Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.
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, the interpretation of the line of best fit, and the general idea of a scatter plot are the important discussion points.
When students articulate what they notice and wonder, they 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.
Monitor for students who use mathematically precise terminology in their responses. In particular, the terms "scatter plot," "linear model," "slope," and "intercept" are important to review during this Warm-up.
Launch
Arrange students in groups of 2. Display the graph for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder.
Activity
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Student Task Statement
What do you notice? What do you wonder?
A scatterplot. Horizontal, from 60 to 80, by 5's, labeled number of people, thousands. Vertical, 105 to 140, by 5’s, labeled maximum noise level, decibels. 12 dots, straight line trending upward and to the right.
Activity Synthesis
Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the graph. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.
If students do not use these terms in their responses, prompt them to recall the vocabulary from grade 8 math:
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?” (We 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.)
“Why should you be careful using a model to predict information far from the collected data?” (Different patterns could show up if the desired information is too far from the data used to make the model. For example, the cost per shirt when buying a few shirts for a school club is different from the bulk pricing when buying hundreds of shirts for the whole school.)
Student Lesson Summary
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 . 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.
A scatterplot. Horizontal, from 0 to 3, by 0 point 5's, labeled weight in pounds. Vertical, 0 to 2 point 5, by 0 point 25s, labeled price in dollars.
12 dots trending upward and to the right. A line of best fit passes through the y axis at 0 comma 0 point 92, and trends upwards and to the right, passing through three dots.
We can interpret the -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 -intercept should always be considered in context to determine if it is reasonable to make sense of the value in that way.
We can 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 prices of packages of broccoli 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 .
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 influence the pricing of packages that far away from the data presented here.
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The vertical intercept appears to be approximately 105 decibels, but the origin is not shown on the graph.
Materials
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Activity Narrative
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).
In the digital version of the activity, students use an applet to find a linear model that could fit some data. The applet allows students to drag points to create a linear model without having to erase and redraw the line multiple times. The digital version may be helpful for trying multiple attempts at finding a linear model and encouraging students to make a guess. It will also familiarize students with creating scatter plots, which will be useful in later lessons.
If students don’t have individual access, projecting the Desmos graph during the Activity Synthesis is helpful.
Launch
Play the video of adding oranges to a box on a scale. The video may need to be paused for students to write down the weights.
A video of an empty box on a scale measuring in grams. Someone adds oranges to the box one by one.
Activity
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Student Task Statement
Watch the video, and record the weight for the number of oranges in the box.
number of oranges
weight in kilograms
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Create a scatter plot of the data.
Draw a line through the data that fits the data well.
Estimate a value for the slope of the line that you drew. What does the value of the slope represent?
Estimate the weight of a box containing 11 oranges. Will this estimate be close to the actual value? Explain your reasoning.
Estimate the weight of a box containing 50 oranges. Will this estimate be close to the actual value? Explain your reasoning.
Estimate the coordinates for the vertical intercept of the line you drew. What might the -coordinate for this point represent?
Which point(s) are best fit by your linear model? How did you decide?
Which point(s) fit the least well with your linear model? How did you decide?
Activity Synthesis
Display the scatter plot, then show the best-fit line.
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.)
“How would the scatter plot and linear model change if larger grapefruits were used instead of oranges?” (The weight would be increased for each point, and the slope of the line would be greater.)
“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.)
“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.)
“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 does not make sense?” (As we 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 0, 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 degrees.)
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Activity Narrative
In this activity, students are asked to interpret the slope and vertical intercept of a linear model in context 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 use these different strategies:
Estimate values from the graph
Use the equation for the linear model to find the values
Plan to have students present in this order to support using more precise answers.
Launch
Arrange students in groups of 2.
Select students who used each strategy described in the Activity Narrative to share later. Aim to elicit both key mathematical ideas and a variety of student voices, especially of students who haven't shared recently.
Activity
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Student Task Statement
The scatter plot shows the sale price of several food items, , and the cost of the ingredients used to produce those items, , as well as a line that models the data. The line is also represented by the equation .
A scatterplot. Horizontal from 0 to 4 by 0 point 5s, labeled ingredient cost, dollars. Vertical, 0 to 12, by 0 point 5's, labeled sale price, dollars. 24 dots trending upward and to the right. Line of best fit trends upwards and to the right, passing through 0 point 41 comma 2 point 19.
What is the predicted sale price of an item that has ingredients that cost $1.50? Explain or show your reasoning.
What is the predicted ingredient cost of an item that has a sale price of $7? Explain or show your reasoning.
What is the slope of the linear model? What does that mean in this situation?
What is the -intercept of the linear model? What does this mean in this situation? Does this make sense?
Student Response
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Building on Student Thinking
Although the grid lines in the graph do not appear to make 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.
Ask previously selected groups to share. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work for all to see.
Connect the different responses to the learning goals by asking questions such as:
“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 ?” (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. It may also be the case that it does not make sense to interpret the -intercept since we have no evidence to believe that the same linear relationship will hold there.)
Action and Expression: Internalize Executive Functions. To support organization, 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 of the alternative method for each question. Supports accessibility for: Language, Organization
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Activity Narrative
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 for each one. 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 for the scatter plots to their partner’s and decide if both their responses are correct for each scatter plot, even if they are different. Follow with a whole-class discussion.
MLR8 Discussion Supports. Display sentence frames to support partner discussions: “_____ represents _____” “Is there another way to say . . . ?” and “It looks like _____ represents . . . .”
Advances: Conversing
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
Activity
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Student Task Statement
Here are several scatter plots.
A.
A scatterplot. Horizontal from 0 to 24 by 2's, labeled age, years. Vertical, 0 to 350, by 25's, labeled reaction time, milliseconds. 21 dots trending linearly downward and right. Dot 1 at approximately 10 comma 310, Dot 21 at approximately 22 comma 185, with line of best fit.
B.
A scatterplot. Horizontal from 0 to 7 by 1's, labeled number of bananas. Vertical, 0 to 3 point 5, by 0 point 5's, labeled price in dollars. 14 dots trending linearly upward and right. Dot 1 at 1 comma 0 point 5, Dot 14 at 6 comma 2 point 53, with line of best fit.
C.
A scatterplot. Horizontal from 0 to 500 by 50's, labeled room size, square feet. Vertical, 0 to 2,200, by 100's, labeled cost to install flooring, dollars. 14 dots trending linearly upward and right. Dot 1 at approximately 150 comma 700, Dot 14 at approximately 450 comma 1,800, with line of best fit.
D.
A scatterplot. Horizontal from 0 to 5 point 5 by 0 point 5's, labeled temperature, degrees Celsius. Vertical, 0 to 24, by 2's, labeled volume, cubic centimeters. 18 dots trending linearly downward and right. Dot 1 at approximately 1 point 5 comma 22, Dot 18 at approximately 4 point 5 comma 15, with line of best fit.
Using the horizontal axis for and the vertical axis for , interpret the slope of each linear model in the situations shown in the scatter plots.
Assume that the linear relationship continues to hold for each of these situations, and interpret the y-intercept of each linear model.
Student Response
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Building on Student Thinking
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 and 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 ?” (A linear model is not exact even for the data it is based on. It is an approximation based on the data represented by 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 0.)
“How do you interpret the slope for each equation?” (The slope is the change in divided by the change in , so look at the labels on the scatter plot, and describe how for each increase of 1 of the -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 -variable.)
“When might it make sense to interpret the -intercept for a linear model?” (It makes sense when -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.)
HSS-ID.C.7
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
Students may struggle with estimating a slope when the scale on the - and -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.
