Students will understand how and why extraneous solutions arise from radical equations.

Solving equations is a form of reasoning, and students learn to critique their reasoning to make sense of unexpected outcomes.

10 minutes

The lesson begins very simply by reviewing the solutions to problems 6 - 10 by asking the students what solutions they arrived at. If the bulk of the class is in agreement on the correct solution, I will affirm it. If a significant number of students have an incorrect solution, I will ask one of them to share their solution method at the board or to explain it while I act as their scribe. (It’s important to note that the students do not yet know whether they are correct or not.) Once the mistake(s) have been identified in a way that respects the student and affirms what is correct in their reasoning, the correct solution will be given.

Once the solutions have been agreed upon, the students to begin answering the concept questions on the reverse of the worksheet. They are *strongly* encouraged to talk with one another about these questions.

15 minutes

As students work together to answer the concept questions, I will circulate around the room checking for understanding. Occasionally, I will stop for a brief 1:1 conversation, but I don't want to give away too much too soon.

One important point for the students to see is that extraneous solutions may arise for a variety of reasons. Division by zero causes them in rational equations, taking roots causes them in radical equations, etc. It's important that my students see how and why they arise in each case, and I'll be asking them to explain it to the class later.

These questions help students to see again the importance of definitions. In order to avoid ambiguity, mathematicians need to carefully define the square root operation. These concept questions are intended to help students see *why* this definition is needed but also how this definition actually leads to extraneous solutions.

15 minutes

5 minutes