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 evaluate algebraic expressions with whole numbers and decimal tenths, and to solve equations with several terms, checking each solution. The museum offers a book whose title gave algebra its name, and some of its worked examples. 1
Curriculum expectations, with their official codes
- C2.2 evaluate algebraic expressions that involve whole numbers and decimal tenths
- C2.3 solve equations that involve multiple terms and whole numbers in various contexts, and verify solutions
2 Before you start
A thought to carry into the lesson.
3 The lesson
How do you know an answer is right?
Put your answer back into the equation and see whether both sides are equal. Then explain, step by step, what you did to each side.
Go deeper
Al-Khwarizmi says that reckoning by al-jabr and al-muqabala needs three kinds of number: roots, squares and simple number 2. A square, he says, is what results from a root multiplied by itself 3. They combine into six types of problem. One type is “five squares equal eighty”, so one square is a fifth of eighty, sixteen 4. In each type, more or less than one square is reduced to a single square, and whatever equals it is treated likewise 3.
Why they did it
In his preface al-Khwarizmi says he wrote the book with the encouragement of the caliph al-Ma'mun, whose love of learning and closeness to scholars he praises 5. He says he made it for what people need in their dealings, such as inheritances and trade 2. These are his words about his own purpose, as the record reports them from the preface of the book.
Think about it
Al-Khwarizmi insisted that everything worked by al-jabr and al-muqabala must bring you to one of six types 6. What do you gain, and what might you lose, when you fit many different problems into a few types, and why?
Into the museum
- Algebra (al-jabr): the three kinds of number, and the two operations that name the science.
- The six types of equation: a worked example for each type.
- The Book of al-Jabr and al-Muqabala: the preface, and the chapters in order.
- al-Khwarizmi: what is known of his life.
Try this
Read the records on algebra and on the six types of equation. Take the example of four roots equal to twenty, write it with a letter for the unknown, solve it, and check your answer.
4 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Algebra (al-jabr) have to do with this lesson?
Compare with the card
Restoring and cancelling. Algebra is the branch of mathematics that solves problems by positing an unknown and handling equations by general methods. The editors of the 1939 Cairo edition of his book note that the name of the science in every language comes from al-jabr, the word al-Khwarizmi used as the title of his book. In his problems al-jabr completes a quantity that has a deficiency by adding its like to both sides, and al-muqabala casts away like terms from both sides. Which of the two do you use when you balance an equation?
In your own words: what does The six types of equation (al-Khwarizmi) have to do with this lesson?
Compare with the card
Six types of equation. Al-Khwarizmi divided everything he needed in al-jabr and al-muqabala into six types. One simple type is "four roots equal twenty", so the root is five and the square twenty-five. Check the answer by putting it back in the equation.
I can now
Notes
- 1. Ontario Ministry of Education, The Ontario Curriculum: Mathematics
- 2. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.30
- 3. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.31
- 4. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.32
- 5. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.29
- 6. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p.43