First Week Idea - Chessboard Squares Extension

I love the YouCubed Week of Inspirational Math ideas!  I love the idea of starting at the very beginning of the year inviting our students to explore patterns, create generalizations, and communicate math!!

With that in mind, I love to stumble on ideas ... some olds ones that need revamping ... like the one I'm sharing today!

I imagine many of us have challenged students with the question - "How many squares are on a Chess board?"  And of course, the first response is often 64, which allows us the opportunity for a sly smile, raised eyebrows, and the comment, "Really?  Are you sure?"

One of the fun things I am doing in retirement is sharing Usborne Books & More with folks ... yes, I'm not just a math lady ... I'm also a book lady!  So I was pleasantly surprised the other day when I was looking through Anna Weltman's, This is Not Another Math Book**, I found a great extension of the question about squares on a Chess board!

Obviously in doing this activity, you want to start with the basic challenge!  And have you ever noticed the pattern??

1    8x8 square

4    7x7 squares

9    6x6 squares

16  5x5 squares ... YES!

The number of subsequent numbers of squares are all square numbers!!  WooHoo!

The extension in Weltman's book above asks students to consider a rectangular board.  How many squares are on a 5 x 8 board or a 4 x 6 board?  Can they not only answer those questions ... but can they generalize the rule??!  Can you make a rule for how many square there are in ANY SIZE rectangular chessboard!

**If you are interested in knowing more about the book I've referenced, let me know.  It is currently on special.  Publisher's Weekly says this about the book:  This companion to THIS IS NOT A MATH BOOK continues to explore the connections between mathematics and art via more than two dozen step-by-step projects. Most of the projects require only the paper templates provided, along with writing utensils, a ruler, tape, and scissors. They include drawing kaleidoscopic patterns (as part of an exploration of symmetry), making a forest of fractals trees, and assembling five-square pentomino shapes (they resemble Tetris pieces) into pictures. There isn’t much discussion of how artists use math (Mondrian is the only fine artist mentioned), but Weltman’s projects ably demonstrate how creative mathematics can be.  

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