(To be photocopied and cut in half – one for every student.)

Teacher’s Guide – Chess Board problem1. Show the chessboard to the class. Ask them: “How many squares are on this

board?”- Students will probably answer “64”. This is the wrong answer.- Eventually, one student may ask if the big square counts, too.

(Yes!) - This will probably get them to realize there are many types of

squares within the grid. You can encourage this thinking, if it does not come out naturally.

2. Hand out the chess boards to each student. Ask them to work in groups to determine how many squares there are in total.

3. Walk around the class, reminding kids that there are 2x2 squares, 3x3 squares, etc.

4. When one group gets the answer, or if all groups are stuck, draw the following chart on the board:

Size of Square 1 x 1 2 x 2Number of each size

64

5. Ask the students to copy the chart and fill it in to find the complete answer.6. The final chart should look like this:

Size of Square 1 x 1 2 x 2 3 x 3 4 x 4 5 x 5 6 x 6

7 x 7 8 x 8

Number of each size

64 49 36 25 16 9 4 1

7. Ask students if they notice a pattern in the last row of numbers. (they should notice that each number is a perfect square).

8. Ask students to write the solution to the chessboard problem as a sum of perfect squares.

82 + 72 + 62 + 52 + 42 + 32 + 22 + 12 = 204

9. Challenge the students to figure out how many squares would be in: a 50 x 50 grid? a 100 x 100 grid? an n x n grid? [this could be homework]