Math – Applications for Living IV

Percents!  Percent discount!  Percent tax!  Percent increase!  What do our students learn about percents?  Not nearly as much as we would like.

One of my students made a comment about percents and taking his family to a restaurant.  As a result, this problem appeared on a worksheet last week in our Math119 class:

I have $20 to go out to dinner.  All restaurant purchases in my state have a 6% sales tax, and I like to leave a tip that is 15% of the total including sales tax.  How much can I spend on food (menu prices) to stay within my $20?

Every one of my students has shown that they can calculate a 6% tax, and find the total price.  Every one of them can find the 15% tip on a dinner, and the total cost.  Less than 10% of them knew what the ‘base’ for the percent is in the problem.  We had already been talking about the net result of a 10% increase (110%, or 1.10 times the base) as well as a 20% decrease (80%, or 0.80 times the base).  In fact, we started off our work with percents by the classic story:

Boss: Bad times; sorry, everybody gets a 10% pay cut.
Boss (next year): Good times are back; everybody gets a 10% pay raise.

Worker: Am I back to where I was?

Many percent situations in the world involve a chain of percent increase or decrease factors operating on a moving base.  In my dinner example, the goal is to see the situation as

1.15(1.06n)=20

The solution here ($16.41) is pretty good, as the rounding happens to work out well; in general, this method is a ‘good approximation’ — an idea that is not brought up in this class.  We are still going through a lot of struggle to identify the base in percent problems.  Later in the semester, we will connect this repeated percent concept to exponential functions; identifying the base correctly will continue to be an issue then.

Whatever you do with percents in your courses … please focus on identifying the base. Being able to calculate a tax or a decrease is nice, but of limited usage.  Percent change is all around us, and we often deal with an unknown base (frequently hidden within the context of the problem).  We don’t need to disguise problems to the point that finding the base is a horrendous exercise for students.  On the other hand, creating cookie-cutter exercises where no thought is needed is a self-defeating practice.  “Percents” and “thinking” go together in the world around us, as they should in our math classes.

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