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# Mathematical processes and applications : Key Stage 4 : Mathematics Content Knowledge

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Mathematical processes and applications
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# 1. How confident are you that you understand what it means to:

## Example

 Example 1 Example 2 A conjecture is a mathematical statement which seems likely to be true but has not been proved to be true under rules of logic. It is usually based on some but not complete evidence. For example: After investigating the products of consecutive odd integers, pupils may conjecture that the product is always 1 less than a multiple of 4. This conjecture can be proved to be true by algebraic arguments. A famous problem in mathematics is Goldbach’s conjecture that every even number greater than 2 can be written as the sum of two primes, e.g. 4 = 2 + 2     10 = 3 + 7 = 5 + 5 6 = 3 + 3     12 = 5 + 7 8 = 3 + 5     14 = 3 + 11 = 7 + 7 and so on. Goldbach’s conjecture has so far never been proved, nor has any counter-example ever been found to disprove the conjecture. Additional user example There are other famous conjectures that are useful to illustrate the point of what a conjecture is and how it is proved.  These are not all necessarily related to the KS4 curriculum. If you google Four Colour Conjecture, you come across the following article: History topic: The four colour theorem This article shows the time scale and methods involved for making conjectures and constructing proofs.  It also illustrates the need for mathematicians to communicate their ideas to each other for verification.

## What this might look like in the classroom

Question
Jamal notices that the difference between two consecutive square numbers is always an odd number.

Does this always work
Can you prove this conjecture

A useful starting point is to look at some consecutive square numbers as follows:

Squares         1    4    9    16   25    36    49
Differences      3     5    7     9     11    13

Tidy up presentation by showing arrows between each pair of square numbers as follows

The conjecture would seem to be true

To prove the conjecture, let the two consecutive square numbers be x2 and (x+1)2

You need to prove that the difference between x2 and (x+1)2 is an odd number

The difference can be written as
(x+1)2x2
= (x2 + 2x + 1) − x2
= 2x +1

Since the number 2x is always an even number then 2x + 1 must be an odd number so that the difference between two consecutive square numbers is always an odd number.

NOTE The conjecture can also be shown to be true diagrammatically as follows

## Taking this mathematics further

There are many well known conjectures which mathematicians have tried to prove or disprove over the years.

Two such conjectures are:

Goldbach's Conjecture which says that "Every even integer greater than two can be expressed as the sum of two primes." For example 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7 or 5 + 5 ...

The Twin Prime Conjecture which suggests that there are an infinite number of twin primes where a twin prime is a pair of prime numbers that have a difference of two. For example 3 and 5 , 5 and 7, 11 and 13, 17 and 19…. are all twin primes.

A list of other conjectures can be found here.

## Making connections

A conjecture is a statement which would appear to be true, but has not yet been formally proven. When a conjecture is proved beyond doubt then it is called a theorem.

Until recently, the most famous conjecture was the (misnamed) Fermat's Last Theorem, which stated that if an integer n is greater than 2, then the equation an + bn = cn has no solutions (providing a, b, and c are not zero). .

The conjecture was finally proven in 1994 by Andrew Wiles, an English research mathematician. See here for further information.

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