I begin class with the simple extraneous example problem and the three possible solutions on board and ask my students to review my work. This is a simple problem so that they can clearly see why it's not a viable solution, but sometimes that results in my students failing to explore the reasoning behind the "correct" solution that doesn't really work. Challenging them to find and explain the flaws in my logic gives them additional reason to be engaged. Since they already love to catch my "mistakes" this generally gets them working to find any problems with my solutions. (MP6) After a few moments I ask for volunteers to share what they've found with the class. If nobody catches the extraneous solution I hint that they should check the problem by putting each solution back into original equation or by graphing each side of the equation to check the intersections. Once everyone recognizes that zero doesn't work as a solution to this problem I tell them that this kind of answer is called an "extraneous solution". The next challenge is to work independently to find all the viable solutions to the rational extraneous example problem, given just the original equation. (MP1) Some students still struggle with rational equations so I help them with questions like "What do the terms in the denominators have in common?" and "How could you find a common denominator for this problem?" When everyone is done, I select students to post their work on the board, based on what I've observed while they were working. I look for at least one student who found a reasonable solution but didn't recognize that it was extraneous, one student who has it all, and any other interesting approaches to solving this problem. Never knowing what methods my students will apply I watch for novel methods that might make sense to other students and help them figure things out. To close this section my students brainstorm other possible reasons why a solution might be mathematically correct but not be a valid solution to the problem. (MP2) We popcorn share the ideas discussed before moving to the next section.
For the main part of this lesson I have my students work independently to build their skill at recognizing extraneous solutions and so that I can more readily identify those who need additional support. I distribute the Extraneous problem set and remind them to show all their work so that they can explain their answers to classmates. I remind them to check their work by substituting their answers back into the problems and/or graphing both sides and checking the intersections. (MP1) While they're working I walk around offering support and redirection as needed. When everyone is done I ask them to work with their right-shoulder partner to compare and critique their solutions. (MP3) I tell them that each team will be asked to share solutions on board so they need to be clear about how they got the solutions and whether or not they are extraneous solutions. (MP2) I allow another 10 -15 minutes for this collaboration, making note of which teams have good explanations for which problems. I give my students a 3-minute warning, then call on teams to present the problems I've selected. As each team presents I encourage the other students to ask questions and critique both the solutions and the reasoning. (MP3)
To wrap up this lesson I ask for volunteers to explain what an extraneous solution is on the board. When several explanations have been posted, I summarize what's been written into a good definition with student input, then ask my students to copy the definition into their notes and include examples. (MP6) My video discusses the importance of directed note-taking.