### Tom Scavo and Byron Petraroja

Contents || Tom's Math Lessons

## Measurement

In anticipation of the primary measurement task described below, we talked about the length and width of our classroom, and how we might go about measuring it. We estimated these measures by sight and wrote down our estimates for future reference. We also discussed what we meant by the area of the room, which we then computed using the above length and width estimates.

Working in pairs, everybody measured the length and width of our classroom. First one student from each team measured while the other recorded, and then they switched roles, measuring the length and width again. The two sets of measurements, a kind of check on their work, were recorded on data sheets (see Figure 1). When all the teams had completed the measurement task, the data were written on the blackboard and compared. Discrepancies between the measurement pairs were noted. Those that could not be attributed to measurement or round-off error were taken again.

We arranged for the students to measure each of the fifth and sixth grade classrooms in our school (there were fourteen altogether). Of course, the other teachers were told of our plans to enter their classroom days in advance. So that the activity would go smoothly, parents and staff were recruited to supervise the classroom visitations. We tried to have one adult for every four students.

At the designated time, the students were again paired off, classrooms were assigned, and meter sticks were handed out. (Meter sticks were chosen as our measuring device because they were plentiful and familiar to the students.) Data sheets were distributed (Figure 1) on which each team recorded its measurement to the nearest centimeter. In fact, each team measured its assigned room twice, as had been done earlier. This served as a check on their work.

Before we could calculate the area of each classroom, the data had to be converted to common units, in this case meters expressed in decimal form. For example, one team found its room to be seven meters and sixty-two centimeters long, but how long was this room in meters only? [Answer: 7.62 meters] Likewise, how wide was the room in meters? These conversions were surprisingly difficult for some students, but we found that teaching the concept in the context of this unit was most effective.

After each team had determined the length and width of its assigned room in meters, teams were given calculators and asked to compute the area of the classroom. This task was easy for the majority of students. As they wrote their answers on the class data sheet, we made sure that they wrote down the units as well (e.g., 64.2 square meters ). In fact, all measurements were accompanied by the appropriate units (that is, a number by itself was not sufficient). See Table 1.

The next class period was spent examining the data. The class data sheet was hung in front of the room and discussion began. Which classroom has the most area? Which room has the least area? We also asked these questions with respect to grade level: for example, which fifth grade class has the largest room? Some of what we discovered we already knew, but there were some facts, such as the size of Room 112, that surprised us.

Note that Room 203 appears twice in Table 1. This particular room, a mixed class of both fifth and sixth graders, presented us with special challenges and opportunities. To facilitate the discussion, we constructed another data sheet (see Figure 2). Our task was to allocate (conceptually) the total area of the room based on the relative numbers of fifth and sixth grade students. This required a knowledge of percentages, and we found ourselves digging into the textbook for some help.

It didn't occur to us at the time, but we could have further investigated the concept of percentages by asking additional questions. For instance, of all fifth and sixth grade students, what percentage are fifth graders? What percentage are sixth graders? What percentage of all students in our school are fifth graders? The latter question would have required additional information we did not have.

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