I wanted to minimize the scaffolding I was providing with this warm and leave as many questions open as possible—so students ask what is supposed to go in the data table, and what they are supposed to graph—and I tell them, “Set it up however you think will help you best understand the problem.” Most students end up setting it up the same way, modeling their work after yesterday, but even just this tiny extra step of asking them to think about how to use these representations helps make the “work” more about thinking and less about “filling in boxes.” I am going to try to continue to this as often as possible—to gradually remove scaffolds in order for students to become more comfortable thinking while working, and to move them away from filling stuff out. This to me is one of the fundamental shifts of the Common Core.
I included the “Important Questions” because I wanted students to take the time to make sure that they actually understood the connections between the tables, the graphs, the equations and the original problem. Even making it so big and obvious, some students still skipped these questions, because they felt like the “first page was easy,” and I directed all students to go back and answer these questions, as a way of showing that they really understood the problems. This was often best done orally, as I circulated.
Again, the levels are built in to make it easy for you to give everyone ample time to think about these problems—for students who fully understand the first problem quickly, there are many levels of challenge available, so you can have everyone work on this until the students who need the most time fully understand the first page.
The Equivalent Expressions Problem Set.pdf asks students to rewrite expressions in many different forms. This is another great opportunity for students to use MP5, because they can easily use the graphing technology to check their answers, if it is available. If they are not sure how to check their answers by graphing, ask them what the graphs look like if the two equations are equivalent and they realize that the graphs will be the same. Some students simply apply the formulas from the example, without understanding the algebra behind it, so when other students are using algebraic methods to check their answers, I ask them to explain this to their peers or write them on the board so that other people could learn from it.
To close the lesson, I want to bring students’ attention to one big idea: the different forms of the equations show different things about the problem. Because this idea will come up throughout the year, it is okay if they don’t immediately understand it. I ask them to look at each form of the equation from the example and to think about what this shows about the original problem. I also ask them to explain how they can determine whether or not two expressions are equivalent both graphically and algebraically.
The idea is to give students the chance to think about these big ideas on a daily basis so that over time they can develop a deeper understanding. After giving them a chance to think about these questions, I ask a few students to share and then highlight important ideas before asking them to write more about the questions.