This is the third day of a long term project started in previous lessons (Checkerboard squares & Squares in a row) 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. In previous lessons students have already begun using data tables to record the different size squares possible and how many of each there are. Students will be at different points in the project and today is a workday for them. Some students will have completed two checkerboard problems from their homework, some one, some neither. Some will have already discovered patterns in the data table which will help them generalize later. As students work on the warmup I am circulating to see at what stage each student is. I mark down who is ready to move on, because those students may need to be paired up and moved. (Groups are being changed next week). I also take note of who is behind so I can work with them.
Students begin the warmup projected on the screen as they enter and continue for about 5-7 minutes. The first problem asks them to look for a pattern in an In/Out table and complete the table. I want them to notice that they are all square numbers, because this will be an important pattern to recognize when we try to generalize the rule for the checkerboard problem. Many of them will notice only that the number going in is multiplied by itself, which I write down under the "out" number. I ask students if there is another way of writing that expression. It is possible that students will not recognize it after just one example, so continue writing 5x5, 7x7, etc. When they see several examples like this they are more likely to come up with 5^2, etc.
The second and third problem relate to the checkerboard problem that we started a couple of days ago and worked on in previous lessons in this unit (Checkerboard squares & Squares in a row ). When they suggest that there are 25 1by1 squares on a 5 by 5 checkerboard and 81 on a 9 by 9 have students explain how they knew that in order to reinforce the use of the area formula. The third question requires that they use this same strategy on the larger squares that can be found on the checkerboard. This is the part where some students may still be stuck and need some scaffolding during this lesson.
Before we begin I tell students that the remainder of the class period will be theirs to work and that on the next class day they will be expected to start work on a Poster for which their checkerboard problems need to be complete. I tell them I am going to introduce the poster task today, because some of us are ready to start planning it since they have finished the problems.
I pass out the poster requirements ( both sides - checkerboard poster requirements & checkerboard poster planning) to those who are ready and tell them they will be given a partner to work with and asigned a checkerboard size that we have worked with (4x4, 6by6, or 8by8). They will be required to create a poster modeling the problem and it's solution and include several elements as outlined in the requirements. They have a choice to create a poster on a checkerboard size we have not worked with, in which case only 2 of the elements are required to be on the poster. They will also be judged by me and by their partner on how well they collaborated as a team. I have created two posters as samples that I post in front of class for them to take a look at. I tell them to prepare by completing the back side of the poster requirements which instructs them on how to look for patterns in the data.
To find out who's truly stuck and who just hasn't gotten it done I ask who would like to take another look at the powerpoint. This helps students reenter at the point where they left off, but it also helps students who are still having trouble "seeing" those overlapping larger squares. It will only take 3-4 minutes to run through it again. Remember to stop on slides 12 and 13 to ask at the end how many total squares of that size will be found before going on. I would keep one of the slides that shows overlapping squares displayed for students to refer to.
I think it is really important for my students to spend time looking for patterns because it is the basis for generalizing those patterns into rules and writing variable expressions.
Before ending class I let students know that it is okay to take their homework home for one more night because it is important that they come prepared to contribute to the poster creation with their partner (they don't know who that is just yet) at the next class.