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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.
