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Mathematicians sometimes cringe at discussion of real-world problems, and write rants decrying the state of mathematics education.
This is because they have full awareness of the beauty of mathematics. What attracted them to mathematics in the first place was not being able to build a better bridge, or track their stock market portfolio, but the simple elegance of, say, Euclid’s proof that the primes are infinite.
However, most students – even if they are taught in an entirely traditional, abstract manner – will never see that beauty. That is because understanding of the subject must be past a certain level before all the pieces fall into place (often this happens at the level of Calculus, which many students never take).
Is there a way to motivate students in the same way a real-life problem might, yet give a direct impression of the beauty of the abstract? I claim it can be done through puzzles.
Puzzle: First, cut the three pieces out. Then making only one more cut (straight or irregular) so you have four pieces, arrange the pieces into a square.
There are multiple solutions, but here’s the one most students come up with:
Arranged they become:
This procedure is precisely what happens with completing the square. Consider the 4x4 square to be x^2, the 2x4 square to be 2x, and the 1x1 square to be 1.
To arrange the pieces into a square, the 2x piece must be cut in half, so each piece is now 1x. The 1 sides in the lower right corner deliniate a missing 1x1 square, which is filled by the last piece.
Now the pieces make a square where each side is x+1, so the area of the square is (x+1)^2.
This is equivalent to completing the square in algebra:
Note that any other solution to the puzzle (for example cutting down the 4x4 piece into a 4x3 and 1x4) can be shown as algebraically equivalent.
Through a simple puzzle the students explore directly an algebraic manipulation which is nothing but an abstraction. The equivalence of the multiple solutions gives a direct (if still perhaps elusive) impression of the beauty of mathematics.
 Jason Dyer holds degrees in Fine Arts Studies and Math and teaches at Pueblo High School in Arizona. His school mascot is the Warriors and his other blog of residence is The Number Warrior.
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