Graphing Functions, Algebra, and Life

In our Applications for Living class, we are taking the last test before the final exam.  The primary topic for this test  is ‘functions and models’, where we cover the use of linear and exponential types — including finding the function from data and graphing functions.  What follows is a list of observations about what students seem to understand and what students tend to struggle with.

SLOPE — Everything is a linear slope to many students.  Even when the problem says specifically “Find the exponential model” [and the reference sheet includes the exponential model y=a(b^x)], a majority of students use the slope formula when we first do this.  After 3 or 4 visits to the idea in class, the majority work with the correct model — however, a third (or more) cling to the slope calculation when starting any ‘find the equation’ problem.  It’s worth noting that our beginning algebra class does not deal with any non-linear graphing.  The students who come from our Math Literacy course have an advantage — they have experience with linear and exponential graphs.

GRAPHING — Plotting points is seldom an issue … if given a pre-scaled coordinate system.  However, students struggle with the concepts of dependent and independent variables; we’ve gotten pretty good at that discrimination since the class dealt with the idea for 4 consecutive classes.  That does not mean that students know that dependent values usually are placed on the horizontal axis; I’ve seen some beautiful graphs which have the dependent variable on the vertical scale.  We talk about graphing equations as being a matter of communication, just like we did for statistical graphs; people expect the dependent to be on the horizontal axes.

GRAPHING — Scaling the axes is not easy.  We learn a routine for determining the scale size (1, 5, or whatever), and that helps.  However, many students do not see a problem with unequal intervals on their graph — especially for the independent variable.  Whether we are using graphing to communicate in a science class or in an article on global warming, equal intervals are critical.

USING MODELS — We are doing both types of problems with both types of models … we are given values of the independent and calculate the dependent, and we are given values of the dependent and solve for the independent.  In the case of exponential models, we solve for the dependent numerically via a calculator program to find the intersection.  Students seem to have a predisposition to calculating independent values; they ask if I will include the word “Intersect” in the problems where that is the correct procedure.  This is a case where the difficulty with dependent vs independent variables collides with selecting a strategy.

EXPONENTIAL FUNCTIONS — We started our work with exponential models in week 4 (12 weeks ago), when we did percent applications.  We did them again when we worked with finance models (like annual compound interest).  We did percents within probability, where we covered repeated probabilities (acting like a basic exponential model).  In the last 3 weeks, we have talked about how we discriminate between linear and exponential change based on descriptions, with ‘percent’ being the most important concept for us.  This spiral definitely improves understanding of percent problems, but students still struggle with exponential functions.  There is a tendency to use the percent as the multiplier (using .03 instead of 1.03), and some students treat the multiplier as a slope value in a linear function.  We make progress, but I would like students to be able to apply exponential equations in other classes and in life.

Here are some problems from the test we are doing:

The price of computer memory is decreasing 5% per year.  Write the exponential model for the price, and use the function to predict the price in 3 years.

The price of fresh oranges is expected to increase by 6¢ per week for the next few months.  The current price is $1.19. Write the linear function for the price, and use this to predict the price in 8 weeks.

I can purchase a motorcycle for $10,504, or I can lease it for a down payment of $750 and monthly payments of $155 per month.  Write the equation that describes the cost of the lease.  Use the equation to find how long I can lease the motorcycle before I pay more than the purchase price.

A rain forest is decreasing at a rate of 12% per year.  In 10 years, what percent of the current rain forest will remain?

A drug follows an exponential model.  After 3 hours, there are 16 mg in the body.  After 4 hours, there are 12 mg in the body.  How much will there be after 5 hours?   [Comment:  This is missed by many students.]

Twenty mg of a drug are administered at 4am, and the function y = 20(0.90^x) shows the amount of the drug in the body after x hours.  When will there be 6 mg of the drug in the body?  (nearest tenth of an hour)

I’ve made several adjustments to how I do the class to help with the struggle points described.  I can see improvements, and I can see individual students improve.  Overall, I am actually pleased with the results.

I hope you will continue to design your classes so that students understand the mathematics in a way that they can apply the ideas.

 
Join Dev Math Revival on Facebook:

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.

 Join Dev Math Revival on Facebook:

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!

 Join Dev Math Revival on Facebook:

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:

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

However, we struggled with this problem:

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:

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:

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.

 Join Dev Math Revival on Facebook: