Lesson 8

The purpose of this warm-up is for students to interpret the rate of change of a function to describe a trend (MP2). It is used to help students recall the role of slope in showing associations for data that can be fit with a linear model in anticipation of looking for associations in this lesson.

Give students 2 minutes of quiet work time followed by a whole-class discussion. 

Ask students to share what the rate of change of this function tells them about the trend. Record and display their responses for all to see. To include more students in the conversation, consider asking some of the following questions:

• “Does anyone agree or disagree with this reasoning? Why?”
• “Did anyone reason about the rate of change in a different way?”
• “Did anyone reason about the rate of change in the same way but would describe the trend differently?”
• “Does anyone want to add on to ____'s reasoning?”

All of the information from this section about scatter plots comes into play as students analyze data about animal body and brain weights. Students begin with a table of data and create a scatter plot. After seeing the scatter plot, students pick out any outliers and fit a line to the scatter plot. Finally, the slope of the line is estimated and its meaning interpreted in context (MP2).

When estimating slope, some students won’t use the scales of the axes correctly, so the slope is reported incorrectly. Some students may not notice the different units of weight used on each axis.

The goal of this discussion is to ensure students can make sense of the data given all the tools from this unit. 

Consider asking some of the following questions:

• "Which data did you consider outliers?" (human and chimpanzee)
• "How did you determine your fitted line?"
• "Let's assume the trend you found continues past the end of the scatter plot. A Tyrannosaurus Rex is a dinosaur that is estimated to have a body weight of about 8,000 kg. What do you expect its brain weight to be?" (About 8,000 g or 8 kg)

In this activity students create another scatter plot to analyze the data they collected about their classmates in a previous lesson (MP4). A suggested linear model is compared to the data and a particular point is identified in both the scatter plot and data table.

Although the scatter plots are left to students to organize, the only linear model considered is \(y = x\) which is symmetric when switching which variable is represented on each axis. If possible, identify any groups who have axes switched to bring up in the discussion.

Note: Some students may be sensitive about their body measurements and providing alternate data allows the class to work with actual values without making students uncomfortable. Depending on your class, consider providing a similar data set to the one collected in the earlier lesson (measurements from the staff, a different class, or invented data that is similar to the data collected).

The goal of this activity is for students to use the methods they have learned in this unit to explore data they have collected. To highlight some of the main points, select 2–3 students to respond to each question:

• “What is the slope of the line that best fit the data?” (Answers vary based on class data, but should be close to 1.)
• If any groups had axes switched, select these groups to show their scatter plots. "What would a point on the line \(y = x\) represent in each graph?" (A person whose arm span and height are identical. It does not matter which way the axes are drawn for this line.)
• “What does it mean about a person whose point is above the line \(y=x\)? What does it mean about a person whose point is below the line?” (Depending on the axes, a person whose point is off of the line has a longer arm span than expected for their height or is taller than expected for their arm span.)
• “Suppose you measure height and arm span for people in you neighborhood. When you make a scatter plot, you notice two clusters of data: one group of data in the lower left of the scatter plot and another group in the upper right. What does this mean? Why do you think there may be these clusters?” (It may mean there are a lot of small children and taller adults.)