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The Magic of Maths!!!!!!!!!!!!!!

Let the games begin

Pascal’s Triangle: Click to see how it’s built: movie

210 210

Colour on top of all the even numbers in Pascal’s triangle …

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Recall: Even and Odd numbers

Even numbers can be divided exactly by 2.

Even numbers end in 0, 2, 4, 6, 8.

Odd numbers yield a remainder of 1 when divided by 2.

Odd numbers end in 1, 3, 5, 7, 9.

Solution: Our very own fractal

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Fractals

For example, Pascal’s triangle was

constructed by always adding two numbers

and writing the result in the space midway

below them.

Fractals tend to occur when we create a

geometric picture by applying the same

rule again and again.

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Fractals

Fractals are geometrical figures whose fragments,

when zoomed, look like the whole picture again.

One of the most famous mathematical fractals is the Mandelbrot set. Click on the link to see it zoomed in a movie.

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Why do fractals occur in nature ?

List 3 things in nature that look a lot like fractals.

Click on the link above to find more.

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Why do fractals occur in nature ?

Some Answers:

Living beings tend to take the shape that best protects them and helps them find energy.

For example, seashells are round because this shape allows them to get the most space inside for smallest shell outside.

Fractals tend to occur when the same rule is repeated again and again.

Once living things have found their best shape to grow, they repeat the same rule for growth every year at larger scales.

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Why do fractals occur in maths ?

Example: Pascal’s triangle

Fractals tend to occur when the same rule is repeated again and again.

In Pascal’s triangle, we always added two numbers and wrote the result midway below them.

Even + Even = Even

Hence in Pascal’s triangle, when we had some even numbers in a row, then an entire triangle below them was even.

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Why do fractals occur in maths ?

Example: Pascal’s triangle

Fractals tend to occur when the same rule is repeated again and again.

In Pascal’s triangle, we always added two numbers and wrote the result midway below them.

Odd + Even = Odd

This rule applies along the slanted edges of the even number triangles.

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Why do fractals occur in maths ?

Example: Pascal’s triangle

Fractals tend to occur when the same rule is repeated again and again.

In Pascal’s triangle, we always added two numbers and wrote the result midway below them.

Odd + Odd = Even

This rule helped create new even triangles under rows of odd numbers in Pascal’s triangle.

You’ll be able to explain more fractal properties of Pascal’s triangle when you’ll learn about binary numbers when you’re older…

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Reading your mind

How to play: 1:Circle a number on a row

2:Then cross off all of the other numbers in the same column and row of the number that you circled.

3:repeat step 1 and 2 until you have 4 numbers.

The sum of the 4 numbers is …………………

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Example

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Example

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Example

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Example

4+6+11+13=34

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Reading your mind: Solution

The sum of the 4 numbers is always 1+2+3+4+4+8+12 = 34

• We pick exactly one number from each row and column.• The second row is 4+ first row• The 3rd row is 8+ the first row.• The 4th row is 12+ the first row.

1 2 3 4

1+4

5

2+4

6

3+4

7

4+4

8

1+8

9

2+8

10

3+8

11

4+8

12

1+12

13

2+12

14

3+12

15

4+12

16

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Now Try it for Yourself

Repeat the same, but this time leave circled 5 numbers.

The sum of the 5 numbers is …………………

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Number Trick

• Choose a two digit number.

• Multiply the tens digit by 5.

• Add 3 to the answer.

• Now double the answer.

• Add the units digit to your answer.

•Tell me your final answer, and I will tell you what number you had chosen.

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Solution:

• Choose a two digit number.

• Multiply the tens digit by 5.

• Add 3 to the answer.

• Now double the answer.

• Add the units digit.

• I subtract 6 to get your number.

A B

5xA

5xA+3

10xA+6

B+10xA+6

B+10xA

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Maths Circles Ireland

Primary Schools Is Maths Fun?

genius

Tessellations:

Fibonacci Numbers:

Sets: Hamilton Circuits

Pentomino Puzzles: