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, represent, compare and order decimal numbers up to thousandths, and to show how fractions and decimals name the same amounts. The museum offers one early account of decimal fractions; the tools, the drawings and most of the practice your class brings. 1
Curriculum expectations, with their official codes
- B1.4 read, represent, compare, and order decimal numbers up to thousandths, in various contexts
- B1.6 describe relationships and show equivalences among fractions and decimal numbers up to thousandths, using appropriate tools and drawings, in various contexts
2 The lesson
How do you know two numbers are equal?
Look at the expectations: they ask for equivalences, shown with tools and drawings. Write a fraction you can split in half, write each half as a decimal, and compare the places you used. A hundred grid or a number line can show that your two names mark the same amount.
Go deeper
The record offers a teaching comparison: a quarter is twenty-five hundredths, and it is also fifteen sixtieths 2. These numbers are the record's own modern explanation, not a line from al-Uqlidisi's book. Equal amounts do not make the two writing systems the same: each decimal place divides the previous unit by ten, while sixty-based places use sixty 2. The question is therefore not only whether an answer is correct, but which scale supports further calculation 2. Saidan distinguishes giving a root on the decimal scale from converting its result into sixty-based notation 2.
Why they did it
The records read here preserve no sentence in which al-Uqlidisi says why he wrote on fractions. They show how he describes his own preparation: as the biography reports, his introduction describes travel, reading and learning from arithmeticians he met 3. The record calls this transmitted testimony, not an independently established itinerary or complete teacher list 3. The biography also stresses the practical value the author attributes to Indian arithmetic 3.
Often misunderstood
It is easy to think that al-Uqlidisi invented the Indian numerals or today's decimal point. The record says the story is neither: it concerns operations on fractional parts and a sign that preserves the units position, with historians disagreeing over how complete the concept was 3. That sign is not the dot that separates whole and fractional parts in our modern image 4 2. And writing whole numbers in base ten does not by itself establish the same treatment of fractions 3.
Think about it
A quarter is twenty-five hundredths and also fifteen sixtieths: one amount in two scales. Which scale would you choose for carrying a calculation below one, and why?
Into the museum
- al-Uqlidisi's decimal-fraction methods: the units sign, repeated halving and the disagreement about how complete the idea was.
- al-Uqlidisi: what is known of the author, and what is not.
- Algorism: the Indian place-value arithmetic that came before his book.
Try this
Read the record on al-Uqlidisi's decimal-fraction methods. Write one half, then half of that, first as fractions and then as decimals, and shade each on a hundred grid.
3 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Al-Uqlidisi's decimal-fraction methods have to do with this lesson?
Compare with the card
Half of a half, in tenths and hundredths. A half can be written as five tenths, and half of that as twenty-five hundredths. In a book composed at Damascus, Abu al-Hasan al-Uqlidisi treats such fractions directly within Indian place-value arithmetic. The record says this is not the invention of the Indian numerals or of today's decimal point, and that historians disagree over how complete his idea was. Try halving a number in 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
- 3. Al-Uqlidisi — MacTutor biography, Biography
- 4. Decimal place-value chart metadata — Wikimedia Commons, Description