In this lesson, students will examine the graphical relationship between the mean and the shape, or distribution, of data. Students will: 
- calculate the mean of a data set in a list or as a graphical representation. 
- create a data set with a specific mean value. 
- graphically represent data sets 
- compare different graphical representations of data sets that have the same mean. 
- recognize when a dataset is symmetric around the mean. 

- How do we use the mean, median, mode, and range to describe a set of data? Why do we need three different measurements of central tendency? 
- How can we use mathematics to create models that help us analyze data, make predictions, and better comprehend the world in which we live, and what are the limitations of these models? 

- Mean: Average; the number found by dividing the sum of a set of numbers by the number of addends. 
- Cluster: Numbers which tend to crowd around a particular point in a set of values. 
- Gap: A space between data points on a graph. 
- Outlier: A value far away from most of the rest in a set of data. 
- Symmetric: An object is symmetrical when one half is a mirror image of the other half. A data set is symmetrical when half the data is above the mean and half is below. 
- Distribution: Arrangement of data to show frequency of occurrence; will form a shape when graphed.

- sticky notes 
- white 8 ½" x 11" paper 
- calculators (optional) 
- chart paper 
- Baseball Wins for the Past Ten Years (M-6-5-3_Baseball Wins and KEY) 
- Points to Consider black line template (M-6-5-3_Points to Consider) 
- 3-2-1 Summary (M-6-5-3_3-2-1 Summary) 
- Price of Milk activity sheet (M-6-5-3_Price of Milk) 
- Lesson 3 Exit Ticket (M-6-5-3_Lesson 3 Exit Ticket and KEY) 
- histograms (M-6-5-3_Histograms)

- Use the 3-2-1 Summary (M-6-5-3_3-2-1 Summary) to determine whether students comprehend the material sufficiently to make relevant observations and draw correct conclusions. 
- The Price of Milk group activity (M-6-5-3_Price of Milk) can be used to motivate students to work together and to compel a good enough understanding of mean so that they can apply it backward, i.e., creating data given the mean. It is also an easy technique to assess students' abilities to graphically represent data. 
- The Lesson 3 Exit Ticket can be used to further evaluate student mastery.

Scaffolding, Active Engagement, Modeling, and Explicit Instruction 
W: Begin the class with data displayed on a histogram. Students will use the data to deepen their understanding and examine the relationships of the data, with a focus on relationships and mean. 
H: Students apply a familiar concept of game scores to understand arithmetic mean. The data sets used provide straightforward mental computations, allowing students to concentrate on central tendencies rather than computing skills. This activity will help students understand that different sets of data can have the same mean value. 
E: Review with students their previous knowledge of stem-and-leaf plots and histograms. Then, students will collaborate in small groups to analyze, compare, and display the data provided. They will discuss the similarities and differences between the data sets, highlighting the fact that various data sets might have the same mean. Recording sheets are used to assist students express their observations and as an informal technique to check for comprehension. 
R: Students work in small groups to collect, analyze, and present their own data sets related to prices for milk. Specific questions are given to help students think about the data, express understanding in their own words, and assist the teacher in assessing progress. 
E: Use Random Reporter for student presentations of the data found and displayed. Teacher observations and the use of Exit Tickets will assist in determining if students comprehend the lesson concepts or require remediation. 
T: Individual student needs are met through the extension activities. Routine activities suggestions allow students to review their understanding of data sets and comparisons on a regular basis. Students who require further practice can benefit from the small-group activity, which reviews lesson concepts. The expansion activity is appropriate for students who are ready to face a higher challenge. 
O: Before teaching this lesson, students should know how to compute the mean and represent data in a stem-and-leaf plot or histogram. Because the purpose of the lesson is to be able to generate data sets that meet specific criteria, the use of calculators may be beneficial. 

"In today's lesson, we will determine the mean value of data sets. We will then investigate whether there is a graphical relationship between the mean and the shape, or distribution, of the data." Post Histograms (M-6-5-3_Histograms) for students to see. "Using the histograms below, we may make observations about how the data is represented. Is it clustered together? Are there any gaps? Are there any outliers? Can the shape indicate a relationship between the data and the mean?"

Give each student some sticky notes. "If the target number is 50, write 5 numbers on a sticky note that, when added together, the sum will be 50. Use only positive whole numbers. Be sure you have 5 numbers in your data set that when added together the total equals 50.” Allow students time to finish this task. "Do you think we'll all have the same numbers when we share our data sets? Why, or why not?" Allow students some time to think-pair-share. Then invite students to express their thoughts aloud. "You each have a data set with numbers that add up to 50. Now you'll create a scenario with your five numbers that sum up to 50. For example, the numbers you wrote down represent the number of points scored by a junior league basketball player in his last 5 games. How can you use the mean to get the average number of points scored per game? Now you come up with a new scenario for your numbers. The question will be the same, but instead of 'points per game', it will be about your scenario." Allow students time to discuss and share their thoughts. Then use the think aloud method to model and record your calculations on the board. 

Have some of students discuss the scenario they created for their numbers. "Are there any scenarios that were the same? Do the numbers make sense based on the scenario?”

"First, add up the numbers in the data set. Assume the data set included 11, 5, 24, 7, 3. If I add the numbers together, the total is 50: 11 + 5 + 24 + 7 + 3 = 50.
To get the mean, I divide the sum by the number of elements in the data set. My data set has five numbers. 50 ÷ 5 = 10. The mean of the data set shown is 10.
That means the average number of points scored per game will be 10." 

"Now it's your time to calculate the mean of your data set. When you're finished, write the mean of your data set on a sticky note. Then discuss your results with the students around you. What do you notice? Why did this happen?" 

"In the previous activity, we discussed how different data sets might have the same mean value. In the case of the problem we used, the mean reflects the average amount of points scored throughout five games, which was 10. When you look at your original data, you'll notice that some games had more points and some game had fewer points. Remember that the mean is the average of a data set." 

Give each student a sheet of white paper and ask them to display their data graphically. Encourage the use of stem-and-leaf plots and histograms. While students are working, monitor their performance and provide verbal guidance as needed. To measure students' understanding, ask them questions like the ones listed below.
How will you represent your data set graphically? 
Will your graphic representation be similar to others in the class? Why? 
What conclusions can you get from analyzing your data graphically? 
Is the mean an actual number in your data set? 
Where is the mean represented on your graphic representation? Does this seem reasonable? Explain. 
Is your data symmetric around the mean? In other words, is roughly half of the data above and half below the mean value? 
Can you discover someone in the classroom with a different graphic representation than you, yet having the same mean? 
How come two people who have the same mean have different graphic representations?

Distribute copies of the Baseball Wins for the Past Ten Years (M-6-5-3_Baseball Wins and KEY) data set to each student. "In the following section of the task, you will explore data sets containing baseball victories over the last 10 years for three different teams. To begin, calculate the mean for each team. Remember to add all of the numbers in the data set and then divide by the number of items. After calculating the mean values for each team, answer the questions on the bottom half of the sheet." Students can calculate the mean with paper and pencil or calculators, depending on their preparation, time, and calculator availability. While students are working, monitor their performance and offer assistance as needed.

Divide students into triads. "Notice that the mean value for all three teams is the same. Also, notice how the data distribution, or shape, changes by team. Talk with your group members about the questions at the bottom of the activity sheet. If necessary, make adjustments to your answers." Students will be able to obtain quick feedback on their performance and clarify any misunderstandings by discussing the answers with their classmates. Be available to answer questions that cannot be answered within the groups.

"Now, with your group members, discuss the data distribution as shown in the stem-and-leaf plots. Then, on chart paper, write your group's observations about the relationship you see between the mean value and the shape of the data for the three teams. Include the similarities and differences. Use mathematical language whenever possible. Points for consideration will be placed on the board. Your responses should not be limited to those posted ideas." Display the Points to Consider black line template on the board for students (M-6-5-3_Points to Consider). Note: The questions listed are from the Points to Consider template.

What conclusions may you draw from the graphical representation of data? 
Is the mean an actual number in the data set?
Where does the mean appear on the graphic representation? Does this seem reasonable? Explain. 
Is the data symmetric around the mean? In other words, is roughly half of the data above and half below the mean value? 
Are there any clusters or gaps in the data? 
Are there any outliers in the data? (Remember, an outlier is a piece of data that stands apart dramatically from the rest of the data.) 
Have students write down their observations on chart paper. When all groups have finished, have students display their group's chart paper around the room. Have groups do a carousel walk from chart paper to chart paper, noting similarities and differences in thinking.

When students return to their original group's chart paper, have them fill in a summary (M-6-5-3_3-2-1 Summary). In the top row, students record three key observations; in the second row, students record two mathematical vocabulary words and an explanation of how those words can be used to help describe data; in the bottom row, students record one realization or question they still have about the concept or activity. Discuss student comments and explain any misunderstandings.

For the remainder of the class, students will remain in groups. "It is your group's responsibility to generate a list of five appropriate pricing for gallons of milk, with an average price of $3.00. Your goal is to create a data set that is not symmetric around the mean." Give each group a copy of the Price of Milk activity sheet (M-6-5-3_Price of Milk). Calculators may be an option for this exercise. Monitor student comprehension and provide assistance as required. Use questions like the ones below to test understanding while students work in groups. 

How did you start this task? 
What strategies did you use to help create your collection of reasonable prices?
What problems did you face while attempting to generate a data set? 
How is knowing the mean and having to calculate a data set different than knowing the data set and having to calculate the mean? 
Are there any clusters or gaps in your data set? Is there an outlier? 
Is your data symmetric around the mean? Explain. 
Do you think it is easier to build a data set that is symmetric or non-symmetric around the mean? Why do you think this way? 
Describe the relationship between the mean value and the shape of the data you generated.
When the students have finished their work, have them present it in groups. Using the random reporter method, ask each group to present data from the Price of Milk. The random reporter method is an approach in which a randomly selected member of each group is invited to present their group's work. This strategy reinforces the idea that all members of the group should learn how to describe their group's work and be ready to convey the information. Keep track of each group's performance and explanations. Clarify any misconceptions and emphasize excellent mathematical thinking and practices. Make the necessary comparisons.

"In this lesson we investigated data sets that had the same mean values but whose shape, or distribution of data, varied from one data set to the next." Have students fill in an exit ticket (M-6-5-3_Lesson 3 Exit Ticket and KEY). Give the exit ticket to students with about five minutes left in class; they must complete it and turn it in before they leave. You can rapidly review the students' responses. The information provided by the exit ticket will help identify who needs more practice and who is proficient. 

Extension: 

Use the ideas and activities below to satisfy your students' needs during the lesson and throughout the year.

Routine: To review the concept that data sets might have the same mean value but differ in shape or distribution, assign one of the problems below to students. Allow students time to find a data set that meets the criteria. Then ask students to compare their data sets to those of other students. Encourage students to discuss how the data sets and distribution of data are similar and different. 
Juan scored an average of 90% on five spelling tests. What are the five possible scores Juan earned on each spelling test? 
The average amount of sugar per serving found in five brands of cereal is 3 grams. How many grams of sugar may each of the five cereal brands contain per serving? 
A ticket to attend a baseball game costs an average of $50, depending on where you sit. What are the four possible ticket prices for each section?

Small Group: Give students a problem to solve. For example: Mason played in six basketball games and scored the following points: 3, 5, 7, 9, 15, 15. What was his average number of points each game? Ask them to determine the mean value of points Mason scored each game, which is 54 ÷ 6 = 9 points. Create a stem-and-leaf plot together. Use Points to Consider (M-6-5-3_Points to Consider) to ask students probing questions. Guide student thinking to ensure that math reasoning and language are being used. Then ask students to create another group of six numbers equal to Mason's overall score of 54 points. Encourage students to try to create a data set that would be reasonable yet has a different shape. Calculators may be useful when students guess and check for data sets that meet the criteria. Check for accuracy. Then, have students independently create a stem-and-leaf plot for their new data set and compare their graphical representations to those of other members of the group. Repeat the process with a similar problem if necessary. For example: Brynn noticed that her last five phone calls on her cell phone lasted 20 minutes, 14 minutes, 7 minutes, 7 minutes, and 7 minutes. How long does each phone call last on average?

Expansion: Students might be given specific criteria and then asked to generate a data set that meet those criteria. Use problems like those listed below. 
Find a set of five numbers with the mean of 5, the median of 5, and the mode of 7. The sum of the digits is 25. (Possible responses: 7, 7, 5, 4, 2) 
Find a set five numbers with the mean of 60 and the median of 25. The sum of the digits is 300. (Potential solutions: 25, 25, 25, 100, 125)
Find a set of four numbers with the mean of 3, the median of 3, and the mode of 3. (Possible response: 1, 3, 3, 5) 
Find a set of seven values with the mean and median are the same but the mode is different. (Possible answers: 3, 6, 8, 10, 13, 15, 15; mean = 10, median = 10, mode = 15) 
Students can also develop similar problems on their own and then solve them with a partner.