Pascal's Triangle in Asia

Task 144 ... Years 2 - 10

Summary

Through this task students have an introduction to mathematics that is not Eurocentric (in spite of the traditional title of the task), an ancient numeral system and Pascal's Triangle before it was named after Pascal. The Chinese Rod system was actually used for doing arithmetic rather than writing and recording. The rods were carried in a bag and used on a counting board in commercial situations. It was a very practical calculating technology, in some ways similar to the Win/Lose A Flat trading game that is often used as part of a place value teaching plan. However, as the challenge suggests, the numerals were recorded in at least one historic document in a way that opens the door to extensive pattern investigations. 		 

Materials

Content

  • history of mathematics
  • recording mathematics
  • writing and interpreting numerals
  • place value
  • exploration of patterns
  • arithmetic skills
Pascal's Triangle in Asia

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card. 		   

In this Chinese system five is the pivotal element. Once a group of five occurs it is collected into 'one bar', either horizontally or vertically. That makes five the 'centre' of a nine digit system and the numeral patterns either side of it grow consistently. Notice that the system is nine digits, not ten like ours, and there are two sets of nine digits making an 18 digit system in all.

The ancient Chinese did not have a symbol for zero. This didn't matter on a counting board because a place was simply left empty. This is what the circle in Question 1c is attempting to show, but the Chinese would not have recorded in this way. They would most likely have left a space if they used the Rod Numeral to record.

With this in mind, the answers to Question 1 using our numerals are a) 49, b) 255,831 and c) 504.

But the Chinese system was subtly different from ours in another way. It was a place value system based on 100 rather than 10. The 18 stick digits can be used to show every whole number from 1 to 99.

Each section of the counting board was divided into two parts. On the right hand part of each section were the Heng rods (top ones on the card) and on the left side were the Tsang rods (bottom ones on the card). So now each section of the board could count as many as 99.

  • The first section on the right counted up to 99 ones.
  • The next section to the left counted up to 99 hundreds.
  • The next section to the left counted up to 99 hundreds of hundreds.
  • ...and so on.
So where we use a comma to show thousands in a number like 255,831, this ancient Chinese system would read the numbers in pairs, more like this 25,58,31 and think of 25 hundreds of hundreds, 58 hundreds and 31 ones. Guided research of Chinese Rod Numerals on the web could be a useful iceberg project.

The Challenge refers to the 1781 diagram on the recording sheet, which is a Japanese modification of a much earlier one, an image of which can be found in Wikipedia at:
http://en.wikipedia.org/wiki/File:Yanghui_triangle.gif.
Click to enlarge it further.

The information about this diagram can be found at:
http://en.wikipedia.org/wiki/Chinese_mathematics#Song_and_Yuan_mathematics,
which shows a date of 1303 AD - considerably before Pascal, in Europe, is credited with 'discovering' it.

(In fairness, it should be recorded that Pascal did not claim that he discovered this triangle pattern. Rather he applied it to calculations in Probability Theory, a branch of mathematics which he is generally credited with inventing. His writing about this was published in 1665, after his death, and, as recorded in Wikipedia, in 1708 a mathematician named Montmort referenced the triangle to Monsieur Pascal for the first time.)

After the first two rows of Pascal's Triangle the number in each circle is the sum of the two numbers directly above it. Print a recording sheet and try this our for yourself. This patten will allow students to continue building the triangle beyond the Murai version. However, there remain many patterns to discover:

  • Look along any diagonal.
  • Sum the numbers on any row.
  • Find Fibonacci Numbers.
  • ...and more
Perhaps student discoveries can be displayed in a special area of the room and grow over time.

Extension

The rows of Pascal's Triangle (Yang Hui's Triangle) sum to powers of two and the whole triangle is symmetric.
  • What happens if you try to build a symmetric triangle with rows that sum to powers of 3? ...or 4? ...or 5?

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works. 		   

Many teachers use Pascal's Triangle in units of work on pattern, and rightly so. In many ways it has a 'magic' feel. However, if the lesson is mostly filling in circles according to a given rule and then searching for patterns, could it be better? The task suggests yes because we can make links with history and links across mathematical content. The pattern unit might be richer by including a review of place value in context (Do we really understand our own system if we haven't explored it's subtleties compared to other systems?) and a taster of probability. Such inter-connected learning in mathematics might be more effective.

It is very easy to collect sufficient matches for each pair - try the art room - and the notes above suggest lesson development.

The probability part might lead you to use Pascal's Pinball machine. This activity physically involves students in working their way through a maze of chairs laid out like the rows of Pascal's Triangle. There are coins (or dice) on each chair - heads go forward right, tails go forward left. The finish row of chairs has no coins, but the students have carried a counter with them to place on the chair where they end. How will these counters be distributed?

The idea is recorded in Chance & Data Investigations Volume 1, Lovitt & Lowe, Curriculum Corporation, 1993, p.88. But it has been developed further in this brief article by Christine Lenghaus, who is happy to share her work with Mathematics Centre visitors.

At this stage, Pascal's Triangle in Asia does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner. 		   

The Pascal's Triangle in Asia task is an integral part of:

  • MWA Number & Computation Years 5 & 6

Green Line
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