Finding the unknown

This lesson asks you to translate between words, algebraic expressions and pictures that describe equal relationships, and to solve equations with whole numbers up to 100 and check the answers. The museum offers one record on al-Khwarizmi's algebra: finding an unknown number through an equation, his drawings of squares and rectangles, and the fact that his algebra was written in words without symbols. The algebraic expressions, the equations and the checking are for your class to bring. ¹

Take a sentence about an unknown number and write it as an equation. Then draw it, for example as a balance, and check that all three say the same thing.

Put your answer back into the equation. Say what it means when both sides match, and what to do when they do not.

Al-Khwarizmi had no symbols, so he gave every step in words. His best-known example is “a square and ten roots equal thirty-nine dirhams”. His rule runs: halve the roots, which gives five; multiply it by itself, twenty-five; add this to thirty-nine, sixty-four; take its root, eight; and subtract half the roots, five, which leaves three. That is the root, and the square is nine ². His algebra was wholly in words, with no symbols, and it accepted only positive roots ³.

Al-Khwarizmi wrote these steps only in words. If you wrote the same steps as an equation with a symbol, which would be easier to check, and why?

• Algebra: the book, the six kinds of problem, and the squares he drew.
• The six types of equation: one worked example for each kind.
• The Book of al-Jabr and al-Muqabala: the book chapter by chapter.

Read about algebra in al-Khwarizmi's work. Write your own problem about a number you do not know, and set it up as an equation.