The purpose of this lesson is for students to explore the structure of equations and play around with changing the different steps of a scenario to see how that changes the resulting equation. I really like the task in this lesson and there are a lot of different ways it can be taken. I find I often come back to the scenario of this lesson to help students understand how to solve equations by "undoing." I also find this to be a great task for practicing the standards of mathematical practice #7 and #8: Look for and make use of structure and Look for and express regularity in repeated reasoning.
We start today's class by reading through the problem that starts half way down the first page. NOTE: we are not doing the trays problem at the top of the page, we looked at this task in the last lesson for an example of how to write about math. Instead, we are starting with the Post-It problem that starts half way down the page. We read the problem aloud together and stop at the end of Question 2.
Students begin working individually or in small groups. Most students will want to solve the original problem right away without first thinking about the order, which I think helps them get a stronger sense of the problem. I look for students who are using guess and check and for students who might try algebra to represent this situation. I want to make sure each of these methods is represented when we share out during the discussion period of class.
Once students have solved the original problem, I give them three post-it notes and ask them to write the middle three steps on the notes. Next the problem asks them to rearrange the notes in different orders and figure out how many students would have started at the original table. I have students who are ready try to write their own equations for these new orders. If they finish this task, they can move on to Question #3.
This task lends itself to lots and lots of discussion. There are many ways to facilitate this discussion and I like to start with students' reasoning as the foundation for any way the discussion might go.
I usually start with a student who has used guess-and-check to solve the original table problem (before changing the order of the post-its). I might ask them to share out the first couple of numbers they tried. One great strategy to introduce students to here is called "Guess, check, generalize." It's a great way for students to practice SMP 8: Look for and express regularity in repeated reasoning. Students should notice they are repeating the same operations each time they guess a different number. First, they multiply their guess by 2, then add 4 to that number, and then divide that number by 3. We can help students to see here that their guess, the unknown, could also be represented by the letter x. From here, students can write the final equation which incorporates all three steps.
I also like to have a student share out who has started with the algebra in the first place. It is also interesting to have students share who have worked backwards through the problem, starting with the final number of students at the table and undoing the operations. Some students may be confused about why we do the opposite operation if we work backwards, for example, we would subtract the 4 students who come to the table rather than add them. This is a great opportunity to connect this work to the algebraic equation because some students may more readily see the "undoing" in the equation itself.
Lastly, we need to talk about the post-it order and how that changes the equations. So much to discuss! Again, students can look at the reverse order and undoing here of the equation in order to understand how the operations affect the number of students who started at the front table.
The discussion in today's class is likely to run long so we likely won't have as much time for reflection as usual. Sometimes I like to end class with appreciations where about three different people share out something they appreciated about today's class. I usually start and focus on something like how much effort or focus students put in to today's lesson. Then I open up the floor to a few other students to share their appreciations. They might appreciation another student who helped them out, or something about the math in particular. I find this to be a positive way to end class when I don't have time for a more in-depth reflection.
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