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Comments on Pre-test (January 21, 2004)
Comments on Post-test (February 2, 2004)
The thoughts expressed on the pre-test fall into two general categories:
- discussion of the meaning of terms, especially the meaning of "area" and
- discussions of the relationship between traingle area and the area of rectangles.
1. About half of the papers made some explicit comments about the
meaning of the word "area" or about what "A" refers to. Some examples (these
are all quotes):
- "area" explains how much paper we are using [when we make a triangle out
of paper]
- area is the space enclosed by a shape
- [area] is a measurement of the inside of a 2-dimensional object
- Suppose we are talking about the floor of your room. The area, or space,
inside the room is a rectangle...
- [area is] the amount of space on the inside of the triangle
- [area is] how much space an object takes up
Five papers talked explicitly about the relationship between the concept of
area and the idea of conting the grid squares (or parts of grid squares) inside
a figure. Four of
these five included diagrams of rectangles or other figures covered by grids.
Only two mentioned the need to choose
a unit of
measurement. The others didn't say anything about how one might know the area
of a single grid square.
2. Almost everyone noted that a rectangle can be sliced along a diagonal to
yeild two congruent right triangles, each having half the area of the square.
There was a lot of variation in the presentnation, but this basic
image was pretty consistent: Several papers were even more limited, in that
they did not talk about slicing rectangles, but limited the imagery to slicing
squares along
a diagonal. Only one paper considered the possibility that
the triange might not have a
right angle.
Some things that few mentioned:
- Given any triangle, if you make a copy of that triangle with
the same orientation (i.e., don't flip it over) and then glue the two copies
together along a common edge you get a parallelogram. On the other hand,
if you slice a parallelogram along a diagonal, you get two traingles. So
the area
formula for a traingle is closely related to the area formula for a parallelogram.
- Sometimes you can see that the area of a parallelogram is equal to the
area of a rectangle with the same base and height by lopping off a triangle
at one
end and moving it to the other end (example). But when the base is very
narrow and the parallegram very "slanted", this doesn't work.
- Euclid Proposition I.35.
Pre-test page