
Answer:
Answers will vary. Sample response: The current would be moving at a velocity of 0.68 ft/sec. The observed velocity was 0.59 ft/sec, so the line predicted a velocity that was 0.09 ft/sec faster than the observed value.

Eureka Math Grade 8 Module 6 Lesson 8 Problem Set Answer Key

Question 1.
The table below shows the mean temperature in July and the mean amount of rainfall per year for 14 cities in the Midwest.

Data Source: http://countrystudies.us/united-states/weather/
a. What do you observe from looking at the data in the table?
Answer:
Answers will vary. Many of the temperatures were in the 70’s, and many of the mean inches of rain were in the 30’s. It also appears that, in general, as the rainfall increased, the mean temperature also increased.

b. Look at the scatter plot below. A line is drawn to fit the data. The plot in the Exit Ticket had the mean July temperatures for the cities on the horizontal axis. How is this plot different, and what does it mean for the way you think about the relationship between the two variables—temperature and rain?

Answer:
This scatter plot has the labels on the axes reversed: (mean inches of rain, mean temperature). This is the scatter plot I would use if I wanted to predict the mean temperature in July knowing the mean amount of rain per year.

c. The line has been drawn to model the relationship between the amount of rain and the temperature in those Midwestern cities. Use the line to predict the mean July temperature for a Midwestern city that has a mean of 32 inches of rain per year.
Answer:
Answers will vary. For 32 in. of rain per year, the line indicates a mean July temperature of approximately 70°F.

d. For which of the cities in the sample does the line do the worst job of predicting the mean temperature? The best? Explain your reasoning with as much detail as possible.
Answer:
Answers will vary. I looked for points that were really close to the line and ones that were far away. The line prediction for temperature would be farthest off for Minneapolis. For 29.41 in. of rain in Minneapolis, the line predicted approximately 67°F, whereas the actual mean temperature in July was 73.2°F. The line predicted very well for Milwaukee.

For 32.95 in. of rain in Milwaukee, the line predicted approximately 73°F, whereas the actual mean temperature in July was 72°F and was only off by about 1°F. The line was also close for Marquette. For 34.81 in. of rain in Marquette, the line predicted approximately 71°F, whereas the actual mean temperature in July was 71.6°F and was only off by about 1°F.

Question 2.
The scatter plot below shows the results of a survey of eighth-grade students who were asked to report the number of hours per week they spend playing video games and the typical number of hours they sleep each night.
Mean Hours Sleep per Night Versus Mean Hours Playing Video Games per Week

a. What trend do you observe in the data?
Answer:
The more hours that students play video games, the fewer hours they tend to sleep.

b. What was the fewest number of hours per week that students who were surveyed spent playing video games? The most?
Answer:
Two students spent 0 hours, and one student spent 32 hours per week playing games.

c. What was the fewest number of hours per night that students who were surveyed typically slept? The most?
Answer:
The fewest hours of sleep per night was around 5 hours, and the most was around 10 hours.

d. Draw a line that seems to fit the trend in the data, and find its equation. Use the line to predict the number of hours of sleep for a student who spends about 15 hours per week playing video games.
Answer:
Answers will vary. A student who spent 15 hours per week playing games would get about 7 hours of sleep per night.

Question 3.
Scientists can take very good pictures of alligators from airplanes or helicopters. Scientists in Florida are interested in studying the relationship between the length and the weight of alligators in the waters around Florida.
a. Would it be easier to collect data on length or weight? Explain your thinking.
Answer:
Answers will vary. You could measure the length from the pictures, but you would have to actually have the alligators to weigh them.

b. Use your answer to decide which variable you would want to put on the horizontal axis and which variable you might want to predict.
Answer:
You would probably want to predict the weight of the alligator knowing the length; therefore, the length would go on the horizontal axis and the weight on the vertical axis.

Question 4.
Scientists captured a small sample of alligators and measured both their length (in inches) and weight (in pounds). Torre used their data to create the following scatter plot and drew a line to capture the trend in the data. She and Steve then had a discussion about the way the line fit the data. What do you think they were discussing, and why?
Alligator Length (inches) and Weight (pounds)

Data Source: James Landwehr and Ann Watkins, Exploring Data, Quantitative Literacy Series (Dale Seymour, 1987).
Answer:
Answers will vary. Sample response: The pattern in the scatter plot is curved instead of linear. All of the data points in the middle of the scatter plot fall below the line, and the line does not really capture the pattern in the scatter plot. A line does not pass through the cluster of points between 60 to 80 in. in length that fit the other points. A model other than a line might be a better fit.

Eureka Math Grade 8 Module 6 Lesson 8 Exit Ticket Answer Key

The plot below is a scatter plot of mean temperature in July and mean inches of rain per year for a sample of midwestern cities. A line is drawn to fit the data.

Data Source: http://countrystudies.us/united-states/weather/
Question 1.
Choose a point in the scatter plot, and explain what it represents.
Answer:
Answers will vary. Sample response: The point at about (72,35) represents a Midwestern city where the mean temperature in July is about 72°F and where the rainfall per year is about 35 inches.

Question 2.
Use the line provided to predict the mean number of inches of rain per year for a city that has a mean temperature of 70°F in July.
Answer:
Predicted rainfall is 33 inches of rain per year. (Some students will state approximately 33.5 inches of rain.)

Question 3.
Do you think the line provided is a good one for this scatter plot? Explain your answer.
Answer:
Yes. The line follows the general pattern in the scatter plot, and it does not look like there is another area in the scatter plot where the points would be any closer to the line.

Engage NY Eureka Math 8th Grade Module 6 Lesson 7 Answer Key

Eureka Math Grade 8 Module 6 Lesson 7 Exercise Answer Key

Example 1.
In the previous lesson, you learned that scatter plots show trends in bivariate data.
When you look at a scatter plot, you should ask yourself the following questions:
a. Does it look like there is a relationship between the two variables used to make the scatter plot?
b. If there is a relationship, does it appear to be linear?
c. If the relationship appears to be linear, is the relationship a positive linear relationship or a negative linear relationship?

To answer the first question, look for patterns in the scatter plot. Does there appear to be a general pattern to the points in the scatter plot, or do the points look as if they are scattered at random? If you see a pattern, you can answer the second question by thinking about whether the pattern would be well described by a line. Answering the third question requires you to distinguish between a positive linear relationship and a negative linear relationship. A positive linear relationship is one that is described by a line with a positive slope. A negative linear relationship is one that is described by a line with a negative slope.

Exercises 1–9
Take a look at the following five scatter plots. Answer the three questions in Example 1 for each scatter plot.

Exercise 1.
Scatter Plot 1

Is there a relationship?
Answer:
Yes

If there is a relationship, does it appear to be linear?
Answer:
Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?
Answer:
Negative

Exercise 2.
Scatter Plot 2

Is there a relationship?
Answer:
Yes

If there is a relationship, does it appear to be linear?
Answer:
Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?
Answer:
Positive

Exercise 3.
Scatter Plot 3

Is there a relationship?
Answer:
No

If there is a relationship, does it appear to be linear?
Answer:
Not applicable

If the relationship appears to be linear, is it a positive or a negative linear relationship?
Answer:
Not applicable

Exercise 4.
Scatter Plot 4

Is there a relationship?
Answer:
Yes

If there is a relationship, does it appear to be linear?
Answer:
No

If the relationship appears to be linear, is it a positive or a negative linear relationship?
Answer:
Not applicable

Exercise 5.
Scatter Plot 5

Is there a relationship?
Answer:
Yes

If there is a relationship, does it appear to be linear?
Answer:
Yes

If the relationship appears to be linear, is it a positive or a negative linear relationship?
Answer:
Negative

Exercise 6.
Below is a scatter plot of data on weight in pounds (x) and fuel efficiency in miles per gallon (y) for 13 cars. Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between x and y.

Answer:
Possible response: There appears to be a negative linear relationship between fuel efficiency and weight. Students may note that this is a fairly strong negative relationship. The cars with greater weight tend to have lesser fuel efficiency.

Exercise 7.
Below is a scatter plot of data on price in dollars (x) and quality rating (y) for 14 bike helmets. Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between x and y.

Answer:
Possible response: There does not appear to be a relationship between quality rating and price. The points in the scatter plot appear to be scattered at random, and there is no apparent pattern in the scatter plot.

Exercise 8.
Below is a scatter plot of data on shell length in millimeters (x) and age in years (y) for 27 lobsters of known age. Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between x and y.

Answer:
Possible response: There appears to be a relationship between shell length and age, but the pattern in the scatter plot is curved rather than linear. Age appears to increase as shell length increases, but the increase is not at a constant rate.

Exercise 9.
Below is a scatter plot of data from crocodiles on body mass in pounds (x) and bite force in pounds (y). Using the questions at the beginning of this lesson as a guide, write a few sentences describing any possible relationship between x and y.

Data Source: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0031781#pone-0031781-t001
(Note: Body mass and bite force have been converted to pounds from kilograms and newtons, respectively.)
Answer:
Possible response: There appears to be a positive linear relationship between bite force and body mass. For crocodiles, the greater the body mass, the greater the bite force tends to be. Students may notice that this is a positive relationship but not quite as strong as the relationship noted in Exercise 6.

Example 2: Clusters and Outliers
In addition to looking for a general pattern in a scatter plot, you should also look for other interesting features that might help you understand the relationship between two variables. Two things to watch for are as follows:
CLUSTERS: Usually, the points in a scatter plot form a single cloud of points, but sometimes the points may form two or more distinct clouds of points. These clouds are called clusters. Investigating these clusters may tell you something useful about the data.
OUTLIERS: An outlier is an unusual point in a scatter plot that does not seem to fit the general pattern or that is far away from the other points in the scatter plot.
The scatter plot below was constructed using data from a study of Rocky Mountain elk (“Estimating Elk Weight from Chest Girth,” Wildlife Society Bulletin, 1996). The variables studied were chest girth in centimeters (x) and weight in kilograms (y).

Exercises 10–12

Exercise 10.
Do you notice any point in the scatter plot of elk weight versus chest girth that might be described as an outlier? If so, which one?
Answer:
Possible response: The point in the lower left-hand corner of the plot corresponding to an elk with a chest girth of about 96 cm and a weight of about 100 kg could be described as an outlier. There are no other points in the scatter plot that are near this one.

Exercise 11.
If you identified an outlier in Exercise 10, write a sentence describing how this data observation differs from the others in the data set.
Answer:
Possible response: This point corresponds to an observation for an elk that is much smaller than the other elk in the data set, both in terms of chest girth and weight.

Exercise 12.
Do you notice any clusters in the scatter plot? If so, how would you distinguish between the clusters in terms of chest girth? Can you think of a reason these clusters might have occurred?
Answer:
Possible response: Other than the outlier, there appear to be three clusters of points. One cluster corresponds to elk with chest girths between about 105 cm and 115 cm. A second cluster includes elk with chest girths between about 120 cm and 145 cm. The third cluster includes elk with chest girths above 150 cm. It may be that age and sex play a role. Maybe the cluster with the smaller chest girths includes young elk. The two other clusters might correspond to females and males if there is a difference in size for the two sexes for Rocky Mountain elk. If we had data on age and sex, we could investigate this further.

Eureka Math Grade 8 Module 6 Lesson 7 Problem Set Answer Key

Question 1.
Suppose data was collected on size in square feet (x) of several houses and price in dollars (y). The data was then used to construct the scatterplot below. Write a few sentences describing the relationship between price and size for these houses. Are there any noticeable clusters or outliers?

Answer:
Answers will vary. Possible response: There appears to be a positive linear relationship between size and price. Price tends to increase as size increases. There appear to be two clusters of houses—one that includes houses that are less than 3,000 square feet in size and another that includes houses that are more than 3,000 square feet
in size.

Question 2.
The scatter plot below was constructed using data on length in inches (x) of several alligators and weight in pounds (y). Write a few sentences describing the relationship between weight and length for these alligators. Are there any noticeable clusters or outliers?

Data Source: Exploring Data, Quantitative Literacy Series, James Landwehr and Ann Watkins, 1987.
Answer:
Answers will vary. Possible response: There appears to be a positive relationship between length and weight, but the relationship is not linear. Weight tends to increase as length increases. There are three observations that stand out as outliers. These correspond to alligators that are much bigger in terms of both length and weight than the other alligators in the sample. Without these three alligators, the relationship between length and weight would look linear. It might be possible to use a line to model the relationship between weight and length for alligators that have lengths of fewer than 100 inches.

Question 3.
Suppose the scatter plot below was constructed using data on age in years (x) of several Honda Civics and price in dollars (y). Write a few sentences describing the relationship between price and age for these cars. Are there any noticeable clusters or outliers?

Answer:
Answers will vary. Possible response: There appears to be a negative linear relationship between price and age. Price tends to decrease as age increases. There is one car that looks like an outlier—the car that is 10 years old. This car has a price that is lower than expected based on the pattern of the other points in the scatter plot.

Question 4.
Samples of students in each of the U.S. states periodically take part in a large-scale assessment called the National Assessment of Educational Progress (NAEP). The table below shows the percent of students in the northeastern states (as defined by the U.S. Census Bureau) who answered Problems 7 and 15 correctly on the 2011 eighth-grade test. The scatter plot shows the percent of eighth-grade students who got Problems 7 and 15 correct on the 2011 NAEP.


a. Why does it appear that there are only eight points in the scatter plot for nine states?
Answer:
Two of the states, New Hampshire and Rhode Island, had exactly the same percent correct on each of the questions, (29,52).

b. What is true of the states represented by the cluster of five points in the lower left corner of the graph?
Answer:
Answers will vary; those states had lower percentages correct than the other three states in the upper right.

c. Which state did the best on these two problems? Explain your reasoning.
Answer:
Answers will vary; some students might argue that Massachusetts at (35,56) did the best. Even though Vermont actually did a bit better on Problem 15, it was lower on Problem 7.

d. Is there a trend in the data? Explain your thinking.
Answer:
Answers will vary; there seems to be a positive linear trend, as a large percent correct on one question suggests a large percent correct on the other, and a low percent on one suggests a low percent on the other.

Question 5.
The plot below shows the mean percent of sunshine during the year and the mean amount of precipitation in inches per year for the states in the United States.

Data source: www.currentresults.com/Weather/US/average-annual-state-sunshine.php
www.currentresults.com/Weather/US/average-annual-state-precipitation.php
a. Where on the graph are the states that have a large amount of precipitation and a small percent of sunshine?
Answer:
Those states will be in the lower right-hand corner of the graph.

b. The state of New York is the point (46,41.8). Describe how the mean amount of precipitation and percent of sunshine in New York compare to the rest of the United States.
Answer:
New York has a little over 40 inches of precipitation per year and is sunny about 45% of the time. It has a smaller percent of sunshine over the year than most states and is about in the middle of the states in terms of the amount of precipitation, which goes from about 10 to 65 inches per year.

c. Write a few sentences describing the relationship between mean amount of precipitation and percent of sunshine.
Answer:
There is a negative relationship, or the more precipitation, the less percent of sun. If you took away the three states at the top left with a large percent of sun and very little precipitation, the trend would not be as pronounced. The relationship is not linear.

Question 6.
At a dinner party, every person shakes hands with every other person present.
a. If three people are in a room and everyone shakes hands with everyone else, how many handshakes take place?
Answer:
Three handshakes

b. Make a table for the number of handshakes in the room for one to six people. You may want to make a diagram or list to help you count the number of handshakes.

Answer:

c. Make a scatter plot of number of people (x) and number of handshakes (y). Explain your thinking.

Answer:

d. Does the trend seem to be linear? Why or why not?
Answer:
The trend is increasing, but it is not linear. As the number of people increases, the number of handshakes also increases. It does not increase at a constant rate.

Eureka Math Grade 8 Module 6 Lesson 7 Exit Ticket Answer Key

Question 1.
Which of the following scatter plots shows a negative linear relationship? Explain how you know.

Answer:
Only Scatter Plot 3 shows a negative linear relationship because the y-values tend to decrease as the value of x increases.

Question 2.
The scatter plot below was constructed using data from eighth-grade students on number of hours playing video games per week (x) and number of hours of sleep per night (y). Write a few sentences describing the relationship between sleep time and time spent playing video games for these students. Are there any noticeable clusters or outliers?

Answer:
Answers will vary. Sample response: There appears to be a negative linear relationship between the number of hours per week a student plays video games and the number of hours per night the student sleeps. As video game time increases, the number of hours of sleep tends to decrease. There is one observation that might be considered an outlier—the point corresponding to a student who plays video games 32 hours per week. Other than the outlier, there are two clusters—one corresponding to students who spend very little time playing video games and a second corresponding to students who play video games between about 10 and 25 hours per week.

Question 3.
In a scatter plot, if the values of y tend to increase as the value of x increases, would you say that there is a positive relationship or a negative relationship between x and y? Explain your answer.
Answer:
There is a positive relationship. If the value of y increases as the value of x increases, the points go up on the scatter plot from left to right.

Engage NY Eureka Math 8th Grade Module 6 Lesson 6 Answer Key

Eureka Math Grade 8 Module 6 Lesson 6 Example Answer Key

Example 1.
A bivariate data set consists of observations on two variables. For example, you might collect data on 13 different car models. Each observation in the data set would consist of an (x,y) pair.
x: weight (in pounds, rounded to the nearest 50 pounds)
and
y: fuel efficiency (in miles per gallon, mpg)
The table below shows the weight and fuel efficiency for 13 car models with automatic transmissions manufactured in 2009 by Chevrolet.

Eureka Math Grade 8 Module 6 Lesson 6 Exercise Answer Key

Exercises 1–8

Exercise 1.
In the Example 1 table, the observation corresponding to Model 1 is (3200,23). What is the fuel efficiency of this car? What is the weight of this car?
Answer:
The fuel efficiency is 23 miles per gallon, and the weight is 3,200 pounds.

Exercise 2.
Add the points corresponding to the other 12 observations to the scatter plot.

Answer:

Exercise 3.
Do you notice a pattern in the scatter plot? What does this imply about the relationship between weight (x) and fuel efficiency (y)?
Answer:
There does seem to be a pattern in the plot. Higher weights tend to be paired with lesser fuel efficiencies, so it looks like heavier cars generally have lower fuel efficiency.

Is there a relationship between price and the quality of athletic shoes? The data in the table below are from the Consumer Reports website.
x: price (in dollars)
and
y: Consumer Reports quality rating
The quality rating is on a scale of 0 to 100, with 100 being the highest quality.

Exercise 4.
One observation in the data set is (110,57). What does this ordered pair represent in terms of cost and quality?
Answer:
The pair represents a shoe that costs $110 with a quality rating of 57.

Exercise 5.
To construct a scatter plot of these data, you need to start by thinking about appropriate scales for the axes of the scatter plot. The prices in the data set range from $30 to $110, so one reasonable choice for the scale of the x-axis would range from $20 to $120, as shown below. What would be a reasonable choice for a scale for the y-axis?

Answer:
Sample response: The smallest y-value is 51, and the largest y-value is 71. So, the y-axis could be scaled from 50 to 75.

Exercise 6.
Add a scale to the y-axis. Then, use these axes to construct a scatter plot of the data.

Exercise 7.
Do you see any pattern in the scatter plot indicating that there is a relationship between price and quality rating for athletic shoes?
Answer:
Answers will vary. Students may say that they do not see a pattern, or they may say that they see a slight downward trend.

Exercise 8.
Some people think that if shoes have a high price, they must be of high quality. How would you respond?
Answer:
Answers will vary. The data do not support this. Students will either respond that there does not appear to be a relationship between price and quality, or if they saw a downward trend in the scatter plot, they might even indicate that the higher-priced shoes tend to have lower quality. Look for consistency between the answer to this question and how students answered the previous question.

Exercises 9–10

Exercise 9.
Data were collected on
x: shoe size
and
y: score on a reading ability test
for 29 elementary school students. The scatter plot of these data is shown below. Does there appear to be a statistical relationship between shoe size and score on the reading test?

Answer:
Possible response: The pattern in the scatter plot appears to follow a line. As shoe sizes increase, the reading scores also seem to increase. There does appear to be a statistical relationship because there is a pattern in the scatter plot.

Exercise 10.
Explain why it is not reasonable to conclude that having big feet causes a high reading score. Can you think of a different explanation for why you might see a pattern like this?
Answer:
Possible response: You cannot conclude that just because there is a statistical relationship between shoe size and reading score that one causes the other. These data were for students completing a reading test for younger elementary school children. Older children, who would have bigger feet than younger children, would probably tend to score higher on a reading test for younger students.

Eureka Math Grade 8 Module 6 Lesson 6 Problem Set Answer Key

Question 1.
The table below shows the price and overall quality rating for 15 different brands of bike helmets.
Data source: www.consumerreports.org

Construct a scatter plot of price (x) and quality rating (y). Use the grid below.

Answer:

Question 2.
Do you think that there is a statistical relationship between price and quality rating? If so, describe the nature of the relationship.
Answer:
Sample response: No. There is no pattern visible in the scatter plot. There does not appear to be a relationship between price and the quality rating for bike helmets.

Question 3.
Scientists are interested in finding out how different species adapt to finding food sources. One group studied crocodilian species to find out how their bite force was related to body mass and diet. The table below displays the information they collected on body mass (in pounds) and bite force (in pounds).

Data Source: http://journals.plos.org/plosone/article?id=10.1371/journal.pone.0031781#pone-0031781-t001
(Note: Body mass and bite force have been converted to pounds from kilograms and newtons, respectively.)
Construct a scatter plot of body mass (x) and bite force (y). Use the grid below, and be sure to add an appropriate scale to the axes.

Answer:

Question 4.
Do you think that there is a statistical relationship between body mass and bite force? If so, describe the nature of the relationship.
Answer:
Sample response: Yes, because it looks like there is an upward pattern in the scatter plot. It appears that alligators with larger body mass also tend to have greater bite force.

Question 5.
Based on the scatter plot, can you conclude that increased body mass causes increased bite force? Explain.
Answer:
Sample response: No. Just because there is a statistical relationship between body mass and bite force does not mean that there is a cause-and-effect relationship.

Eureka Math Grade 8 Module 6 Lesson 6 Exit Ticket Answer Key

Energy is measured in kilowatt-hours. The table below shows the cost of building a facility to produce energy and the ongoing cost of operating the facility for five different types of energy.

Question 1.
Construct a scatter plot of the cost to build the facility in dollars per kilowatt-hour (x) and the cost to operate the facility in cents per kilowatt-hour (y). Use the grid below, and be sure to add an appropriate scale to the axes.

Answer:

Question 2.
Do you think that there is a statistical relationship between building cost and operating cost? If so, describe the nature of the relationship.
Answer:
Answers may vary. Sample response: Yes, because it looks like there is a downward pattern in the scatter plot. It appears that the types of energy that have facilities that are more expensive to build are less expensive to operate.

Question 3.
Based on the scatter plot, can you conclude that decreased building cost is the cause of increased operating cost? Explain.
Answer:
Sample response: No. Just because there may be a statistical relationship between cost to build and cost to operate does not mean that there is a cause-and-effect relationship.

Engage NY Eureka Math 8th Grade Module 6 Lesson 5 Answer Key

Eureka Math Grade 8 Module 6 Lesson 5 Exercise Answer Key

Exercises 1–2

Exercise 1.
In your own words, describe what is happening as Aleph is running during the following intervals of time.
a. 0 to 15 minutes
Answer:
Aleph runs 2 miles in 15 minutes.

b. 15 to 30 minutes
Answer:
Aleph runs another 2 miles in 15 minutes for a total of 4 miles.

c. 30 to 45 minutes
Answer:
Aleph runs another 2 miles in 15 minutes for a total of 6 miles.

d. 45 to 60 minutes
Answer:
Aleph runs another 2 miles in 15 minutes for a total of 8 miles.

Exercise 2.
In your own words, describe what is happening as Shannon is running during the following intervals of time.
a. 0 to 15 minutes
Answer:
Shannon runs 1.5 miles in 15 minutes.

b. 15 to 30 minutes
Answer:
Shannon runs another 0.6 mile in 15 minutes for a total of 2.1 miles.

c. 30 to 45 minutes
Answer:
Shannon runs another 0.5 mile in 15 minutes for a total of 2.6 miles.

d. 45 to 60 minutes
Answer:
Shannon runs another 0.4 mile in 15 minutes for a total of 3.0 miles.

Exercises 3–5

Exercise 3.
Different breeds of dogs have different growth rates. A large breed dog typically experiences a rapid growth rate from birth to age 6 months. At that point, the growth rate begins to slow down until the dog reaches full growth around 2 years of age.
a. Sketch a graph that represents the weight of a large breed dog from birth to 2 years of age.

Answer:
Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.
Answer:
The function is nonlinear because the growth rate is not constant.

c. Is the function represented by the graph increasing or decreasing? Explain.
Answer:
The function is increasing but at a decreasing rate. There is rapid growth during the first 6 months, and then the growth rate decreases.

Exercise 4.
Nikka took her laptop to school and drained the battery while typing a research paper. When she returned home, Nikka connected her laptop to a power source, and the battery recharged at a constant rate.
a. Sketch a graph that represents the battery charge with respect to time.

Answer:
Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.
Answer:
The function is linear because the battery is recharging at a constant rate.

c. Is the function represented by the graph increasing or decreasing? Explain.
Answer:
The function is increasing because the battery is being recharged.

Exercise 5.
The long jump is a track-and-field event where an athlete attempts to leap as far as possible from a given point. Mike Powell of the United States set the long jump world record of 8.95 meters (29.4 feet) during the 1991 World Championships in Tokyo, Japan.
a. Sketch a graph that represents the path of a high school athlete attempting the long jump.

Answer:
Answers will vary

b. Is the function represented by the graph linear or nonlinear? Explain.
Answer:
The function is nonlinear. The rate of change is not constant.

c. Is the function represented by the graph increasing or decreasing? Explain.
Answer:
The function both increases and decreases over different intervals. The function increases as the athlete begins the jump and reaches a maximum height. The function decreases after the athlete reaches maximum height and begins descending back toward the ground.

Exercises 6–9

Exercise 6.
Use the graph from Example 3 to answer the following questions.
a. Is the function represented by the graph linear or nonlinear?
Answer:
The function is nonlinear. The rate of change is not constant.

b. Where is the function increasing? What does this mean within the context of the problem?
Answer:
The function is increasing during the following intervals of time: 0 to 4 seconds, 8 to 12 seconds, 16 to 20 seconds, 24 to 28 seconds, and 32 to 36 seconds. It means that Lamar and his sister are rising in the air.

c. Where is the function decreasing? What does this mean within the context of the problem?
Answer:
The function is decreasing during the following intervals of time: 4 to 8 seconds, 12 to 16 seconds, 20 to 24 seconds, 28 to 32 seconds, and 36 to 40 seconds. Lamar and his sister are traveling back down toward the ground.

Exercise 7.
How high above the ground is the platform for passengers to get on the Ferris wheel? Explain your reasoning.
Answer:
The lowest point on the graph, which is at 5 feet, can represent the platform where the riders get on the Ferris wheel.

Exercise 8.
Based on the graph, how many revolutions does the Ferris wheel complete during the 40-second time interval? Explain your reasoning.
Answer:
The Ferris wheel completes 5 revolutions. The lowest points on the graph can represent Lamar and his sister at the beginning of a revolution or at the entrance platform of the Ferris wheel. So, one revolution occurs between 0 and 8 seconds, 8 and 16 seconds, 16 and 24 seconds, 24 and 32 seconds, and 32 and 40 seconds.

Exercise 9.
What is the diameter of the Ferris wheel? Explain your reasoning.
Answer:
The diameter of the Ferris wheel is 40 feet. The lowest point on the graph represents the base of the Ferris wheel, and the highest point on the graph represents the top of the Ferris wheel. The difference between the two values is 40 feet, which is the diameter of the wheel.

Eureka Math Grade 8 Module 6 Lesson 5 Problem Set Answer Key

Question 1.
Read through the following scenarios, and match each to its graph. Explain the reasoning behind your choice.
a. This shows the change in a smartphone battery charge as a person uses the phone more frequently.
b. A child takes a ride on a swing.
c. A savings account earns simple interest at a constant rate.
d. A baseball has been hit at a youth baseball game.
Scenario: ____

Answer:
Scenario: c

The savings account is earning interest at a constant rate, which means that the function is linear.

Scenario: ____

Answer:
Scenario: d

The baseball is hit into the air, reaches a maximum height, and falls back to the ground at a variable rate.

Scenario: ____

Answer:
Scenario: b

The distance from the ground increases as the child swings up into the air and then decreases as the child swings back down toward the ground.

Scenario: ____

Answer:
Scenario: a

The battery charge is decreasing as a person uses the phone more frequently.

Question 2.
The graph below shows the volume of water for a given creek bed during a 24-hour period. On this particular day, there was wet weather with a period of heavy rain.

Describe how each part (A, B, and C) of the graph relates to the scenario.
Answer:
A: The rain begins, and the volume of water flowing in the creek bed begins to increase.
B: A period of heavy rain occurs, causing the volume of water to increase.
C: The heavy rain begins to subside, and the volume of water continues to increase.

Question 3.
Half-life is the time required for a quantity to fall to half of its value measured at the beginning of the time period. If there are 100 grams of a radioactive element to begin with, there will be 50 grams after the first half-life, 25 grams after the second half-life, and so on.
a. Sketch a graph that represents the amount of the radioactive element left with respect to the number of half-lives that have passed.

Answer:
Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.
Answer:
The function is nonlinear. The rate of change is not constant with respect to time.

c. Is the function represented by the graph increasing or decreasing?
Answer:
The function is decreasing.

Question 4.
Lanae parked her car in a no-parking zone. Consequently, her car was towed to an impound lot. In order to release her car, she needs to pay the impound lot charges. There is an initial charge on the day the car is brought to the lot. However, 10% of the previous day’s charges will be added to the total charge for every day the car remains in the lot.
a. Sketch a graph that represents the total charges with respect to the number of days a car remains in the impound lot.

Answer:
Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.
Answer:
The function is nonlinear. The function is increasing.

c. Is the function represented by the graph increasing or decreasing? Explain.
Answer:
The function is increasing. The total charge is increasing as the number of days the car is left in the lot increases.

Question 5.
Kern won a $50 gift card to his favorite coffee shop. Every time he visits the shop, he purchases the same coffee drink.
a. Sketch a graph of a function that can be used to represent the amount of money that remains on the gift card with respect to the number of drinks purchased.

Answer:
Answers will vary.

b. Is the function represented by the graph linear or nonlinear? Explain.
Answer:
The function is linear. Since Kern purchases the same drink every visit, the balance is decreasing by the same amount or, in other words, at a constant rate of change.

c. Is the function represented by the graph increasing or decreasing? Explain.
Answer:
The function is decreasing. With each drink purchased, the amount of money on the card decreases.

Question 6.
Jay and Brooke are racing on bikes to a park 8 miles away. The tables below display the total distance each person biked with respect to time.

a. Which person’s biking distance could be modeled by a nonlinear function? Explain.
Answer:
The distance that Jay biked could be modeled by a nonlinear function because the rate of change is not constant. The distance that Brooke biked could be modeled by a linear function because the rate of change is constant.

b. Who would you expect to win the race? Explain.
Answer:
Jay will win the race. The distance he bikes during each five-minute interval is increasing, while Brooke’s biking distance remains constant. If the trend remains the same, it is estimated that both Jay and Brooke will travel about 7.2 miles in 30 minutes. So, Jay will overtake Brooke during the last 5 minutes to win the race.

Question 7.
Using the axes in Problem 7(b), create a story about the relationship between two quantities.
a. Write a story about the relationship between two quantities. Any quantities can be used (e.g., distance and time, money and hours, age and growth). Be creative! Include keywords in your story such as increase and decrease to describe the relationship.
Answer:
Answers will vary.
A person in a car is at a red stoplight. The light turns green, and the person presses down on the accelerator with increasing pressure. The car begins to move and accelerate. The rate at which the car accelerates is
not constant.

b. Label each axis with the quantities of your choice, and sketch a graph of the function that models the relationship described in the story.

Answer:
Answers will vary based on the story from part (a).

Eureka Math Grade 8 Module 6 Lesson 5 Exit Ticket Answer Key

Question 1.
Lamar and his sister continue to ride the Ferris wheel. The graph below represents Lamar and his sister’s distance above the ground with respect to time during the next 40 seconds of their ride.

a. Name one interval where the function is increasing.
Answer:
The function is increasing during the following intervals of time: 40 to 44 seconds, 48 to 52 seconds, 56 to 60 seconds, and 72 to 76 seconds.

b. Name one interval where the function is decreasing.
Answer:
The function is decreasing during the following intervals of time: 44 to 48 seconds, 52 to 56 seconds, 64 to 66 seconds, 70 to 72 seconds, and 76 to 80 seconds.

c. Is the function linear or nonlinear? Explain.
Answer:
The function is both linear and nonlinear during different intervals of time. It is linear from 60 to 64 seconds and from 66 to 70 seconds. It is nonlinear from 40 to 60 seconds and from 70 to 80 seconds.

d. What could be happening during the interval of time from 60 to 64 seconds?
Answer:
The Ferris wheel is not moving during that time, so riders may be getting off or getting on.

e. Based on the graph, how many complete revolutions are made during this 40-second interval?
Answer:
Four revolutions are made during this time period.

Engage NY Eureka Math 8th Grade Module 6 Lesson 4 Answer Key

Eureka Math Grade 8 Module 6 Lesson 4 Exercise Answer Key

Exercise 1.
Read through each of the scenarios, and choose the graph of the function that best matches the situation. Explain the reason behind each choice.
a. A bathtub is filled at a constant rate of 1.75 gallons per minute.
b. A bathtub is drained at a constant rate of 2.5 gallons per minute.
c. A bathtub contains 2.5 gallons of water.
d. A bathtub is filled at a constant rate of 2.5 gallons per minute.

Scenario:
Explanation:
Answer:
Scenario: c

Explanation: The amount of water in the tub does not change over time; it remains constant at 2.5 gallons.

Scenario:
Explanation:
Answer:
Scenario: b

Explanation: The bathtub is being drained at a constant rate of 2.5 gallons per minute. So, the amount of water is decreasing, which means that the slope of the line is negative. The graph of the function also shows that there are initially 20 gallons of water in the tub.

Scenario:
Explanation:
Answer:
Scenario: d

Explanation: The tub is being filled at a constant rate of 2.5 gallons per minute, which implies that the amount of water in the tub is increasing, so the line has a positive slope. Based on the graph, the amount of water is also increasing at a faster rate than in choice (a).

Scenario:
Explanation:
Answer:
Scenario: d

Explanation: The tub is being filled at a constant rate of 2.5 gallons per minute, which implies that the amount of water in the tub is increasing, so the line has a positive slope. Based on the graph, the amount of water is also increasing at a faster rate than in choice (a).

Exercise 2.
Read through each of the scenarios, and sketch a graph of a function that models the situation.
a. A messenger service charges a flat rate of $4.95 to deliver a package regardless of the distance to the destination.
Answer:

Answer:
The delivery charge remains constant regardless of the distance to the destination.

b. At sea level, the air that surrounds us presses down on our bodies at 14.7 pounds per square inch (psi). For every 10 meters that you dive under water, the pressure increases by 14.7 psi.

Answer:
The initial value is 14.7 psi. The function increases at a rate of 14.7 psi for every 10 meters, or 1.47 psi per meter.

c. The range (driving distance per charge) of an electric car varies based on the average speed the car is driven. The initial range of the electric car after a full charge is 400 miles. However, the range is reduced by 20 miles for every 10 mph increase in average speed the car is driven.

Answer:
The initial value of the function is 400. The function is decreasing by 20 miles for every 10 mph increase in speed. In other words, the function decreases by 2 miles for every 1 mph increase in speed.

Exercise 3.
The graph below represents the total number of smartphones that are shipped to a retail store over the course of 50 days.

Match each part of the graph (A, B, and C) to its verbal description. Explain the reasoning behind your choice.
i. Half of the factory workers went on strike, and not enough smartphones were produced for normal shipments.
Answer:
C; if half of the workers went on strike, then the number of smartphones produced would be less than normal. The rate of change for C is less than the rate of change for A.

ii. The production schedule was normal, and smartphones were shipped to the retail store at a constant rate.
Answer:
A; if the production schedule is normal, the rate of change of interval A is greater than the rate of change of interval C.

iii. A defective electronic chip was found, and the factory had to shut down, so no smartphones were shipped.
Answer:
B; if no smartphones are shipped to the store, the total number remains constant during that time.

Exercise 4.
The relationship between Jameson’s account balance and time is modeled by the graph below.

a. Write a story that models the situation represented by the graph.
Answer:
Answers will vary.
Jameson was sick and did not work for almost a whole week. Then, he mowed several lawns over the next few days and deposited the money into his account after each job. It rained several days, so instead of working, Jameson withdrew money from his account each day to go to the movies and out to lunch with friends.

b. When is the function represented by the graph increasing? How does this relate to your story?
Answer:
It is increasing between 6 and 9 days. Jameson earned money mowing lawns and made a deposit to his account each day. The money earned for each day was constant for these days. This is represented by a straight line.

c. When is the function represented by the graph decreasing? How does this relate to your story?
Answer:
It is decreasing between 9 and 14 days. Since Days 9–14 are represented by a straight line, this means that Jameson spent the money constantly over these days. Jameson cannot work because it is raining. Perhaps he withdraws money from his account to spend on different activities each day because he cannot work.

Eureka Math Grade 8 Module 6 Lesson 4 Problem Set Answer Key

Question 1.
Read through each of the scenarios, and choose the graph of the function that best matches the situation. Explain the reason behind each choice.
a. The tire pressure on Regina’s car remains at 30 psi.
b. Carlita inflates her tire at a constant rate for 4 minutes.
c. Air is leaking from Courtney’s tire at a constant rate.

Scenario:
Explanation:
Answer:
Scenario: c

Explanation: The tire pressure decreases each minute at a constant rate.

Scenario:
Explanation:
Answer:
Scenario: a

Explanation: The tire pressure remains at 30 psi.

Scenario:
Explanation:
Answer:
Scenario: b

Explanation: The tire pressure is increasing each minute at a constant rate for 4 minutes.

Question 2.
A home was purchased for $275,000. Due to a recession, the value of the home fell at a constant rate over the next 5 years.
a. Sketch a graph of a function that models the situation.

Answer:
Graphs will vary; a sample graph is provided.

b. Based on your graph, how is the home value changing with respect to time?
Answer:
Answers will vary; a sample answer is provided.
The value is decreasing by $25,000 over 5 years or at a constant rate of $5,000 per year.

Question 3.
The graph below displays the first hour of Sam’s bike ride.

Match each part of the graph (A, B, and C) to its verbal description. Explain the reasoning behind your choice.
i. Sam rides his bike to his friend’s house at a constant rate.
Answer:
A; the distance from home should be increasing as Sam is riding toward his friend’s house.

ii. Sam and his friend bike together to an ice cream shop that is between their houses.
Answer:
C; Sam was at his friend’s house, but as they start biking to the ice cream shop, the distance from Sam’s home begins to decrease.

iii. Sam plays at his friend’s house.
Answer:
B; Sam remains at the same distance from home while he is at his friend’s house.

Question 4.
Using the axes below, create a story about the relationship between two quantities.
a. Write a story about the relationship between two quantities. Any quantities can be used (e.g., distance and time, money and hours, age and growth). Be creative. Include keywords in your story such as increase and decrease to describe the relationship.
Answer:
Answers will vary. Give students the freedom to write a basic linear story or a piecewise story.
A rock climber begins her descent from a height of 50 feet. She slowly descends at a constant rate for 4 minutes. She takes a break for 1 minute; she then realizes she left some of her gear on top of the rock and climbs more quickly back to the top at a constant rate.

b. Label each axis with the quantities of your choice, and sketch a graph of the function that models the relationship described in the story.

Answer:
Answers will vary based on the story from part (a).

Eureka Math Grade 8 Module 6 Lesson 4 Exit Ticket Answer Key

Question 1.
The graph below shows the relationship between a car’s value and time.

Match each part of the graph (A, B, and C) to its verbal description. Explain the reasoning behind your choice.
i. The value of the car holds steady due to a positive consumer report on the same model.
Answer:
B; if the value is holding steady, there is no change in the car’s value between years.

ii. There is a shortage of used cars on the market, and the value of the car rises at a constant rate.
Answer:
C; if the value of the car is rising, it represents an increasing function.

iii. The value of the car depreciates at a constant rate.
Answer:
A; if the value depreciates, it represents a decreasing function.

Question 2.
Henry and Roxy both drive electric cars that need to be recharged before use. Henry uses a standard charger at his home to recharge his car. The graph below represents the relationship between the battery charge and the amount of time it has been connected to the power source for Henry’s car.

Answer:

a. Describe how Henry’s car battery is being recharged with respect to time.
Answer:
The battery charge is increasing at a constant rate of 10% every 10 minutes.

b. Roxy has a supercharger at her home that can charge about half of the battery in 20 minutes. There is no remaining charge left when she begins recharging the battery. Sketch a graph that represents the relationship between the battery charge and the amount of time on the axes above. Assume the relationship is linear.
Answer:
See the above graph.

c. Which person’s car will be recharged to full capacity first? Explain.
Answer:
Roxy’s car will be completely recharged first. Her supercharger has a greater rate of change compared to Henry’s charger.

Engage NY Eureka Math 8th Grade Module 7 Lesson 1 Answer Key

Eureka Math Grade 8 Module 7 Lesson 1 Example Answer Key

Example 1.
Write an equation that allows you to determine the length of the unknown side of the right triangle.

Answer:
Write an equation that allows you to determine the length of the unknown side of the right triangle.
Note: Students may use a different symbol to represent the unknown side length.
Let b cm represent the unknown side length. Then, 5² + b² = 13².
Verify that students wrote the correct equation; then, allow them to solve it. Ask them how they knew the correct answer was 12. They should respond that 13²-5² = 144, and since 144 is a perfect square, they knew that the unknown side length must be 12 cm.

Example 2.
Write an equation that allows you to determine the length of the unknown side of the right triangle.

Answer:
→ Write an equation that allows you to determine the length of the unknown side of the right triangle.
Let c cm represent the length of the hypotenuse. Then, 4² + 9² = c².

→ There is something different about this triangle. What is the length of the missing side? If you cannot find the length of the missing side exactly, then find a good approximation.
Provide students time to find an approximation for the length of the unknown side. Select students to share their answers and explain their reasoning. Use the points below to guide their thinking as needed.

→ How is this problem different from the last one?
The answer is c² = 97. Since 97 is not a perfect square, the exact length cannot be represented as an integer.

→ Since 97 is not a perfect square, we cannot determine the exact length of the hypotenuse as an integer; however, we can make an estimate. Think about all of the perfect squares we have seen and calculated in past discussions. The number 97 is between which two perfect squares?
The number 97 is between 81 and 100.
If c² were 81, what would be the length of the hypotenuse?
The length would be 9 cm.
If c² were 100, what would be the length of the hypotenuse?
The length would be 10 cm.

→ At this point, we know that the length of the hypotenuse is somewhere between 9 cm and 10 cm. Think about the length to which it is closest. The actual length of the hypotenuse is determined by the equation c² = 97. To which perfect square number, 100 or 81, is 97 closer?
The number 97 is closer to the perfect square 100 than to the perfect square 81.

→ Now that we know that the length of the hypotenuse of this right triangle is between 9 cm and 10 cm, but closer to 10 cm, let’s try to get an even better estimate of the length. Choose a number between 9 and 10 but closer to 10. Square that number. Do this a few times to see how close you can get to the number 97.

Provide students time to check a few numbers between 9 and 10. Students should see that the length is somewhere between 9.8 cm and 9.9 cm because 9.8² = 96.04 and 9.9² = 98.01. Some students may even check 9.85; 9.85² = 97.0225. This activity shows students that an estimation of the length being between 9 cm and 10 cm is indeed accurate, and it helps students develop an intuitive sense of how to estimate square roots.

Example 3.
Write an equation to determine the length of the unknown side of the right triangle.

Answer:
→ Write an equation to determine the length of the unknown side of the right triangle.
Let c cm represent the length of the hypotenuse. Then, 3² + 8² = c².
Verify that students wrote the correct equation, and then allow them to solve it. Instruct them to estimate the length, if necessary. Then, let them continue to work. When most students have finished, ask the questions below.

→ Could you determine an answer for the length of the hypotenuse as an integer?
No. Since c² = 73, the length of the hypotenuse is not a perfect square.
Optionally, you can ask, “Can anyone find the exact length of the hypotenuse as a rational number?” It is important that students recognize that no one can determine the exact length of the hypotenuse as a rational number at this point.

→ Since 73 is not a perfect square, we cannot determine the exact length of the hypotenuse as a whole number. Let’s estimate the length. Between which two whole numbers is the length of the hypotenuse? Explain.
Since 73 is between the two perfect squares 64 and 81, we know the length of the hypotenuse must be between 8 cm and 9 cm.
→ Is the length closer to 8 cm or 9 cm? Explain.
The length is closer to 9 cm because 73 is closer to 81 than it is to 64.
→ The length of the hypotenuse is between 8 cm and 9 cm but closer to 9 cm.

Example 4.
In the figure below, we have an equilateral triangle with a height of 10 Inches. What do we know about an equilateral triangle?

Answer:
→ In the figure below, we have an equilateral triangle with a height of 10 inches. What do we know about an equilateral triangle?
Equilateral triangles have sides that are all of the same length and angles that are all of the same degree, namely 60°.
Let’s say the length of the sides is x inches. Determine the approximate length of the sides of the triangle.

→ What we actually have here are two congruent right triangles.
Trace one of the right triangles on a transparency, and reflect it across the line representing the height of the triangle to convince students that an equilateral triangle is composed of two congruent right triangles.

With this knowledge, we need to determine the length of the base of one of the right triangles. If we know that the length of the base of the equilateral triangle is x inches, then what is the length of the base of one of the right triangles? Explain.

Eureka Math Grade 8 Module 7 Lesson 1 Exercise Answer Key

Exercise 1.
Use the Pythagorean theorem to find a whole number estimate of the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let x cm be the length of the unknown side.
6² + x² = 11²
36 + x² = 121
x² = 85
The length of the unknown side of the triangle is approximately 9 cm. The number 85 is between the perfect squares 81 and 100. Since 85 is closer to 81 than 100, then the length of the unknown side of the triangle is closer to 9 cm than it is to 10 cm.

Exercise 2.
Use the Pythagorean theorem to find a whole number estimate of the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let c in. be the length of the hypotenuse.
6² + 10² = c²
36 + 100 = c²
136 = c²
The length of the hypotenuse is approximately 12 in. The number 136 is between the perfect squares 121 and 144. Since 136 is closer to 144 than 121, the length of the unknown side of the triangle is closer to 12 in. than it is to 11 in.

Exercise 3.
Use the Pythagorean theorem to find a whole number estimate of the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let x mm be the length of the unknown side.
9² + x² = 11²
81 + x² = 121
x² = 40
The length of the hypotenuse is approximately 6 mm. The number 40 is between the perfect squares 36 and 49. Since 40 is closer to 36 than 49, then the length of the unknown side of the triangle is closer to 6 mm than it is to 7 mm.

Eureka Math Grade 8 Module 7 Lesson 1 Problem Set Answer Key

Question 1.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let x in. be the length of the unknown side.
13² + x² = 15²
169 + x² = 225
x² = 56
The number 56 is between the perfect squares 49 and 64. Since 56 is closer to 49 than it is to 64, the length of the unknown side of the triangle is closer to 7 in. than it is to 8 in.

Question 2.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let x cm be the length of the unknown side.
x² + 12² = 13²
x² + 144 = 169
x² = 25
x = 5
The length of the unknown side is 5 cm. The Pythagorean theorem led me to the fact that the square of the value of the unknown length is 25. Since 25 is a perfect square, 25 is equal to 5²; therefore, x = 5, and the unknown length of the triangle is 5 cm.

Question 3.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let c in. be the length of the hypotenuse.
4² + 12² = c²
16 + 144 = c²
160 = c²
The number 160 is between the perfect squares 144 and 169. Since 160 is closer to 169 than it is to 144, the length of the hypotenuse of the triangle is closer to 13 in. than it is to 12 in.

Question 4.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let x cm be the length of the unknown side.
x² + 11² = 13²
x² + 121 = 169
x² = 48
The number 48 is between the perfect squares 36 and 49. Since 48 is closer to 49 than it is to 36, the length of the unknown side of the triangle is closer to 7 cm than it is to 6 cm.

Question 5.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let c in. be the length of the hypotenuse.
6² + 8² = c²
36 + 64 = c²
100 = c²
10 = c
The length of the hypotenuse is 10 in. The Pythagorean theorem led me to the fact that the square of the value of the unknown length is 100. We know 100 is a perfect square, and 100 is equal to 10²; therefore, c = 10, and the length of the hypotenuse of the triangle is 10 in.

Question 6.
Determine the length of the unknown side of the right triangle. Explain how you know your answer is correct.

Answer:
Let c cm be the length of the hypotenuse.
7² + 4² = c²
49 + 16 = c²
65 = c²
The number 65 is between the perfect squares 64 and 81. Since 65 is closer to 64 than it is to 81, the length of the hypotenuse of the triangle is closer to 8 cm than it is to 9 cm.

Question 7.
Use the Pythagorean theorem to estimate the length of the unknown side of the right triangle. Explain why your estimate makes sense.

Answer:
Let x mm be the length of the unknown side.
3² + x² = 12²
9 + x² = 144
x² = 135
The number 135 is between the perfect squares 121 and 144. Since 135 is closer to 144 than it is to 121, the length of the unknown side of the triangle is closer to 12 mm than it is to 11 mm.

Question 8.
The triangle below is an isosceles triangle. Use what you know about the Pythagorean theorem to determine the approximate length of the base of the isosceles triangle.

Answer:
Let x ft. represent the length of the base of one of the right triangles of the isosceles triangle.
x² + 7² = 9²
x² + 49 = 81
x² = 32
Since 32 is between the perfect squares 25 and 36 but closer to 36, the approximate length of the base of the right triangle is 6 ft. Since there are two right triangles, the length of the base of the isosceles triangle is approximately 12 ft.

Question 9.
Give an estimate for the area of the triangle shown below. Explain why it is a good estimate.

Answer:
Let x cm represent the length of the base of the right triangle.
x² + 3² = 7²
x² + 9 = 49
x² = 40
Since 40 is between the perfect squares 36 and 49 but closer to 36, the approximate length of the base is 6 cm.
A = \(\frac{1}{2}\)(6)(3) = 9
So, the approximate area of the triangle is 9 cm². This is a good estimate because of the approximation of the length of the base. Further, since the hypotenuse is the longest side of the right triangle, approximating the length of the base as 6 cm makes mathematical sense because it has to be shorter than the hypotenuse.

Eureka Math Grade 8 Module 7 Lesson 1 Exit Ticket Answer Key

Question 1.
Determine the length of the unknown side of the right triangle. If you cannot determine the length exactly, then determine which two integers the length is between and the integer to which it is closest.

Answer:
Let x in. be the length of the unknown side.
9² + x² = 15²
81 + x² = 225
x² = 144
x = 12
The length of the unknown side is 12 in. The Pythagorean theorem led me to the fact that the square of the value of the unknown length is 144. We know 144 is a perfect square, and 144 is equal to 12²; therefore, x = 12, and the unknown length of the triangle is 12 in.

Question 2.
Determine the length of the unknown side of the right triangle. If you cannot determine the length exactly, then determine which two integers the length is between and the integer to which it is closest.

Answer:
Let x mm be the length of the unknown side.
2² + 7² = x²
4 + 49 = x²
53 = x²
The number 53 is between the perfect squares 49 and 64. Since 53 is closer to 49 than 64, the length of the unknown side of the triangle is closer to 7 mm than 8 mm.

Engage NY Eureka Math 8th Grade Module 5 Lesson 8 Answer Key

Eureka Math Grade 8 Module 5 Lesson 8 Exploratory Challenge/Exercise Answer Key

Exercise 1.
Consider the function that assigns to each number x the value x².
a. Do you think the function is linear or nonlinear? Explain.
Answer:
I think the function is nonlinear. The equation describing the function is not of the form y = mx + b.

b. Develop a list of inputs and outputs for this function. Organize your work using the table below. Then, answer the questions that follow.

Answer:

c. Plot the inputs and outputs as ordered pairs defining points on the coordinate plane.

Answer:

d. What shape does the graph of the points appear to take?
Answer:
It appears to take the shape of a curve.

e. Find the rate of change using rows 1 and 2 from the table above.
Answer:
\(\frac{25 – 16}{ – 5 – ( – 4)}\) = \(\frac{9}{ – 1}\) = – 9

f. Find the rate of change using rows 2 and 3 from the table above.
Answer:
\(\frac{16 – 9}{ – 4 – ( – 3)} = \) = \(\frac{7}{ – 1}\) = – 7

g. Find the rate of change using any two other rows from the table above.
Answer:
Student work will vary.
\(\frac{16 – 25}{4 – 5}\) = \(\frac{ – 9}{ – 1}\) = 9

h. Return to your initial claim about the function. Is it linear or nonlinear? Justify your answer with as many pieces of evidence as possible.
Answer:
This is definitely a nonlinear function because the rate of change is not a constant for different intervals of inputs. Also, we would expect the graph of a linear function to be a set of points in a line, and this graph is not a line. As was stated before, the expression x² is nonlinear.

Exercise 2.
Consider the function that assigns to each number x the value x³.
a. Do you think the function is linear or nonlinear? Explain.
Answer:
I think the function is nonlinear. The equation describing the function is not of the form y = mx + b.

b. Develop a list of inputs and outputs for this function. Organize your work using the table below. Then, answer the questions that follow.

Answer:

c. Plot the inputs and outputs as ordered pairs defining points on the coordinate plane.

Answer:

d. What shape does the graph of the points appear to take?
Answer:
It appears to take the shape of a curve.

e. Find the rate of change using rows 2 and 3 from the table above.
Answer:
\(\frac{ – 8 – ( – 3.375)}{ – 2 – ( – 1.5)}\) = \(\frac{ – 4.625}{ – 0.5}\) = 9.25

f. Find the rate of change using rows 3 and 4 from the table above.
Answer:
\(\frac{ – 3.375 – ( – 1)}{ – 1.5 – ( – 1)}\) = \(\frac{ – 2.375}{ – 0.5}\) = 4.75

g. Find the rate of change using rows 8 and 9 from the table above.
Answer:
\(\frac{1 – 3.375}{1 – 1.5}\) = \(\frac{ – 2.375}{ – 0.5}\) = 4.75

h. Return to your initial claim about the function. Is it linear or nonlinear? Justify your answer with as many pieces of evidence as possible.
Answer:
This is definitely a nonlinear function because the rate of change is not a constant for any interval of inputs. Also, we would expect the graph of a linear function to be a line, and this graph is not a line. As was stated before, the expression x³ is nonlinear.

Exercise 3.
Consider the function that assigns to each positive number x the value \(\frac{1}{x}\).
a. Do you think the function is linear or nonlinear? Explain.
Answer:
I think the function is nonlinear. The equation describing the function is not of the form y = mx + b.

b. Develop a list of inputs and outputs for this function. Organize your work using the table below. Then, answer the questions that follow.

Answer:

c. Plot the inputs and outputs as ordered pairs defining points on the coordinate plane.

Answer:

d. What shape does the graph of the points appear to take?
Answer:
It appears to take the shape of a curve.

e. Find the rate of change using rows 1 and 2 from the table above.
Answer:
\(\frac{10 – 5}{0.1 – 0.2}\) = \(\frac{5}{ – 0.1}\) = – 50

f. Find the rate of change using rows 2 and 3 from the table above.
Answer:
\(\frac{5 – 2.5}{0.2 – 0.4}\) = \(\frac{2.5}{ – 0.2}\) = – 12.5

g. Find the rate of change using any two other rows from the table above.
Answer:
Student work will vary.
\(\frac{1 – 0.625}{1 – 1.6}\) = \(\frac{0.375}{ – 0.6}\) = – 0.625

h. Return to your initial claim about the function. Is it linear or nonlinear? Justify your answer with as many pieces of evidence as possible.
Answer:
This is definitely a nonlinear function because the rate of change is not a constant for any interval of inputs. Also, we would expect the graph of a linear function to be a line, and this graph is not a line. As was stated before, the expression \(\frac{1}{x}\) is nonlinear.

Exercises 4–10
In each of Exercises 4–10, an equation describing a rule for a function is given, and a question is asked about it. If necessary, use a table to organize pairs of inputs and outputs, and then plot each on a coordinate plane to help answer the question.

Exercise 4.
What shape do you expect the graph of the function described by y = x to take? Is it a linear or nonlinear function?
Answer:

I expect the shape of the graph to be a line. This function is a linear function described by the linear equation y = x. The graph of this function is a line.

Exercise 5.
What shape do you expect the graph of the function described by y = 2x² – x to take? Is it a linear or nonlinear function?
Answer:

I expect the shape of the graph to be something other than a line. This function is nonlinear because its graph is not a line. Also the equation describing the function is not of the form y = mx + b. It is not linear.

Exercise 6.
What shape do you expect the graph of the function described by 3x + 7y = 8 to take? Is it a linear or nonlinear function?
Answer:

I expect the shape of the graph to be a line. This function is a linear function described by the linear equation 3x + 7y = 8. The graph of this function is a line. (We have y = – \(\frac{3}{7}\) x + \(\frac{8}{7}\).)

Exercise 7.
What shape do you expect the graph of the function described by y = 4x³ to take? Is it a linear or nonlinear function?
Answer:

I expect the shape of the graph to be something other than a line. This function is nonlinear because its graph is not a line. Also the equation describing the function is not of the form y = mx + b. It is not linear.

Exercise 8.
What shape do you expect the graph of the function described by \(\frac{3}{x}\) = y to take? Is it a linear or nonlinear function? (Assume that an input of x = 0 is disallowed.)
Answer:

I expect the shape of the graph to be something other than a line. This function is nonlinear because its graph is not a line. Also the equation describing the function is not of the form y = mx + b. It is not linear.

Exercise 9.
What shape do you expect the graph of the function described by \(\frac{4}{x^{2}}\) = y to take? Is it a linear or nonlinear function? (Assume that an input of x = 0 is disallowed.)
Answer:

I expect the shape of the graph to be something other than a line. This function is nonlinear because its graph is not a line. Also the equation describing the function is not of the form y = mx + b. It is not linear.

Exercise 10.
What shape do you expect the graph of the equation x² + y² = 36 to take? Is it a linear or nonlinear function? Is it a function? Explain.
Answer:

I expect the shape of the graph to be something other than a line. It is nonlinear because its graph is not a line. It is not a function because there is more than one output for any given value of x in the interval ( – 6, 6). For example, at x = 0 the y – value is both 6 and – 6. This does not fit the definition of function because functions assign to each input exactly one output. Since there is at least one instance where an input has two outputs, it is not a function.

Eureka Math Grade 8 Module 5 Lesson 8 Problem Set Answer Key

Question 1.
Consider the function that assigns to each number x the value x² – 4.
a. Do you think the function is linear or nonlinear? Explain.
Answer:
The equation describing the function is not of the form y = mx + b. It is not linear.

b. Do you expect the graph of this function to be a straight line?
Answer:
No

c. Develop a list of inputs and matching outputs for this function. Use them to begin a graph of the function.

Answer:

Question 2.
Consider the function that assigns to each number x greater than – 3 the value \(\frac{1}{x + 3}\).
a. Is the function linear or nonlinear? Explain.
Answer:
The equation describing the function is not of the form y = mx + b. It is not linear.

b. Do you expect the graph of this function to be a straight line?
Answer:
No

c. Develop a list of inputs and matching outputs for this function. Use them to begin a graph of the function.

Answer:

d. Was your prediction to (b) correct?
Answer:
Yes, the graph appears to be taking the shape of some type of curve.

Question 3.
a. Is the function represented by this graph linear or nonlinear? Briefly justify your answer.

Answer:
The graph is clearly not a straight line, so the function is not linear.

b. What is the average rate of change for this function from an input of x = – 2 to an input of x = – 1?
Answer:
\(\frac{ – 2 – 1}{ – 2 – ( – 1)}\) = \(\frac{ – 3}{ – 1}\) = 3

c. What is the average rate of change for this function from an input of x = – 1 to an input of x = 0?
Answer:
\(\frac{1 – 2}{ – 1 – 0}\) = \(\frac{ – 1}{ – 1}\) = 1
As expected, the average rate of change of this function is not constant.

Eureka Math Grade 8 Module 5 Lesson 8 Exit Ticket Answer Key

Question 1.
The graph below is the graph of a function. Do you think the function is linear or nonlinear? Briefly justify your answer.

Answer:
Student work may vary. The plot of this graph appears to be a straight line, and so the function is linear.

Question 2.
Consider the function that assigns to each number x the value \(\frac{1}{2}\) x². Do you expect the graph of this function to be a straight line? Briefly justify your answer.
Answer:
The equation is nonlinear (not of the form y = mx + b), so the function is nonlinear. Its graph will not be a straight line.

Engage NY Eureka Math 8th Grade Module 6 Lesson 2 Answer Key

Eureka Math Grade 8 Module 6 Lesson 2 Example Answer Key

Example 1: Rate of Change and Initial Value
The equation of a line can be interpreted as defining a linear function. The graphs and the equations of lines are important in understanding the relationship between two types of quantities (represented in the following examples by x and y).
In a previous lesson, you encountered an MP3 download site that offers downloads of individual songs with the following price structure: a $3 fixed fee for a monthly subscription plus a fee of $0.25 per song. The linear function that models the relationship between the number of songs downloaded and the total monthly cost of downloading songs can be written as
y = 0.25x + 3,
where x represents the number of songs downloaded and y represents the total monthly cost (in dollars) for
MP3 downloads.
a. In your own words, explain the meaning of 0.25 within the context of the problem.
Answer:
In the example on the previous page, the value 0.25 means there is a cost increase of $0.25 for every 1 song downloaded.

b. In your own words, explain the meaning of 3 within the context of the problem.
Answer:
In the example on the previous page, the value of 3 represents an initial cost of $3 for downloading 0 songs. In other words, there is a fixed cost of $3 to subscribe to the site.

The values represented in the function can be interpreted in the following way:

Eureka Math Grade 8 Module 6 Lesson 2 Exercise Answer Key

Exercises 1–6: Is It a Better Deal?
Another site offers MP3 downloads with a different price structure: a $2 fixed fee for a monthly subscription plus a fee of $0.40 per song.
Exercise 1.
Write a linear function to model the relationship between the number of songs downloaded and the total monthly cost. As before, let x represent the number of songs downloaded and y represent the total monthly cost (in dollars) of downloading songs.
Answer:
y = 0.4x + 2

Exercise 2.
Determine the cost of downloading 0 songs and 10 songs from this site.
Answer:
y = 0.4(0) + 2 = 2.00. For 0 songs, the cost is $2.00.
y = 0.4(10) + 2 = 6.00. For 10 songs, the cost is $6.00.

Exercise 3.
The graph below already shows the linear model for the first subscription site (Company 1): y = 0.25x + 3. Graph the equation of the line for the second subscription site (Company 2) by marking the two points from your work in Exercise 2 (for 0 songs and 10 songs) and drawing a line through those two points.

Answer:

Exercise 4.
Which line has a steeper slope? Which company’s model has the more expensive cost per song?
Answer:
The line modeled by the second subscription site (Company 2) is steeper. It has the larger slope value and the greater cost per song.

Exercise 5.
Which function has the greater initial value?
Answer:
The first subscription site (Company 1) has the greater initial value. Its monthly subscription fee is $3 compared to only $2 for the second site.

Exercise 6.
Which subscription site would you choose if you only wanted to download 5 songs per month? Which company would you choose if you wanted to download 10 songs? Explain your reasoning.
Answer:
For 5 songs: Company 1’s cost is $4.25 (y = 25(5) + 3); Company 2’s cost is $4.00 (y = 0.4(5) + 2). So, Company 2 would be the better choice. Graphically, Company 2’s model also has the smaller y – value when x = 5.
For 10 songs: Company 1’s cost is $5.50 (y = 0.25(10) + 3); Company 2’s cost is $6.00 (y = 0.4(10) + 2). So, Company 1 would be the better choice. Graphically, Company 1’s model also has the smaller y – value at x = 10.

Exercises 7–9: Aging Autos

Exercise 7.
When someone purchases a new car and begins to drive it, the mileage (meaning the number of miles the car has traveled) immediately increases. Let x represent the number of years since the car was purchased and y represent the total miles traveled. The linear function that models the relationship between the number of years since purchase and the total miles traveled is y = 15000x.
a. Identify and interpret the rate of change.
Answer:
The rate of change is 15,000. It means that the mileage is increasing by 15,000 miles per year.

b. Identify and interpret the initial value.
Answer:
The initial value is 0. This means that there were no miles on the car when it was purchased.

c. Is the mileage increasing or decreasing each year according to the model? Explain your reasoning.
Answer:
Since the rate of change is positive, it means the mileage is increasing each year.

Exercise 8.
When someone purchases a new car and begins to drive it, generally speaking, the resale value of the car (in dollars) goes down each year. Let x represent the number of years since purchase and y represent the resale value of the car (in dollars). The linear function that models the resale value based on the number of years since purchase is
y = 20000 – 1200x.
a. Identify and interpret the rate of change.
Answer:
The rate of change is – 1,200. The resale value of the car is decreasing $1,200 every year since purchase.

b. Identify and interpret the initial value.
Answer:
The initial value is $20,000. The car’s value at the time of purchase was $20,000.

c. Is the resale value increasing or decreasing each year according to the model? Explain.
Answer:
The slope is negative. This means that the resale value decreases each year.

Exercise 9.
Suppose you are given the linear function y = 2.5x + 10.
a. Write a story that can be modeled by the given linear function.
Answer:
Answers will vary. I am ordering cupcakes for a birthday party. The bakery is going to charge $2.50 per cupcake in addition to a $10 decorating fee.

b. What is the rate of change? Explain its meaning with respect to your story.
Answer:
The rate of change is 2.5, which means that the cost increases $2.50 for every additional cupcake ordered.

c. What is the initial value? Explain its meaning with respect to your story.
Answer:
The initial value is 10, which in this story means that there is a flat fee of $10 to decorate the cupcakes.

Eureka Math Grade 8 Module 6 Lesson 2 Problem Set Answer Key

Question 1.
A rental car company offers the following two pricing methods for its customers to choose from for a
one – month rental:
Method 1: Pay $400 for the month, or
Method 2: Pay $0.30 per mile plus a standard maintenance fee of $35.
a. Construct a linear function that models the relationship between the miles driven and the total rental cost for Method 2. Let x represent the number of miles driven and y represent the rental cost (in dollars).
Answer:
y = 35 + 0.30x

b. If you plan to drive 1,100 miles for the month, which method would you choose? Explain your reasoning.
Answer:
Method 1 has a flat rate of $400 regardless of miles. Using Method 2, the cost would be $365
(y = 35 + 0.3(1100)). So, Method 2 would be preferred.

Question 2.
Recall from a previous lesson that Kelly wants to add new music to her MP3 player. She was interested in a monthly subscription site that offered its MP3 downloading service for a monthly subscription fee plus a fee per song. The linear function that modeled the total monthly cost in dollars (y) based on the number of songs downloaded (x) is
y = 5.25 + 0.30x.
The site has suddenly changed its monthly price structure. The linear function that models the new total monthly cost in dollars (y) based on the number of songs downloaded (x) is y = 0.35x + 4.50.
a. Explain the meaning of the value 4.50 in the new equation. Is this a better situation for Kelly than before?
Answer:
The initial value is 4.50 and means that the monthly subscription cost is now $4.50. This is lower than before, which is good for Kelly.

b. Explain the meaning of the value 0.35 in the new equation. Is this a better situation for Kelly than before?
Answer:
The rate of change is 0.35. This means that the cost is increasing by $0.35 for every song downloaded. This is more than the download cost for the original plan.

c. If you were to graph the two equations (old versus new), which line would have the steeper slope? What does this mean in the context of the problem?
Answer:
The slope of the new line is steeper because the new linear function has a greater rate of change. It means that the total monthly cost of the new plan is increasing at a faster rate per song compared to the cost of the old plan.

d. Which subscription plan provides the better value if Kelly downloads fewer than 15 songs per month?
Answer:
If Kelly were to download 15 songs, both plans will cost the same ($9.75). Therefore, the new plan is cheaper if Kelly downloads fewer than 15 songs.

Eureka Math Grade 8 Module 6 Lesson 2 Exit Ticket Answer Key

In 2008, a collector of sports memorabilia purchased 5 specific baseball cards as an investment. Let y represent each card’s resale value (in dollars) and x represent the number of years since purchase. Each card’s resale value after 0, 1, 2, 3, and 4 years could be modeled by linear equations as follows:
Card A: y = 5 – 0.7x
Card B: y = 4 + 2.6x
Card C: y = 10 + 0.9x
Card D: y = 10 – 1.1x
Card E: y = 8 + 0.25x

Question 1.
Which card(s) are decreasing in value each year? How can you tell?
Answer:
Cards A and D are decreasing in value, as shown by the negative values for rate of change in each equation.

Question 2.
Which card(s) had the greatest initial value at purchase (at 0 years)?
Answer:
Since all of the models are in slope – intercept form, Cards C and D have the greatest initial values at $10 each.

Question 3.
Which card(s) is increasing in value the fastest from year to year? How can you tell?
Answer:
Card B is increasing in value the fastest from year to year. Its model has the greatest rate of change.

Question 4.
If you were to graph the equations of the resale values of Card B and Card C, which card’s graph line would be steeper? Explain.
Answer:
The Card B line would be steeper because the function for Card B has the greatest rate of change; the card’s value is increasing at a faster rate than the other values of other cards.

Question 5.
Write a sentence explaining the 0.9 value in Card C’s equation.
Answer:
The 0.9 value means that Card C’s value increases by 90 cents per year.

Engage NY Eureka Math 8th Grade Module 6 Lesson 3 Answer Key

Eureka Math Grade 8 Module 6 Lesson 3 Example Answer Key

Example 1: Rate of Change and Initial Value Given in the Context of the Problem
A truck rental company charges a $150 rental fee in addition to a charge of $0.50 per mile driven. Graph the linear function relating the total cost of the rental in dollars, C, to the number of miles driven, m, on the axes below.

Answer:

a. If the truck is driven 0 miles, what is the cost to the customer? How is this shown on the graph?
Answer:
$150, shown as the point (0, 150). This is the initial value. Some students might say “b.” Help them to use the term initial value.

b. What is the rate of change that relates cost to number of miles driven? Explain what it means within the context of the problem.
Answer:
The rate of change is 0.5. It means that the cost increases by $0.50 for every mile driven.

c. On the axes given, sketch the graph of the linear function that relates C to m.
Answer:
Students can plot the initial value (0, 150) and then use the rate of change to identify additional points as needed. A 1, 000-unit increase in m results in a 500-unit increase for C, so another point on the line is (1000, 650).

d. Write the equation of the linear function that models the relationship between number of miles driven and total rental cost.
Answer:
C = 0.5m + 150

Eureka Math Grade 8 Module 6 Lesson 3 Exercise Answer Key

Exercises
Jenna bought a used car for $18, 000. She has been told that the value of the car is likely to decrease by $2, 500 for each year that she owns the car. Let the value of the car in dollars be V and the number of years Jenna has owned the car be t.

Answer:

Exercise 1.
What is the value of the car when t = 0? Show this point on the graph.
Answer:
$18, 000. Shown by the point (0, 18000)

Exercise 2.
What is the rate of change that relates V to t? (Hint: Is it positive or negative? How can you tell?)
Answer:
-2, 500. The rate of change is negative because the value of the car is decreasing.

Exercise 3.
Find the value of the car when:
a. t = 1
Answer:
$18000 – $2500 = $15500

b. t = 2
Answer:
$18000 – 2($2500) = $13000

c. t = 7
Answer:
$18000 – 7($2500) = $500

Exercise 4.
Plot the points for the values you found in Exercise 3, and draw the line (using a straightedge) that passes through those points.
Answer:
See the graph above.

Exercise 5.
Write the linear function that models the relationship between the number of years Jenna has owned the car and the value of the car.
Answer:
V = 18000 – 2500t or V = -2500t + 18000

An online bookseller has a new book in print. The company estimates that if the book is priced at $15, then 800 copies of the book will be sold per day, and if the book is priced at $20, then 550 copies of the book will be sold per day.

Answer:

Exercise 6.
Identify the ordered pairs given in the problem. Then, plot both on the graph.
Answer:
The ordered pairs are (15, 800) and (20, 550). See the graph above.

Exercise 7.
Assume that the relationship between the number of books sold and the price is linear. (In other words, assume that the graph is a straight line.) Using a straightedge, draw the line that passes through the two points.
Answer:
See the graph above.

Exercise 8.
What is the rate of change relating number of copies sold to price?
Answer:
Between the points (15, 800) and (20, 550), the run is 5, and the rise is -(800-550) = -250. So, the rate of change is \(\frac{-250}{5}\) = -50.

Exercise 9.
Based on the graph, if the company prices the book at $18, about how many copies of the book can they expect to sell per day?
Answer:
650

Exercise 10.
Based on the graph, approximately what price should the company charge in order to sell 700 copies of the book per day?
Answer:
$17

Eureka Math Grade 8 Module 6 Lesson 3 Problem Set Answer Key

Question 1.
A plumbing company charges a service fee of $120, plus $40 for each hour worked. Sketch the graph of the linear function relating the cost to the customer (in dollars), C, to the time worked by the plumber (in hours), t, on the axes below.

Answer:

a. If the plumber works for 0 hours, what is the cost to the customer? How is this shown on the graph?
Answer:
$120 This is shown on the graph by the point (0, 120).

b. What is the rate of change that relates cost to time?
Answer:
40

c. Write a linear function that models the relationship between the hours worked and the cost to the customer.
Answer:
C = 40t + 120

d. Find the cost to the customer if the plumber works for each of the following number of hours.
i) 1 hour
Answer:
$160

ii) 2 hours
Answer:
$200

iii) 6 hours
Answer:
$360

e. Plot the points for these times on the coordinate plane, and use a straightedge to draw the line through the points.
Answer:
See the graph on the previous page.

Question 2.
An author has been paid a writer’s fee of $1, 000 plus $1.50 for every copy of the book that is sold.
a. Sketch the graph of the linear function that relates the total amount of money earned in dollars, A, to the number of books sold, n, on the axes below.

Answer:

b. What is the rate of change that relates the total amount of money earned to the number of books sold?
Answer:
1.5

c. What is the initial value of the linear function based on the graph?
Answer:
1, 000

d. Let the number of books sold be n and the total amount earned be A. Construct a linear function that models the relationship between the number of books sold and the total amount earned.
Answer:
A = 1.5n + 1000

Question 3.
Suppose that the price of gasoline has been falling. At the beginning of last month (t = 0), the price was $4.60 per gallon. Twenty days later (t = 20), the price was $4.20 per gallon. Assume that the price per gallon, P, fell at a constant rate over the twenty days.

Answer:

a. Identify the ordered pairs given in the problem. Plot both points on the coordinate plane above.
Answer:
(0, 4.60) and (20, 4.20); see the graph above.

b. Using a straightedge, draw the line that contains the two points.
Answer:
See the graph above.

c. What is the rate of change? What does it mean within the context of the problem?
Answer:
Using points (0, 4.60) and (20, 4.20), the rate of change is -0.02 because \(\frac{4.20-4.60}{20-0}\) = \(\frac{-0.4}{20}\) = -0.02. The price of gas is decreasing $0.02 each day.

d. What is the function that models the relationship between the number of days and the price per gallon?
Answer:
P = -0.02t + 4.6

e. What was the price of gasoline after 9 days?
Answer:
$4.42; see the graph above.

f. After how many days was the price $4.32?
Answer:
14 days; see the graph above.

Eureka Math Grade 8 Module 6 Lesson 3 Exit Ticket Answer Key

Question 1.
A car starts a journey with 18 gallons of fuel. Assuming a constant rate, the car consumes 0.04 gallon for every mile driven. Let A represent the amount of gas in the tank (in gallons) and m represent the number of miles driven.

Answer:

a. How much gas is in the tank if 0 miles have been driven? How would this be represented on the axes above?
Answer:
There are 18 gallons in the tank. This would be represented as (0, 18), the initial value, on the graph above.

b. What is the rate of change that relates the amount of gas in the tank to the number of miles driven? Explain what it means within the context of the problem.
Answer:
-0.04; the car consumes 0.04 gallon for every mile driven. It relates the amount of fuel to the miles driven.

c. On the axes above, draw the line that represents the graph of the linear function that relates A to m.
Answer:
See the graph above. Students can plot the initial value (0, 18) and then use the rate of change to identify additional points as needed. A 50-unit increase in m results in a 2-unit decrease for A, so another point on the line is (50, 16).

d. Write the linear function that models the relationship between the number of miles driven and the amount of gas in the tank.
Answer:
A = 18 – 0.04m or A = -0.04m + 18

Question 2.
Andrew works in a restaurant. The graph below shows the relationship between the amount Andrew earns in dollars and the number of hours he works.

a. If Andrew works for 7 hours, approximately how much does he earn in dollars?
Answer:
$96

b. Estimate how long Andrew has to work in order to earn $64.
Answer:
3 hours

c. What is the rate of change of the function given by the graph? Interpret the value within the context of the problem.
Answer:
Using the ordered pairs (7, 96) and (3, 64), the slope is 8. It means that the amount Andrew earns increases by $8 for every hour worked.

Engage NY Eureka Math 8th Grade Module 6 Lesson 1 Answer Key

Eureka Math Grade 8 Module 6 Lesson 1 Example Answer Key

Example 2: Another Rate Plan
A second wireless access company has a similar method for computing its costs. Unlike the first company that Lenore was considering, this second company explicitly states its access fee is $0.15, and its usage rate is $0.04 per minute.
Total Session Cost = $0.15 + $0.04 (number of minutes)
Answer:
→ How is this plan presented differently?
In this case, we are given the access fee and usage rate with an equation. In the first example, just data points were given.

→ Based on the work with the first set of problems, how do you think the two plans are different?
The values for the access fee and usage charge per minute are different, or the initial value and the rate of change are different.

Eureka Math Grade 8 Module 6 Lesson 1 Exercise Answer Key

Exercises 1–6

Exercise 1.
Lenore makes a table of this information and a graph where number of minutes is represented by the horizontal axis and total session cost is represented by the vertical axis. Plot the three given points on the graph. These three points appear to lie on a line. What information about the access plan suggests that the correct model is indeed a linear relationship?

Answer:
The amount charged for the minutes connected is based upon a constant usage rate in dollars per minute.

Exercise 2.
The rate of change describes how the total cost changes with respect to time.
a. When the number of minutes increases by 10 (e.g., from 10 minutes to 20 minutes or from 20 minutes to 30 minutes), how much does the charge increase?
Answer:
When the number of minutes increases by 10 (e.g., from 10 minutes to 20 minutes or from 20 minutes to 30 minutes), the cost increases by $0.30 (30 cents).

b. Another way to say this would be the usage charge per 10 minutes of use. Use that information to determine the increase in cost based on only 1 minute of additional usage. In other words, find the usage charge per minute of use.
Answer:
If $0.30 is the usage charge per 10 minutes of use, then $0.03 is the usage charge per 1 minute of use (i.e., the usage rate). Since the usage rate is constant, students should use what they have learned in Module 4.

Exercise 3.
The company’s pricing plan states that the usage rate is constant for any number of minutes connected to the Internet. In other words, the increase in cost for 10 more minutes of use (the value that you calculated in Exercise 2) is the same whether you increase from 20 to 30 minutes, 30 to 40 minutes, etc. Using this information, determine the total cost for 40 minutes, 50 minutes, and 60 minutes of use. Record those values in the table, and plot the corresponding points on the graph in Exercise 1.
Answer:
Consider the following table and graphs.

Exercise 4.
Using the table and the graph in Exercise 1, compute the hypothetical cost for 0 minutes of use. What does that value represent in the context of the values that Lenore is trying to figure out?
Answer:
Since there is a $0.30 decrease in cost for each decrease of 10 minutes of use, one could subtract $0.30 from the cost value for 10 minutes and arrive at the hypothetical cost value for 0 minutes. That cost would be $0.10. Students may notice that such a value follows the regular pattern in the table and would represent the fixed access fee for connecting. (This value could also be found from the graph after completing Exercise 6.)

Exercise 5.
On the graph in Exercise 1, draw a line through the points representing 0 to 60 minutes of use under this company’s plan. The slope of this line is equal to the constant rate of change, which in this case is the usage rate.
Answer:

Exercise 6.
Using x for the number of minutes and y for the total cost in dollars, write a function to model the linear relationship between minutes of use and total cost.
Answer:
y = 0.03x + 0.10

Exercises 7–16

Exercise 7.
Let x represent the number of minutes used and y represent the total session cost in dollars. Construct a linear function that models the total session cost based on the number of minutes used.
Answer:
y = 0.04x + 0.15

Exercise 8.
Using the linear function constructed in Exercise 7, determine the total session cost for sessions of 0, 10, 20, 30, 40, 50, and 60 minutes, and fill in these values in the table below.

Answer:

Exercise 9.
Plot these points on the original graph in Exercise 1, and draw a line through these points. In what ways does the line that represents this second company’s access plan differ from the line that represents the first company’s access plan?
Answer:
The second company’s plan line begins at a greater initial value. The same plan also increases in total cost more quickly over time; in other words, the slope of the line for the second company’s plan is steeper.

Exercises 10–12

MP3 download sites are a popular forum for selling music. Different sites offer pricing that depends on whether or not you want to purchase an entire album or individual songs à la carte. One site offers MP3 downloads of individual songs with the following price structure: a $3 fixed fee for a monthly subscription plus a charge of $0.25 per song.
Exercise 10.
Using x for the number of songs downloaded and y for the total monthly cost in dollars, construct a linear function to model the relationship between the number of songs downloaded and the total monthly cost.
Answer:
Since $3 is the initial cost and there is a 25 cent increase per song, the function would be
y = 3 + 0.25x or y = 0.25x + 3.

Exercise 11.
Using the linear function you wrote in Exercise 10, construct a table to record the total monthly cost (in dollars) for MP3 downloads of 10 songs, 20 songs, and so on up to 100 songs.

Answer:

Exercise 12.
Plot the 10 data points in the table on a coordinate plane. Let the x-axis represent the number of songs downloaded and the y-axis represent the total monthly cost (in dollars) for MP3 downloads.
Answer:

A band will be paid a flat fee for playing a concert. Additionally, the band will receive a fixed amount for every ticket sold. If 40 tickets are sold, the band will be paid $200. If 70 tickets are sold, the band will be paid $260.
Exercise 13.
Determine the rate of change.
Answer:
The points (40,200) and (70,260) have been given.
So, the rate of change is 2 because \(\frac{260-200}{70-40}\) = 2.

Exercise 14.
Let x represent the number of tickets sold and y represent the amount the band will be paid in dollars. Construct a linear function to represent the relationship between the number of tickets sold and the amount the band will be paid.
Answer:
Using the rate of change and (40,200):
200 = 2(40) + b
200 = 80 + b
120 = b
Therefore, the function is y = 2x + 120.

Exercise 15.
What flat fee will the band be paid for playing the concert regardless of the number of tickets sold?
Answer:
The band will be paid a flat fee of $120 for playing the concert.

Exercise 16.
How much will the band receive for each ticket sold?
Answer:
The band receives $2 per ticket.

Eureka Math Grade 8 Module 6 Lesson 1 Problem Set Answer Key

Question 1.
Recall that Lenore was investigating two wireless access plans. Her friend in Europe says that he uses a plan in which he pays a monthly fee of 30 euro plus 0.02 euro per minute of use.
a. Construct a table of values for his plan’s monthly cost based on 100 minutes of use for the month, 200 minutes of use, and so on up to 1,000 minutes of use. (The charge of 0.02 euro per minute of use is equivalent to 2 euro per 100 minutes of use.)
Answer:

b. Plot these 10 points on a carefully labeled graph, and draw the line that contains these points.

Answer:

c. Let x represent minutes of use and y represent the total monthly cost in euro. Construct a linear function that determines the monthly cost based on minutes of use.
Answer:
y = 30 + 0.02x

d. Use the function to calculate the cost under this plan for 750 minutes of use. If this point were added to the graph, would it be above the line, below the line, or on the line?
Answer:
The cost for 750 minutes would be €45. The point (750,45) would be on the line.

Question 2.
A shipping company charges a $4.45 handling fee in addition to $0.27 per pound to ship a package.
a. Using x for the weight in pounds and y for the cost of shipping in dollars, write a linear function that determines the cost of shipping based on weight.
Answer:
y = 4.45 + 0.27x

b. Which line (solid, dotted, or dashed) on the following graph represents the shipping company’s pricing method? Explain.

Answer:
The solid line would be the correct line. Its initial value is 4.45, and its slope is 0.27. The dashed line shows the cost decreasing as the weight increases, so that is not correct. The dotted line starts at an initial value that is too low.

Question 3.
Kelly wants to add new music to her MP3 player. Another subscription site offers its downloading service using the following: Total Monthly Cost = 5.25 + 0.30(number of songs).
a. Write a sentence (all words, no math symbols) that the company could use on its website to explain how it determines the price for MP3 downloads for the month.
Answer:
“We charge a $5.25 subscription fee plus 30 cents per song downloaded.”

b. Let x represent the number of songs downloaded and y represent the total monthly cost in dollars. Construct a function to model the relationship between the number of songs downloaded and the total monthly cost.
Answer:
y = 5.25 + 0.30x

c. Determine the cost of downloading 10 songs.
Answer:
5.25 + 0.30(10) = 8.25
The cost of downloading 10 songs is $8.25.

Question 4.
Li Na is saving money. Her parents gave her an amount to start, and since then she has been putting aside a fixed amount each week. After six weeks, Li Na has a total of $82 of her own savings in addition to the amount her parents gave her. Fourteen weeks from the start of the process, Li Na has $118.
a. Using x for the number of weeks and y for the amount in savings (in dollars), construct a linear function that describes the relationship between the number of weeks and the amount in savings.
Answer:
The points (6, 82) and (14, 118) have been given.
So, the rate of change is 4.5 because \(\frac{118-82}{14-6}\) = \(\frac{36}{8}\) = 4.5.
Using the rate of change and (6, 82):
82 = 4.5(6) + b
82 = 27 + b
55 = b
The function is y = 4.5x + 55.

b. How much did Li Na’s parents give her to start?
Answer:
Li Na’s parents gave her $55 to start.

c. How much does Li Na set aside each week?
Answer:
Li Na is setting aside $4.50 every week for savings.

d. Draw the graph of the linear function below (start by plotting the points for x = 0 and x = 20).

Answer:

Eureka Math Grade 8 Module 6 Lesson 1 Exit Ticket Answer Key

A rental car company offers a rental package for a midsize car. The cost comprises a fixed $30 administrative fee for the cleaning and maintenance of the car plus a rental cost of $35 per day.
Question 1.
Using x for the number of days and y for the total cost in dollars, construct a function to model the relationship between the number of days and the total cost of renting a midsize car.
Answer:
y = 35x + 30

Question 2.
The same company is advertising a deal on compact car rentals. The linear function y = 30x + 15 can be used to model the relationship between the number of days, x, and the total cost in dollars, y, of renting a compact car.
a. What is the fixed administrative fee?
Answer:
The administrative fee is $15.

b. What is the rental cost per day?
Answer:
It costs $30 per day to rent the compact car.

Engage NY Eureka Math 8th Grade Module 5 Lesson 6 Answer Key

Eureka Math Grade 8 Module 5 Lesson 6 Exercise Answer Key

Exercise
A function assigns to the inputs shown the corresponding outputs given in the table below.

a. Do you suspect the function is linear? Compute the rate of change of this data for at least three pairs of inputs and their corresponding outputs.
Answer:
\(\frac{2 – ( – 1)}{1 – 2}\) = \(\frac{3}{ – 1}\)
= – 3

\(\frac{ – 7 – ( – 13)}{4 – 6}\) = \(\frac{6}{ – 2}\)
= – 3

\(\frac{2 – ( – 7)}{1 – 4}\) = \(\frac{9}{ – 3}\)
= – 3
Yes, the rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is equal to – 3. Since the rate of change is the same, then I know it is a linear function.

b. What equation seems to describe the function?
Answer:
Using the assignment of 2 to 1:
2 = – 3(1) + b
2 = – 3 + b
5 = b
The equation that seems to describe the function is y = – 3x + 5.

c. As you did not verify that the rate of change is constant across all input/output pairs, check that the equation you found in part (a) does indeed produce the correct output for each of the four inputs 1, 2, 4, and 6.
Answer:
For x = 1 we have y = – 3(1) + 5 = 2.
For x = 2 we have y = – 3(2) + 5 = – 1.
For x = 4 we have y = – 3(4) + 5 = – 7.
For x = 6 we have y = – 3(6) + 5 = – 13.
These are correct.

d. What will the graph of the function look like? Explain.
Answer:
The graph of the function will be a plot of four points lying on a common line. As we were not told about any other inputs for this function, we must assume for now that there are only these four input values for the function.
The four points lie on the line with equation y = – 3x + 5.

Eureka Math Grade 8 Module 5 Lesson 6 Problem Set Answer Key

Question 1.
A function assigns to the inputs given the corresponding outputs shown in the table below.

a. Does the function appear to be linear? Check at least three pairs of inputs and their corresponding outputs.
\(\frac{9 – 17}{3 – 9}\) = \(\frac{ – 8}{ – 6}\)
= \(\frac{4}{3}\)

\(\frac{17 – 21}{9 – 12}\) = \(\frac{ – 4}{ – 3}\)
= \(\frac{4}{3}\)

\(\frac{21 – 25}{12 – 15}\) = \(\frac{ – 4}{ – 3}\)
= \(\frac{4}{3}\)
Yes. The rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is equal to \(\frac{4}{3}\). Since the rate of change is the same, the function does appear to be linear.

b. Find a linear equation that describes the function.
Answer:
Using the assignment of 9 to 3
9 = \(\frac{4}{3}\) (3) + b
9 = 4 + b
5 = b
The equation that describes the function is y = \(\frac{4}{3}\) x + 5. (We check that for each of the four inputs given, this equation does indeed produce the correct matching output.)

c. What will the graph of the function look like? Explain.
Answer:
The graph of the function will be four points in a row. They all lie on the line given by the equation
y = \(\frac{4}{3}\) x + 5.

Question 2.
A function assigns to the inputs given the corresponding outputs shown in the table below.

a. Is the function a linear function?
Answer:
\(\frac{2 – 0}{ – 1 – 0}\) = \(\frac{2}{ – 1}\)
= – 2

\(\frac{0 – 2}{0 – 1}\) = \(\frac{ – 2}{ – 1}\)
= 2
No. The rate of change is not the same when I check the first two pairs of inputs and corresponding outputs. All rates of change must be the same for all inputs and outputs for the function to be linear.

b. What equation describes the function?
Answer:
I am not sure what equation describes the function. It is not a linear function.

Question 3.
A function assigns the inputs and corresponding outputs shown in the table below.

a. Does the function appear to be linear? Check at least three pairs of inputs and their corresponding outputs..
Answer:
\(\frac{2 – 6}{0.2 – 0.6}\) = \(\frac{ – 4}{ – 0.4}\)
= 10

\(\frac{6 – 15}{0.6 – 1.5}\) = \(\frac{ – 9}{ – 0.9}\)
= 10

\(\frac{15 – 21}{1.5 – 2.1}\) = \(\frac{ – 6}{ – 0.6}\)
= 10
Yes. The rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is equal to 10. The function appears to be linear.

b. Find a linear equation that describes the function.
Answer:
Using the assignment of 2 to 0.2:
2 = 10(0.2) + b
2 = 2 + b
0 = b
The equation that describes the function is y = 10x . It clearly fits the data presented in the table.

c. What will the graph of the function look like? Explain.
Answer:
The graph will be four distinct points in a row. They all sit on the line given by the equation y = 10x.

Question 4.
Martin says that you only need to check the first and last input and output values to determine if the function is linear. Is he correct? Explain.
Answer:
No, he is not correct. For example, consider the function with input and output values in this table.

Using the first and last input and output, the rate of change is
\(\frac{9 – 12}{1 – 3}\) = \(\frac{ – 3}{ – 2}\)
= \(\frac{3}{2}\)
But when you use the first two inputs and outputs, the rate of change is
\(\frac{9 – 10}{1 – 2}\) = \(\frac{ – 1}{ – 1}\)
= 1
Note to teacher: Accept any example where the rate of change is different for any two inputs and outputs.

Question 5.
Is the following graph a graph of a linear function? How would you determine if it is a linear function?

Answer:
It appears to be a linear function. To check, I would organize the coordinates in an input and output table. Next, I would check to see that all the rates of change are the same. If they are the same rates of change, I would use the equation y = mx + b and one of the assignments to write an equation to solve for b. That information would allow me to determine the equation that represents the function.

Question 6.
A function assigns to the inputs given the corresponding outputs shown in the table below.

a. Does the function appear to be a linear function?
Answer:
\(\frac{ – 6 – ( – 5)}{ – 6 – ( – 5)}\) = \(\frac{1}{1}\)
= 1

\(\frac{ – 5 – ( – 5)}{ – 5 – ( – 5)}\) = \(\frac{1}{1}\)
= 1

\(\frac{ – 4 – ( – 2)}{ – 4 – ( – 2)}\) = \(\frac{2}{2}\)
= 1
Yes. The rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is equal to 1. Since the rate of change is constant so far, it could be a linear function.

b. What equation describes the function?
Answer:
Clearly the equation y = x fits the data. It is a linear function.

c. What will the graph of the function look like? Explain.
Answer:
The graph of the function will be four distinct points in a row. These four points lie on the line given by the equation y = x.

Eureka Math Grade 8 Module 5 Lesson 6 Exit Ticket Answer Key

Question 1.
Sylvie claims that a function with the table of inputs and outputs below is a linear function. Is she correct? Explain.

Answer:
\(\frac{ – 25 – (10)}{ – 3 – 2}\) = \(\frac{ – 35}{ – 5}\)
= 7

\(\frac{10 – 31}{2 – 5}\) = \(\frac{ – 21}{ – 3}\)
= 7

\(\frac{31 – 54}{5 – 8}\) = \(\frac{ – 23}{ – 3}\)
= \(\frac{23}{3}\)
No, this is not a linear function. The rate of change was not the same for each pair of inputs and outputs inspected, which means that it is not a linear function.

Question 2.
A function assigns the inputs and corresponding outputs shown in the table to the right.
a. Does the function appear to be linear? Check at least three pairs of inputs and their corresponding outputs.

Answer:
\(\frac{3 – ( – 2)}{ – 2 – 8}\) = \(\frac{5}{ – 10}\) = – \(\frac{1}{2}\)
\(\frac{ – 2 – ( – 3)}{8 – 10}\) = \(\frac{1}{ – 2}\) = – \(\frac{1}{2}\)
\(\frac{ – 3 – ( – 8)}{10 – 20}\) = \(\frac{5}{ – 10}\) = – \(\frac{1}{2}\)
Yes. The rate of change is the same when I check pairs of inputs and corresponding outputs. Each time it is equal to – \(\frac{1}{2}\) . Since the rate of change is the same for at least these three examples, the function could well be linear.

b. Can you write a linear equation that describes the function?
Answer:
We suspect we have an equation of the form y = – \(\frac{1}{2}\) x + b. Using the assignment of 3 to – 2:
3 = – \(\frac{1}{2}\) ( – 2) + b
3 = 1 + b
2 = b
The equation that describes the function might be y = – \(\frac{1}{2}\) x + 2.
Checking: When x = – 2, we get y = – \(\frac{1}{2}\) ( – 2) + 2 = 3. When x = 8, we get y = – \(\frac{1}{2}\) (8) + 2 = – 2. When x = 10, we get y = – \(\frac{1}{2}\) (10) + 2 = – 3. When x = 20, we get y = – \(\frac{1}{2}\) (20) + 2 = – 8.
It works.

c. What will the graph of the function look like? Explain.
Answer:
The graph of the function will be four distinct points all lying in a line. (They all lie on the line with equation y = – \(\frac{1}{2}\) x + 2 ).

Eureka Math Grade 8 Module 5 Lesson 6 Multi – Step Equations I Answer Key

Set 1:
3x + 2 = 5x + 6
4(5x + 6) = 4(3x + 2)
\(\frac{3x + 2}{6}\) = \(\frac{5x + 6}{6}\)
Answer:
Answer for each problem in this set is x = – 2.

Set 2:
6 – 4x = 10x + 9
– 2( – 4x + 6) = – 2(10x + 9)
\(\frac{10x + 9}{5}\) = \(\frac{6 – 4x}{5}\)
Answer:
Answer for each problem in this set is x = – \(\frac{3}{14}\).

Set 3:
5x + 2 = 9x – 18
8x + 2 – 3x = 7x – 18 + 2x
\(\frac{2 + 5x}{3}\) = \(\frac{7x – 18 + 2x}{3}\)
Answer:
Answer for each problem in this set is x = 5.

Engage NY Eureka Math 8th Grade Module 5 End of Module Assessment Answer Key

Eureka Math Grade 8 Module 5 End of Module Assessment Task Answer Key

Question 1.
a. We define x as a year between 2008 and 2013 and y as the total number of smartphones sold that year, in millions. The table shows values of x and corresponding y values.

i) How many smartphones were sold in 2009?
Answer:
17.3 Million Smartphones were sold in 2009

ii) In which year were 90 million smartphones sold?
Answer:
90 Million Smartphones were sold in 2011

iii) Is y a function of x? Explain why or why not.
Answer:
Yes, It is a function because for each input there is exactly one output. Specifically only one number will be assigned to represent the number of smartphones sold in the given year.

b. Randy began completing the table below to represent a particular linear function. Write an equation to represent the function he was using and complete the table for him.

Answer:

y = 3x + 4

c. Create the graph of the function in part (b).

Answer:

d. At NYU in 2013, the cost of the weekly meal plan options could be described as a function of the number of meals. Is the cost of the meal plan a linear or nonlinear function? Explain.

8 meals: $125/week
10 meals: $135/week
12 meals: $155/week
21 meals: $220/week
Answer:
\(\frac{125}{8}\) = 15.625
\(\frac{135}{10}\) = 13.5
\(\frac{155}{12}\) = 12.917
\(\frac{220}{21}\) = 10.476
The cost of the meal plan is a nonlinear function. The cost per meal is different based on the plan. Chosen for example, one plan charges almost $16 per meal while another is about $10. Also, when the data is graphed, The points do not fall on a line.

Question 2.
The cost to enter and go on rides at a local water park, Wally’s Water World, is shown in the graph below.

A new water park, Tony’s Tidal Takeover, just opened. You have not heard anything specific about how much it costs to go to this park, but some of your friends have told you what they spent. The information is organized in the table below.

Each park charges a different admission fee and a different fee per ride, but the cost of each ride remains the same.
a. If you only have $14 to spend, which park would you attend (assume the rides are the same quality)? Explain.
Answer:
Let x represent the number of rides
Let w represent the total cost at wally’s water world
W = 2x + 8
Wally’s
W = 2x + 8
14 = 2x + 8
6 = 2x
3 = x

Tony’s
T = 0.75x + 12
14 = 0.75x + 12
2 = 0.75x
2.67 ≈ x
At wally’s you can go in 3 rides with $14, At tony’s just 2 rides. I would go to wally’s because i could go on more rides.

b. Another water park, Splash, opens, and they charge an admission fee of $30 with no additional fee for rides. At what number of rides does it become more expensive to go to Wally’s Water World than Splash? At what number of rides does it become more expensive to go to Tony’s Tidal Takeover than Splash?
Answer:
Let S represent total cost at splash, S = 30
Wally’s
30 = 2x + 8
22 = 2x
11 = x

Tony’s
30 = 0.75x + 12
18 = 0.75x
24 = x
At Wally’s you can go on 11 rides with $30. The 12th ride makes wally’s more expensive than splash.
At Tony’s you can go on 24 rides with $30. The 25th ride makes tony’s more expensive than splash.

c. For all three water parks, the cost is a function of the number of rides. Compare the functions for all three water parks in terms of their rate of change. Describe the impact it has on the total cost of attending each park.
Answer:
Wally’s rate of change is 2, $2 per ride.
Tony’s rate of change is 0.75, $0.75 per ride.
Splash’s rate of change is 0, $0 extra per ride.
Wally’s has the greatest rate of change that means that the total cost at wally’s will increase the fastest as we go on more rides. At tony’s the rate of change is just 0.75 so the total cost increases with the number of rides we go on, but not as quickly as wally’s. Splash has a rate of change of zero, The number of rides we go on does not impact the total cost at all.

Question 3.
For each part below, leave your answers in terms of π.
a. Determine the volume for each three-dimensional figure shown below.

Answer:
V = \(\frac{1}{3}\) π (16)(9)
= (16)(3)π
= 48 π
The volume is 48 π mm²

V = π (4)(5.3)
= π (21.2)
= 21.2 π
The volume is 21.2 π cm³.

V = \(\frac{4}{3}\) π (3²)
= 4(9) π
= 36 π
The volume is 36 πin³

b. You want to fill the cylinder shown below with water. All you have is a container shaped like a cone with a radius of 3 inches and a height of 5 inches; you can use this cone-shaped container to take water from a faucet and fill the cylinder. How many cones will it take to fill the cylinder?

Answer:
Volume of cylinder = π (64) (3)
= 192 π
Volume of cone = \(\frac{1}{3}\) π (9) (5)
= \(\frac{45}{3}\) π
= 15 π
The volume of cylinder is 192 π in³
The volume of cone is 15 π in³
\(\frac{192 \pi}{15 \pi}\) = \(\frac{192}{15}\) = 12.8
It takes 12.8 cone of the given size to fill the cylinder.

c. You have a cylinder with a diameter of 15 inches and height of 12 inches. What is the volume of the largest sphere that will fit inside of it?

Answer:
The cylinder has radius of 7.5 cm, but the height is just 12 cm. That means the maximum radius for the sphere is 6 cm. Anything larger would not fit in the cylinder. Then the volume of the largest sphere that will fit in the cylinder is 288 in³.
V= \(\frac{4}{3}\) π (6³)
= \(\frac{4}{3}\) π (216)
= 288 π

Engage NY Eureka Math 8th Grade Module 5 Lesson 11 Answer Key

Eureka Math Grade 8 Module 5 Lesson 11 Example Answer Key

Example 1.
Compute the exact volume for the sphere shown below.

Answer:
Provide students time to work; then, have them share their solutions.
Sample student work:
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π(4³ )
= \(\frac{4}{3}\) π(64)
= \(\frac{256}{3}\) π
= 85 \(\frac{1}{3}\) π
The volume of the sphere is 85 \(\frac{1}{3}\) π cm³.

Example 2.
A cylinder has a diameter of 16 inches and a height of 14 inches. What is the volume of the largest sphere that will fit into the cylinder?

Answer:
→ What is the radius of the base of the cylinder?
The radius of the base of the cylinder is 8 inches.

→ Could the sphere have a radius of 8 inches? Explain.
No. If the sphere had a radius of 8 inches, then it would not fit into the cylinder because the height is only 14 inches. With a radius of 8 inches, the sphere would have a height of 2r, or 16 inches. Since the cylinder is only 14 inches high, the radius of the sphere cannot be 8 inches.

→ What size radius for the sphere would fit into the cylinder? Explain.
A radius of 7 inches would fit into the cylinder because 2r is 14, which means the sphere would touch the top and bottom of the cylinder. A radius of 7 means the radius of the sphere would not touch the sides of the cylinder, but would fit into it.

→ Now that we know the radius of the largest sphere is 7 inches, what is the volume of the sphere?
Sample student work:
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π(7³ )
= \(\frac{4}{3}\) π(343)
= \(\frac{1372}{3}\) π
= 457 \(\frac{1}{3}\) π
The volume of the sphere is 457 \(\frac{1}{3}\) π cm³.

Eureka Math Grade 8 Module 5 Lesson 11 Exercise Answer Key

Exercises 1–3

Exercise 1.
What is the volume of a cylinder?
Answer:
V = πr² h

Exercise 2.
What is the height of the cylinder?
Answer:
The height of the cylinder is the same as the diameter of the sphere. The diameter is 2r.

Exercise 3.
If volume(sphere) = 2/3 volume(cylinder with same diameter and height), what is the formula for the volume of a sphere?
Answer:
Volume(sphere) = \(\frac{2}{3}\) (πr²h)
Volume(sphere) = \(\frac{2}{3}\) (πr²2r)
Volume(sphere) = \(\frac{4}{3}\) (πr³)

Exercises 4–8

Exercise 4.
Use the diagram and the general formula to find the volume of the sphere.

Answer:
V = \(\frac{4}{3}\) πr³
V = \(\frac{4}{3}\) π(6³ )
V ≈ 288π
The volume of the sphere is about 288π in³.

Exercise 5.
The average basketball has a diameter of 9.5 inches. What is the volume of an average basketball? Round your answer to the tenths place.
Answer:
V = \(\frac{4}{3}\) πr³
V = \(\frac{4}{3}\) π(4.75³ )
V = \(\frac{4}{3}\) π(107.17)
V ≈ 142.9π
The volume of an average basketball is about 142.9π in³.

Exercise 6.
A spherical fish tank has a radius of 8 inches. Assuming the entire tank could be filled with water, what would the volume of the tank be? Round your answer to the tenths place.
Answer:
V = \(\frac{4}{3}\) πr³
V = \(\frac{4}{3}\) π(8³ )
V = \(\frac{4}{3}\) π(512)
V ≈ 682.7π
The volume of the fish tank is about 682.7π in³.

Exercise 7.
Use the diagram to answer the questions.

a. Predict which of the figures shown above has the greater volume. Explain.
Answer:
Student answers will vary. Students will probably say the cone has more volume because it looks larger.

b. Use the diagram to find the volume of each, and determine which has the greater volume.
Answer:
V = \(\frac{1}{3}\) πr² h
V = \(\frac{1}{3}\) π(2.5²)(12.6)
V = 26.25π
The volume of the cone is 26.25π mm³.
V = \(\frac{4}{3}\) πr³
V = \(\frac{4}{3}\) π(2.8³)
V = 29.269333…π
The volume of the sphere is about 29.27π mm³. The volume of the sphere is greater than the volume of the cone.

Exercise 8.
One of two half spheres formed by a plane through the sphere’s center is called a hemisphere. What is the formula for the volume of a hemisphere?

Answer:
Since a hemisphere is half a sphere, the volume(hemisphere) = \(\frac{1}{2}\) (volume of sphere).
V = \(\frac{1}{2}\) (\(\frac{4}{3}\) πr³ )
V = \(\frac{2}{3}\) πr³

Eureka Math Grade 8 Module 5 Lesson 11 Problem Set Answer Key

Question 1.
Use the diagram to find the volume of the sphere.

Answer:
V = \(\frac{4}{3}\) πr³
V = \(\frac{4}{3}\) π(9³)
V = 972π
The volume of the sphere is 972π cm³.

Question 2.
Determine the volume of a sphere with diameter 9 mm, shown below.

Answer:
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π(4.5³ )
= \(\frac{364.5}{3}\) π
= 121.5π
The volume of the sphere is 121.5π mm³.

Question 3.
Determine the volume of a sphere with diameter 22 in., shown below.

Answer:
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π(11³ )
= \(\frac{5324}{3}\) π
= 1774 \(\frac{2}{3}\) π
The volume of the sphere is 1774 \(\frac{2}{3}\) π in³.

Question 4.
Which of the two figures below has the lesser volume?

Answer:
The volume of the cone:
V = \(\frac{1}{3}\) πr² h
= \(\frac{1}{3}\) π(16)(7)
= \(\frac{112}{3}\) π
= 37 \(\frac{1}{3}\) π

The volume of the sphere:
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π(2³ )
= \(\frac{32}{3}\) π
= 10 \(\frac{2}{3}\) π
The cone has volume 37 \(\frac{1}{3}\) π in³ and the sphere has volume 10 \(\frac{2}{3}\) π in³. The sphere has the lesser volume.

Question 5.
Which of the two figures below has the greater volume?

Answer:
The volume of the cylinder:
V = πr² h
= π(3²)(6.2)
= 55.8π

The volume of the sphere:
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π(5³)
= \(\frac{500}{3}\) π
= 166 \(\frac{2}{3}\) π
The cylinder has volume 55.8π mm³ and the sphere has volume 166 \(\frac{2}{3}\) π mm³. The sphere has the greater volume.

Question 6.
Bridget wants to determine which ice cream option is the best choice. The chart below gives the description and prices for her options. Use the space below each item to record your findings.

A scoop of ice cream is considered a perfect sphere and has a 2-inch diameter. A cone has a 2-inch diameter and a height of 4.5 inches. A cup, considered a right circular cylinder, has a 3-inch diameter and a height of 2 inches.
Answer:

a. Determine the volume of each choice. Use 3.14 to approximate π.
Answer:
First, find the volume of one scoop of ice cream.
Volume of one scoop = \(\frac{4}{3}\) π(1³)
The volume of one scoop of ice cream is \(\frac{4}{3}\) π in³, or approximately 4.19 in³.
The volume of two scoops of ice cream is \(\frac{8}{3}\) π in³, or approximately 8.37 in³.
The volume of three scoops of ice cream is \(\frac{12}{3}\) π in³, or approximately 12.56 in³.
Volume of half scoop = \(\frac{2}{3}\) π(1³)
The volume of half a scoop of ice cream is \(\frac{2}{3}\) π in³, or approximately 2.09 in³.
Volume of cone = \(\frac{1}{3}\) (πr²)h
V = \(\frac{1}{3}\) π(1²)4.5
V = 1.5π
The volume of the cone is 1.5π in³, or approximately 4.71 in³. Then, the cone with half a scoop of ice cream on top is approximately 6.8 in³.
V = πr² h
V = π1.5²(2)
V = 4.5π
The volume of the cup is 4.5π in³, or approximately 14.13 in³.

b. Determine which choice is the best value for her money. Explain your reasoning.
Answer:
Student answers may vary.
Checking the cost for every in³ of each choice:
\(\frac{2}{4.19}\) ≈ 0.47723…
\(\frac{2}{6.8}\) ≈ 0.29411…
\(\frac{3}{8.37}\) ≈ 0.35842…
\(\frac{4}{12.56}\) ≈ 0.31847…
\(\frac{4}{14.13}\) ≈ 0.28308…
The best value for her money is the cup filled with ice cream since it costs about 28 cents for every in³.

Eureka Math Grade 8 Module 5 Lesson 11 Exit Ticket Answer Key

Question 1.
What is the volume of the sphere shown below?

Answer:
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π(3³ )
= \(\frac{108}{3}\) π
= 36π
The volume of the sphere is 36π in³.

Question 2.
Which of the two figures below has the greater volume?

Answer:
V = \(\frac{4}{3}\) πr³
= \(\frac{4}{3}\) π(4³)
= \(\frac{256}{3}\) π
= 85 \(\frac{1}{3}\) π
The volume of the sphere is 85 \(\frac{1}{3}\) π mm³.
V = \(\frac{1}{3}\) πr² h
= \(\frac{1}{3}\) π(3²)(6.5)
= \(\frac{58.5}{3}\) π
= 19.5π
The volume of the cone is 19.5π mm³. The sphere has the greater volume.