Contributor: Laura Bracken, Lewis-Clark State College. 

In my elementary algebra classes, we use a problem-solving organizer called The Five Steps that I developed based on the work of George Polya. Since this is an algebra class, I emphasize the use of equations to represent and solve problems. In one of my classes earlier this semester, my students practiced using more than one property of equality to solve a linear equation in one variable. We began to do the following problem together as a class.

A trucker has fixed annual costs of $27,600. The average non-fixed costs are . If the total annual cost to operate the truck is limited to $80,000, find the number of miles that the trucker can drive. Round to the nearest whole number.Step 1. Understand the problem.
We discussed the difference of fixed costs and non-fixed costs and then assigned a variable: m = number of miles that the trucker can drive.Step 2. Make a plan.
I asked students to write a word equation that represents the relationships in the problem. With their input, I wrote the equation ‘fixed annual cost + (non-fixed costs)(number of miles) = total cost’.

Step 3. Carry out the plan.
The students helped to identify the fixed annual cost of $27,600, the non-fixed costs or $0.45 per mile, and the total costs of $80,000, which led to the equation . Removing the units, we solved .Rounding, the solution was .

Step 4. Look Back.
In this step, we ask students to first think about whether the solution is possible. For example, an answer of 11 mi or 1 million miles is ridiculous. Then, we ask them to explain why the answer is a possible solution. They usually work backwards, show that the answer is in the ballpark, or estimate. In this problem, a student suggested that we estimate. The answer of 116,444 mi is about 120,000 miles. The non-fixed costs are ($120,000)($0.45) which is $54,000. Since $27,600 plus $54,000 is $81,600, which is close to the given total of $80,000, the answer seemed reasonable.

Step 5. Final Answer.
To limit the costs to $80,000, the trucker can drive about 116,444 mi.

After working through this problem at the board with the class, I send them to work in pairs on this problem: A sales representative has an annual base salary of $24,000. He makes a $20 commission on each product he sells. Find the number of products he must sell each year to have a total annual salary of $60,000.

Walking around the room, I looked at what my students were doing and listened to their conversations. After a while, I asked for their equations, wrote them on the board, and asked them to volunteer to explain their thinking. One student responded that since we had just finished the trucking problem, she thought this was probably like it and so she tried to write an equation that looked like the one we had used to solve that problem. An older student, who wrote the fourth equation (in the image to the right), said that he just figured out how much money the person had to make in commission and then divided it by the amount earned from selling one product. I explained to the class that there are many ways to write an equation with a solution that is the correct answer. I encouraged them to write an equation that makes sense to them.

We then talked about the last incorrect equation. I said that it was a good attempt. We worked through the problem to the solution of 720,000 products and I asked why this answer was wrong. One student said that selling 720,000 products would make much more money than the person needed to make $60,000. “Exactly right,” I said. So then I asked the class what they should do when this happens. One student said, “Start over.” Another student very seriously replied, “Give up.” which prompted some laughter. I talked about the persistence that we all need to apply to problem solving and that it is normal not to get the right equation on the first try. Time was up.

When I tried this again…

My next class came in and I taught the lesson again. However this time, when I taught the trucking problem, I suggested two ways to write the equation: , and .

When this class did the commission problem, the results were interesting. All but one student chose to set the problem up using the second equation.

This experience is similar to something I did a few years back at the beginning of the semester. With the only directions being to write an equation to find the answer, I asked a group of math majors in their real analysis class to solve three application problems and I also asked three classes of elementary algebra students to do the same. The math majors typically wrote equations like while almost all of the elementary algebra students wrote equations in which the variable was isolated.

Why do math textbooks present the problem like the math majors do? Of course, textbooks are written by mathematicians who were once math majors, but also the applications are in a section where students are learning to solve equations using two properties of equality. Since the equation requires two properties of equality to solve, writing and solving the equation in this way reinforces this algebra. Perhaps we can also argue that this prepares students for linear modeling with equations such as.

However, I wonder what is more important at this point in the course. Should we insist that students write equations in which the variable is not isolated? Should we concentrate on building their confidence in problem solving, no matter which equation they choose? As developmental math students often prefer to learn one way to solve a problem, should we continue to show alternatives and emphasize that there is more than one way?

This is one reason why I do not find teaching developmental mathematics boring after 20 years in the classroom. There is much more to the art of teaching than just presenting examples from the textbook. It is a balancing act of tools, strategies, textbooks, and persistence to find the correct equation for student success.

Laura Bracken, co-author of Investigating Prealgebra and Investigating Basic College Mathematics, teaches developmental mathematics at Lewis-Clark State College. As developmental math coordinator, Laura led the process of developing objectives, standardizing assessments, and enforcing placement including a mastery skill quiz program. She has worked collaboratively with science faculty to make connections between developmental math and introductory science courses. Laura has presented at numerous national and regional conferences and currently serves as the regional representative for the AMATYC Placement and Assessment Committee. Read more from Laura Bracken at her blog, Dev Math Diary.

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