I start today's lesson by letting students know that we will be building on the previous lesson (Border Tiles) and looking at more complicated pattern. Like in the previous lesson, I will be asking students to find the number of shaded tiles needed without just counting them up. We read through Checkerboard Borders together and I ask them to come up with two different ways to "figure out" the number of tiles. It is important to differentiate this task from the Border Tiles problem by pointing out that in this pattern, the size of the square is inside of the border work. In Border Tiles, the shaded tiles were included in the length of the square.
Next, students get to work. I like to give students the option to work together or alone on this problem. Some students need help getting started. I might ask them how they are looking at the shaded tiles. Do they see them as an inner and an outer boarder? Do they seem them as covering the four sides outside of the square? Depending on their answer, I'll encourage them to see if they can find a way to keep track of them.
Once they find one way to determine the number of shaded tiles, I'll ask them to see if they can look at the problem in a different way and come up with another representation. I find that asking them to find more than one way can help them have more flexibility with the problem but may also help them see how their peers view the problem in a different way from them.
Once everyone has two representations, we'll group up to discuss what they found.
I like to have different students come up the Smartboard and "mark up" the Checkerboard graphic to show how they figured out the number of shaded tiles. As we look at each method, I'll ask students how they think their numbers might change if they had a 7 x 7 square instead of a 5 x 5 square? I ask them to predict what the new numbers would look like.
For example, if a student looked at the shaded tiles as having an inner and an outer border, she might have said she multiplied 3 x 4 because there are three shaded tiles on each of the four sides and then added that number to 4 x 4 for the outer border. I'll ask her how she thinks those numbers will change if she's working with a 7 x 7 square on the interior instead of a 5 x 5 square.
I like to have as many students as possible share their methods before moving on. This discussion section of the class is also a good opportunity for students to make connections between the different methods. Some will be similar, but just slightly different from the way another student looked at the same problem.
This is usually as far as I get in one class period of 60 minutes. Depending on how far we get with this discussion, I might ask students to predict whether they think their pattern will work for a different sized square (focusing on odd numbers) or assign them a homework assignment of looking at a different size. Ideally, the more times they can examine their pattern on different sized squares, they easier it will be for them to generalize in the next lesson.
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