Engaging students: Equations of two variables

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Madison duPont. Her topic, from Algebra: equations of two variables.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Problem: It’s tax free weekend (clothes are tax free) and you want to spend exactly \$15 (so you can get \$5 back from a \$20 bill) on only shirts and shorts. Shirts are on sale for \$4 and shorts are on sale for \$3.

1. Write an equation to model this situation.
2. Determine how many shorts and shirts you should buy to spend exactly \$15.

This problem does a good job of introducing a relatable and realistic situation that can be written as an equation with 2 unknowns. The mathematical portion of solving this is also approachable using conceptual strategies such as drawings, counting in groups, or more calculative tactics like trial and error with multiplication and addition, or even more advanced concepts like knowledge of division algorithm. The use of traditional variables is not even necessary to write an equation as the students can use pictures or words next to the coefficients to represent the unknowns. Because there are multiple levels of approaching the problem both in creating an equation and in finding the unknowns, this is a good exercise to have them explore the topic and gain conceptual understanding.

How could you as a teacher create an activity or project that involves your topic?

Activity: Have students sit in groups (2-4). Have 10 di-cut images of apples and 10 di-cut images of bananas (or oranges, etc.) in the center of the group to serve as manipulatives. On each of the apple di-cuts write \$.10 in the center and on each of the banana (or other fruit) write \$.20. Tell the students they need to find a way to spend exactly \$1.00 (using at least one of each fruit).

This activity allows students to explore the concept of considering two unknowns in the same situation in a tactile and conceptual way before encountering the mysterious algebraic equation. Students sharing answers can demonstrate that there are different possibilities and therefore the number of fruits is truly variable and can be written as an equation.

How can this topic be used in your students’ future courses in mathematics or science?

An equation of two variables will be the stepping stone to linear equations and functions. When the equation is solved for “y” in terms of “x” you will get a linear function. Having a decent conceptual understanding of two-variable equations and being familiar with manipulating the equations will help students begin to understand notions of inputs and outputs and to see that having one variable will allow you to find the other. All of those topics will lead to the graphing of functions and taking algebraic work to a visual type of mathematics. Equations of three variables will also be a future topic related to this one as well as solving systems of equations for both two variable and three variable equations. Knowing how much will be built off of this topic makes equations of two variables much more appealing for teachers to teach the topic well and for students to learn conceptual and mathematical components of this topic well.