How many squares are there in a 8X8 chessboard?

8X8 Chessboard

This question may be confusing for students because it is actually about the total number of any possible squares in a 8X8 chessboard, not simply the number of 1X1 squares in a 8X8 chessboard. To help students understand better, the teacher can break it into smaller chessboards and find patterns.

Number of squares in 1 X 1 chessboard -> 1 square

Number of squares in 2 X 2 chessboard:

1X1 : 4 squares,   2X2: 1 square   -> total number of squares=4+1=5 squares

Number of squares in 3 X 3 chessboard:

1X1: 9 squares

2X2: 4 squares (look below)

3X3: 1 square

è  Total number of squares = 9+4+1=14 squares.

We can look at the total numbers of squares in each chessboard and see a pattern.

1X1: 1 = 1^2

2X2: 5 = 1^2+ 2^2

3X3: 14 = 1^2+ 2^2+3^2

...

8x8: 1^2+2^2+3^2+4^2+5^2+6^2+7^2+8^2=1+4+9+16+25+36+49+64=204

è  There are 204 squares in a 8X8 chessboard.

To extend the puzzle for students:
I will first let students solve the problem by themselves and see the point where they have difficulties in common. Then, I will show them a clear definition of rectangles and squares to help them find possible outcomes in a 8X8 chessboard.
Or, I can draw a different picture of the equilateral triangle, which is divided into smaller equilateral triangles. Then, I will show them how to find possible equilateral triangles within the big equilateral triangle.