Solving linear systems has always been one of my favorite units to teach. That probably has a lot to do with the fact that I love solving systems. Elimination is straight up my JAM! So I was really excited to work on the Algebra 2 Systems of Equations unit. I thought choosing the contexts for the lessons would be easy. After all, I had a ton of linear system word problems from many years of teaching.
But as I looked through my old notes, I saw those story problems were much less helpful than I had hoped. The example problems in my notes pretty much always looked like this:
Solve a fairly simple system of equations.
Solve a more challenging system of equations.
Solve a STORY PROBLEM by writing a system of equations and THEN solving it the exact same way as the previous problems.
The contexts in the story problems were not highlighting the mathematics, they were disguising it. The student process for solving these problems became 1) Pick out the needed information to write a system, then 2) Use a familiar procedure to solve the system you wrote. While there are times when these types of problems are worthwhile, they were not helpful in getting students to figure out how to use elimination or why to use elimination without me showing them how to do it first.
So what kind of context would work? One of the most important aspects is making sure there is a low entry point for students. All of your students should be able to start the problem. They should understand the context even if they don’t understand the math yet.
Next, there should be more to solving the problem than just extracting the information given. The context should highlight something we want students to notice. In the case of solving with elimination, we want students noticing the structure of the equations.
The goal is to make the problems into a puzzle students want to figure out rather than a code to decipher. They should be able to use intuition and apply prior knowledge to generate their own solution path.
After lots of brainstorming and collaboration, we came up with some activities with contexts that we think fit the bill. Here are a few highlights. We hope you try them out with your students!
Context: How Much is a Taco?
Concept: Elimination
Concept: Consistent and Inconsistent Systems
Note how these lessons motivate the need for elimination by focusing on what’s different between two orders. If only one quantity has changed, then the difference in price can be solely attributed to the change in that quantity. This helps illuminate to students why we want the coefficients of the variables to be the same.
In the Panera Bread lesson, students discuss what it means to charge fairly and how this is related to having a consistent system. There exists a bagel price and a cream cheese price that works for both orders (i.e. satisfies both equations). If prices charged to customers was not consistent, then not only would the pricing not be fair, but there would be no way of finding the exact price of a bagel or tub of cream cheese.
Additionally, the idea of a dependent system is often taught only graphically or analytically—two lines that coincide and thus have infinitely many solutions. In this lesson, students see that dependent equations are equivalent equations. Of course someone that buys double the amount of each item would pay double the price! This is fair and expected, and for that reason, it becomes a redundant equation—it does not provide any new information that would allow you to solve for the cost of a bagel or a tub of cream cheese.