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 read and represent whole numbers up to one million, to describe how they are used every day, and to use estimation and written algorithms with whole numbers and decimal tenths. The museum offers the story of one book of written arithmetic; the numbers and the practice are yours. 1
Curriculum expectations, with their official codes
- B1.1 read and represent whole numbers up to and including one million, using appropriate tools and strategies, and describe various ways they are used in everyday life
- B2.4 represent and solve problems involving the addition and subtraction of whole numbers and decimal numbers, using estimation and algorithms
- B2.7 represent and solve problems involving the multiplication of three-digit whole numbers by decimal tenths, using algorithms
- B2.8 represent and solve problems involving the division of three-digit whole numbers by decimal tenths, using appropriate tools, strategies, and algorithms, and expressing remainders as appropriate
2 Before you start
A thought to carry into the lesson.
3 The lesson
Why do the places matter?
Write a number such as 305 000 and say what each digit is worth. Then estimate before you calculate, and check whether your written answer is close to your estimate.
Go deeper
What is known about the man behind the book? The Cairo editors quote Ibn al-Nadim's Fihrist: he came from Khwarazm, devoted himself to al-Ma'mun's treasury of wisdom, was one of the astronomers, and people relied on his two tables known as the Sindhind 2. From the Fihrist and the preface of the algebra they show that he was al-Ma'mun's contemporary, though the years of his birth and death cannot be fixed 3. TDV adds that he died after 232 AH (847 CE) 4. These are the facts behind the algorism record's picture of al-Khwarizmi.
Why they did it
The record preserves no sentence in which al-Khwarizmi says why he wrote his book of arithmetic, and its Arabic original is lost 4. It does show where he worked: devoted to al-Ma'mun's treasury of wisdom 2. Ibn Khaldun says the craft of arithmetic arose from the need of reckoning in dealings, and that it was passed on in the cities by teaching it to children 5. The preface of his algebra names practical needs of reckoning as the reason for that other book 6.
Often misunderstood
"Al-Khwarizmi's book" can mean more than one book. TDV places the algebra before the Indian-reckoning book, and names another work of his on addition and subtraction that, from al-Baghdadi's quotations, appears concerned with finger reckoning; shared operations do not make these works one text 4. The Cairo editors list the Book of Arithmetic among four books of his that Ibn al-Nadim does not mention 3.
Think about it
The word "algorithm" comes from al-Khwarizmi's name, but he wrote more than one book about calculating. Why is it a mistake to say that whatever one of his books says must also be in the others, and what would you check before saying so?
Into the museum
- Algorism: the numerals, the lost original and the Latin translation.
- al-Khwarizmi: the sources for his life, and the dates the sources do not fix.
- al-Khwarizmi's Book of Arithmetic: the book, and the words that came from his name.
Try this
Read the record on algorism. Choose an addition and a multiplication, write the ordered steps you follow, and ask a partner to follow your steps exactly.
4 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Algorism (reckoning with Indian decimal numerals in al-Khwarizmi's book of arithmetic) have to do with this lesson?
Compare with the card
Nine numerals and a zero. Algorism is a way of writing and working numbers with nine numerals and a zero, each place taking its value from its position: units, tens, hundreds, thousands. Al-Khwarizmi wrote a book on it in Baghdad through which, says the TDV encyclopaedia, the Indian numerals and the decimal system entered the Islamic world. Numbers are added, subtracted, multiplied and divided in writing by ordered steps. Which steps do you follow when you add by hand?
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. 22
- 3. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p. 23
- 4. HÂRİZMÎ, Muhammed b. Mûsâ — TDV İslâm Ansiklopedisi (İhsan Fazlıoğlu), pp. 224–227
- 5. Muqaddimat Ibn Khaldun, p. 495
- 6. al-Khwarizmi, Kitab al-Jabr wal-Muqabala, ed. Mosharafa and Ahmad (Cairo, 1939), p. 30