Lesson 22

The purpose of this warm-up is to elicit the idea that data can appear in clusters, which will be useful when students work with a variety of scatter plots in the future. While students may notice and wonder many things about these graphs, how the data are clustered is the important discussion point.

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.

Keep students in groups of 2. Allow 2 minutes quiet think time followed by partner and whole-class discussion.

Display the scatter plots 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 the things they notice and wonder with their partner, followed by a whole-class discussion.

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 scatter plot. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. 

If the clustering of the data in the scatter plots does not come up during the conversation, ask students to discuss this idea. Tell students that when data seems to have more than one pattern, it is called clustering. Clustering of the data like in graph B, C, and D can reveal hidden patterns. Usually, clustering means there are subgroups within our data that may represent different trends.

For example, in Plot B, the data may represent body measurements of a certain species of bird. Although the data originally came from a group that made sense (a single species), there appear to be subgroups that have a large influence on the data as well. The lower half of the data may represent females and the upper half may represent males, so we can see that there are different patterns within the different subgroups.

When clustering is present, it may be helpful to investigate the cause of the separation and analyze the data within the subgroups rather than as a whole.

In previous lessons, data was fit with a linear model, but is not appropriate in all cases. In this activity, students individually sort cards of different scatter plots then discuss with a partner. They then re-sort the cards based on linearity and then again by positive or negative association (MP4).

Adjust the time of the activity to 15 minutes. If students do not bring up the non-linear nature of some of the scatter plots, make sure to discuss them during the synthesis. Tell students that, even when a linear association seems to be present, it may only fit the data very close to the data that is present. For example, if we cover up the scatter plot to the right of 6 on Card A, there may appear to be a linear association that would fit the data, but it would not apply to the data on the right side of the scatter plot.

Arrange students in groups of 2, and give each student a set of cards. Give 5 minutes of quiet work time for the first question followed by partner discussion. For the final two questions, ask students to take turns sorting the cards by putting one card into one of the categories, then explaining their reasoning for putting it there. After both partners have agreed on the placement of that card, the other partner may repeat these steps to sort one more card. Repeat this process until all cards are sorted.

Tell the students that there is a third category for the last two parts of the activity: neither. It is possible for data to have absolutely no association and appear completely random which would not be a linear association or a non-linear association. It is also possible for data to not have a strictly positive or negative association if, for example, the scatter plot seems to increase and then decrease.

To help clarify the meaning of linear and non-linear associations as well as positive and negative associations, consider asking some of the following questions:

• “How did you originally group the cards?”
• “Which graphs did you sort into the 'positive association' group?” (F and G)
• “Are there any cards that did not fit with either positive or negative association?” (Yes, the points for card A form an arc that first goes up and then down.)
• “Which graphs did you sort into the 'linear association' group?” (C and E)
• “If a scatter plot has no association at all (neither linear nor non-linear), what might its scatter plot look like?” (A lot of random dots scattered all around the plot with no obvious trend.)

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)