Lesson 22

Observing More Patterns in Scatter Plots

22.1: Notice and Wonder: Clustering (5 minutes)

Warm-up

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.

Launch

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.

Student Facing

What do you Notice? What do you Wonder?

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 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.

22.2: Scatter Plot City (20 minutes)

Activity

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).

Teacher Notes for IM 6–8 Math Accelerated
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.

Launch

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.

Representation: Internalize Comprehension. Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity by beginning with fewer cards. For example, give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
Supports accessibility for: Conceptual processing; Organization
Speaking: MLR8 Discussion Supports. To support students' explanations for sorting the scatter plots in the way they chose, display sentence frames for students to use when they are working with their partner. For example, “I think ____ because _____ .” or “I (agree/disagree) because ______ .”
Design Principle(s): Support sense-making; Optimize output for (explanation)

Student Facing

Your teacher will give you a set of cards. Each card shows a scatter plot.

1. Sort the cards into categories and describe each category.
2. Explain the reasoning behind your categories to your partner. Listen to your partner’s reasoning for their categories.
3. Sort the cards into two categories: positive associations and negative associations. Compare your sorting with your partner’s and discuss any disagreements.
4. Sort the cards into two categories: linear associations and non-linear associations. Compare your sorting with your partner’s and discuss any disagreements.

Activity Synthesis

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.)

22.3: Animal Brains (15 minutes)

Activity

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).

Launch

Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by 5 minutes of partner discussion and 5 minutes of whole-class discussion.

If using the digital activity, students can still work independently to analyze the scatter plot and answer the prompts. Then students can discuss their thinking in groups of 2 followed by whole-class discussion.

Student Facing

Is there an association between the weight of an animal’s body and the weight of the animal’s brain?

animal body weight (kg) brain weight (g)
cow 465 423
grey wolf 36 120
goat 28 115
donkey 187 419
horse 521 655
potar monkey 10 115
cat 3 26
giraffe 529 680
gorilla 207 406
human 62 1320
rhesus monkey 7 179
kangaroo 35 56
sheep 56 175
jaguar 100 157
chimpanzee 52 440
pig 192 180

Use the data to make a scatter plot.  Are there any outliers?

Experiment with the line to fit the data. Drag the points to move the line. You can close the expressions list by clicking on the double arrow.

1. Without including any outliers, does there appear to be an association between body weight and brain weight? Describe the association in a sentence.
2. Adjust the line by moving the green points, fitting the line to your scatter plot, and estimate its slope. What does this slope mean in the context of brain and body weight?

Launch

Arrange students in groups of 2. Give students 5 minutes of quiet work time followed by 5 minutes of partner discussion and 5 minutes of whole-class discussion.

If using the digital activity, students can still work independently to analyze the scatter plot and answer the prompts. Then students can discuss their thinking in groups of 2 followed by whole-class discussion.

Student Facing

Is there an association between the weight of an animal’s body and the weight of the animal’s brain?

Use the data in the table to make a scatter plot. Are there any outliers?

animal body weight (kg) brain weight (g)
cow 465 423
grey wolf 36 120
goat 28 115
donkey 187 419
horse 521 655
potar monkey 10 115
cat 3 26
giraffe 529 680
gorilla 207 406
human 62 1,320
rhesus monkey 7 179
kangaroo 35 56
sheep 56 175
jaguar 100 157
chimpanzee 52 440
pig 192 180
1. After removing the outliers, does there appear to be an association between body weight and brain weight? Describe the association in a sentence.
2. Using a piece of pasta and a straightedge, fit a line to your scatter plot, and estimate its slope. What does this slope mean in the context of brain and body weight?

Student Facing

Use one of the suggestions or find another set of data that interested you to look for associations between the variables.

• Number of wins vs number of points per game for your favorite sports team in different seasons
• Amount of money grossed vs critic rating for your favorite movies
• Price of a ticket vs stadium capacity for popular bands on tour

After you have collected the data,

1. Create a scatter plot for the data.
2. Are any of the points very far away from the rest of the data?
3. Would a linear model fit the data in your scatter plot? If so, draw it. If not, explain why a line would be a bad fit.
4. Is there an association between the two variables? Explain your reasoning.

Anticipated Misconceptions

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.

Activity Synthesis

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)
Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each response or observation that is shared, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they hear to a partner, before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement.
Design Principle(s): Support sense-making

Lesson Synthesis

Lesson Synthesis

The goal of this discussion is to help students reflect on all of the things they have learned about bivariate data in this unit. Here are some possible questions for discussion:

• “In your own words, what is a non-linear association?” (The points in a scatter plot do not suggest a linear model would fit the data well.)
• “In your own words, what are clusters in data?” (The points in a scatter plot are clumped together in different groups.)
• “What do clusters usually mean?” (There may be multiple patterns present within the data. Perhaps there are subgroups that show different patterns.)
• “What does a point in a scatter plot tell you?” (Two measurements about an individual in a population).
• “What is an association between variables?” (A trend that suggests that as one variable increases, the other variable tends to increase if it is a positive association or decrease if it is a negative association.)
• “What does a fitted line tell you about the data?” (It represents a model that can be used to make predictions about the dependent variable based on the value of the independent variable.)
• “What does the slope of a fitted line tell you about the data?” (The amount the dependent variable will increase [or decrease] for a one-unit increase in the independent variable.)

Student Lesson Summary

Student Facing

Sometimes a scatter plot shows an association that is not linear:

We call such an association a non-linear association. In later grades, you will study equations that can be models for non-linear associations.

Sometimes in a scatter plot we can see separate groups of points.

We call these groups clusters.

People often collect data in two variables to investigate possible associations between two numerical variables and use the connections that they find to predict more values of the variables. Data analysis usually follows these steps:

1. Collect data.
2. Organize and represent the data, and look for an association.
3. Identify any outliers and try to explain why these data points are exceptions to the trend that describes the association.
4. Find n equation that fits the data well.

Although computational systems can help with data analysis by graphing the data, finding an equation that might fit the data, and using that equation to make predictions, it is important to understand the process and think about what is happening. A computational system may find an equation that does not make sense or use a line when the situation suggests that a different model would be more appropriate.