I begin this lesson with a word game, which tends to pique my students' interest since the words are not specific to math and the game is actually pretty simple. I write this challenge on my board "Change the word "wood" to "loan" in 4 moves or less by changing only one letter per move and making real words each time." Most of my students are ready to tackle this challenge and I share clues with the few who are reluctant or struggling. I allow about 5 minutes and if nobody has solve it by then, I walk them through a possible solution. (see below)
wood to loan in 4 moves, 1 letter change per move, must be real word each time.
wood - good - goad - load - loan
This sets us up to try the same kind of challenge with a rational expression. I explain that the rules are very similar; try to change the expression in 5 moves or less with only one change per move and each expression has to be equivalent to the original. I know this sounds like a strange way to work into rewriting rational expressions, but I've had good luck with most of my students over the years because they understand the step-by-step process better this way. The biggest stumbling block is students who want to make multiple changes and count it as just one move. I give my students a chance to solve some Example problems by themselves (MP1) while I walk around and observe, then either have one or more students demonstrate their answer on the board, or walk the class through my solution as shown in the example problems.
You will want to print out the Rational Expressions Cards on card stock and cut them out as cards, with one set per three or four students. I put each card set in a baggie to keep them separate and make it easier to distribute them. You will also need copies of the Rational Target Game Rules. For the main activity of this lesson I have my students play a target game to rewrite rational expressions using the basic rules we've just practiced with the example. (MP1, MP2, MP7) I group my students into threes and distribute the game rules. I generally select teams randomly with an eye out for any students who are really struggling. Those students I may place together so that I can more closely observe their game and offer support as needed. I tell the teams to review the rules carefully and ask if there are any questions before distributing the card decks. As they play through the game, I walk around offering encouragement and serving as the final arbiter for disputes. (I try not to intercede unless absolutely necessary to keep the game moving!) As the final ten minutes of class approach I advise my students to finish up the problem they're on and put away the materials.
To close this lesson I have my students work as a class to summarize their own rules of rational rewriting based on their practice with the game. (MP7) This gives them a chance to really think about how they did the rewriting and also gives them greater ownership of the process (instead of just writing down what I say). I explain more about why I think this is appropriate in my Rational video. After about five minutes I ask for a volunteer "scribe" to write on the board and invite my students to share their rules. When everyone has had an opportunity to share (including the scribe) I help my students summarize the information into a list of Rational Rules we can use throughout the year. I suggest (but don't require because this again encourages responsibility and ownership and allows students to keep their own set of rules in their own words it they choose) that they copy the rules into their notes. I will be posting a copy of what this year's class comes up with, but in the mean time I've summarized previous responses in the Rational Rules resource. For some of you this may not need to be a full day's lesson, but I've found that my students do better as the year progresses if I take the time now to help them reinforce their mathematical foundations and make connections.