Materials & Prep
Prepare a board display or handout with the data sets used below and a short independent problem set. Students need paper and pencils. Draw a number line from 0 to 40 for the opening example.
Opening
Display seven cards labeled 7, 8, 8, 9, 9, 10, and 40 in a row. Tell students these are the numbers of minutes seven students waited for a bus. Ask: “Without calculating anything, which number or small range seems most typical of the group? What makes 40 different?” Students first point to or write their answer, then turn and talk. Accept observations such as “most values are close to 9” and “40 is far away.”
ConnectExplain that students just used the shape of the data to decide what represents a typical value. Today they will compare two numerical ways to describe the center, the mean and the median, and decide which one gives the more useful picture.
Direct instruction
DefineThe mean is found by adding all values and dividing by the number of values. The median is the middle value after the data are ordered. Both describe center, but they respond differently to extreme values.
Worked exampleUse 7, 8, 8, 9, 9, 10, 40. Model the thinking aloud: “The values are already ordered. There are seven values, so the fourth value is the middle: the median is 9. For the mean, I must include every value: 7 + 8 + 8 + 9 + 9 + 10 + 40 = 91, and 91 ÷ 7 = 13. The mean is 13.”
ConfrontAsk, “If I report 13 minutes as typical, does that match most students’ waits?” Students should notice that six of the seven values are between 7 and 10. Explain that 40 pulls the mean upward, while the median remains at 9 because the extreme value is only one position in the ordered list. The common error is choosing the mean automatically because it is familiar. The correct choice depends on whether the data contain an extreme value or are fairly balanced.
DisplayRecord this decision guide: “Extreme or strongly uneven data: median usually describes a typical value better. Fairly balanced data without extreme values: mean can use all the values and is often a good choice.” Emphasize that students still calculate both when asked, then justify which better represents the data.
Guided practice
Work through these sets with students. For each, students calculate the mean and median, circle the better description of a typical value, and give one reason.
A. 12, 14, 15, 16, 18. Mean = 75 ÷ 5 = 15. Median = 15. The data are balanced with no extreme value, so the mean is a reasonable choice.
B. 5, 6, 6, 7, 8, 30. Mean = 62 ÷ 6 = 10⅓. Median = (6 + 7) ÷ 2 = 6.5. The median better describes the typical value because 30 is far from the other values.
Check for understandingHave students hold up M for mean or D for median for each set, then explain their choice to a partner. Ask specifically, “Why is 10⅓ not a good description of most values in Set B?” Listen for “the extreme value pulls the mean up” rather than only “the median is smaller.”
ScaffoldGive students a three-column organizer labeled “Ordered data,” “Mean and median,” and “Which better describes typical data? Why?” Provide the sentence frame: “The is better because the data are , and the value affects the .”
Independent work
Students solve this focused set independently, then compare answers with a partner. For each set, find the mean and median, choose the better measure, and justify the choice in one sentence.
- 18, 19, 20, 21, 22
- 2, 3, 3, 4, 25
- 14, 14, 15, 15, 16, 16
- 6, 7, 8, 8, 9, 10, 35
Expected results: 1. Mean 20, median 20, either is reasonable, with the mean using all values. 2. Mean 7⅖, median 3, median is better because 25 is extreme. 3. Mean 15, median 15, either is reasonable because the data are balanced. 4. Mean 83 ÷ 7 = 11 6/7, median 8, median is better because 35 is extreme.
CirculateAsk students to point to the value they believe affects the choice and to explain why. If a student always chooses the mean, cover the calculations and ask, “Which measure is closer to where most of the data sit?”
ExtensionStudents change Set 2 by replacing 25 with a value that would make the mean and median equal, then explain how the shape of the data changed.
Closing
Exit ticketGive students the data set 10, 11, 11, 12, 13, 50. They must calculate the mean and median, decide which better describes a typical value, and justify the choice in two sentences. Expected result: mean = 107 ÷ 6 = 17⅚, median = (11 + 12) ÷ 2 = 11.5, so the median is better because 50 is an extreme value that pulls the mean upward. Collect the response as the closing check for the objective.