Transcript Connections

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 2
Make 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=
OxO=
OxE=
ExE=
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