Chapter 9 – Proof

Exercise 9.1

1. 36 in = 1 yd; 9 in = 0.25 yd (divide both sides by 4); $$\sqrt{9}$$  in = $$\sqrt{0.25}$$  yd; 3 in = 0.5 yd (positive square root of both sides). Is it true that 3 inches equals half a yard? What is wrong with this “proof”?

Exercise 9.2

1. Use a dynamic geometry program to create a setting like the medians of a triangle discussed in the chapter. Move the figure to confirm, visually, that an invariant has been created.
2. Graphing circles centered at (2, -5) and (5, -2), each with a radius of 2, and the line y = -x yields a figure in which the line appears tangent to the circles. Zooming shows that is not the case. Create a similar environment using symbolic manipulating, function-plotting software.

Exercise 9.3

1. Create a lesson for an algebra class that includes a historical proof.
2. Create a lesson for a geometry class that includes a historical proof.
3. Create a lesson for a pre-calculus class that includes a historical proof.
4. Create a lesson for a calculus class that includes a historical proof.
5. Create a lesson for a pre-algebra class that includes a historical proof.

Exercise 9.4

1. Show that sin²x + cos²x = 1 graphically. Describe how this development could be used in the secondary curriculum.
2. Use software/application or a graphing calculator to show two different identities graphically, and describe the results for each.

Exercise 9.5

1. Prove that 2 + 4 + 6 + . . . + 2n = n(n + 1)
2. Prove that

$$\frac{1}{1\times 3}+\frac{1}{3\times 5}+\frac{1}{5\times 7}+\dots \frac{1}{(2n-1)\times (2n+1)}=\frac{1}{2n+1}$$

Exercise 9.6

1. Do a proof that shows how the divisibility rule for 9 would work with a four-digit number WXYZ.
2. Should students be required to prove a divisibility rule? Why or why not?
3. Why does the 6 rule break into an even 3 rule? Explain why a similar rule could or could not be devised for divisibility by 15.
4. Describe a divisibility rule for some number other than those discussed here in the text and prove why it works.
5. Are divisibility rules limited to integers?

Exercise 9.7

Note that in today’s technological world, you can find each of these solved somewhere on the Internet. The challenge is to avoid doing that, and YOU do the proof. THAT is how you grow your own abilities.

1. Prove that

$$1^2+2^2+3^2+\dots +n^2=\frac{n(n+1)(2n+1)}{6}$$

2. Prove that r^mrⁿ = r^{m + n} for r, a real number, and m and n as counting numbers.
3. Prove that if n is a natural number, a is a real number and a > – 1, then
   (1 + a)ⁿ  >  1 +na.
4. Find and prove a unique statement using PMI.

1. Suppose that the surface of the Earth is smooth and spherical and that the distance around the equator is 25,000 miles. A steel band is made to fit tightly around the Earth at the equator, then the band is cut and a piece of band 18 feet long is inserted. Assuming the equator is a circle and the band is a concentric circle, to the nearest foot, what will be the gap, all the way around, between the band and the Earth’s surface? (Use 3.14 as an approximate value of Pi.)  

Hint: Try a smaller problem with a smaller Earth.

Answer

Answer/solution: 3 feet

Circumference = Pi x diameter

25,000 miles = 132,000,000 ft

132,000,000 = 3.14 x D

D = 42,038,216.56 ft.

If 18 feet is added to the circumference, then

132,000,018 = 3.14 x D

D = 42,038,222.29 ft

The new diameter – old diameter = 42038222.29 – 42038216.56 = 5.73

5.73 is the added length for the new diameter. Half of this would be the gap on each side of the Earth between the Earth and the new band, which is 2.865 and that rounds to 3 feet.

2. Two = One                                               

Observe the following algebraic proof.

Given:

     A and B are real numbers;

     A = B

A = B

                                  A² = AB

                            A² – B² = AB – B²

                 (A + B)(A – B) = B(A – B)

         Dividing both sides by (A – B) yields

                           (A + B) = B

          Since A = B, substituting B for A yields

                             B + B = B

                                 2B = B

                Dividing both sides by B yields

                                   2 = 1

Since we know that 2 does not equal 1, clearly state the mistake made in the above algebraic proof.

Hint:  What happens when you divide by zero?

Answer

Answer/solution: The error occurs when division is done with (A – B).  Since A = B, A – B = 0 and the step involves division by zero which is undefined in the set of real numbers.

There is no Additional Learning Activities for this chapter.