Count the Rectangles Problem

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This can be used as an extension of 1st Grade, Lesson 2 or as a stand alone problem.

Assume that the figure below is a 3×3 grid of squares (It may not actually display this way on your screen).  If this doesn’t print out right, then please redraw the grid for your students.

Here are two challenging questions:

How many square are in the figure?

How many rectangles are there in the figure?


Answer for Number of Squares: There are nine 1×1 squares, four 2×2 squares, and one 3×3 square, for a total of 14 squares.

Answer for Number of Rectangles: There are 14 squares plus six 1×2 rectangles, six 2×1 rectangles, two 2×3 rectangles and two 3×2 rectangles for a total of 30 rectangles

The correct answers are not as important as the process of discovery.