Intent This is the second POW of the course. The primary purpose of this, as for all POWs, is to give students the opportunity to solve a significant problem outside of class, to generalize their solutions, and to prepare a formal written account of their work.

Mathematics This POW will draw on and strengthen students’ ability to visualize geometrically, to collect and organize a complex set of information, and to generalize their solutions. Students are asked to generalize their methods for counting the number of squares of different sizes on an 8-by-8 checkerboard to produce a method for counting the squares on a board with dimensions n by n. This activity is connected to an important mathematical theme of this unit: patterns. In counting all the squares of different sizes on a checkerboard, students will have to be systematic to ensure they have accounted for all the squares. To do so, they will have to recognize patterns in the locations of squares of various sizes.

Progression This POW is posed toward the end of Communicating About Mathematics, and students will work on it into Investigations. Unlike The Broken Eggs, the student book includes no follow-up activities in support of the various components of students’ write-ups of this POW. Students will be introduced to summation notation in another activity, Add It Up, as they are working outside of class on this POW. This notation can then be brought into Checkerboard Squares as appropriate.

Approximate Time 10 minutes for introduction 10 minutes for discussion 1 to 3 hours for activity (at home) 20 minutes for presentations

Classroom Organization Whole-class introduction, concluding with presentations and class discussion

Doing the Activity The first part of every POW write-up is the student’s statement of the problem. Having students work on and share their problem statements soon after the POW is assigned, in addition to helping students to learn how to write a problem statement, will help clarify for many students what the problem is. Announce when the write-up is due. Solicit presenters immediately, or nearer the due date, either way reminding the presenters of basic expectations and providing them with transparencies and pens.

Discussing and Debriefing the Activity Discussion of this POW can possibly extend over two days. Before students turn in their write-ups, you might offer them an opportunity to review the work of other students. This is their second POW, so they will have formed some idea of what is expected, but seeing each other’s work may nonetheless be of great value. As they read other students’ work, you might have students focus on what makes a good paper, what makes an adequate paper, and what makes a poor paper. After the sharing of POW write-ups is complete, you might want to ask students to do focused free-writing on this topic: What makes a good POW write-up?(see “Focused Free-Writing” [link to Overview section on writing] in the Overview to the Interactive Mathematics Program). After they have written for about five minutes, let students share their ideas. They can read aloud from their written work or simply discuss what they wrote about. Have the assigned students give their presentations, limiting each to about five minutes. Encourage presenters to speak about their investigation process at least as much as they speak about their findings. When findings overlap, presenters may wish to emphasize slight nuances they saw, questions they explored, and the like. In the discussion that grows out of the presentations, focus on the patterns that students have discovered. Bring out that finding patterns helps us to analyze mathematical situations. Student interest may offer opportunities to extend the exploration. For example, this POW lends itself to trying to explain why the square numbers appear. Students, or you, may raise such questions as these: Why is the number of squares of each size itself a square number? Why is it the particular square that it is? The problem can also be another opportunity to use summation notation. You might inquire, Do you see a way to express your findings using summation notation?

Key Questions Why is the number of squares of each size itself a square number? Why is it the particular square that it is? Do you see a way to express your findings using summation notation?

Supplemental Activities Different Kinds of Checkerboards (extension) is a follow-up to POW 2: Checkerboard Squares in which students find the number of squares on nonsquare checkerboards and search for a general rule for checkerboards of dimensions m-by-n. Lots of Squares (extension) is a substantial investigation in which students are asked to divide a square into different numbers of smaller squares. The goal is to determine which numbers of smaller squares are impossible and which are possible and to prove their results.

Samples of Student Work Sample 1: Page 1 [Link to Checkerboard Squares Student 1a] Page 2 [Link to Checkerboard Squares Student 1b] Sample 2: Page 1 [Link to Checkerboard Squares Student 2a] Page 2 [Link to Checkerboard Squares Student 2b] Sample 3: Page 1 [Link to Checkerboard Squares Student 3]