Wednesday, October 29, 2014

Conversations on the Rifle Range 13: Percents, Simple Interest, and Ninja Warriors

Barry Garelick, who wrote various letters under the name Huck Finn, published here, is at work writing what will become "Conversations on the Rifle Range". This will be a documentation of his experiences teaching math as a long-term substitute. OILF proudly presents episode number 13:

During my second week of teaching, in my second class of the day, I was having the class do review problems prior to a chapter test the next day, when Robert, a mildly autistic and very bright boy, started crying. The review problems contained many computations and it was easy to make calculation errors. (I didn’t allow calculators) He was continuing to get wrong answers. I heard him muttering “I can’t do this; I’m getting them all wrong. I’m going to fail, I’m going to fail!”

I came over to him. “What’s wrong, Robert?”

“I keep getting the wrong answers. I’m going to fail this test.” He started hitting his forehead.

“You need to take your time,” I said. I searched his paper and found a problem that he had done correctly. “Look, you got that one right. So just keep doing them. Take your time.” He calmed down and continued with the review. Some minutes later he had completed the problems and was busy at work on a drawing of a Ninja warrior that he frequently drew.

When I inherited Mrs. Halloran’s classes (the teacher for whom I was subbing), she gave me a calendar of assignments for last year’s second semester. She said I should transcribe the previous year’s lessons into a blank planner to use this semester, and told me to take care to not give tests or quizzes on a Monday. “Students tend to forget things over the weekend,” she told me. “And I always include a review of the material the day before a test or quiz.”

My first three classes were all pre-algebra. This allowed me to try an approach in my first class, adjust it in the second one, and supposedly perfect it in the third class. In theory this made sense; in practice not always.

The pre-algebra classes were halfway through the chapter on percentages and how to apply them when I came on board. The teacher had taught them to solve percentage problems using the method of proportion. Thus, a problem like “35 is 40% of what number?” was to be solved by restating it as a proportion using the “part/whole” technique. Since 35 is part of a whole, 35 is the numerator. The “whole” is the denominator—“x” in this case. The left hand side is then 35/x. The right hand side is the percentage 40/100. You’re left with 35/x = 40/100. Students solve for x via cross multiplying and dividing.

During my short orientation with Mrs. Halloran, she mentioned this method. I told her that when I was student teaching, teachers had the same book (Holt, “Prealgebra”—as bad as if not worse than Holt’s “Algebra 1”) but had taught students to translate such problems into algebraic equations. The above problem would take the form of 35 = 0.4x, where the “is” translates to the equal sign, 40% to 0.4, and “of what” to “0.4 multiplied by x” or 0.4x. Mrs. Halloran very firmly told me: “We don’t do that; they’ve learned the proportion method. Use that method only; they’ll get confused.”

There are advantages to both methods, of course. But since they’ve already had the “part/whole” technique in sixth grade, I would have preferred having them use the equation form. But since Mrs. Halloran was quite firm about sticking with the proportional method, I honored her request.

For other types of problems for which I was not given an edict, I elaborated. When calculating discounts, students multiply the price by the discount rate, and subtract the result from the price. For example, to find the price of a $200 bicycle discounted 20%, they multiply $200 by 0.2 to get $40, and then subtract to get $160.

I gave the class an alternative—a shortcut—to use if they felt like it. I explained that a discount is like the store offering some money towards your purchase. “Suppose your parents tell you that they’ll kick in 20% towards the purchase of a new bike. What percentage will you pay?”

A few hands went up. Someone volunteered: “80 percent”. Then a little back and forth on how they came up with that—something you’d think would be easy, and in fact perhaps it’s SO easy that they have trouble articulating it, but eventually someone would say “100 minus 20 equals 80” and it’s on from there. Interestingly, only a few students used the short-cut. The rest preferred the long way. Other detours from the main course proved equally interesting. When talking about interest (simple interest—Mrs. Halloran had indicated to skip compound interest, which, in retrospect I wish I had taught), the problems were stated in terms of savings. I brought up the fact that interest can also be something that you pay.

“When you borrow money from a bank, to buy a car, or a house, the bank charges you interest.”

We worked through a problem. Then a girl asked: “Why would anyone want to do that? You end up paying more for what you’re buying than if you just bought it without borrowing.”

Ordinarily such questions are used as an occasion to warn of the dangers of credit, getting in over your head, and so forth, which I had done for my first pre-algebra class. But I decided to try a different approach with the second group to see how it worked out.

“Some items like cars and houses cost more money than people may have,” I said. “Like a house, for example. So a loan is one way you can afford to pay for it.” “But you’re paying more for it,” the girl insisted. “Why would you do that?” A rather astute boy named Brian who tended to offer expert opinions at the drop of a hat said “The going price for a house in this area is about $700,000, and not too many people have that kind of money.” I figured someone in his family was in real estate.

“Can’t you just save up?” someone else asked.

“How long would it take you to save up that kind of money?” I asked. “It depends on how much you make,” Robert said without looking up from a drawing of a Ninja warrior that he was working on. “If you invent something that everyone wants, you can make millions of dollars.”

I had similar ideas when I was that age. I let Robert’s statement go unchallenged--and stand as another alternative to consider, like the shortcut for calculating discounted prices.

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