Applications for Living: Growth, Decay, and Algebra

We are taking the second test today in our Applications for Living class.  The first test covered ideas of quantities and geometry, including dimensional analysis.  This second test covers a variety of ideas, mostly related to percents.

Everybody is the class has completed two courses which include some work with percents.  For a few, it’s been several years — however, that does not matter.  All students get a fresh start on percents in this class.  The work is intense for this test, and then we return to percents towards the end of the semester when we cover exponential models.

The initial struggle starts when we deal with unknown base numbers, when we know the new value after a known percent change.  like this:

In 2014, there were 20000 people voting.  This was a 10% increase from 2012.  Find the number voting in 2012.

Students really want to find 10% of that number in the problem, and then subtract.  Initially, they argue that this method is valid.  After doing a few where we know the base, most accept that we can’t do the multiply and subtract method.  To help, we take a growth and decay approach (though the words are not emphasized).  For the voting problem, we work on seeing a 1.10 multiplier on the base (unknown).  For a discount problem, we work on expressing “15% off” as a 0.85 multiplier.

We then use a sequence of percent changes.  One of the early ones is:

We have $50 to take our family out to dinner.  There is a 6% sales tax on the total price of the food, and we always leave a 15% tip (15% of the total with the tax).  How much can we spend (for the price of food)?

The success rate for this problem is very low; even with help, few students can see how to work this problem.  Over the next few days, we see other expressions like  “1.15(1.06n)=50”.

When we start compound interest, we begin with a basic idea:  A = P(1 +APR)^Y.  We talk about compound interest as a sequence of percent increases.  Without any preparation, students encounter this problem:

At one point, home prices were increasing 10% per year.  What would the price of a $100,000 home be after 5 years?

The majority of the students saw the basic relationship, and used the compound interest formula (for annual compounding) on a problem that did not involve interest.  That type of transfer is a good sign.

This does not mean that students really get the percent to multiplier idea.  One of the questions on today’s test is one that I have mentioned before:

The retail cost of a computer is 27% more than its wholesale cost.  Determine which of these statements is true.

The options for this question include ‘retail cost is 27% of the wholesale cost’, and ‘the retail cost is 127% more than the wholesale cost’; these are frequently selected in preference to the correct “the retail cost is 127% of the wholesale cost”.  Building new pathways in the brain is easier than repairing old ones.

Our course is not heavily algebraic … except to use algebra as a way to express relationships.  Our work with growth and decay culminates in exponential models, where students need to go from “the prices are falling 4% per year, and the current price is $50” to the model y = 50(0.96)^x.  We like to look at this model as being related to the compound interest formula.

This strand of “percent change” (growth, decay) runs through our course, which I am pleased with — the world involves many exponential models, which means that students will need to be adept in using them in science classes.  One of the later problems we look at is:

If there are 50,000 cheetahs today, and the population is declining 8% per year, how many cheetahs will there be in 10 years?  How long will it take to have just 500 cheetahs?

We look at both of those questions from a numeric perspective.  The first is a simple calculation; the second is solved numerically from the graphs.  [We also learn how to graph such models, including the design of appropriate scales.]

A single post can not tell the entire story of any topic.  However, I’ve tried to include some basic benchmarks in the story of ‘growth, decay, and algebra’ in an applied math course.

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