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As this course comes to a close and you reflect on ways to bring your new understanding of data analysis, statistics, and probability into your teaching, you have both a challenge and an opportunity: to enrich the mathematical conversations you have with your students around data. As you are well aware, some students will readily grasp the statistical ideas being studied, and others will struggle.

The problems in Part D describe scenarios from a classroom case study involving children’s developing statistical ideas. Some student comments are given for each scenario. For each student, comment on the following:

**• **Understanding: What does the statement reveal about the student’s understanding or misunderstanding of statistical ideas? Which statistical ideas are embedded in the student’s observations?

**• **Next Instructional Moves: If you were the teacher, how would you respond to each student? What questions might you ask so that students would ground their comments in the context? What further tasks and situations might you present for each child to investigate? See Note 6 below.

**Problem D1
**Mr. Shapple teaches two sections of seventh-grade math. Students in both groups were trying to determine the typical height of a seventh-grade girl and a seventh-grade boy. They measured their heights in inches, combined their data, and then displayed the data on the line plots below:

After comparing the two data sets, here is what the students had to say:

**a. **Gregory: “The boys are taller than the girls.”

**b. **Marie: “Some of the boys are taller than the girls, but not all of them.”

**c. **Arketa: “I think we should make box plots so it would be easier to compare the number of boys and girls.”

**d. **Michael: “The median for the girls is 63 and for the boys it’s 65, so the boys are taller than the girls, but only by two inches.”

**e. **Paul [reacting to Michael’s statement]: “I figured out that the boys are two inches taller than the girls, too, but I figured out that the median is 62 for the girls and 64 for the boys.”

**f. **Kassie: “The mode for the girls is 62, but for the boys, there are three modes — 61, 62, and 65 — so they are taller and shorter, but some are the same.”

**g. **DeJuan: “But if you look at the means, the girls are only 62.76 and the boys are 64.5, so the boys are taller.”

**h. **Carl: “Most of the girls are bunched together from 62 to 65 inches, but the boys are really spread out, all the way from 61 to 68.”

**Problem D2**

The class then made box plots, building on Arketa’s suggestion. This new representation gave them another way to compare and discuss the variance between the two data sets as they analyzed the heights of seventh-grade boys and girls. Here are the box plots they made:

As they compared the box plots, students made the following comments:

**a. **Arketa: “There is a lot of overlap in heights between the boys and girls.”

**b. **Michael: “We can see that the median for the boys is higher than for the girls.”

**c. **Monique: “It looks like just 12.5% of the boys are taller than all of the girls, and maybe about 10% of the girls are shorter than the shortest boy.”

**d. **Gregory: “The boys are taller than the girls, because 50% of the boys are taller than 75% of the girls.”

**e. **Morgan: “You can see that the middle 50% of the girls are more bunched together than the middle 50% of the boys, so the girls are more similar in height.”

**f.** Janet: “Why isn’t the line in the box for the boys in the middle like it is for the girls? Isn’t that supposed to be for the median, and the median is supposed to be in the

middle?”

**Problem D3
**To encourage students to discuss ideas of sampling and population, Mr. Shapple asked them to think about what they could say about other classes of seventh graders. Here are some of their responses:

b.

**Note 6**

If you’re working in a group, make a two-column chart with the labels “Understanding” and “Next Instructional Moves” for recording the group’s responses. You might also want to review the process of making a box plot.

Session 4, Part D: The Box Plot

**Problem D1
a. **Gregory does not quantify his statement and may only be looking at the upper extreme value. The teacher could ask Greg to determine “how much taller” the boys are than the girls.

b.

c.

d.

e.

f.

g.

h.

**Problem D2
a. **Arketa is comparing the variation by looking at the range of each data set. The teacher might ask her to quantify her response.

b.

c.

d.

e.

f.

**Problem D3**

a. |
Kassie thinks that her class’s sample is representative of the district if the district were defined as the population. The teacher could ask the other students to give reasons for supporting or refuting Kassie’s conjecture. |

b. |
Nichole uses personal judgment about her class’s data probably being “more in the middle,” but she is correct in thinking that a larger sample would increase the range of the data, as a larger sample might reveal that her classmates are more homogenous in comparison to other seventh-grade classes, if the population were defined as the country. The teacher might ask Nichole why she thinks her class is “more in the middle,” and then ask the rest of the class to react to her conjecture. |

c. |
Charles is thinking about measures of center. The teacher might ask Charles to explain further what he means by “on average.” Is he referring to the mode, the median, or the mean? |

d. |
Carl thinks that their sample is representative of the country if the country were defined as the population, but not if the population included seventh-graders from other countries. The teacher might use this as an opportunity to discuss further reasons for defining the population being investigated. |