I do a quick opening today to remind students of the work we did in the last lesson. In the last lesson, students shared out how they found the number of shaded tiles needed for the checkerboard problem. Our next step is to try to generalize this work. I let students know they'll be looking at a different sized square to see if their same method for finding the number of shaded tiles will apply. This work follows the Building More Checkerboard Borders task, but the worksheet may be unnecessary or actually confusing to students.
Next, I send students back to work to try out another sized square. I often have students try to replicate their counting method on a 7 x 7 square. Some students will choose other sizes. I try to steer students away from looking at an even by sized square for now. The pattern is hard to recreate on an even square so we look at that case toward the end of the lesson. Even in the odd by odd square, some students will have difficulty figuring out where the shade the tiles. I like to encourage them to start at the corners and work from there. Once they have a new sized design, I ask them to apply their 5 x 5 method to the new square. Does it still work? How have the numbers changed (the operations should stay the same)? From here, I can ask students to think about an n x n square. Can they generalize their work and come up with an algebraic expression to represent the number of shaded tiles they'll need?
This work is a great opportunity to emphasize the Standard for Mathematical Practice 7: Look for and Make Use of Structure. Students generate their own ways to understand the border patterns and then work on representing that pattern algebraically.
We come back together to see what students found. From here, we start to write algebraic expressions to match the methods that the students used to keep track of the shaded tiles. This checkerboard squares student work shows four different ways that students were able to represent the shaded tiles. Once we have worked through each method, I write a list of all the algebraic expressions we have generated and ask students how we can simplify them. We spend some time talking about equivalent expressions and how all of the expressions are equivalent even though they show a different underlying structure to the problem.
What about an even sized square? I finish class today by asking students if their method will work on an even sided square. I encourage students to start by looking at a 6 x 6 square and seeing how they would shade the border tiles. We have a discussion about how the math will work with a 6 x 6 square (their algebraic expressions will still hold), but the design of the cafeteria floor doesn't really hold. We talk about what this means and if the math problem should have been written this way. Ultimately, I ask them if it's fair to students to include an even by even sided square in this problem? There is no real "right" answer to this question, and I let students weigh in with their own opinions. This year, students reported that the even by even square only really serves to confuse students. We talked about how we could restrict the problem by looking at it as a function and determining that the domain of the square should be to have odd sides.
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