Two sessions of 45-60 minutes each

Set out materials.
Find a place to store the materials between sessions.

Pairs

(per pair)
Butcher paper (a few feet)
One film canister
One large, flat sheet of graph paper (centimeter paper works well)
Scissors
Scotch tape
One pound of sand (for use in second session)
One plastic picnic plate, aluminum pie pan or the like
Optional: tagboard or old file folder
Optional:compass
One blank cylinders chart (available here)

Cleanup is simplified if students work on paper when using the sand. The spilled sand can then easily be emptied into the wastebasket.

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Students need to see the patterns of scaled growth in different shapes before they can understand the universality of scaling patterns. In this activity, they can investigate the relative growth of lengths, areas, and volumes as cylinders are scaled up. In this case, we scale up by changing the height of the cylinder and the diameter (or circumference) of the base. In the activity, students will measure the three linear dimensions of the cylinder. This is so that they will see that all linear dimensions increase at the same rate as you scale up proportionately. Similarly, they will be asked to measure two areas of the cylinder.

Students connect their experience with cylinders to the world outside the classroom by giving examples of where they encounter cylinders in their everyday life. These examples can include cans, poles, bottles, and pipes. Having students make a list of cylinders makes a nice homework assignment.

This activity requires a good deal of exacting work. It is best to do it in two sessions. In the first session, students will measure the cylinder's linear dimensions; in the second, they will measure for area and volume dimensions.

After illustrating how the cylinders are made, have students predict what kind of pattern of growth they might expect as they scale up. How much like or unlike the patterns of cube growth do you expect the cylinder's pattern to be? Then, ask the students to make their cylinders in two steps. To build the cylinders, the students will draw and cut out rectangles of paper which will form the cylinder sides. To create a uniform basis for measurement in scaling, the students should use film canisters as the basis for what will be called the unit rectangle. The length and width of this unit rectangle will be doubled, tripled, quadrupled, etc., in the scaling-up process.