Overview of the Lesson / Preparing for the Lesson / Students' Ways of Thinking / Assessment

1. Measuring

Each student makes a guess about the height of the flagpole and whether or not he or she will get the same measurement each time. [Teacher note: Be sure that students explain the reason for their anticipations.]

Each student measures the height of the flagpole from 2 different positions and records the distance from the base of the flagpole. Then each student estimates the height of the pole by adding his or her height to the distances recorded. Each student's estimates are collected and the estimates for all students are listed.

If for some reason, students cannot engage in this measurement activity, student data sets are included as Tinkerplots Ô files and as text files.

Teacher Note: If students use the inclinometer and paces without much in the way of preparation, their measurements are apt not to be normally (bell shaped) distributed. The resulting distribution reflects a blend of different methods of measure, individual differences in interpreting the task, and other chaos. This is fine, because it leaves room later for wondering why the shape of the data changes when the measurements are conducted more uniformly and/or with greater accuracy.

Thought-Revealing Questions: Pre Measurement 1. How could we use this tool to measure the height of the flagpole? 2. What should we keep in mind when we measure? Why is it important to keep this in mind? 3. What do you think will happen when we measure? What do you think we will get? Why? 4. Compared to the real length of the height of flagpole, where will our measurements be? (Less than the real length, more than the real length, or the real length) How much less/more do you think our measurements will be than the real length? ( A lot less/more, a little less/ more)

 

Thought-Revealing Questions: Post Measurement Activity 1. What were you trying to learn by measuring? 2. Why did we get different measurements? 3. Do you think your measurement is close to the real length of her arm span? Why? 4. What could we do to make our measurements closer to the real length? Why are our values are not exactly the real length?

2. Designing a Display

Students work in pairs to design a display on chart paper of the measurements.

Directions for Students Can we say that everyone got the same measurement? Why or why not? If you look at the whole collection of measurements, what do they tell you? Make a display on the chart paper - a picture or a chart or a graph-that shows other people all of our measurements at a glance. If there are any trends or relationships, anything important or something else that you notice, then the display should help other people see this quickly. You can use the cards to plan out your display. You might want to move the measurements around until you find just the way you want to arrange them. Then make your chart by writing the measurements on the graph paper, so that you can pick up the display and bring it to the front of the class. Write large enough so everyone can see.

Teacher note . Students tend to design displays that are not conventional, but, as described below, discussions about the variations in design help develop an appreciation of different senses of the “shape” of the data. Although it is tempting to use computer tools to create the display, paper-and-pencil tools lead to more invention. Paper and pencil often make the important issue of interval more visible. For example, some students may create “bins” for values, thus creating intervals that affect the “shape” of the data. However, they may also juxtapose them without regard to the entire range of the interval. That is, students arrange values in order, such as 10's, 20's, and then juxtapose 40's, if there are no values in the 30's bin. The resulting display highlights clumps of values but makes “holes” in the data invisible.

Thought-Revealing Questions: As Students Design their Displays 1. What do you think is interesting in the data set? (A trend, a shape, strange values etc.) 2. What is the first thing you want someone else to notice in your display? How have you shown that?

3. Comparing Displays

Students hold up their displays and other students either write, or report verbally, one aspect of the measurements that the display helps make more visible, as well as one aspect of the measurements not evident from just looking at the display. Alternatively, students give their own display to another student, and then each student (or student pair) describes the qualities of the display that they notice. The emphasis in these discussions is on what each display allows us to “see” and what each display “hides.” The aim is to help students recognize that different senses of the data are tied to how the data are displayed, and that representational choices entail trade-offs.. [*If no display uses an interval, then the teacher will need to introduce the notion of developing “bins” of similar values. Students should compare at least one interval-based display with those they invent.]

Thought-Revealing Questions: As Students Compare Displays 1. What does the display do a good job of helping other people notice? What does it make less noticeable? 2. How did the designer show/hide that <feature>? 3. What can you see with this display that you can't see as easily with this other display? 4. Of all the displays, which do you think helps us see what the real height of the <attribute, such as height of flagpole> might be? Why?