Saturday, November 26, 2005

Pascal's triangle (in Chinese)

I like to give my students the following illustration and ask them for the pattern:

The illustration "dates from a book of 1303 CE written by Chu Shï-kié. The earliest known use of the pattern was by Yang Hui, whose books date from 1261 & 1275 CE. Chu Shï-kié refers to the triangle as already being old. Jamshid Al-Kashi, who died around 1436 CE, was an astronomer at the court of Ulugh Beg in Samarkand in the 15th Century. Al-Kashi was the first known Arabic author to consider 'Pascal's' Triangle.", to quote the BBC.

They quickly come up with the pattern (every number is the sum of the two above), and thereby can also agree on the interpretation of the chinese numerals. Then they are ready to start exploring some of the wonderful "hidden" patterns in the triangle. These are some of the ones they may find - an additional exercise is of course to explain the pattern:

- adding the numbers in a row gives a power of 2.
- start from the top row, and move downwards - every step either down to the left or down to the right. In how many ways can you get to the number 20 (in the middle of the seventh row)? (20)
- coloring the even numbers gives us some triangles. Why? And what about coloring the numbers divisible by three? Or the ones that give a remainder of 1 when divided by 3?
- the rows with prime numbers in have a special property - which?
- when ordering a pizza, you get a choice of 8 toppings. In how many ways can you choose two toppings? Three? Four? Why are these numbers found in one and the same row of Pascal's triangle?
- choose any number in the triangle (though not on the edge). Color the six numbers circling your chosen number red-blue-red-blue-red-blue. Multiply the red numbers. Then multiply the blue numbers. What do you see? Why is this so?
- calculate and so on (if you have the patience). Do you see a connection with Pascal's triangle?
- n people meet and everybody wants to shake hands with everybody else. How many handshakes does that make? Where do you find those numbers in the triangle?
- choose a row. Let the number to the far right be ones, the next tens, the next hundreds and so on. Factorize the combined number. What do you see?
- draw a circle and place five dots on the periphery of the circle. How many lines can you make between the dots? How many triangles can you make with corners in the dots? How many quadrilaterals? And so on...

These small exercises show some connections in mathematics between geometry, algebra, combinatorics and numbers - and most of them could easily be used in primary or lower secondary school. Given in the right way, they may also work as a "landscape of investigation" (Skovsmose).

Please tell me if you know of more patterns in the triangle, by the way. (I know that the Fibonacci numbers are there, but I always find the "diagonals" where they occur too odd to point them out to my students...)


Blogger Gene Chase said...

One pattern ould be to find the error in the triangle!

6:32 PM  
Blogger Bjørn said...

Good point!

7:14 PM  
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