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### Reflecting on the How Many Squares on a Checkerboard? Problem

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So, how did things go in your classroom with the How Many Squares on a Checkerboard task? I’m told that we’re still getting a good number of visits to the blog, but few visitors are taking the next step and leaving a comment. The hope is that the blog will become an interactive way for members to share thoughts, comments, ideas, and so on. Please realize that a comment need not be long—a simple “Tried it with my class—they really liked it” is great. Maybe your students solved the task in a really interesting way that you hadn’t thought of before; maybe you had to remove roadblocks before students got going on the task. We would love to hear from you!

All right, so let’s have a look at the How Many Squares on a Checkerboard task and some approaches to solving it. A common strategy is to start with a simpler problem:

How many squares on a 1 × 1 square? 1

How many squares on a 2 × 2 square? 5
(four 1 × 1 squares and one 2 × 2 square)

How many squares on a 3 × 3 square? 14

For this last one, solvers must see that the board has not only squares of different sizes but also overlapping squares, so a 3 × 3 square has 9 (nine) 1 × 1 squares; 4 different 2 × 2 squares (overlapping, as demonstrated in the checkerboard examples below; and 1 (one) 3 × 3 square), and so on, until eventually arriving at the following solution:

How many squares on an 8 × 8 checkerboard? 204

 64 1 × 1 squares 49 2 × 2 squares 36 3 × 3 squares 25 4 × 4 squares 16 5 × 5 squares 9 6 × 6 squares 4 7 × 7 squares 1 8 × 8 square

Strategies for identifying and extending patterns, drawing diagrams, making a table, and so on soon come into play. Often the final result for the Checkerboard problem is presented in a table like the one below.

Students and teachers readily identify patterns that emerge. The problem can then be extended to determine the total number of squares on any size square board, with the corresponding algebra being introduced as appropriate.

In the higher grades, an extension might be to find the total number of rectangles that can be found on an 8× 8 checkerboard. (Warning: This is not at all trivial!)

So what did you think of the Checkerboard task? How did your students respond to the task? What strategies did they use? Did they encounter any stumbling blocks? How did the student resolve the stumbling blocks? You are welcome to share class photos or student work samples. Hope to hear from you!

Ralph Connelly is Professor Emeritus in the Faculty of Education at Brock University in Ontario, where he taught elementary math methods courses for 30+ years. He is active in both NCTM, where he’s served on several committees, currently the Editorial Panel of TCM, and NCSM, where he’s served two terms as Canadian Director as well as on numerous committees.

 This Student Math Notes activity is a great tool for exploring the checkerboard task.http://www.nctm.org/publications/article.aspx?id=22347Posted by: ElizabethS_05318 at 8/13/2014 10:49 AM

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