CONJECTURES

A conjecture is a statement that must be proved or disproved.

To DISPROVE a statement, provide an example where it is NOT true.

Mathematics students are male.

Men achieve higher marks in Mathematics than women.

Justin Bieber is a responsible adult.

Prime numbers are odd.

Real numbers are counting numbers.

Numbers that are divisible by 4 are also divisible by 8.

What about proving a conjecture true?

Conjecture: Chocolate bars contain peanuts.

Proof: Snickers is a chocolate bar. Snickers bars contain peanuts. The conjecture is true.

What is the problem with that argument?

Chocolate bars contain peanuts.

ALL chocolate bars contain peanuts.

SOME chocolate bars contain peanuts.

VERY IMPORTANT

1 example can DISPROVE.

To PROVE we need to show it is true for ALL cases.

ALGEBRA (your favourite!!)

Let’s try it!

• Investigate the type of number you get if you find the sum of an odd number and an even number.

• Express your answer in words as a conjecture.• Use algebra to prove this conjecture.

THINGS YOU SHOULD KNOW…• Review of Number

Real, integer, whole, counting, rational, irrational, prime, composite…

TWIN PRIME?

SQUARE NUMBER?

TRIANGULAR NUMBER?

How would you write these using algebra?

Even numbers:

Odd numbers:

3 consecutive even numbers:

3 consecutive odd numbers:

How could you represent a two digit number using

algebra?

LOCATING ERRORS

1) Squaring Solve for x: x + 9 - x + 3 = 0

2) Square rooting

- 2 = - 2 1 – 3 = 4 – 6 1 – 3 + 9/4 = 4 – 6 + 9/4 (1 – 3/2)2 = (4 – 3/2)2

1 – 3/2 = 4 – 3/2 1 = 4

3) Dividing by 0

m = p m2 = mp m2 – p2 = mp – p2

(m – p)(m + p) = p (m – p) m + p = p m + m = m 2m = m 2 = 1

4) Inequalities and multiplying or dividing by a negative

10 < 20 - 10 < - 20

SNAP TO IT!!!

Pick a number.Square it.Add ten times the original number.Add 25.Take the square root (rounding to the nearest whole number).Subtract your original answer.What do you get?

Consider the following examples of multiplying any two-digit number by 11. What pattern can you see?

11 x 42 = 46211 x 71 = 78111 x 45 = 495

1. Express your answer as a conjecture.2. Can you find an example that disproves your

initial conjecture? If so, reconsider your conjecture.

3. Use algebra to prove your conjecture.

THIS IS YOUR CHANCE TO BE FAMOUS!!

• Twin Prime conjecture There are an infinite number of twin primes.

• Goldbach’s conjecture Every even number greater than two can be written as a sum of two prime numbers.