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Task
Some routine problems dealing with area
and perimeter evoke unexpected responses. What to measure,
how to measure it, and how to anticipate variations in area
and perimeter are often difficult. "If we measure the
area of rectangles by dividing them into squares, then what
shape should we use to measure area for triangles?" Not
such a bad question: Why don't we use "triangular units"
to measure triangular areas?
Part 1
A 3 x 5 index card has an area of 15
square units. You can see that the shape of this card is rectangular.
If you want to create a square card with exactly the same
area, 15 square units, the dimensions of the side will change.
What this means is that the card has the same area as a square
that also has 15 square units. It should be possible to cut
up the 3 x 5 card into (several) pieces in such a way that
these pieces can be put together to form a square card.
There are two problems
now:
- How do you construct a square whose sides are the square
root of 15, given only a 3 x 5 card?
- The card allows you to measure 3 units and 5 units easily.
Put two cards together and 2 units can be measured off
quite easily. But how do you measure the square root of
15 units?
- More generally, how do you measure the square root of
(x) units, where x can be anything?
- Still more generally, what numbers can be 'constructed'?
For instance, can you draw a line that is pi units long?
Or a cube root? Now that you have made the square, how do
you cut up the 3 x 5 card so that the pieces can be re-assembled
to make the square? What are the minimum number of cuts
that you need to make?
Part 2
- Now what if the card isn't 3 x 5, but some other dimensions.
Like 7 x 9. Then how do you cut up the card so that the
pieces form a square?
- What's the general recipe for cutting up a rectangle to
create a square?
Part 3
- What is the recipe for cutting up a triangle to make
a square?
- Create two squares S1 and S2. How do you cut up S1 and
S2 to make a square whose area is the sum of the original
areas? What does this have to do with the Theorem of Pythagoras?
Part 4
- Now the big problem...take a convex polygon, how do you
cut up the polygon so that when you put the pieces back together
you get a square?
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