Note: If students need a review in types of data displays, an optional review is available (M-6-5-1_Optional Review of Data Displays.docx). If students need practice with line plots, an optional worksheet is available (M-6-5-1_Line Plot Practice and KEY.docx).
Prior to the lesson, print and cut apart the Measures of Central Tendency Match Game cards (M-6-5-1_Match Game Cards and KEY.doc). Distribute cards to six different students. Have these students come up to the front of the room and find their match. Check for accuracy and then post the matched cards for students to see as a visual reminder of the meanings of each measure of central tendency. Have a quick discussion about these terms. “In today’s lesson we are going to look at data and determine which measure of central tendency, mean, median, or mode, is the best descriptor of the data set.” [IS.8 - Struggling Learners and ELL Students]
Give to each student a copy of the Sports Stadium Capacity Data (M-6-5-1_Sports Stadium Data and KEY.doc). If available, allow students to use a calculator for this part of the lesson. “Let’s look at the data for the North American Stadiums. What do you notice about the data? What observations can you make?” Allow students think-pair-share time to discuss their observations. Then have students share their responses and record on chart paper. Encourage students to use math language when possible. [IS.9 - All Students] Guide students through looking at the different measures of central tendency by thinking aloud. “Is there a mode for this data set? Remember earlier in our Match Game we reviewed that mode is the value that occurs most frequently. There is no mode since no values are repeated in the data set. Mode would not be a good descriptor for this data set. [IS.10 - All Students] (Record this information on the board.) If I wanted to determine the value for median, would a calculator be helpful? No. Median is the value that divides the data set in half. [IS.11 - All Students] I need to first arrange the items in the data set sequentially.”(Record this information on the board.)
65,000 80,000 87,000 92,000 102,000 110,000
“Then I determine the middle value.” Record on the board.
65,000 80,000 87,000 92,000 102,000 110,000
“Since there are two middle values, I average them. I add them together and divide by 2.”
87,000 + 92,000 = 179,000 ÷ 2 = 89,500
“The median value for this data set is 89,500. This might be a good descriptor for this data set.”
“If I wanted to find the mean value for this data set a calculator would be helpful. I first have to add all the numbers in the data set together. I also could add these values together with paper and pencil.” Record on the board.
65,000 + 80,000 + 87,000 + 92,000 + 102,000 + 110,000 = 536,000
“Then I have to divide the sum by the number of items in the data set. The sum is 536,000 and there are six items in the data set.” Record on the board.
536,000 ¸ 6 = 89,333
“The mean value for this data set is 89,333. This value is similar to the median.”
“Now let’s look at the South American sports stadium capacity data. How is the data similar to the North American sports stadium capacity data and how is it different? Take a few moments with a partner to discuss the data.” Allow students time to pair-share and then ask students to share their observations aloud. Add responses to the chart paper that was used at the beginning of the lesson. “With a partner, calculate the measures of central tendency for the South American sports stadium capacities. When you are finished calculating the values, check with another set of partners to see if your calculations agree. [IS.12 - Struggling Learners and ELL Students] If there is a discrepancy please re-calculate the values together. Then decide which measure of central tendency would best represent this data set and support your choice with mathematical reasoning. Complete the questions on the Sports Stadium activity sheet.” When students are finished with this part of the activity discuss student findings. While students are working, monitor student performance and assess student understanding using questions similar to those listed below. [IS.13 - All Students]
- How did you begin this task?
- What do you predict will be the best measure of central tendency for this data set? Why do you think this way?
- How do outliers of data affect a measure of central tendency?
- How do you find the mode of a data set?
- Why don’t you think there is a mode for this data set?
- How is calculating the median value in this data set different than calculating the median value for the North American sports stadium capacities?
- What do you have to remember when determining the median value? (Organize the numbers sequentially. If there is an even number of items in a data set, find the mean value of the two middle numbers. That will be the median for that data set.)
- How do you find the mean of a data set?
- Is the mean value you calculated an actual value in the data set?
- Why might the mean, median, and mode be considered the measures of the “center” of the data? (Median is the center, literally, since it is the middle value that divides a data set in half. Mean can be considered the center of the data since it balances out the highs with the lows. Mode can be considered the center of the data since it is the value that occurs most frequently.)
- In your opinion which measure of central tendency best represents this data set and why?
“Depending on the data set, a certain measure of central tendency may be better to use than the others. [IS.14 - Struggling Learners and ELL Students] When we use mode for a measure of central tendency, that means the value occurs most frequently in a data set. What are some situations where we might want to know a value that occurs most frequently?” Allow students time to think-pair-share and share their ideas aloud. Discuss the validity of each response. Then record valid student responses on chart paper labeled: When to Use Mode, When to Use Median, When to Use Mean. Repeat the process for median and mean. Guide student thinking and provide verbal prompting if necessary. The chart below provides examples for each measure of central tendency. [IS.15 - All Students]
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When to Use Mode
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When to Use Median
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When to Use Mean
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- jersey number most worn by players of a franchise
- typical shoe size for a basketball player
- responses to a multiple choice question
- typical jersey size for players
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- cost of cars
- cost of college tuition
- cost of houses in a certain state
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- average height of boys and girls at a certain age
- average score on a given test
- average monthly electric bill for a household
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Post problems similar to those listed on the Pick the Best Measure of Central Tendency activity sheet (M-6-5-1_Pick the Best Measure and KEY.doc). In groups have students calculate the mean, median, and mode. The use of calculators is suggested. Then have students determine which measure of central tendency would be the best descriptor. Have students record the measure of central tendency they chose on an index card and display once all students are finished. This can help provide immediate feedback and identify which students demonstrate proficiency and which students need additional instructional support. [IS.16 - All Students] Have students share the reasoning for their choice and ask probing questions to ensure students are using mathematical reasoning and math language to defend their answers. [IS.17 - All Students]
- “In your opinion which measure of central tendency best represents this data set and why?”
- “Is there a different perspective where you would choose one of the other measures of central tendency to best represent the data?”
- “Why might the mean, median, and mode be considered measures of the ‘center’ of the data?” (Median is the center, literally, since it is the middle value that divides a data set in half. Mean can be considered the center of the data since it balances out the highs with the lows. Mode can be considered the center of the data since it is the value that occurs most frequently.)
Have students do a Quick Write following the activity to synthesize their learning about when a certain measure of central tendency may be a better descriptor than others for a data set. A Quick Write is a three- to five-minute activity when students can reflect on their learning. Students can complete their Quick Write in a math journal or on an index card. [IS.18 - All Students]
“In this lesson we investigated data sets and which measure of central tendency would best represent a data set. Sometimes different measures of central tendency are intentionally used to represent the data in a manner that is favorable to a certain perspective.” Have students complete an exit ticket (M-6-5-1_Lesson 1 Exit Ticket.doc). Give the exit ticket to students with about 5 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 let you know who may need additional practice and who has mastered the skill. [IS.19 - All Students]
Extension:
Use the following strategies and activities to meet the needs of your students during the lesson and throughout the year. [IS.20 - All Students]
- Routine: To review the concept of measures of central tendency give each student a Three Circle Venn Diagram (M-6-5-1_Venn Diagram and KEY.doc) and have students use various colored pencils (if available). Students can record as much information as possible in 30 seconds on their own Venn diagram and then switch to the person on their right. Students then can read what the student(s) before them recorded and add new information to the Venn diagram in the next 45 seconds. The use of different colors per student will help facilitate accountability and evaluations. Have students repeat the process three more times encouraging students to record real-world contexts where a certain measure of central tendency may be the best descriptor for that data as well. Then have students return the Venn diagram to the original owner after a total of four rotations. Students then can read the new information that was recorded and adjust any misinformation. Then have each student share one piece of information that was added to his/her Venn diagram that s/he did not think about originally.
- Small Group: For those students who are having a difficult time determining which measure of central tendency may best represent a set of data, have them complete a Measures of Central Tendency Sort (M-6-5-1_Central Tendency Sort and KEY.doc). These are taken from the chart listed early in the lesson. Have students add their own ideas in the blank spaces. Then have students switch with a partner and re-sort for fluency. Students also can complete a Three Circle Venn Diagram (M-6-5-1_Venn Diagram and KEY.doc) to help organize information in a different structure. Students can have an additional practice using the Measures of Central Tendency Review (M-6-5-1_Central Tendency Review and KEY.doc).
- Expansion: Activity 1. Students who demonstrate proficiency may calculate measures of central tendency for data in current events. Have students research data on the Internet or in newspapers. Once a valid set of data is found, students can work together to find the measures of central tendency or they can work individually and then compare answers. Have students draw conclusions about the data set based on their findings.
- Activity 2: Hand out the Misleading Data Displays worksheet (M-6-5-1_Misleading Data Displays.docx) and have students work together or individually on it. The goal is to practice identifying what causes a data display to be misleading. This may also be a fun activity to try using current events if time and resources are available.
