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Pascal’s triangle

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Pascal’s triangle

Pascal’s triangle

Where’s the Maths?

Pascal’s Triangle Animation

Natural numbers

Triangular numbers

square numbers?

pentagonal numbers

hexagonal numbers

Triangular numbers

1

3

6

10

15

21

square numbers

1

4

9

16

25

36

pentagonal numbers

1

5

12

22

35

51

hexagonal numbers

1

6

15

28

45

66

tetrahedral numbers

What is a

pentatope

number?

Pentatope numbers

Hockey Stick / Christmas Stocking pattern

Powers of 2

Powers of 11

Eugène Charles Catalan defined the Catalan Numbers as the number of ways to disect a polygon with non intersecting diagonals.

Catalan numbers

Catalan numbers

Catalan numbers

1

1

2

3

5

8

13

Fibonacci numbers

1

1

2

3

5

8

Fibonacci numbers

prime numbers

odd numbers

even numbers

Multiples of 3

Multiples of 4

Multiples of 5

Multiples of 6

Multiples of 7

Is it a cube?

Combining triangles

What happens if I….

Use a different number at the top of the triangle?

Use different numbers down the sides of the triangle?

Subtract the numbers instead of adding them?

Multiply the numbers instead of adding them?

Use fractions?

Use decimals?

What happens if I….

Look at the triangle from a different angle?

Harmonic triangle

Yang hui’s triangle

Pascal’s petals

Pascal petals

Antonio’s Pizza Palace

It's Friday night and the Pizza Palace is more crowded than usual. At the counter the Pascalini's are trying to order a large pizza, but can't agree on what topping(s) to select.

Antonio, behind the counter, says, "I only have 8 different toppings. It can't be that hard to make up your mind. How many different pizzas could that be?"

"Well, we could get a plain pizza with no toppings," says Mr. Pascalini.

"Or we could get a pizza with all 8 toppings," says Mrs. Pascalini.

"What about a pizza with extra cheese and green peppers?" asks Pepe.

"You're not helping!" Antonio yells at Pepe. "Get back to work."

As Pepe starts to clear off the nearest table, he mumbles to himself, "or a pizza with anchovies, extra cheese, mushrooms, and olives."

Antonio hands an order pad to Mr. Pascalini and says, "When you decide, write it down and I'll make it."

Then he helps the next people in line, who know what they want: a large pizza with mushrooms, green peppers and tomatoes.

How many different pizzas can be ordered at the Pizza Palace if a pizza can be selected with any combination of the following toppings:

anchovies, extra cheese, green peppers, mushrooms, olives, pepperoni, sausage, and tomatoes?

How many different pizzas can you order with only one topping?

How many different pizzas can you order each with seven toppings?

Are the number of one-topping pizzas and the number of seven-topping pizzas related? (Why or why not?)

How many different pizzas can you order with two toppings?

How many different pizzas can you order with six toppings?

Are the number of two-topping pizzas and the number of six-topping pizzas related? (Why or why not?)

Can you find these numbers in Pascal's triangle?

Can you use Pascal's triangle to help you find the number of pizzas that can be ordered with three, four, or five toppings?

In all, how many different pizzas can be ordered?

Quincunx machine

Quincunx

Animation

Link

Quincunx

On YouTube

How does this work?

Points on a circle

sierspinski

Is there a way to predict the number of downward triangles you will have after n iterations?

Twelve days of christmas

On the twelfth day of Christmas my true love sent to me

Twelve drummers drumming, Eleven pipers piping,

Ten lords a-leaping, Nine ladies dancing,

Eight maids a-milking, Seven swans a-swimming,

Six geese a-laying, Five gold rings, Four calling birds,

Three French hens, Two turtle doves

and a partridge in a pear tree.

PNC Total Christmas Price Index Interactive Charts

City grid

Starting in the top left corner of a 2 x 2 grid, there are 6 routes

(without backtracking) to the bottom right hand corner.

How many routes are there through a 20 x 20 grid?

Heads and tails

How many different combinations are there when you flip a coin once?

How many different combinations are there when you flip a coin twice?

How many different combinations are there when you flip a coin ten times?

Heads and tails

Blaisepascal

Blaise Pascal was a French mathematician who lived from 1623-1662.

A pascaline