Primes, Polygrams and Pool Tables

WMA Curriculum EveningNumber 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 primesStrand: 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

Distribution of number of representations

01234567

0 20 40 60 80

Even number

Nu

mb

er

of

co

mb

ina

tio

ns

Number Combinations # of combinations

2   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

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?

C

BA

D

Pocket D after 5 bounces

• Ratio, common factors, primes, similar shapes

• Use tables and rules to describe linear relationships

• Conjecture, justify and verify

• Structure and organise work

• Communicate

Pool Table Problem

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?

• How many stars can be drawn using 15 points and what angle sums will you get?

pentagram

hexagram