Connections

Jim Hogan

Sec Mat Advisor 2007

We are but one

Things derive their being and nature by mutual dependence and are nothing in themselves.

Nagarjuna, a buddist sage of long ago.

From The Tao of Physics

By Fritjof Capra PhD (Theoretical Physics)

Mathematical Connections

This .ppt illustrates the way in which numbers are connected to each other and geometry as a starting point to understanding connection,

and develops new insights to curriculum interpretation and scheme development for the classroom,

and attempts to guide teachers to the real purposes of learning mathematics

and to the power of modelling ideas.

N = {1, 2, 3, …}

The natural numbers are made up of odd and even numbers.

Task 1

Make a model of an odd number.

N = {1, 2, 3, …}

The natural numbers are made up of odd and even numbers.

An odd number.Why is it a model? Is it a good model?Does your model convey the essence of

oddness? Why?

N = {1, 2, 3, …}

The natural numbers are made up of odd and even numbers.

Task 2Make a model of an even number.

N = {1, 2, 3, …}

The natural numbers are made up of odd and even numbers.

An even number.Why is it a model? Is it a good model?Does your model convey the essence of

evenness? Why?

Properties

Use your models to explore the properties of

ODD and EVEN

O + O = O + E = E + E =

O x O = O x E = E x E =

Other properties?

What are properties?

Imagine a smelly little fish.

Now take away the smell, the colour, the length, the width, the depth, the weight, the feel, the sound it makes and any other attribute it might have.

What is left?

An Odd Connection

The set of odd numbers

O = { 1, 3, 5, 7, 9, …}

The sum of the odd numbers are square.

S = {1, 4, 9, 16, 25, …}

Task 2. Make a model to show this connection.

The Triangular Numbers

N = {1, 2, 3, 4, 5, …}

The sum of the natural numbers are the triangular numbers.

T = { 1, 3, 6, 10, 15, …)

Task 3. Make models of T1, T2, T3, and T4.

A Triangular Connection

Notice that

T2 + T3 = S3

T3 + T4 = S4

The triangular numbers are connected to the square numbers.

If A = B and B = C then A = C

The odds are connected to the squares and the square are connected to the triangular numbers.

The triangle numbers must be connected to the odd numbers!

Task 4. Make a model to show this connection.

Odds and Evens

But the odd numbers and even numbers are very closely related. If you are not one then you are the other!

So the even numbers must be connected to both the squares and the triangular numbers.

Task 5. Make models to illustrate these connections.

Handshakes and Chords

If there are 5 people in a room and they all shake hands, how many handshakes happened? Why? Explain!

How many lines can join 5 dots?

How are these problems connected to number?

And on…and on…and on…

…to the multiples of 3 and 4 and …, the powers of 2, and 3 and 4…, the factors, the primes, the shapes, the angles, the decimals, the fractions, the mean, the combinations, …

Mathematics is connected