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

1. Measurement Preambles

Each student predicts the value of the measurement he or she believes will result from measuring the length of three different persons' arm-span (a student or teacher), head circumference (a student or teacher) and (optional) the area of a person's hand. Each student measures each body part ,and each measurement is recorded on an index card or sticky note.

Teacher Note: If students do not agree about method of measure or units of measure, the resulting distribution of measurements 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. First tools for measurement might include:

For head circumference: A 15-cm. ruler.

For wing span: A 15-cm. ruler

For area of hand: Square grid paper, where what is at hand is large vs. small squares. Such grids are included at the end of this lesson. Let students use the large grids for the first measure, because this will make keeping track of the parts more difficult. Each square in the large grid is 3 cm. x 3 cm., and each square in the small-square grid is 1 cm. x 1 cm.

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. (See Lesson 1B for further discussion of these issues.)

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 invented.]