Category: math reasoning and applications

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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Just For Fun …

We have a traditional intermediate algebra course, and my classes are currently working on factoring.  Of course, these topics are only appropriate if a student is headed towards a STEM-type field; most of my students are done with this class, so there is a basic mismatch.  [That problem relates to the current work on the Michigan Transfer Agreement, which may take intermediate algebra out of the general education mix.]

However, we try to always have fun in class, and my students know that I don’t mind looking at other ideas.  One of those ‘ideas’ happened today; this is not radical, nor important in our class — but it was just plain fun.

We were working on factoring by use of formulas.  This particular problem dealt with a perfect square trinomial, with fractional coefficients.  Like this:

¼(a²) – (2/3)a + (4/9)

I’ve already told students that we are doing this much factoring just because it is on our departmental final; we are looking at them as puzzles.  This problem got us into looking for squares of fractional terms.  We got through it, and showed the factored form.

So, one of the students says:

Can we clear fractions?

Of course, I said.  “What would you do?” The student replied “Multiply by 36″.  Now, we have been focusing on what I call the 3 big rules of factoring — write as an equivalent product, use integers unless the problem had fractions, and each factor must be prime.  Since multiplying by 36 clearly changes the value, we need to do something to ‘keep it balanced’.  The solution is to show a division by 36:

(1/36) * 36[¼(a²) - (2/3)a + (4/9)]

So, we distributed the 36 and factored the resulting non-fractional trinomial … and kept the (1/36) factor in front.    To me, this was just plain fun; I know most students don’t agree — but at least they got to see somebody have fun with algebra.

This particular issue has been a problem; it seems like a few students would ‘clear fractions’ but without keeping the balance on the assessments for this material.  These students tended to be those I expect to do better — willing to think and reason, trying to connect information, etc.  I’ve not felt okay about just bringing up the clearing fractions method, because most students do not think of it in this context.

I just hope that I have more students like this one, who will be willing to ask a good question … and we can have some fun!

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Applications for Living — Geometric Reasoning

We are taking a test in our Applications for Living class, and I am struck by two things.  First, students have made major improvements in how they deal with converting rates (like pounds per second into grams per hour). Second, how bad geometric reasoning is, before and after our work on it.

Just about the simplest idea in all of geometry is ‘perimeter’.  Students have very little trouble with a rectangle as a stand alone object.  This problem created a speed-bump:

perimeter trapezoid math119





As a class, we ‘passed’ on that item (in terms of proportion with correct work).

However, we struggled with this problem:

perimeter rect plus 2 triangles math119





We did not pass on this item, as a class.  The most common error, of course, was counting the ’12 inches’ (which is completely internal to the figure).  Not as many included the ’8 inches’, which is also internal.  We always say that perimeter is the distance around a figure, but that is not internalized as strongly as the “2L + 2W” rule.

A bonus question on the test looks like this:

perimeter rect plus 1 triangle find sides Math119



This problem combines the reasoning about perimeter with some understanding of right triangles as components of shapes.  A few students got this one right.

We spent parts of 3 classes working on our reasoning and problem solving.  These compound geometric shapes are common objects in our environment (at least in the USA).  I’d like to think that our students would be able to find the amount of trim or edging to install.

We are a bit too eager to pull out a formula for perimeter (where it is never required for sided-figures); when we talk about circles, it’s not connected well enough to other ideas like perimeter.  One of the problems we did in class caused a lot of struggle:

perimeter rect plus circle Math119



We used this problem as a tool to work on reasoning about perimeter (and area).  Much scaffolding was needed; since we only spent 3 classes on geometry, we did not overcome prior mis-conceptions in most cases.  Our better results with dimensional analysis (rate conversions) is due mostly to the fact that students had few things to unlearn.

Let’s do a little less variety in geometry, with more focus on reasoning.  Formulas are fine for area and all-things-circular, but have no business in the perimeter of sided figures.

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Easy or Worthwhile?

I was walking by our copy machine this week, and saw a handout for the same material that I was about to work on in a class.  I took a look, and reacted a bit strongly to what I saw.

The basic idea on the handout was this:

The easy way to solve these equations is to enter one side as Y1, the other side as Y2, and have the calculator find the intersection.

I have to admit that using the calculator can be easy … not as easy as just looking up the answer, but sometimes easier than a human being solving the problem.  The question is:  Is it just easy, but not worthwhile?  Do students gain anything from using a built in program to solve a problem?

I face this issue in our Applications for Living class.  A bit later in the semester, we will talk about medians and then about quartiles.  Students discover that the calculator will find all of that for them.  Should students start to use the calculator to find the quartiles and median right away, to avoid the tedious work of ordering sets of 12 to 20 numbers?

In this statistics example, the material is worthwhile if the student can answer this question easily:

A set of 100 numbers has a median of 40, a lower quartile of 25 and an upper quartile of 70.  How many of those numbers are between 25 and 70?

A basic understanding of quartiles gives a good approximation (50); I’d be thrilled if a student said ‘about 50 but we don’t know for sure’.  In the practice of statistics, technology is always used to find the calculated parameters … and we need to know how to interpret those values.

The content for the handout I saw was ‘solving absolute value equations’, one of my least favorite topics because it tends to be hard to understand while there are a relatively small number of places where this needs to be applied.  However, the understanding of absolute value statements contributes to some common themes in mathematics — multiple representations in general, symmetry in particular.  Technology (as used for an ‘easy way’) avoids all of this stuff that makes it worthwhile.

A focus on the ‘easy way’ presumes that the only purpose for a topic is to get the corresponding correct answers.  To me, a student that uses the calculator to solve |x-5|=7 is just as dependent as a student who uses a calculator for 8 + 5.  The solution is simple enough that it can be done mentally; even writing out all steps gets it done quicker than a calculator process.  If all we do is show students how to obtain correct answers, what is the value that we have added to their education?  If we need to solve |25.8x + 4/3|=8.52, I will certainly tell students … ‘well, we understand how to solve this problem ourselves, so let’s set it up that way — and here is how to check that on a calculator’.  Of course, I know of no place, outside of an algebra textbook, where such a problem would be needed.

Easy is not the primary goal.  Worthwhile learning, and education, are the main things.  Every time we avoid learning we detract from our students’ education.  Technology has a role to play; ‘easy’ does not.  Understanding is a lot more valuable than a hundred correct ‘answers’.

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