Pascal's Triangle

Task 1.

Take a blank piece of paper and a pen. Imagine you have three coins, better still, get three coins. See how many ways you can place the three coins heads up or tails up. For example, one way is HHH, then HHT or HTH and so on. See if you can find a way of doing this systematically. How many different arrangement can you find? (Big Hint: There are 8!) Count how many of these have zero heads, how many have 1 head, how many have 2 heads and how many have three heads. Write out the first few lines of Pascal's triangle. Can you see the connection?

Now try four coins. Notice anything? Investigate. Does this always work?

Task 2.

Now take your three coins and arrange them in a triangle. Add another row to your triangle. How many coins are there in total? Add another row and count the total. These are called triangle numbers. Put your numbers into a table. See if you can find them in Pascal's triangle.

Task 3.

This is a bit harder. You can do it with coins if you like. Imagine you are on a ship stacking canon balls into a triangular pyramid. There would be a triangular frame to hold the bottom triangle in place. Put three balls (coins!) in a triangle and one on top. Four Balls in the pyramid. Add a layer by making the bottom row bigger. How many canon balls all together? Keep going! Make a table of results. Can you see a connection in Pascal's triangle?

Task 4.

This is where you need to be able to do a bit of proper maths.

Because of the limitations of this software, I will write "To the power of" as a "^", so (x + y)^0 means "raised to the power zero".

To start with, this will sound a bit odd! You know anything to the power zero is 1. So...

(x + y)^0 = 1

Anything raised to the power 1 is itself. So...

(x + y)^1 = x + y

You should be able to work out (x + y)^2 by multiplying out the algebra. Organise your result so that you start with an x^2 term, then an xy term, then a y^2 term. (We call this descending powers of x, because the powers of x go x^2, x^1 (or x), and x^0 (or 1).

Now, the next bit is a bit harder. You need to do this carefully. See if you can find

(x + y)^3 note, we can multiply our previous result (x + y)^2 by (x + y) to save a bit of time.

Again, write the result in descending powers of x. Each of the terms will have a coefficient (That is, a number at the front). If the number is a 1, you don't need to include it because 1 x anything is itself.

You should notice a pattern arising that is linked to the work in the previous tasks. Can you predict (using descending powers of x and ascending powers of y) what the results will be if you do (x + y)^4?

We can actually use Pascals triangle to short-cut a whole load of algebra.

This maths leads to a strand of the A Level maths syllabus called binomial expansion, and leads on to a really important bit of maths called "Taylor's Expansion" and "MacLaurin's expansion", which become important in A Level Further Maths and degree level Maths. This leads to an idea that ALL of the major functions in maths can be written as a function in ascending powers of x. Which is a bit weird.

If you had trouble getting a result for this last task, don't worry, you will have an opportunity to ask questions in an on line session soon......