Cylinder and Scale

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What It Is (Continued)

However, since students are measuring for purposes of comparison and to find growth patterns, it may be more useful to measure in another way. In the second method, students measure cylinder height in terms of h, the height of the smallest cylinder (which is also the short length of the unit rectangle); measure circumference in terms of c, the circumference of the smallest cylinder; and measure diameter of the base in terms of d, the diameter of the smallest cylinder. Given the way that the larger cylinders were constructed, measuring height and circumference in these units makes scaling patterns very clear. Diameters are determined by marking a unit diameter on the edge of a piece of paper held against the mouth of the film canister and placing marks adjacent to both of the inner sides. Measurement of the diameter of the base for larger cylinders then becomes equivalent to asking, "How many times does the unit diameter length fit into a line across the middle of a cylinder base?" Measuring the area of the cylinder side using this method is equivalent to asking the question, "How many small rectangles does it take to cover the larger rectangle which will form the side of the cylinder?" Results of all of these measurements should be recorded on the chart. Students answer this question using the manipulatives. We call these small rectangles can surfaces, since they represent the inner surface area of the film canister. These can surfaces are our unit of area.

Measuring for Area and Volume Dimensions

In the second session, the students will measure the area of the base and the volume and analyze and discuss results.

Since the base of the cylinder is a circle, areas are not simple. The best way to measure the area of the base is to outline the base on a piece of centimeter graph paper and count the squares enclosed.

Students measure volume using film canisters as their measuring cups. Volume is measured in terms of the number of film canisters full of sand (cans) that it takes to fill the cylinder. Start by placing the paper cylinder on a plastic picnic plate or something similar (so that the sand does not spill everywhere). One partner stands the cylinder on the plate, so that the plate forms a bottom, holding it in as round a shape as possible, and the other begins to pour in film canisters full of sand. As the sand goes in, its weight actually helps to make the cylinder round. If you are careful, very little sand will leak out of the bottom. Fill the cylinder to the top and record the number of cans that it holds.

Sometimes the question "If it isn't round does it change the volume?" will come up. Students can answer this by trying it. More noticeable changes in volume come from bigger changes in shape. In the extreme, the sides of the cylinder can be pushed together, reducing the volume to close to nothing.