Given a percent and another number, do we multiply … divide … or something else?
A few years ago, I was at a presentation about a pre-algebra course where the presenter classified percent problems as either growth or decay. My initial response was that these are concepts too advanced for that course; after a few minutes, I liked the idea, and my experiences since have strengthened that opinion.
Within a few days, I had a chance to work on percents in both a beginning algebra course and in our applications course. In both courses, the percent problems are varied; one thing that was constant — students ‘wanted’ to multiply a percent and the other number in a problem, regardless of the context. Sales tax rate and marked price … multiply and add. Sales tax rate and final price … multiply and subtract (wrong). Percent decrease and old amount … multiply and maybe add. Percent decrease and new amount … multiply and add (wrong).
We seem to have reinforced overly simplistic rules about percents to the point where students are impervious to a need to change; 40% wrong answers is not enough (even if I asked ’8 is 40% of what?’). It’s really that 100% value that is the problem. The connection between a growth rate of 3% and a multiplier of 1.03 is a challenge.
In the applications course, I had students work in small groups on a sequence of problems to make a transition from a simple percent value to a multiplier. They worked hard, explained to each other, and seemed to do well. The next day … a quiz on percents where they could use the multiplier; result — not so good. In the applications course, we use this multiplier again — in our finance work (1 + APR) and in our exponential models [y = a(1+r)^x]. I suspect that a deliberate focus on the multiplier in 3 chapters might result in some improvement.
I actually fault our presentations on percents as the root of this ‘evil’. We do “2 places to the left”, “is over of”, and mechanical use of “a is n% of b”; sure, we include problems where students need to find the base (divide), but the work is too superficial. Students do not generally understand the contexts where percents are used. An initial approach on growth or decay, which means seeing the multiplier, might just help.
The most common uses of percents in developmental courses is usually in that pre-algebra course. Based on long-term goals of understanding, if we are not going to cover the whole story of percents (with the multiplier), we should omit percents entirely. However, percents are one of the richest zones of overlap between math and the world students experience that we need to see percents as a good thing — and do it right. Fewer tricks, a lot more understanding!
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