Draft: this lesson was written with AI help and has not yet been read by a human editor.
1 What your class asks
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
Curriculum expectations, with their official codes
- C2.1 translate among words, algebraic expressions, and visual representations that describe equivalent relationships
- C2.3 solve equations that involve whole numbers up to 100 in various contexts, and verify solutions
2 The lesson
How can one relationship be said in words and in symbols?
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.
How do you know a solution is right?
Put your answer back into the equation. Say what it means when both sides match, and what to do when they do not.
Go deeper
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.
Think about it
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?
Into the museum
- 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.
Try this
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.
3 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Algebra (al-jabr wal-muqabala as a discipline in al-Khwarizmi's work) have to do with this lesson?
Compare with the card
Finding an unknown number. Algebra, al-jabr wal-muqabala, is the art of finding an unknown number from known ones through an equation. Al-Khwarizmi set it out in Baghdad under the caliph al-Ma'mun in a concise book, for people's needs in inheritance, bequests, trade and land measurement. The TDV biography of al-Khwarizmi describes his drawing with squares and rectangles as a geometric representation and verification of the solution, and TDV notes that his algebra was wholly in words, with no symbols. How could you show the same unknown in words, in a drawing and in symbols?
I can now