This is day two of what will probably be a week long lesson in which students will be expected to determine the number of squares on an 8 by 8 checkerboard, a 6 by 6, a 4 by 4, and a square checkerboard of any dimension. This lesson began with the previous lesson in this unit (checkerboard squares). Students will need to determine what different sizes of squares are possible and record the number of each size are on the checkerboard and then find the sum.
They will find patterns in the sums of the different sized checkerboards and generalize a rule in a variable expression to determine the number of squares on any sized checkerboard. For most of them this is the first time they have been asked a question like this so I wanted one with a pretty simple entry point and one in which there were easy points of scaffolding. Prior to this lesson students have already calculated the number of 1 by 1 squares on the standard checkerboard using the area formula. They have also discovered other sized squares, but may be having difficulty counting them at this point. Last night's homework was to try to find a good strategy for counting the larger possible squares of each size.
As students enter I hand them a half sheet with the 4 by 4 checkerboard on it. They need to find how many squares are on this checkerboard. As they spend some time working on it in their groups I circulate to see how far they got on their homework which was to find a counting strategy for the number of squares on a standard 8 by 8 checkerboard. In addition to working on the warmup groups are naturally sharing ideas they came up with from their homework since they apply to solving this problem as well.
As groups seem to be finishing up their warmup I get their attention and ask them how this warmup problem is similar to and different from the homework they worked on last night and will continue working on tonight. They will likely say that it is exactly the same only simpler or smaller. I ask why they think I would have them back up and work on a simpler problem. They will likely suggest that it helps to find a strategy that will work on the more difficult one. If they don't come up with that I just tell them. Going over the warmup will be part of the exploration section.
The point of this lesson is to get them organized, to display the data in a table so that patterns can begin to emerge. So, before we go over the solution for the 4 b y4 warmup problem I ask them to discuss what information they needed to first find the total number of all squares on the warmup problem. I want them to come up with two things:
At this point I ask them what different sized squares are possible on the 4 by 4 checkerboard. I project the 4 by 4 checkerboard onto the whiteboard so students can come up and highlight the different sizes using markers. I set up a table on the board to record the different possible sizes.
Then, starting with the 1 by 1 squares have students explain how they figured out how many there were. Reinforce the area formula here by counting sets of 4. Ask them how they knew there would be 4 along the other side without having to count them. This will help later. Fill in the table with 16 and 4x4.
Now the tricky part...a strategy for counting the larger overlapping squares. Ask students to share how they counted the 2by2 squares, have some come up if they are willing. Some students are still having trouble seeing the overlapping squares so the more ways students have of "seeing" them the better.
Some kids really like this last method because they only see one at a time and are not distracted by the overlap.
I really want a student to come up with the idea that if 3 squares fit along the top side of the board then the same number will fit along the other side of the board (because it is a square) and they can just multiply. It is possible no one will. If they don't ask them how they knew how many 1by1s would fit along the side without having to count them. Some other guiding questions that might help are:
Now I return to the 4by4 checkerboard and say that it was a little tricky counting those overlapping 2by2 squares in just one row and I really don't want to continue this way, does anyone have a suggestion. Hopefully someone will make the suggestion, but they still might not. Scaffold some more....
"I know that there are 3 of the 2by2 squares along the top side. I know that the board is a square. So, how many should fit along the other side?" Continue working out the solution for the 4by4 checkerboard in this way until you have completed the table. When adding up the totals to find the final answer remind them that they have strategies to add these numbers mentally and have them share some of those strategies (from earlier lessons in this unit "Let's talk addition", "What were they thinking", "Delightful decimals")
Before handing out their new homework sheet to work on I want to sum it up using the powerpoint we started in the last lesson (Checkerboard Squares). You may quickly start from the beginning slide or just start at slide 10. If you start at slide 10 make sure you review how they figured out how many 1by1 small squares were on the big checkerboard. When you get to a question be sure to pause before clicking to animate to get responses from students. When you get to the end of slides 12 and 13 be sure to ask how many total 2by2 and 3by3 squares there are and ask how they figured that before moving on to the next slide. This will reinforce use of the area formula and help surface patterns. Many students may notice the pattern right away, some won't see it until tomorrow.
As you pass out the new homework sheet tell them you have included a table for them to keep track of the data and point out that they may want to use one for the second problem as well, which is a 6by6 checkerboard. Tell them that many of them will find a final solution to both problems tonight, some may find just one of them. If they get stuck on the big one, remind them that working on the simpler problem first (6by6) might help. Any additional time can be spent working on homework together.