Pascal's Triangle | Interactive Investigation

Building the Triangle

1

Describe how each number in Pascal’s Triangle is formed from the numbers above it. Can you explain why the edges are always 1?

2

Print a blank triangle and fill it in by hand. How far can you get without making a mistake?

Shading Patterns

3

Shade the odd and even numbers. What pattern do you see? Does it remind you of a famous fractal?

4

Shade multiples of 3. What pattern emerges? Now try multiples of 5. How do the patterns compare?

5

Try multiples of 4, 6, 7. Which ones create the most interesting patterns? Why do you think prime numbers create different patterns to composite numbers?

6

Use custom colouring to shade all the numbers that are multiples of both 2 and 3 (i.e. multiples of 6). How does this relate to the separate patterns?

Diagonals

7

Highlight the first diagonal (all 1s). Now the second diagonal (1, 2, 3, 4, 5…). What sequence is this?

8

Highlight the third diagonal (1, 3, 6, 10, 15…). What are these numbers called? Can you see why?

Hint: try drawing them as dots arranged in triangles.

9

The fourth diagonal gives 1, 4, 10, 20, 35… These are the tetrahedral numbers. Can you see why they might be called that?

Row Sums

10

Turn on Row Sums. What pattern do the sums follow? Can you express this as a formula?

Hint: row 0 sums to 1, row 1 sums to 2, row 2 sums to 4…

11

Can you explain why each row sum is double the previous row sum?

Hidden Sequences

12

Can you find the square numbers hiding in Pascal’s Triangle?

Hint: look at what happens when you add two adjacent numbers in the second diagonal.

13

Can you find the Fibonacci sequence? Try adding the numbers along shallow diagonals (going up-right from the left edge).

Hint: the sums go 1, 1, 2, 3, 5, 8, 13…

14

Write down the digits in each row as a single number: 1, 11, 121, 1331, 14641… What are these numbers? What happens at row 5 and beyond?

Hint: think about powers of 11.

Hockey Stick Pattern

15

Start at any 1 on the edge. Move diagonally inward, adding numbers as you go. When you stop, look at the number one step further in the opposite diagonal direction. What do you notice?

16

For example: 1 + 3 + 6 + 10 = 20. Where is 20 in the triangle? Can you find more hockey sticks?

Probability and Combinatorics

17

If you toss a coin 4 times, there is 1 way to get 0 heads, 4 ways to get 1 head, 6 ways to get 2 heads, 4 ways to get 3 heads, and 1 way to get 4 heads. Which row of Pascal’s Triangle is this?

18

The numbers in Pascal’s Triangle are called binomial coefficients. Can you find out what this means and why (a + b)² = 1a² + 2ab + 1b² uses the numbers from row 2?

Create Your Own Triangles

19

What if you started with 1, 2 in the first row instead of 1, 1? Or with 1, 3, 1 in the second row? How does changing the starting numbers change the patterns?

20

What if instead of adding two numbers to get the one below, you multiplied them? Or found the difference? What triangle do you get?