Let's take a look at the last problem from the previous lesson (or come up with a new one that also leads to an extraneous solution). Write the equation on the board and ask the class to walk you through the steps of the solution, beginning with the solutions that must be excluded from the domain.
As the students propose a step, be sure to ask them to justify it, and then write out all of the steps on the board (it should look something like this). In the end, everyone should see that one of the solutions is a number that was excluded from the domain. This seems like a contradiction!
Now, ask the question, "What did we do wrong? Where did we make a mistake or an invalid assumption?" If they aren't sure what to say, I'd go back to the original equation and ask, "If I plug in this number, what do I get." The class should reply, "An undefined number." Then proceed through each of the intermediate steps asking the same thing. They should soon identify exactly the point at which the equation goes from being undefined to defined. (MP 3) This is where the extraneous solution was created! Since our task was to find numbers that make the original equation true, this solution must be extraneous. It may make the final equation true, but not the original. (See the video for more.)
Make sure that students clearly understand both how to identify an extraneous solution and how it is created.
For the remainder of class, the students should work individually or in small groups to complete Solving Rational Equations.
The first few problems are very similar to the one just completed as a class. Observe the students' work and ask questions to make sure that they comfortable both identifying extraneous solutions and explaining why they are extraneous. Encourage the students to examine the original equation to determine which numbers cannot belong to the solution set. Doing this at the beginning will make it easier to identify extraneous solutions later. (MP 7)
This worksheet provides some practice simplifying rational expressions, as well as graphing rational equations. Students are first asked to solve a system of equations graphically, estimating the intersection point, then they are asked to solve the same system algebraically. This helps them to check their previous work!
As class winds down, I find it's important to call the class together for one final conversation. As students have been working, I've been taking note of their progress and common mistakes. Now is the time to point some of these things out, to ask students what aspects of the problems they find troublesome, and to provide some encouragement.
For homework, I will ask everyone to complete the remainder of the worksheet. I'll also point out that the graphical solution and the algebraic solution should agree with one another. If they don't something has gone wrong and you should go back and check your work!