What is Pascal's triangle?
Pascals triangle is a triangular array (which just means triangular shaped group of numbers) that have many special patterns and properties.
Pascal's triangle is named after a French mathematician called Blaise Pascal, although mathematicians from around the world for centuries before him had been using the patterns contained within the triangle. The triangle is constructed using a summation process.
We start by setting out our triangular array, the template at the end of this page might help you if you want to print it out.
Put 1's all the way down the left and right sides.
The third row, has a blank gap. We get the value for this by adding the two directly above it.
which is
The next row continues in the same manner, adding the two numbers directly above it...
which becomes 
We can continue on and on in this pattern. This creates our Pascal's triangle.
Fill in the first 10 rows of Pascal's triangle.
Find the sum of the numbers in each row. Present your findings in a suitable table and write a statement about what you notice about the sum of the numbers in the row of Pascal's triangle.
Imagine you are a secret agent and your method of writing code to other spies in your agency consists of using dots and dashes.
We would only be able to have 4 two letter words. They would be
Your turn now.
- Make a list of all the 3 letter words you could make, how many are there?
- Make a list of all the 4 letter words you could make, how many are there?
- Make a list of all the 5 letter words you could make, how many are there?
What is the connection between that activity and the idea of counting techniques we have been studying?
Can you see how these numbers might appear in our triangle? (write down your thoughts on this or discuss with a friend, we don't need to confirm it just yet!)
- Can you calculate how many 6 letter words you could make without writing out the exhaustive list?
Looking at your list of 4 letter words make from only dots and dashes.
- How many start with a dash?
- How many start with a dot?
- Remind me again how many 3 letter words there were?
Can you see how these numbers might appear in our triangle? (write down your thoughts on this or discuss with a friend, we don't need to confirm it just yet!)
Does this same pattern occur when looking at the transition from 2 letter words, to 3 letter words that start with a dot or a dash?
Looking at your list of 4 letter words.
- How many are made of 4 dashes?
- How many have 1 dot and 3 dashes?
- How many have 2 dots and 2 dashes?
- How many have 3 dots and 1 dash?
- How many have 4 dots?
- How many 4 letter words again?
Can you see how these numbers might appear in our triangle? (write down your thoughts on this or discuss with a friend, we don't need to confirm it just yet!)
How can we test and confirm our thoughts on this? Maybe with 5 letter words?
Let's look now at calculating and checking our thoughts on activities 1-5, with an investigation into the 6 letter words possible.
- From Pascals triangle, how many 6 letter words do you think there are?
- Calculate the number of 6 letter words using counting techniques. (Hint- each of the positions of the 6 letter word can be filled in one of two ways).
- How many of the 6 letter words start with a dash? (can we get this information from the triangle?)
- Calculate how many start with a dash using counting techniques (Hint - the first position has to be a dash, but how many ways can each of the other 5 positions be filled?)
- Can you now use the triangle to determine -
- how many have 2 dashes and 4 dots
- how many have 3 dashes and 3 dots
- how many have 4 dashes and 2 dots
- how many have 5 dashes and 1 dot
- How many 9 letter words are there with 6 dots and 3 dashes?
- How many 9 letter words are there with 3 dots and 6 dashes?
- How many 8 letter words are there with 2 dots and 6 dashes?
- How many 8 letter words are there with 6 dots and 2 dashes?
Why are those pairs of questions connected?
Write down the following (on one line like I have)
4C0 4C1 4C2 4C3 4C4
Then underneath that write the answers....
What connection do you see to the triangle?
TEMPLATE
This template is made available in case you would like to print it out to complete this investigation.
Reflect: Are there other patterns with the triangle? We have focused on patterns relating to combinations but what else can be found within the triangle? There are a surprising number of patterns that can be found. Here are a few to get started: Pascal's Triangle.