To begin this lesson I refer back to an earlier lesson about closure called "Open and Shut". I ask for volunteers to summarize what we learned about polynomials and offer the hint "What operations are polynomials closed for?" if needed. Once we've completed that review by correctly identifying which operations are closed and what "closure" means I tell my students that today they get to work with rational numbers to determine what operations they're closed for. I hand out notecards and tell my students that they need to state their opinion about closure for each operation (addition, subtraction, multiplication, division) and give supporting evidence for their opinion. (MP2) When everyone has written their opinion I open the floor for a class discussion about how to prove it. My goal is that my students feel ready to develop proofs for their opinions about closure for each operation. I explain more about why I have my students state and support their opinions in my Just Do It video. This is a plus standard but I feel that it gives my students an opportunity to reinforce their understanding of closure in general and polynomial operations more specifically.
Teamwork 20 minutes: Today's lesson is a little unusual because I'm not giving my students any problems or examples to work with. I have them get into groups of 2-3 with partners sharing the same opinions about the system of rational expressions and what operations it is closed for. I tell them that their challenge today is to attempt to prove or disprove their own conjectures about whether rational expressions are closed for addition, subtraction, multiplication, or division and that they will have to present their evidence to the class. (MP1, MP2, MP8) While my students are working I walk around offering encouragement and redirection as needed. I particularly watch for teams that are struggling with how to prove/disprove their opinions and ask leading questions like "Have you tried to find any polynomial examples that work or don't work?" or "What examples have you tried so far?"
After about twenty minutes I tell my students to prepare to present. I randomly select teams to present their evidence/proofs to class for discussion and critique. Students who are not presenting are expected to offer comments, questions, or critiques about each presentation. (MP3) When all the teams have presented I guide the class in a discussion of what they now believe. I help my students reach a class consensus about what operations rational expressions are closed for and what evidence they've used to reach that consensus.
To close this lesson I ask my students to summarize individually in their notebooks what operations rational expressions are closed for and how they know. I tell them it's not okay to say "we did it in class". (MP7) This closure piece ensures that every student has a record of what the day's discussion was about and it also gives them the opportunity to express themselves in words rather than algebraic or numeric symbols.