Lesson 24

The purpose of this warm-up is for students to answer questions about relative frequency of items after finding missing information in a two-way table.

Monitor for students who find the percentages for the final two questions using different strategies to share during the whole-class discussion. 

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

Ask students to share the missing information they found for the table. Record and display their responses for all to see.

Select students previously identified to explain how they found the percentages for the final two questions and what that percentage represents.

1. Students who find a percentage using the values given (for example 31% since \(\frac{5}{16} \approx 0.31\)), then subtract from 100% (for example 69% since \(100 - 31 = 69\)) to answer the question.
2. Students who find the actual values first by subtracting (for example \(16 - 5 = 11\)) then compute the percentage (for example 69% because \(\frac{11}{16}=0.6875\)).

Ask the rest of the class if they agree or disagree with the strategies and give time for any questions they have.

Now that students are more familiar with two-way tables showing relative frequency, they are ready to create their own segmented bar graphs. In this activity, students create two segmented bar graphs based on the same two-way table by considering percentages of the rows and columns separately. After creating the segmented bar graphs, they are analyzed to determine if there is an association present in the data.

Arrange students in groups of 2. After a brief introduction, give 5–10 minutes of quiet work time. Ask students to compare their answers with their partner and try to resolve any differences. Finish with a whole-class discussion. 

Display the two-way table from the previous lesson's cool-down activity containing the data collected about the class's playing sports and musical instruments. If the data is unavailable, the data from this lesson's warm-up can be used.

Tell students they should work with their partners to each work on one of the graphs. One student should work on problems 1 and 2 while their partner should work on 3 and 4. After they have completed their graphs, they should work together to understand their partners graphs and complete the last problem together.

Students may draw the segmented bar graph incorrectly. Most likely, they will accidentally graph frequency instead of relative frequency. They may also graph relative frequencies, but without stacking them. Both segmented bars should go from 0 to 100.

To clarify how to create and interpret segmented bar graphs, ask:

• “What different information can be seen by the two segmented bar graphs?”
• “Why are the numbers in the top left box in the two tables different? What do they mean?” (In the first table it represents the percentage who also play musical instruments out of all the people who play sports. In the second table it represents the percentage of people who also play sports out of all the people who play musical instruments.)
• “Is there an association between the two variables? Explain or show your reasoning.” (The answer will depend on class data, but the reasoning should include an analysis of the relative frequencies within categories. There is an association if the percentages within one category are very different from the percentages in another category.)

If there is an association, ask what the segmented bar graphs would look like if there was no association. If there is not an association, ask what the segmented bar graphs would look like if there was one.

This activity provides students less structure for their work in creating segmented bar graphs to determine an association (MP4). In addition, the data in this activity is split into more than two options. Students work individually to create a segmented bar graph based on either columns or rows and then share their information with a partner who has created the other segmented bar graph. Together, partners discuss the segmented bar graphs to determine if there is an association between the variables (MP3). In particular, students should notice that there is evidence of an association is the relative frequencies within a category are very different from the relative frequencies in another category.

As students work, identify groups that use the different segmented bar graphs to explain why there is an association between the color of the eraser and flaws.

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

Provide students access to colored pencils. Either assign or have partners choose which will make a graph for each row and which will make a graph for each column.

The purpose of this discussion is to identify strategies for creating segmented bar graphs and for analyzing them to determine if there is an association among variables.

Ask, “What strategies did you use to create the segmented bar graphs?” (First, we created a new table of the relative frequencies. Then we approximated the heights of the segments based on the percentages from the table.)

Select previously identified groups to share their explanation for noticing an association.

1. Groups that use the segmented bar graph based on rows.
2. Groups that use the segmented bar graph based on columns.

After both explanations are shared, ask students, "Do you think that noticing the association was easier with one of the graphs?" (Likely the segmented bar graph based on rows is easier since there are only 2 segments and it is easier to see that the yellow and green erasers are more flawed.)

Finally, ask students, "If there was not an association between color and flaws, what might the segmented bar graph based on the rows look like? What might the segmented bar graph based on the columns look like?” (The segmented bar graph based on the rows would have each segmented bar look about the same. That is, the line dividing the two segments would be at about the same height in each bar. The segmented bar graph based on the columns would have segments that are all approximately equal. That is, each segment should represent about 25% of the entire bar.)