This lesson continues to build on the idea that complicated mathematical expressions and equations (like the rational expression in my Example resource) can be broken into component parts and reassembled without changing the value of the expression/equation. I also continue the focus on appropriate vocabulary for accurate communication.
I begin class with an Example on the board and let my students discuss it for the first few minutes while I'm getting attendance (and listening!) Usually they talk about how to simplify the expression because they think that's what I'll be asking them to do. That discussion wanders all over the place as students suggest combining like terms, reducing the fraction to lowest terms and even canceling the denominator (somehow?) but I value the insights it gives me and also the fact that my students are willing to try to figure out how to find an answer. (MP1) Their comments help me decide what to emphasize as I bring the discussion to a close so it changes a bit with each class but the focus mentioned at the beginning of this lesson stays the same. I begin by asking for a volunteer to identify at least one term in the expression and write it on the board. I continue to ask for terms until my students assure me they have found them all (sometimes they still have one or two left, but I'll let them "discover" them as we move forward). I then ask if anyone has any other ideas about how we can label parts of the expression. This is where the missing terms, if any are generally caught. Again, I anticipate at least a few students mentioning that it is a rational expression with a numerator and a denominator. I also expect them to mention coefficients and exponents because that's what we did at the beginning of this unit and it should still be fresh for them. When my students have identified all the components they can think of I tell them that we'll be reassembling the parts like puzzle pieces and like we did in the lesson called Puzzle it Out.
You will need copies of the Creating Rational Puzzles directions, heavy cardstock (or blank index cards) and scissors for each team. I ask students to pair-share ideas about how to reassemble the components we've identified. (MP7) While they're working I walk around and ask specific students to post their idea on the board. I select students who have either a unique approach or who need some positive reinforcement. Once we have several ideas posted I ask my students to review them to see if there are any that they did not think of, do not understand or believe are not equivalent to the original expression. (MP3) When all questions have been answered, I tell them that they will now get to create their own rational puzzles. I distribute the directions then have my students work in teams of two to create at least five rational expressions/equations and then decompose them into component parts, making their own rational "puzzle sets" on heavy cardstock. I discuss why this is a good lesson for manipulatives in my video. When every team is done, I have them swap puzzles with another team and work to reassemble those new sets. (MP1, MP7) I tell each team that when they come up with a new way to put the puzzle pieces together they need to make sure that their new expression/equation is equivalent to the original. I then say that once they've confirmed that each new "puzzle" works, they need to write down both the original expression/equation and the new one they've made on a separate sheet of paper. I walk around while my students are working offering encouragement and redirection as necessary. You may have students more advanced than mine who don't need to have a whole day for this lesson, but I have found that if I skip this or try to shorten it too much, I just have to spend more time reteaching later.
Because I had my students work in teams for most of this lesson, I chose to have the wrap up piece be independent. Each student must identify components of a new example (like the one given below) to demonstrate their individual competence at decomposing rational expressions. I give each student a notecard and ask them to copy the expression, label all the components, and write at least one new equivalent expression. (MP1, MP7) This is their ticket out the door today!
(1 + 3/r)/( 2 - 5/r^2)
or any other iterations of regrouping!