Lesson 5

The purpose of this Math Talk is to elicit strategies and understandings students have for finding the mean of a small data set. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to find the mean of larger sets.

Monitor for students who use the traditional algorithm for finding mean, as well as those who use use the concept of fair share.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. Connect the solutions by asking students to notice any patterns in the data sets. In particular, focus student attention on how a single value farther from the typical values tends to more significantly impact the mean than single values closer to typical values.

To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

The purpose of this activity is for students to differentiate between the steps to take to calculate each measure. Give students time to work with a partner to match a measure with a list of steps. Review the correct answers as a whole group, then allow students to calculate each measure. 

If students aren’t sure about the meaning of “variability,” tell them that it means how spread out the data are. For example, in a typical grade 9 class, the ages of students usually have low variability since they are mostly the same age, but the ages of people in the school parking lot after school has great variability since there are older adults as well as younger siblings and students.

The goal of this activity is for students to recall how to calculate each measure and to practice the calculations. Discuss any complications or confusions that arose. Time permitting, discuss how to use the measures to interpret data. Here are sample questions to promote class discussion:

• “What kind of insights will the principal get from knowing these statistics about the tenth-grade students?” (The principal would know if tenth graders are generally qualified for the advanced literature class, and can use this to make decisions about who to place in the class. Since the median is 500, the principal knows that more than half of the tenth grade students are qualified. On the other hand, the IQR of 205 means that there is a wide range of scores.)
• “What is one action you would take as principal, if you had these statistics?” (Everyone with a score of 600 or higher would take the advanced course. I would pay extra attention to any students below the median of 500 by providing tutorials.)
• “What does the mean say about the tenth graders? Does the median say something different?” (The mean could tell us that the typical tenth grader is not qualified for the advanced class (\(478.59 < 490\)). However, the median tells us that more than half are qualified. So, some of the lower scores that are well below the level of qualification are pulling the mean score below 490.)
• “Why is it important to have multiple measures of center and measures of variability?” (Having multiple measures can provide more information about the data than just using one measure. For example, when the mean is affected by really high or low scores, the median can provide more insight about the data’s center. The mean uses every data value and gives a sense of the typical student score. The median divides the group in half.)

The purpose of this activity is for students to practice calculating measures of center and measures of variability. Students will work in pairs and each partner is responsible for answering the questions in either column A or column B. Although each row has two different problems, they share the same answer. Ensure that students work their problems out independently and collaborate with one another when they do not arrive at the same answers. Students construct viable arguments and critique the reasoning of others (MP3) when they resolve errors by critiquing their partner’s work or explaining their reasoning.

Arrange students in groups of two. In each group, ask students to decide who will work on column A and who will work on column B. If students are unfamiliar with the row game structure, demonstrate the protocol before they start working.

Consider asking students to explain why their solutions might be the same for each row.

Sample responses for each row:

1. Sharing a lot of the value from 100 to the other small numbers is similar to sharing a small amount of the value from the 25 to the other closer values.
2. Most of the values from column B (except the 49) are greater than the values from column A. Since 49 is so much less then the other values, it will influence the mean value lower to match the mean from column A.
3. When rearranged in order, both data sets have 6 in the middle. Since the actual values on either side do not matter for median, the solution is the same.
4.  The values from column A are the same as the values from column B, only 10 greater. Since IQR measures the range of the middle half of the values, it is the same for both sets of data.
5. Although the data sets are different, the values that are used to calculate Q1 (\(\frac{36+40}{2} = 38\)) and Q3 (\(\frac{50+52}{2} = 51\)) for the two data sets are the same.
6. The values in column B are each one greater than the values in column A. Since MAD measures variability and both data sets are spread out the same amount, the MADs are the same.

The goal of the lesson is to ensure students know the difference between each measure and how to calculate them. Discuss any misconceptions students encountered while working, and how each measure can be used. Here are sample questions to promote a class discussion:

• “In situations when you and your partner did not arrive at the same answer, how did you agree on an answer?” (We talked about what we did to get our answers, and checked our steps with a calculator. If we did not agree, we had to convince one another about our thinking.)
• “What is the difference between measures of center and measures of variability?” (Measures of center are median and mean, and they summarize the data set with a number that describes a typical value of the data set, which can also be thought of as a center of its distribution. Measures of variability here are IQR and MAD, and they tell us how spread out the data set is.)