If the Pythagorean theorem is mysterious to you, try this activity. I can’t guarantee it will make everything clear, but I hope it will at least help.

You need paper, pencil, scissors, and a ruler; probably it wouldn’t hurt to have an L-shaped ruler to get right angles, or you could use graph paper. First, make five paper squares. One should be 5″ by 5″, two should be 4″ by 4″, and two should be 3″ by 3″. On one of the 4x4s and one of the 3x3s, draw 1″ grid lines to divide the big square into 1″ by 1″ squares (that is, draw lines that split the 3×3 into thirds both directions, and the 4×4 into fourths both directions).

Pause and observe first that you have just illustrated the fact that a square of side length A has area A² — the 3×3 is split into 9 square inches and the 4×4 into 16 square inches. We could call this an arithmetic abstraction of a geometric fact.

Take your 5×5 and one of each of your other size squares and arrange them to outline a triangle. Each square will be corner-to-corner with each of the other two, and the triangle they make in the negative space will be a right triangle with side lengths 3″, 4″, and 5″. The right angle will be made by the shorter two sides.

Now, take your 5×5, your unmarked 4×4, and your gridded 3×3. Place the 4×4 on top of the 5×5, matching one corner of each. Cut the 3×3 first into three 3×1 strips. Take one of those strips and cut it at the one-third mark, into a 1×1 and a 2×1. Now, take your two 3x1s, your 2×1 and your 1×1 and arrange them like puzzle pieces around the 4×4 to fill the rest of the 5×5. They should fit exactly (“exactly” given that these are not precision-machined parts). This illustrates that the area of the square made by the longest side of a right triangle is the sum of the areas of the squares made by the two shorter sides. Combining with the arithmetic abstraction of area above, we can represent this as the equation a² + b² = c², where a, b, and c are the side lengths of the triangle, with c the largest.

Conversely, we could also see this as a geometric representation of the arithmetic fact that 9 + 16 = 25. Most of the time we use arithmetic to represent geometry, though, because we can do the arithmetic without scissors.

The Pythagorean theorem says that if you have a right triangle, no matter what the side lengths are specifically, the square you make on the longest side will be exactly coverable by the squares made on the shorter two sides. You may have to be creative in how you cut up the squares, but there will be a way. For example, now take your 5×5, your unmarked 3×3, and your gridded 4×4. Cut the 4×4 in half into two 2x4s, and then cut one inch off the end of one of those so you have a 2×4, a 2×3, and a 1×2. If you now tuck the 3×3 into a corner of the 5×5, the pieces you just cut from the 4×4 should fit around it, but the way you cut up the 4×4 was different from the 3×3.