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Essays

Almost everyone is taught how to *do* many procedures (multiplication of big numbers, long division, etc.) without being taught *why* these procedures work.

Learning to truly understand memorized math procedures is an important part of Math 20 and Math 25. So is learning to communicate effectively about math topics, both in writing and when speaking.

Your task is to write an answer to **four** of the following essay questions. Imagine you are writing to an inquisitive third-grader who understands how to do that sort of math but not why it works.

You can turn in your work as a written assignment (written on paper, or typed and sent by e-mail). Or you can turn it in as an in-class presentation, if no other student has already presented that essay question. In either case your explanation will probably be much clearer if you include some pictures and/or example problems.

**1.** Explain why the sum of two even numbers is always even. Then explain why the sum of two odd numbers is always even.

**2.** Explain why 5 × 3 = 3 × 5, but 5 − 3 ≠ 3 − 5.

**3.** Explain why it does not make sense to divide by zero.

**4.** Explain why we make a "triangle of zeroes" when we multiply large numbers.

**5.** Explain why two consecutive whole numbers greater than three cannot both be prime.

**6.** Explain why the area of a triangle is ^{1}⁄_{2} × base × height, and the area of a parallelogram is base × height.

**7.** Give three lengths which you cannot make a triangle out of. Explain why you cannot.

**8.** Explain why we need common denominators when adding or subtracting fractions, but not when multiplying fractions

**9.** Explain why we divide fractions by "flipping the second fraction and multiplying".

**10.** Explain why the product of two numbers is equal to their GCF times their LCM.

**11.** We learned the definition of a **rate** in chapter five: *A rate is a kind of ratio in which the two numbers have labels.* Yet in real-life language we sometimes call percentages rates (a commission rate, interest rate, sales tax rate, etc.). Explain how these percentages are also rates that do indeed fit the definition. (What numbers are on the top and bottom of the ratio? What labels are on the top and bottom of the ratio?)

**12.** Explain the difference between these two problems: 15% of 70 equals what? and 70 is 15% of what?.

**13.** Once upon a time Daisy Duck invests money that decreases by 3% the first year and then increases by 3% the second year. After those two years, is her investment less, equal, or more than its original value? How do you know, even without knowing how much she originally invested?

**14.** In Big Question #4, when Janice needed to set aside 25% of her $50 gift certificate for tax and tip, it did not help her to think, "What is 25% of $50?" Why did that question lead to an incorrect answer?

**15.** When the interest rate doubles or triples, why does simple interest have a matching increase? Also, why does compound interest *not* have a matching increase when the interest rate doubles or triples?

**16.** Why do the payments listed on the Amortization Table decrease as the duration of the loan increases? Also, why is there less total interest on a shorter-duration loan than on a longer-duration loan of the same size, even though the monthly payments are larger?

**17.** In Big Question #10, why does Cindy end up with the most retirement savings, even though she saves for the fewest years? If the annual interest rate decreases, her lead also decreases—at what annual interest rate does her lead finally disappear?

**Harder #1** Prove that 11 is the only palindromic prime with an even number of digits. (You will want to find a math department faculty member willing to help you or your homework group write a carefully worded proof.)