Key Stone Problems…Key Stone Problems…

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Set 9© 2007 Herbert I. Gross

You will soon be assigned problems to test whether you have internalized the material

in Lesson 9 of our algebra course. The Keystone Illustrations below are

prototypes of the problems you'll be doing. Work out the problems on your own.

Afterwards, study the detailed solutions we've provided. In particular, notice that several different ways are presented that could be used to solve each problem.

Instructions for the Keystone Problems

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© 2007 Herbert I. Gross

As a teacher/trainer, it is important for you to understand and be able to respond

in different ways to the different ways individual students learn. The more ways

you are ready to explain a problem, the better the chances are that the students

will come to understand.

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© 2007 Herbert I. Gross

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#1a. Some geometry textbooks define two regions to be equal if they have the same area. In this context, is it true or is it false

that the relation “is equal to” is an equivalence relation?

Keystone Illustrations for Lesson 9

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Answer: True© 2007 Herbert I. Gross

Answer: TrueSolution for #1a:

For “is equal to” to be an equivalence relation, three things have to be true…

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© 2007 Herbert I. Gross

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(1) It must be reflexive.

That is, every geometric region must have the same area as itself.

This is obviously true.

Solution for #1a:

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© 2007 Herbert I. Gross

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(2) It must be symmetric.

That means that if the first region has the same area as the second region, it is also

true that the second region hasthe same area as the first region.

This is also a truism. For example: if the first region has an area of 36 square inches, then the second region also has

an area of 36 square inches.

Solution for #1a:

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© 2007 Herbert I. Gross

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(3) It must be transitive.

That means if the first region has the same area as the second region, and the second region has the same area as the third region, then the first region must

have the same area as the third region.

Solution for #1a:

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© 2007 Herbert I. Gross

For example, suppose the first region has an area of 36 in2. The fact that it is equal to the second region means that the area of the second region is also 36 in2. The

fact that the second region has the same area as the third region means that the area of the third region is also 36 in2.

Therefore, the first region is equal to the third region.

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#1b. Some geometry textbooks define two regions to be equal if they have the same

area.

Is it also true if two regions are equal (as defined above), they also have

the same perimeter? Explain.

Keystone Illustrations for Lesson 9

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Answer: False© 2007 Herbert I. Gross

• There are different ways to define what we mean by equal. For example, some geometry books define two regions to be congruent if they have identical size and shape.

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© 2007 Herbert I. Gross

Notenext

• Even then however, the two regions can be different with respect to their position

in space.

We can show that if two regions (for example, two rectangles) are congruent, they have to have the same perimeter.

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© 2007 Herbert I. Gross

Discussionnext

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Answer: FalseSolution for #1b:

Getting back to the question at hand, all we have to do to show that the statement is false is to find two regions that have the same area but not the same perimeter.

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© 2007 Herbert I. Gross

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So, for example, consider the rectangle R1 whose dimensions are 4 in. by 9 in.

and the rectangle R2 whose dimensions are 3 in. by 12 in.

Solution for #1b:

The area of R1 is 4 in. × 9 in., or 36 square inches and the area of R2 is 3 in. × 12 in.

which is also 36 square inches.

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© 2007 Herbert I. Gross

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On the other hand, the perimeter of R1 is

2 × (4 in. + 9 in.), or 26 inches

while the perimeter of R2 is

2 × (3 in. + 12 in.) or 30 inches.

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Solution for #1b:

R1 has an area of 36 square in.

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© 2007 Herbert I. Gross

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A = 4 × 9 = 36 square in.

= 26 inches

while the perimeter of R1 is…

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P = 2 × (4 + 9)

Solution for #1b:

The area of R2 is 36 square inches

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© 2007 Herbert I. Gross

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A = 3 × 12 = 36 square inches

= 30 inches

while the perimeter is…

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P = 2 × (3 + 12)

As an aside, a square is a special case of a rectangle. A square whose dimensions are 6 in. by 6 in. also has, like R1 and R2, an area of 36 in2. However since each of the square’s

sides is 6 in., its perimeter is 4 × 6 in. or 24 in. (unlike R1’s perimeter which is 26 inches or

R2’s perimeter, which is 30 in.). The interesting point is that, of all the rectangles that have the same area, the square has the least (smallest)

perimeter.

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© 2007 Herbert I. Gross

Digression

However, to get back to the main point, we want to emphasize that when we talk about equal or equivalent, we always mean with

respect to a specific relation.

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© 2007 Herbert I. Gross

Equivalence

For example, when we write 3 + 2 = 9 – 4, we do not mean that there is no difference

between the symbols 3 + 2 on the left and 9 – 4 on the right side; but rather that

they name the same number ― five.

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The main point, then, that we must always keep in mind is that in

mathematics when we use the term equal, we mean equal with respect to a

certain property.

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© 2007 Herbert I. Gross

Final Note