Systems of Equations – Sample Lessons and Resources

For this months lesson feature, I am going to focus on Systems of Equations. I chose this topic because I just did a workshop with Algebra 1 teachers in NJ, and this is where they were in their pacing guide, so I am making an assumption that many algebra teachers might also be focusing on this content as well this time of year. I am using a problem from Fostering Algebraic Thinking with Casio Technology in order to provide a real-world problem-solving experience (and I have the resource), but I have altered the problem so that I can utilize the all-in-one capabilities of (tables, graphs, equations, geometry, text).

The Problem

In 2010, there were approximately 950,000 doctors in the United States, and approximately 350,000 of them were primary care doctors. It was estimated that more than 45,000 new primary care doctors will be needed by 2020, but the number of medical school students entering family practice decreased by more than 25 percent from 2002 to 2007. With laws reforming health care, many more people will be insured in the United States. 

For many reasons, including a growing and aging population, the demand for doctors will likely increase in future years. The number of doctors available is also expected to increase. But, due to the high cost of insurance and the fear of malpractice lawsuits, many have predicted that the increase in the number of practicing doctors will not keep up with the increase in demand for doctors.

The table to the right provides data from a study conducted in the state of Michigan. These data approximate the number of doctors that were or will actually be licensed and practicing in Michigan, called the supply, and the number of doctors that were or will be needed by the people of Michigan, called demand.

The question is, will there be enough doctors to provide all the services? The shortage of doctors is a problem that challenges the entire country, not just Michigan.

The Lesson

A shared paper has been created in called Systems of Equations Help! Not Enough Doctors, which you can access by clicking on the title. The idea behind this problem is to provide a real-world context where students can use tables, graphs, and equations (along with calculations) to create a system of equations. They can solve these using methods such as substitution, elimination, and graphing. Students will also be practicing how to model with mathematics, applying what they know about relationships and being able to create a system of equations that fits the context of the situation in order to find a reasonable solution.

In the activity, there is obviously some focus first on getting students to really understand the problem and what the numbers represent, and then the idea is to have them look for patterns and relationships as they look for a solution. First in the table, then by looking at a scatter plot of the data, where they again try to determine a solution based on a visual. Continuing to look for trends, they use prior knowledge to recognize linear relationships, create equations that model the data, and then graph those equations to find a more precise solution. Then, as a check, they solve their system of equations algebraically. It’s all about multiple representations and helping students see the connections between all the representations, and depending on whether you want a specific, precise answer or just a generalized answer, you might choose a different representation. – Lesson In Action

The video below shows the activity and does a brief walk through of some of the components and what it would be like doing the activity from a student perspective. I am a big believer in the think-pair-share approach, so I would suggest having students do the Notice and Wonder individually first, then pair up, then share so that you can make sure that any misunderstandings about the context, and clarification about the numbers is figured out before students start solving. Then I would suggest small groups for working on the problem itself.

Other System of Equation Activities and/or video links