Let's see what other authors have to say about this very important concept. We begin, as usual, with David Joyce:
Chapter 5 is about areas, including the Pythagorean theorem. It begins with postulates about area: the area of a square is the square of the length of its side, congruent figures have equal area, and the area of a region is the sum of the areas of its nonoverlapping parts. A theorem follows: the area of a rectangle is the product of its base and height. There is no proof given, not even a "work together" piecing together squares to make the rectangle. An actual proof is difficult. It would depend either on limiting processes (which are inappropriate at this level), or the construction of a square equal to a rectangle (which could be done much later in the text). It would be just as well to make this theorem a postulate and drop the first postulate about a square.
So far, Chapter 5 of the Prentice-Hall text sounds very similar to Chapter 8 of the U of Chicago text, as the Pythagorean Theorem is included in the area chapters of both texts. But I've already discussed my treatment of the Pythagorean Theorem.
According to Joyce, the Prentice-Hall text assumes as a postulate the area of a square, and claims without proof the area of a rectangle. Joyce states that there are two ways to prove the theorem. The first is by "limiting processes." You may wonder, why are "limiting processes needed" -- why can't we just divide the rectangle into squares? After all, isn't that what "square units" are -- a rectangle with area 32 square units (the first rectangle shown in the U of Chicago text) can be divided into 32 unit squares.
The problem is, what if the sides of the rectangle have irrational lengths? Dividing the rectangle into unit squares only works for the rational numbers, not the real numbers. If we had a rectangle with dimensions 1 and sqrt(2), we wouldn't be able to divide it into squares so easily. The well-known paper size A4 is supposed to be a rectangle whose ratio of length to width is about sqrt(2). Another famous irrational rectangle is called a golden rectangle. This rectangle defies division into squares -- cut off the largest possible square, and the remaining rectangle is similar to the original, so that cutting off another square leaves a third rectangle similar to the original, and so on. The ratio of its length to width is (1 + sqrt(5)) / 2 -- a well-known irrational number known by the Greek letter phi.
So limiting processes are needed to show that this works for irrational side lengths. We'll revisit the idea of limiting processes later on when we discuss another famous irrational number -- pi.
Joyce also mentions that one can construct a square that has the same area as a rectangle. We can imagine how to accomplish this construction. Think about it -- if the length and width of a rectangle are x and y, then the square must have area sqrt(xy). Notice that this is the geometric mean of x and y, so it implies that the geometric means of Section 14-1 are relevant. Once we have the square, we must then prove that it and the rectangle really have the same area. Most likely, what will happen is that both the square and the rectangle are divisible into triangles -- and we don't need to know how to find the triangles' area, as the triangles making up both figures will be congruent. So the proof would be similar to the area proof of the Pythagorean Theorem.
We move on to Dr. Franklin Mason. As it turns out, Dr. M provides a clever way to derive the rectangle formula from that of the square -- and no limiting processes are needed! He notes that a square with side length x + y must have area (x + y)^2, which is x^2 + 2xy + y^2 according to the polynomial manipulation of Algebra I. This square can be divided into four rectangles -- two of them squares of sides x and y, the other two rectangles x by y. Subtracting x^2 and y^2 leaves 2xy as the combined area of the two rectangles, so that xy must be the area of the each of these rectangles with dimensions x and y. QED
Dr. Hung-Hsi Wu, meanwhile, discusses the area of rectangles with rational sides in the context of multiplying fractions. He points out that, for example, the area of a rectangle with dimensions 1/2 by 1/3 must be 1/6, because clearly six congruent copies of them fill the unit square. Actually, Dr. M does the same with squares with fractional side lengths before giving his clever derivation of the rectangle formula. Wu then points out that by a form of hand-waving used to avoid limits, The Fundamental Assumption of School Mathematics -- any formula that works for rational numbers also works for irrational numbers, so we have the rectangle formula.
After all of this, what does the U of Chicago do? It provides us with an Area Postulate:
Area Postulate:
a. Uniqueness Property: Given a unit region, every polygonal region has a unique area.
b. Rectangle Formula: The area of a rectangle with dimensions l and w is lw.
c. Congruence Property: Congruent figures have the same area.
d. Additive Property: The area of the union of two nonoverlapping regions is the sum of the areas of the regions.
Therefore the U of Chicago, in part b, does exactly what Joyce suggests:
It would be just as well to make this theorem [the rectangle formula -- dw] a postulate and drop the first postulate about a square.
All sources provide the Congruence and Additive Properties, albeit with different names. For example, Dr M. calls the Additive Property the Area Addition Postulate -- to parallel other postulates such as "Segment Addition Postulate" and "Angle Addition Postulate."
Notice that the U of Chicago text states, "To find the area of a region, cover it with congruent copies of a fundamental region [emphasis mine]." The phrase "fundamental region" appeared in yesterday's lesson -- the fundamental region is the region that is repeated in a tessellation. So we can see another reason for including yesterday's lesson on tessellations -- one can find the area of a region by tessellating it with squares.
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