I use the opening few minutes of this lesson to discuss the most important points from the previous lesson. In particular, it's important to discuss why the simplified form of a rational algebraic expression might not be perfectly equivalent to the original. The pair of graphs that the students all generated should illustrate this nicely. The original and simplified forms are perfectly equivalent at all but one point, and that point is determined by the common factor that was eliminated in the simplification process.
So, is it appropriate to simplify rational expressions? Yes, but we have to pay attention to what's been eliminated because it might be important.
The other focus of the discussion should be on how students were able to use the structure of the expression to simplify it. Students should have noticed things like the difference of two squares, the usefulness of factoring out a -1, etc. This is a good opportunity to emphasize the ways in which we can make use of structure in mathematics. (MP 7)
Keeping with the rhythm that's been established over the previous two lessons, it's now time for students to begin solving a handful of problems individually. These are somewhat more challenging; some involve cubic polynomials, some contain complex fractions.
The aim during these 15 minutes is for students to successfully simplify the four expressions on the front of Simplifying Rational Expressions 3. I'll circulate around the room and help students to quickly catch errors. If necessary, we'll discuss appropriate strategies for simplifying the expressions more efficiently. For instance, once students have factored the denominators, it's only necessary to check the numerators for common factors. If common factors do not exist, there is no need to completely factor the numerators.
Now, students should work either individually or in small groups to solve the problems on the back of the worksheet. Rather than simplifying expressions, they are now solving equations.