Lesson 4

Linear Models

4.1: Notice and Wonder: Crowd Noise (5 minutes)

Warm-up

The purpose of this warm-up is to help students recall information about scatter plots, which will be useful when students expand their understanding in a later activity.

While students may notice and wonder many things about these images, the relationship between the number of people and the maximum noise level, interpreting the line of best fit, and a general understanding of a scatter plot are the important discussion points.

Through articulating things they notice and things they wonder about the scatter plot and the linear model, students have an opportunity to attend to precision in the language they use to describe what they see (MP6). They might first propose less formal or imprecise language, and then restate their observation with more precise language in order to communicate more clearly.

Notice students who use correct terminology in their responses. In particular, the terms scatter plot, linear model, slope, and intercept are important to review during this warm-up.

Launch

Tell students their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Student Facing

What do you notice? What do you wonder?

$y = 1.5x+22.7$

Scatter plot.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

The vertical intercept appears to be approximately 105 decibels, but the origin is not shown on the graph.

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If students do not use these terms in their responses, ensure they recall the vocabulary from grade 8 math:

  • scatter plot
  • linear model
  • slope
  • intercept

4.2: Orange You Glad We’re Boxing Fruit? (15 minutes)

Activity

The mathematical purpose of this activity is for students to create a scatter plot from data given in context, to informally find a line they think does a good job of describing the relationship, to interpret the slope and vertical intercept of the linear model, and to use the linear model to make predictions. In creating a model of the data, students are modeling with mathematics (MP4).

This activity works best when each student has access to devices that can run Desmos or other graphing technology, because the level of precision is important. If students don’t have individual access, projecting the Desmos graph is helpful during the synthesis.

Launch

Play the video of adding oranges to a box on a scale. You may need to pause the video for students to write down the weights.

Students will be using the applet at desmos.com/calculator/zree3xeqja, embedded in the activity.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts to address the questions about estimation. Ask students to make predictions about similar scenarios that may become more variable as the sample size increases. For example, discussing the predictability of the weight of a box of fruit as a given sample size increases may be a relatable discussion that will encourage students to think about the difference between a grape and a grapefruit.
Supports accessibility for: Conceptual processing; Memory

Student Facing

Use the Desmos graphing calculator to help answer the questions. 

 
  1. Watch the video and record the weight for the number of oranges in the table.
  2. Adjust the window settings as needed to fit the scatter plot of the data.
  3. Drag the moveable points to reposition the line through the points so it fits the data well.
  4. Estimate a value for the slope of the line. What does the value of the slope represent?
  5. Estimate the weight of a box containing 11 oranges. Will this estimate be close to the actual value? Explain your reasoning.
  6. Estimate the weight of a box containing 50 oranges. Will this estimate be close to the actual value? Explain your reasoning.
  7. Estimate the coordinates for the vertical intercept of the line you drew. What does the \(y\)-coordinate for this point represent?
  8. Which point(s) are best fit by your linear model? How did you decide?
  9. Which point(s) are fit the least well by your linear model? How did you decide?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Play the video of adding oranges to a box on a scale. You may need to pause the video for students to write down the weights.

Representation: Internalize Comprehension. Begin the activity with concrete or familiar contexts to address the questions about estimation. Ask students to make predictions about similar scenarios that may become more variable as the sample size increases. For example, discussing the predictability of the weight of a box of fruit as a given sample size increases may be a relatable discussion that will encourage students to think about the difference between a grape and a grapefruit.
Supports accessibility for: Conceptual processing; Memory

Student Facing

  1. Watch the video and record the weight for the number of oranges in the box. 

    number of oranges weight in kilograms
    3  
    4  
    5  
    6  
    7  
    8  
    9  
    10  
  2. Create a scatter plot of the data.

    Blank coordinate plane, no grid. Horizontal axis from 0 to 10 by 1’s, labeled “number of oranges”. Vertical axis from 0 to 4 by 0 point 25’s, labeled “weight in kilograms”.
 
  3. Draw a line through the data that fits the data well.
  4. Estimate a value for the slope of the line that you drew. What does the value of the slope represent?
  5. Estimate the weight of a box containing 11 oranges. Will this estimate be close to the actual value? Explain your reasoning.
  6. Estimate the weight of a box containing 50 oranges. Will this estimate be close to the actual value? Explain your reasoning.
  7. Estimate the coordinates for the vertical intercept of the line you drew. What might the \(y\)-coordinate for this point represent?
  8. Which point(s) are best fit by your linear model? How did you decide?
  9. Which point(s) are fit the least well by your linear model? How did you decide?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may struggle with estimating a slope when the scale on the \(x\) and \(y\) axes are different. Ask students to find the coordinates for a couple of points on or near the line and find the slope between those points.

Activity Synthesis

Display the scatter plot, then show the best-fit line.

Scatter plot. Weight in kilograms. Number of oranges.

$y = 0.216x + 0.345$

Scatter plot. Weight in kilograms. Number of oranges.

Tell students to keep this data and scatter plot for a future lesson.

Ask:

  • “How would the scatter plot and linear model change if the box itself was heavier?” (The dots and line would shift up.)
  • “Can you think of reasons why the real weight of 50 oranges might be different from the answer we get by using an estimate from the linear model?” (50 oranges may not fit in one box, so additional boxes may be needed, which will change the linear pattern in the data. Many of the additional oranges may be very small or very large and not fit the general trend seen with these 10 oranges. While the estimate from the line is interesting and may be better than a wild guess, it should not be considered a very good estimate.)
  • “How many oranges did we measure?” (10)
  • “Since we measured 10 oranges, how does that affect your confidence in the estimate for 50 oranges?” (It makes me very skeptical that the estimate will be accurate. Although the data may look like a line for this section, there may be very different things happening farther away.)
  • “How would the scatter plot and linear model change if grapefruits were used instead of oranges?” (The weight would be increased for each point, and the slope of the line would be greater.)
  • “In this case, there might be some meaning attached to the \(y\)-intercept. That is not always the case. Why might interpreting the \(y\)-intercept not make sense in some situations? Can you think of a situation in which an interpretation of the \(y\)-intercept might not make sense?” (As you get farther from the collected data, the linear model may not make sense any more. Especially in situations in which the amount of something is approaching zero, very different things can be happening. For example, if we have data about water at temperatures around 60 to 70 degrees, that will be very different from what happens when the temperature is at 0.)

4.3: Food Markup (5 minutes)

Activity

In this activity, students are asked to interpret in context the slope and vertical intercept of a linear model given a scatter plot and the equation for a linear model that fits the data well. The linear model is also used to interpolate and extrapolate information about the data in context.

Monitor for students who:

  1. estimate values based on the graph
  2. use the equation for the linear model to find the values

Launch

Arrange students in groups of 2.

Student Facing

The scatter plot shows the sale price of several food items, \(y\), and the cost of the ingredients used to produce those items, \(x\), as well as a line that models the data. The line is also represented by the equation \(y = 3.48x + 0.76\).

Scatterplot and line of best fit.
  1. What is the predicted sale price of an item that has ingredients that cost \$1.50? Explain or show your reasoning.
  2. What is the predicted ingredient cost of an item that has a sale price of \$7? Explain or show your reasoning.
  3. What is the slope of the linear model? What does that mean in this situation?
  4. What is the \(y\)-intercept of the linear model? What does this mean in this situation? Does this make sense?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Although the grid in the picture does not look like squares, each line in both the horizontal and vertical directions are 0.5 apart. Tell students who are confused to look at the values listed on the axes and identify the values attached to a few of the grid lines.

Activity Synthesis

The purpose of this discussion is for students to describe where to find the slope and vertical intercept from a scatter plot and interpret the values in terms of the context.

Select previously identified students to share in the order listed in the narrative. Ask students:

  • “Are the values obtained from looking at the graph close to the values calculated using the equation?” (Yes, the values are close.)
  • “Which method will you use to predict values based on the model?” (Maybe a mixture of both methods. Using the equation produces a more precise value from the model, but it is also important to look at the graph to get a sense of how well the model fits the data where I am predicting.)
  • “Why do you think the intercept for the model is not \((0,0)\)?” (Companies may still try to charge money for items that cost nothing to produce. They need to pay for the salaries of their workers, research and development, as well as other things that require the company to make money more than just the cost of making the item. It may also be the case that it does not make sense to interpret the \(y\)-intercept, since we have no evidence to believe that the same linear relationship will hold there.)
Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their interpretations of the slope, present an ambiguous explanation. For example: “The slope means that as sales increase, the cost of ingredients increases.” Ask students to identify inaccuracies, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who identify and clarify the ambiguous language in the statement. Invite students to share their critiques and corrected explanations with the class. Listen for and amplify the language students use to describe what happens to the sale price when the cost of ingredients increases by \$1. This helps students evaluate, and improve on, the written mathematical arguments of others, as they clarify the meaning of slope in context.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Action and Expression: Internalize Executive Functions. Provide students with a two-column graphic organizer. Label one column “graph” and another column “equation.” Ask students to transfer their answers into the corresponding column on their organizer. When the whole group shares, they can continue to use the organizer capturing the answer if anyone used the alternative method for each question.
Supports accessibility for: Language; Organization

4.4: The Slope is the Thing (10 minutes)

Activity

The mathematical purpose of this activity is for students to interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Students are given scatter plots for different pairs of variables and the equation of a line of best fit. Students use the line of best fit and its equation to describe the meaning of the vertical intercept and slope.

Launch

Arrange students in groups of 2. Ask students to compare their responses to their partner’s for the scatter plots and decide if both their responses are correct for each scatter plot, even if they are different. Follow with a whole-class discussion.

Conversing: MLR8 Discussion Supports. In pairs, encourage students to take turns sharing their interpretation of the slope and vertical intercept for each scatter plot. Display the following sentence frames for all to see: “_____ represents _____”, “Is there another way to say . . .?”, and “It looks like _____ represents . . . .” Encourage students to challenge each other when they disagree. This will help students clarify their understanding of the meaning of slope and the vertical intercept in context.
Design Principle(s): Support sense-making; Maximize meta-awareness
Action and Expression: Develop Expression and Communication. To help students get started, display sentence frames such as: “What does this part of _____ mean?” and “I predict _____ because . . . .” Encourage students to annotate their graphs while using the sentence frames.
Supports accessibility for: Language; Organization

Student Facing

  1. Here are several scatter plots.

    A. $y = \text{-}9.25x+400$

    Scatter plot with line of best fit.

    B. $y = 0.44x+0.04$

    Scatterplot with line of best fit.

    C. $y = 4x+87$

    Scatter plot with line of best fit.

    D. $y = \text{-}2.4x+25.0$

    Scatter plot with line of best fit.
    1. Using the horizontal axis for \(x\) and the vertical axis for \(y\), interpret the slope of each linear model in the situations shown in the scatter plots.
    2. If the linear relationship continues to hold for each of these situations, interpret the \(y\)-intercept of each linear model in the situations provided.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Clare, Diego, and Elena collect data on the mass and fuel economy of cars at different dealerships. Clare finds the line of best fit for data she collected for 12 used cars at a used car dealership. The line of best fit is \(y=\frac{\text-9}{1000}x + 34.3\) where \(x\) is the car’s mass, in kilograms, and \(y\) is the fuel economy, in miles per gallon.

Diego made a scatter plot for the data he collected for 10 new cars at a different dealership.

A scatter plot.

Elena made a table for data she collected on 11 hybrid cars at another dealership.

mass (kilograms)

fuel economy (miles per gallon)

1,100

38

1,200

39

1,250

35

1,300

36

1,400

31

1,600

27

1,650

28

1,700

26

1,800

28

2,000

24

2,050

22

  1. Interpret the slope and \(y\)-intercept of Clare’s line of best fit in this situation.

  2. Diego looks at the data for new cars and used cars. He claims that the fuel economy of new cars decreases as the mass increases. He also claims that the fuel economy of used cars increases as the mass increases. Do you agree with Diego’s claims? Explain your reasoning.

  3. Elena looks at the data for hybrid cars and correctly claims that the fuel economy decreases as the mass increases. How could Elena compare the decrease in fuel economy as mass increases for hybrid cars to the decrease in fuel economy as mass increases for new cars? Explain your reasoning.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

The purpose of this discussion is for students to describe the rate of change and the vertical intercept using the context in each graph.

For each question, give students time to think individually, then share their response with their partner, then select a student or pair of students to respond to the question. Ask students:

  • “Why is the intercept for the bananas not \((0,0)\)?” (A linear model is not exact even for the data it is based on. It is an approximation based on the data in the scatter plot. It is possible that it represents the weight of the bag that bananas were placed in. It is also possible that this value does not make sense since there is no evidence to believe the same linear relationship will hold near zero.)
  • “How do you interpret the slope for each equation?” (The slope is the change in \(y\) divided by the change in \(x\), so look at the labels on the scatter plot and talk about for each increase of one on the \(x\) variable, on average, there is a decrease (if the slope is negative) or increase (if the slope is positive) in the variable represented by the \(y\) direction.)
  • “When might it make sense to interpret the \(y\)-intercept for a linear model?” (When \(x\) values around 0 are in the range of the data used to create the model. In other cases, care should be taken to put too much faith in the answer since the linear trend may not continue to hold farther from the collected data.)

Lesson Synthesis

Lesson Synthesis

The goal of this discussion is for students to make connections between bivariate data, a linear model, and the context of the data.

  • “How do you represent bivariate numerical data? How do you represent bivariate categorical data?” (You could use a two-way table for categorical data and a scatter plot for numerical data.)
  • “Why is creating a linear model useful?” (A linear model allows us to make predictions for the data in a range near the given data. It also helps to describe the relationship between the two variables quantitatively.)
  • “What are some situations in which you have encountered a scatter plot or a line of best fit previously? What is the meaning of the slope and vertical intercept of the line of best fit in this context?” (In science class, we graphed the relationship between the temperature and the time it took for a reaction to take place. The slope represents how much the reaction time changes, on average, for each unit increase in temperature. The vertical intercept represents the reaction time when the temperature is 0.)

4.5: Cool-down - Roar of the Crowd (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

While working in math class, it can be easy to forget that reality is somewhat messy. Not all oranges weigh exactly the same amount, beans have different lengths, and even the same person running a race multiple times will probably have different finishing times. We can approximate these messy situations with more precise mathematical tools to better understand what is happening. We can also predict or estimate additional results as long as we continue to keep in mind that reality will vary a little bit from what our mathematical model predicts.

For example, the data in this scatter plot represents the price of a package of broccoli and its weight. The data can be modeled by a line given by the equation \(y = 0.46x + 0.92\). The data does not all fall on the line because there may be factors other than weight that go into the price, such as: the quality of the broccoli, the region where the package is sold, and any discounts happening in the store.

$y = 0.46x+0.92$

A scatter plot with a line of best fit.

We can interpret the \(y\)-intercept of the line as the price for the package without any broccoli (which might include the cost of things like preparing the package and shipping costs for getting the vegetable to the store). In many situations, the data may not follow the same linear model farther away from the given data, especially as one variable gets close to zero. For this reason, the interpretation of the \(y\)-intercept should always be considered in context to determine if it is reasonable to make sense of the value in that way.

We can also interpret the slope as the approximate increase in price of the package for the addition of 1 pound of broccoli to the package.

The equation also allows us to predict additional values for the price of a package of broccoli for packages that have weights near the weights observed in the data set. For example, even though the data does not include the price of a package that contains 1.7 pounds of broccoli, we can predict the price to be about $1.70 based on the equation of the line, since \(0.46 \boldcdot 1.7 + 0.92 \approx 1.70\).

On the other hand, it does not make sense to predict the price of 1,000 pounds of broccoli with this data, because there may be many more factors that will influence the pricing of packages that far away from the data presented here.