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.

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