Al-Khwarizmi's Algebra — Complete the Square with Paper Tiles

Around 820 AD, in Baghdad, al-Khwarizmi wrote a book on solving equations by 'al-jabr' -- restoring. From that word we get ALGEBRA, and from the Latin form of his name, ALGORITHM. His method has a beautiful picture you can cut out.

To solve x-squared plus 10x equals 39, cut a square of unknown side x (call it a biggish square), and two long strips each 5 wide (together they are the 10x). Arrange the square with a strip along two of its sides -- the shape is almost a bigger square, but missing a corner.

The missing corner is a 5 by 5 square -- 25. Cut it and add it in: now you have a perfect big square whose area is the original 39 plus the added 25, which is 64. A square of area 64 has side 8. But that side is x + 5, so x + 5 = 8 and x = 3. You solved a quadratic by rearranging paper.

What your tiles teach. (1) 'Completing the square' works for EVERY quadratic: add the square of half the x-coefficient to make a perfect square, then take a square root. (2) Doing those steps once in general gives the quadratic formula every student learns. (3) Al-Khwarizmi's leap -- treating an unknown as a symbol you manipulate by fixed rules -- is the foundation of all later mathematics, science and engineering. (4) His name lives on twice: in 'algebra', the language of the unknown, and in 'algorithm', the step-by-step recipe at the heart of every computer program.