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Thought for the day: The simplest pictures may sometimes lead to surprisingly deep discoveries.

Concepts: length and area; properties of angles; square roots; connecting number patterns to algebra (possibly the Pythagorean theorem)

 
 

Examples of noticing and wondering

I notice two overlapping squares of different sizes, one tilted at a 45° angle to the other.
I notice that a side of the large square is a diagonal of the small square.
I notice that the diagonal of the smaller square looks about 1 and a half times the length of its side.
I notice that the orange region where the squares overlap is half the area of the small square.
I notice that four copies of the triangular overlap fit into the larger square.

I wonder what the area of the larger square is.
I wonder what the side lengths of the larger square are.
I wonder how the other lengths and areas would change if the side length of the smaller square were 2.
I wonder what would happen if I made a picture like this beginning with a non-square rectangle.
I wonder if it would help to use graph paper to make my new drawings.

 
 

Notes

This image has to do with ideas related to the Pythagorean theorem.

The area of the small square is 1 square unit. The area of the triangular overlap is half of a square unit. Since you may join four of the orange triangles to create the larger square, the area of the larger square is 1/2 • 4 = 2 square units. Finally, because the area of a square is its side-length-squared, the side length is the square root of its area. Consequently, the side length of the larger square (also the diagonal of the smaller square) is the square root of 2, which is approximately 1.414. This is a reasonable answer. Before making this discovery, most students tend to estimate that a diagonal is about 1.5 time as long as the side length of square.

Notice that students were able to find the length of the diagonal without using the Pythagorean theorem! If they continue to construct "tilted squares" from the diagonals of other rectangles, they have an opportunity to discover a pattern: the area of the tilted square is always the sum of the squares of the side-lengths of the rectangle, thus leading to a further discovery—the Pythagorean theorem! For a more structured set of problems built around this type of image, see Exploration 4: Geoboard Squares and Exploration 6: A New Slant on Measurement from my book, Advanced Common Core Math Explorations: Ratios, Proportions, and Similarity.

I would strongly recommend that students make any new drawings on graph paper in order to help them determine areas. Regardless, non-square rectangles are more challenging to work with than squares for at least a couple of reasons: (1) it may be difficult to ensure that the angles in the tilted shape are actually right angles, and (2) four copies of the triangular overlaps will not generally completely cover the tilted square. 
 

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