Transcript Frank Kane curriculum presentation

Primes, Polygrams and Pool
Tables
WMA Curriculum Evening
Number and Algebra strand
Frank Kane – Onslow College
(NZC) Why study mathematics and statistics?
“…students develop the ability to think
creatively, critically, strategically and logically.
They learn to structure and organise, to carry out
procedures flexibly and accurately, to process
and communicate information, and to enjoy
intellectual challenge.
… other important thinking skills. They learn to
create models and predict outcomes, to
conjecture, to justify and verify, and to seek
patterns and generalisations….”
Doing mathematics
• “OK, let’s see if I can do this without making a mistake.”
• “Hmmm…which technique do I have to use here?”
• “How can I describe this situation using maths?”
• “Hmmm…interesting…I wonder if this works for other
cases.”
Pairs of primes
Strand: Number and Algebra
Level: 5
Key Competencies: Thinking, using symbols, relating to others
Objectives: Reinforcement of prime numbers, structuring and
presenting an investigation, appreciation that mathematics has
unanswered questions, notion of proof.
20 = 3 + 17
Can you find any other pairs of prime numbers
that add to 20?
So, which numbers can be written as the sum of
two primes?
Suggested guidelines for setting out an investigation
Aim: a clear statement of the problem
Method: diagrams, working
Results: clearly summarised e.g. table
Conclusions: answer to the question(s),
formulae, explanations
Number
Combinations
2
Distribution of number of
representations
# of combinations
0
4
2+2
1
6
3+3
1
8
3+5
1
10
3+7, 5+5
2
12
5+7
1
14
3+11, 7+7
2
Number of
combinations
•
•
•
•
7
6
5
4
3
2
1
0
0
20
40
Even number
60
80
Distribution for number of representations for even numbers
up to 1 million
http://en.wikipedia.org/wiki/Goldbach's_conjecture
Lemoine’s Conjecture (1895)
Every odd number greater than 5 can be
expressed as the sum of a prime number
and 2 times a prime number
e.g. 23 = 13 + 2 × 5
For all n > 2,
2n + 1 = p + 2q
Sums and Products – a logic puzzle
Two integers, A and B, each between 2 and 20 inclusive,
have been chosen.
The product, A×B, is given to Peter. The sum, A+B, is
given to Sally. They each know the range of numbers.
Their conversation is as follows:
Peter: "I don't know what your sum is, Sally"
Sally: "I already knew that you didn't know. I don't know
your product."
Peter: "Aha, NOW I know what your sum must be!"
Sally: "And I have now figured out your product!!"
What are the numbers?
Pool Table Problem
A ball is struck from
the bottom left corner
so that it travels at a
45° angle to the
sides.
• In which pocket will
the ball end up?
• How many bounces
will it make with the
sides of the table?
D
C
A
B
Pool Table Problem
A ball is struck from
the bottom left corner
so that it travels at a
45° angle to the
sides.
• In which pocket will
the ball end up?
• How many bounces
will it make with the
sides of the table?
D
C
A
B
Pocket D after 5 bounces
Pool Table Problem
• Ratio, common factors, primes, similar
shapes
• Use tables and rules to describe linear
relationships
• Conjecture, justify and verify
• Structure and organise work
• Communicate
Polygrams
• What do the angles at the 5 vertices of a
pentagram add up to?
• What about a star made from 6 points
(hexagram)?
• Using 7 points, two stars can be drawn.
What are the two angle sums?
pentagram
hexagram
• How many stars can be drawn using 15
points and what angle sums will you get?