Another Rational Approach to Solving: Graphing

Today students will be working with their graphing calculators. If you do not have access to a whole class set, I think this activity would also work nicely with students working in pairs and sharing calculators. I am going to have all of my students use their own calculator, but work in teams to discuss their findings.

Today’s activity is very prescriptive in telling students how to solve a rational equation on the graphing calculator. The goal is not for students to establish their own understanding and their own procedure for how to solve by graphing. The goals are as follows:

• To help students develop a deeper understanding of what it means to solve a rational equation
• To push to students to remember what the graph of an equation represents,
• To see if students can use their understanding of systems of equations to rewrite an equation as a system in order to solve it by graphing

As students work to make these connections they are engaging in MP2. The strategy of writing a system is powerful for students. They can use it to check (or estimate) solutions to an equation. This strategy is even more powerful when students leverage graphing technology (MP5).

Today's lesson may seem a bit procedural, but I think there are some deeper understandings that students are developing. To get students thinking in the right direction, I begin by posing these questions to students:

• What does it mean to solve an equation? 
• What does it mean if there are variables on both sides of the equation?

It is important that students can articulate that they are looking for the values of x that make both sides of equation equal. Then I will pose the problem in the discovery activity and ask students, "How do you think that I cab solve this equation by graphing?"

I am hoping to lead students into the big idea here: We can graph each side of the equation as a separate function and use the graph to estimate the solution of the resulting system. Once this idea is raised, I will pass out Solving Rational Equations by Graphing and have students work through these examples. There are many practice problems here, so I don’t necessarily expect my student to finish them all.

As students work through this activity, there are some secondary learning targets that I also hope are hit:

• I hope students learn while completing this activity is how to put the rational functions in the calculator with correct grouping symbols. It is important that students understand that there are implied parentheses around the numerator and denominator of a rational expression. To help establish this, I think I am going to put the answers to this activity on the board in a random order and ask students to make sure their answers are there. If students are graphing incorrectly, they should quickly realize that none of their answers are correct and will hopefully seek help from a teammate or me.
• Students will have their first look at asymptotes. Although, I do not plan on formally introducing asymptotes here, I do plan on mentioning to students that the vertical lines they see are not part of the graph, but actually where the graph is undefined. 
• Since we have already established the idea that dividing by zero causes our function to be undefined, I think it is appropriate to just leave it at that for now.