Lesson 4

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.

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.

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

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

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.

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.

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

 

\(y = 0.216x + 0.345\)

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.)

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

Arrange students in groups of 2.

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.

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.)

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.

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.

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.)