The Best Intro to Systems of Equations

This year I have been working with some talented educators from the University of Kansas.  They are creating a resource called Enhanced Learning Maps.  The gist of it is that it is a formative assessment tool aligned with the Common Core State Standards.  The maps unpack the standards and shows all the connected concepts, helping me identify what to work on with students at all levels of understanding.  Along with these maps are assessments to identify misconceptions, summaries of research, and lesson plans with activities.  Follow this link for more information on the Enhanced Learning Maps Project.

In the past, my systems of equations lessons followed linear equations and graphing so it seemed logical to start with graphs and having students create graphs from situations, which is standard 8.EE.8.a: Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Today, I introduced systems of equations with a lesson from the Enhanced Learning Maps project.  It starts with a problem to get students thinking about these types of situations.

You are in charge of buying snacks for a class party. After going to the store, you determine that purchasing 20 bags of chips and 10 packages of fruit snacks would cost $23. Thinking this may not be enough, you also determine that 20 bags of chips and 20 packages of fruit snacks would cost $29. How much is each bag of chips and each package of fruit snacks?

I encouraged students to come up with more than one way to solve it.  We discussed different strategies.  We didn't linger too long on this problems.

Then we did a jigsaw.  My students work in teams of 4.  One person from each team went to a different corner of the room to work out a problem with their "new" group.  Each group was given the same situation, but given a different representation. (Full documents here).

Each group was asked to answer the following questions:

• Which option is the better choice based on cost?
• Is there a number of people for which the costs would be equal?
• Use any mathematics or representations that are useful to your decision making process.

After they answered the questions they returned to their original groups and shared their problem (they did not know they were the same) and their method for solving.  Afterwards we had a great class discussion.  Here are things we noticed/realized:

• The graph groups solved it the quickest by simply drawing lines and finding the point of intersection. 
• The group with the coupons took the longest.
• The group with the table felt theirs was relatively easy to solve once the table was filled in.
• The strategies were different depending on the representation they were given.  
• None of the groups without a graph created one to help them solve the problem.
• None of the groups without a table created one to help them solve the problem

After discussing the different strategies, students were given another problem.

You are shopping for a new cell phone and have pricing from two different stores, Cellular City and Mobile Mart. Cellular City sells a phone for $300 and charges $45.50 per month for service. Mobile Mart sells the same phone for $260 and charges $48 per month for service. Which store is the better choice based on cost? Is there a point in time at which the total costs would be equal for both stores? If so, when would this occur and what would the total cost be?

After the jigsaw activity, it was amazing to listen to their different strategies to solve. There was more variety than I have seen students come up with in the past because now they were focusing on all of their background knowledge of all the different representations.  It was clear they had a better understanding of the problem and the relationships within it.   Some groups realized that their two equations could be set equal to each other (equal values method/subsitution).  One group found the difference in the starting prices and the difference in the monthly charge (elimination).  I was blown away by the thinking, strategies, and discourse that occurred during this lesson.  It makes so much sense to pull from their knowledge of the 4 representations (graph, table, equation, situation) instead of focusing on just one at a time.  The math was more intuitive and less contrived compared to how it has felt in previous years.  

I encourage you to try this lesson.  Let me know how it goes in the comments.  

Thank you to the Enhanced Learning Maps Team.  Please note that this lesson and the linked document is Copyright © 2016 by The University of Kansas.