“In today’s lesson we are going to calculate the mean value of data sets. [IS.4 - Struggling Learners and ELL Students] We then will look to see if there is a graphical relationship between the mean and the shape, or distribution, of the data.” Post Histograms (M-6-5-3_Histograms.doc) for students to see. “By using the following histograms, we can make observations about the way the data is represented. Is it clustered together? Are there gaps? Are there outliers? [IS.5 - All Students] Can the shape help show a relationship between the data and the mean?”
Give every student a few sticky notes. “If the target number is 50, record 5 numbers on your sticky note that when added together the sum will be 50. Use positive whole numbers only. Be sure you have 5 numbers in your data set that when added together the total equals 50.” [IS.6 - All Students] Allow students some time to complete this task. “Before we share our data sets, do you think we will all have the same numbers? Why or why not?” Allow students some time to think-pair-share. Then allow students to share their thinking aloud. [IS.7 - All Students] “You each have a data set that when the numbers are added together the sum will be 50. Now, you will come up with a scenario for your five numbers that add up to 50. One example is that the numbers you wrote down are the number of points a junior league basketball player scored in his last 5 games. How can you calculate the average number of points per game using the mean? Now you come up with a different scenario for your numbers. The question will be the same question but instead of ‘points per game’ it will be what your scenario is about.” Allow students some time to discuss and share their thinking. Then model using the think aloud strategy and record the calculations on the board. [IS.8 - Struggling Learners and ELL Students]
Have some of the students share the scenario they came up with for their numbers. “Were there any scenarios that were the same? Do the numbers make sense based on the scenario?”
- “First you have to add up the numbers in the data set. Let’s say the data set included 11, 5, 24, 7, 3. If I add the numbers together the sum equals 50: 11 + 5 + 24 + 7 + 3 = 50.
- To find the mean I divide the sum by the number of items in the data set. There are five numbers in my data set. 50 ¸ 5 = 10. The mean of the data set shown is 10.
- That means the average number of points scored per game would be 10.”
“Now it is your turn to calculate the mean for your data set. When you are finished, you can record the mean for your data set on a sticky note. Then share your findings with students around you. What do you notice? Why did this happen?”
“In the last activity we saw how different data sets can have the same mean value. In the context of the problem we used, the mean represents the average number of points scored in five games, which was 10. When you look at your original data you can see how in some of the games more points were scored and in some of the games fewer points were scored. Remember mean is the average for a data set.”
Give each student a sheet of white paper and ask students to represent their data graphically. Encourage the use of stem-and-leaf plots or histograms. [IS.9 - Struggling Learners and ELL Students] While students are working monitor student performance and provide verbal guidance where necessary. Ask students questions similar to those listed below to help assess understanding. [IS.10 - All Students]
- How are you going to represent your data set graphically?
- Will your graphic representation be similar to others in the classroom? Why?
- What observations can you make by looking at your data represented 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 about half of the data above the mean value and about half of the data below the mean value?
- Can you find someone in the classroom who has a different graphic representation than you even if you have the same mean?
- How can two people with the same mean have different graphic representations?
Distribute copies to each student of the data set Baseball Wins for the Past Ten Years (M-6-5-3_Baseball Wins and KEY.doc). “In the next part of the activity you are going to investigate data sets that show baseball wins for the past ten years for three different teams. To begin you will calculate the mean for each team. Remember to add up the numbers in the data set and then divide by the number of items in the data set. After calculating the mean values for each team finish the questions on the bottom half of the sheet.” Students can calculate the mean using paper/pencil or calculators depending on student readiness, time, and availability of calculators. While students are working, monitor student performance and provide support if necessary. [IS.11 - All Students]
Group students into triads. “Notice how the mean value for all three teams is the same. Also notice how the distribution, or shape, of the data varies from team to team. With the members in your group discuss the questions at the bottom of the activity sheet. Adjust answers if necessary.” This will allow students to receive immediate feedback on their performance and clarify any misunderstandings by discussing the answers with classmates. Be available for any questions that arise that cannot be answered within the groups. [IS.12 - All Students]
“Now with the members of your group discuss the distribution of data as seen in the stem-and-leaf plots. Then record on chart paper your group’s observations about the relationship you observe between the mean value and the shape of the data for the three teams. Include similarities and differences. Use mathematical language when possible. [IS.13 - All Students] Points to consider will be posted on the board. [IS.14 - All Students] 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.doc). Note: Questions listed are from Points to Consider template.
- What observations can you make by looking at the data represented graphically?
- Is the mean an actual number in the data set?
- Where is the mean represented on the graphic representation? Does this seem reasonable? Explain.
- Is the data symmetric around the mean? In other words is about half of the data above the mean value and about half of the data below the mean value?
- Are there clusters or gaps in the data?
- Are there outliers in the data? (Remember an outlier is a piece of data that stands apart significantly from the rest of the data.)
Have students record their observations on chart paper. When all groups are finished have students post their group’s chart paper around the room. Have groups do a carousel walk from chart paper to chart paper and notice similarities and differences in thinking. When students return to their original group’s chart paper have each student complete a summary (M-6-5-3_3-2-1 Summary.doc). In the top row students record three key observations they have made; in the second row students record two mathematical vocabulary words along with an explanation of how those words can be used to help describe data; in the bottom row students record one realization made or one question they may still have about the concept or activity. Discuss student responses and clarify any misunderstandings. [IS.15 - All Students]
For the next part of the lesson students will remain in groups. “It is your group’s job to generate a collection of 5 reasonable prices for gallons of milk where the average price per gallon is $3.00. Your goal is to construct 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.doc). The use of calculators may be an option for this activity. Monitor student understanding and provide support if needed. To assess understanding while students are working in groups use questions similar to those listed below.
- How did you begin this task?
- What strategies did you use to help generate your collection of reasonable prices? [IS.16 - All Students]
- What problems did you encounter when trying to create 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?
- Do you have 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 create 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 students are finished, have groups present their work. Using the random reporter method, [IS.17 - All Students] ask each group to present the information from the Price of Milk. The random reporter method is a strategy where a random member of each group is asked to present that group’s work. This method reinforces the idea that all members of the group should understand how to explain their group’s work and be prepared to present the information. Monitor groups’ performances and explanations. Clarify any misconceptions and highlight good mathematical thinking and processes. [IS.18 - All Students] Make appropriate 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 complete an exit ticket (M-6-5-3_Lesson 3 Exit Ticket and KEY.doc). Give the exit ticket to students with about five minutes left in class; they must complete it and hand it in before leaving. You can quickly review students’ responses. Information provided by the exit ticket will identify who may need additional practice and who demonstrates proficiency.
Extension:
Use the following strategies and activities to meet the needs of your students during the lesson and throughout the year. [IS.19 - Struggling Learners and ELL Students]
- Routine: To review the concept that data sets can have the same mean value yet different shapes, or distribution of data, give students one of the problems below. Allow students time to find a data set to fit the criteria. Then have students compare their data sets with other students. Have students discuss how the data sets and distribution of data are similar and different.
- Juan earned an average score of 90% on five spelling tests. What are five possible scores Juan earned on each of his spelling tests?
- The average amount of sugar per serving found in five brands of cereal is 3 grams. What are the possible grams of sugar per serving in each of the five brands of cereal?
- The average price of a ticket to attend a baseball game depending on where you sit is $50. What are four possible ticket prices for each section?
- Small Group: Create a problem for students to solve. For example: Mason played in six basketball games and scored the following points: 3, 5, 7, 9, 15, 15. What is the average number of points he scored per game? Ask students to calculate the mean value of points Mason scored each game which is 54¸ 6 = 9 points. Construct a stem-and-leaf plot together. Using Points to Consider (M-6-5-3_Points to Consider.doc), ask students probing questions. Guide student thinking to ensure math reasoning and math language are being utilized. Then ask students to generate another group of six numbers that equals the same number of total points Mason scored which is 54. Also encourage students to try to generate a data set that would be reasonable yet have a different shape. Use of calculators may be beneficial as students guess and check for a data set that fits the criteria. Check for accuracy. Then have students independently construct a stem-and-leaf plot for their new data set and compare their graphical representations with other members of the group. Repeat the process using a similar problem if necessary. For example: Brynn saw that her last five phone calls on her cell phone lasted 20 minutes, 14 minutes, 7 minutes, 7 minutes, and 7 minutes. What is the average number of minutes per phone call?
- Expansion: Students can be given certain criteria and then asked to generate a data set that fits the given criteria. Use problems similar to those listed below.
- Find a set of five numbers where the mean is 5, the median is 5, and the mode is 7. The sum of the digits is 25. (Possible answer: 7, 7, 5, 4, 2)
- Find a set of five numbers where the mean is 60 and the median is 25. The sum of the digits equals 300. (Possible answer: 25, 25, 25, 100, 125)
- Find a set of four numbers where the mean is 3, the median is 3, and the mode is 3. (Possible answer: 1, 3, 3, 5)
- Find a set of seven numbers where the mean and the median have the same value but the mode has a different value. (Possible answer: 3, 6, 8, 10, 13, 15, 15 ~ mean = 10, median = 10, mode = 15)
Students also can create similar problems of their own and then exchange their problems with a partner to solve.
