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 determine pattern rules and use them to extend patterns, to predict and justify predictions, and to find missing elements. The museum offers one rule for building a table of numbers from the row before it; it does not serve growing, shrinking or linear patterns, graphs or algebraic expressions, which your class brings from elsewhere. 1
Curriculum expectations, with their official codes
- C1.3 determine pattern rules and use them to extend patterns, make and justify predictions, and identify missing elements in repeating, growing, and shrinking patterns, and use algebraic representations of the pattern rules to solve for unknown values in linear growing patterns
2 The lesson
How can you state a rule so that someone else can continue the pattern?
Write the rule in words first, then in a table of values. Predict the next two rows, and check your prediction by applying the rule.
Go deeper
Al-Karaji's rule builds each new row of numbers from the row before it. The sources describe it so: the square gives one, two and one 2. To get the next list, keep a one at each end, add one and two to get three, and then two and one to get another three 2. The page he wrote is not in front of us: the record says that al-Samawal transmits the account of an original computation that no longer survives 2 3.
Why they did it
The records preserve no sentence in which al-Karaji says why he worked out these numbers, and the museum does not supply one. They show where the rule belongs: early-eleventh-century Baghdad algebra 2 3. MacTutor reports that he calculated the cube of a sum in al-Fakhri, and the cube of a difference and fourth power of a sum in al-Badi 2. Al-Samawal admired him but also identified shortcomings and developed improvements 4 3.
Often misunderstood
The table is often called Pascal's triangle. The record says the name does not imply that the triangle was unknown before Pascal, and that MacTutor says so expressly 5. The history of Jaina mathematics describes a triangle in a tenth-century commentary on an older work 6. For China, MacTutor describes Jia Xian's table 7. The record calls this comparable work, not something shown to have been borrowed from al-Karaji 7 2.
Think about it
The record checks the rule with chosen numbers, but says this does not alone prove the rule for every power 2. Why is an example that agrees not enough, and what else would you want to see before you trusted the rule for every row?
Into the museum
- Al-Karaji's coefficient rule: how each row is made from the row before, and what is lost.
- al-Karaji: his works, his name, and what is not known about him.
- Al-Bahir fi al-jabr: the book that preserves the calculation.
- al-Samaw'al al-Maghribi: the later writer who passes it on.
Try this
Read the record on al-Karaji's binomial-coefficient recurrence. Build the next rows of the table by adding neighbours, and write your rule as one sentence.
3 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Al-Karaji's binomial-coefficient recurrence have to do with this lesson?
Compare with the card
A table built by adding neighbours. Start with a row of numbers, 1, 2, 1. The next row keeps a one at each end and adds each pair of neighbours: one and two make three, two and one make three, so the row is 1, 3, 3, 1. The sources attribute this rule of adding adjacent entries in a table to Abu Bakr al-Karaji; al-Samawal al-Maghribi transmits it from a computation that no longer survives. The record calls it a bounded improvement, not the world's first arithmetic triangle. Can you state the rule for the next row in your own words?
I can now
Notes
- 1. Ontario Ministry of Education, The Ontario Curriculum: Mathematics
- 2. Al-Karaji — MacTutor mathematical biography, Biography
- 3. Al-Samawal — Adel Anbouba, Dictionary of Scientific Biography, algebra
- 4. Al-Samawal — MacTutor biography, Biography
- 5. Pascal — MacTutor: triangle and later powers, Biography
- 6. Jaina mathematics — MacTutor, combinations
- 7. Jia Xian — MacTutor, Biography