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 build whole numbers by their places, and to read, compare and order decimal numbers up to hundredths. The museum offers two records: how nine numerals and a zero give each place its value, and an early treatment of tenths and hundredths. Large whole numbers up to 100 000, comparing and ordering decimals, and everyday uses of numbers are for your class to bring. 1
Curriculum expectations, with their official codes
- B1.1 read, represent, compose, and decompose whole numbers up to and including 100 000, using appropriate tools and strategies, and describe various ways they are used in everyday life
- B1.5 read, represent, compare, and order decimal numbers up to hundredths, in various contexts
2 Before you start
A thought to carry into the lesson.
3 The lesson
How does a digit's place change its value?
Write a number with the same digit in several places and say what each one is worth. Then break the number apart by its places and put it back together.
What happens to place value below one?
Look at how each place to the right is worth less than the one before it. Compare two decimal numbers place by place, starting from the largest place.
Go deeper
Start with 13 and halve it again and again. The historian Saidan describes the list 13, 6.5, 3.25, 1.625, 0.8125, and says the steps can be undone by multiplying by two 2. The record writes the numbers with our modern dot, not as al-Uqlidisi drew them: he put a line over the units place and insisted that it be used 2. Keeping the units place marked is what lets us read the places below one, instead of mistaking small parts for a large whole number 2.
Think about it
Each time you halve 13 the number gets smaller and needs more places. Why would a writer need to mark where the ones place is before writing 0.8125, and what could go wrong without a mark?
Into the museum
- al-Uqlidisi's decimal-fraction methods: the halving list and the units mark, with the historians' disagreement about them.
- Algorism: the nine numerals and the zero that give each place its value.
- al-Uqlidisi: what is known of the author, and what is not.
Try this
Read about al-Uqlidisi's decimal-fraction methods. Write a half, a quarter and a tenth of a unit as tenths and hundredths.
4 Carry it further
Where this lesson leads in the museum.
5 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
Numbers that take their value from their place. Algorism is a way of writing and working numbers with nine numerals and a zero, in which each place takes its value from its position: units, then tens, then hundreds, then thousands. Muhammad ibn Musa al-Khwarizmi wrote a book on it in Baghdad, through which the Indian numerals and the decimal system entered the Islamic world. In the numbers 5, 50 and 500, what is the digit 5 worth in each?
In your own words: what does Al-Uqlidisi's decimal-fraction methods have to do with this lesson?
Compare with the card
Tenths and hundredths. A half can be written as five tenths, and half of that as twenty-five hundredths. A book composed by Abu al-Hasan al-Uqlidisi at Damascus in 341/952–953 contains an early direct treatment of such fractions within Indian place-value arithmetic. The record adds that this is not the invention of Indian numerals or of today's decimal point. Which place holds the five in five tenths?
I can now
Notes
- 1. Ontario Ministry of Education, The Ontario Curriculum: Mathematics
- 2. Al-Uqlidisi — A. S. Saidan's DSB biography, decimal-fraction discussion