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 represent and solve problems that add, subtract, multiply and divide whole numbers and decimal tenths, using tools and strategies that include algorithms and arrays. The museum offers the history of how such written steps reached the Islamic world and where the word "algorithm" comes from; the problems, the arrays and the practice are your class's. 1
Curriculum expectations, with their official codes
- B2.4 represent and solve problems involving the addition and subtraction of whole numbers that add up to no more than 10 000 and of decimal tenths, using appropriate tools and strategies, including algorithms
- B2.5 represent and solve problems involving the multiplication of two- or three-digit whole numbers by one-digit whole numbers and by 10, 100, and 1000, using appropriate tools, including arrays
- B2.6 represent and solve problems involving the division of two- or three-digit whole numbers by one-digit whole numbers, expressing any remainder as a fraction when appropriate, using appropriate tools, including arrays
2 The lesson
Which operation is the problem asking for, and in what steps?
Read each problem and decide whether it joins, separates, joins equal groups or shares equally. Then write your steps in order so that someone else could follow them.
Go deeper
Ibn Khaldun counts four jobs in arithmetic. Joining one by one is adding, he says, and joining by multiples is multiplying; separating one by one is taking away, and separating into equal parts is dividing 2. The record on algorism says that with the Indian numerals these four operations become steps repeated place by place 2. From al-Khwarizmi's name comes the word "algorithm", for an ordered way of solving problems 3.
Think about it
Ibn Khaldun describes multiplying as joining by multiples and dividing as separating into equal parts. Why might it help to see multiplying as a kind of joining and dividing as a kind of separating when you decide which operation a problem asks for?
Into the museum
- Algorism: the numerals, the steps and the word "algorithm".
- Ibn Khaldun: the author of the Muqaddima, where the passage on arithmetic comes from.
- al-Khwarizmi's Book of Arithmetic: the words that came from his name.
Try this
Read the record Algorism. Choose one addition and one division problem from your class, solve each in ordered written steps, and name the operation each step does.
3 Carry it further
Where this lesson leads in the museum.
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
Joining and separating. The record gives Ibn Khaldun's account of arithmetic as reckoning numbers by joining and separating: joining one by one is addition, and joining by multiples is multiplication; separating one by one is subtraction, and separating into equal parts is division. Indian reckoning, it says, is doing these four operations with numerals written in their places. Which of the four does each of your problems ask for?
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
From al-Khwarizmi to "algorithm". Where does the word "algorithm" come from? The record says al-Khwarizmi's Arabic book on Indian reckoning is lost, but a twelfth-century Latin translation survives that opens "Dixit Algorizmi", "al-Khwarizmi said". The editors of the Cairo edition of his algebra say that Westerners learned arithmetic from this Latin translation and from books built on it; his Latin name, "Algorismus", became a name for the numerals themselves, and from it comes the word "algorithm". Which steps does your own method for adding follow?
I can now