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Thought for the day: Very simple 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)
Beginning
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 wonder what the area of the larger square is.
I wonder what the side lengths of the larger square are.
Exploring
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 how the other lengths and areas would change if the side length of the smaller square were 2 (or some other number).
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.
Creating
As they notice and wonder, students may create their own pictures like the one in this prompt. Many of their strategies may flow out of things that they have wondered about. For example, they may
Add more segments (or other features) to the prompt in order to understand it better.
Change the length of the side of the smaller square (or even make it a variable!).
Begin by defining the length of a side of the larger square instead.
Begin with a non-square rectangle, and create a square from its diagonal.
Draw many pictures like these. Look for patterns, and create a formula for the area of a square formed from the diagonal of a rectangle.
Begin with a non-square rectangle, and create another non-square rectangle from one of its diagonals.
Continue creating larger and larger squares from the diagonals in a single picture.
For any of these drawings, calculate lengths and areas, look for patterns, describe patterns algebraically, etc. Make any discoveries you can.
Reflecting and Extending
I notice that the area of the larger square is twice the area of the smaller square.
I notice that the length of a side of a square is always the square root of its area.
I notice that the length of a side of the larger square is the square root of 2.
I notice that I can also make squares from the diagonals of non-square rectangles.
I notice that when the original rectangle is not a square, I can decompose the square into four right triangles with a square in the middle.
I notice a rule for predicting the area of the large square from the side lengths of the smaller rectangle. (Add the squares of lengths of its sides.)
i wonder if I could use this rule to create a formula for “slanted” segments.
I wonder if I can prove that this rule is always true.
Notes
This image has to do with the Pythagorean theorem and related ideas.
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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