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# National Curriculum - Reasoning and problem solving : Key Stage 2 : Mathematics Content Knowledge

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National Curriculum - Reasoning and problem solving
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# 1. How confident are you that, in terms of KS2 mathematics, you understand what it means to:

## Example

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 finding the sum of the digits of different multiples of 9, we can conjecture that the sum of the digits is always divisible by 9. 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.

When investigating problems children may start to notice patterns or relationships that can be expressed as conjectures. Some further examples of conjectures that children might explore:

Multiplying any whole number - odd or even - by an even number gives an even answer.

The total of any 3 consecutive numbers will always equal 3 multiplied by the middle number.

In investigating the number of handshakes that would take place if all the children in a group were to shake hands, a table would need to be developed to show the findings.

 Number of people (n) Number of handshakes 2 1 3 3 4 6 5 10 n

We can then conjecture that the number of handshakes for any group size could be calculated by multiplying one less than the group size (n − 1) by the group size (n) and dividing by 2.

## What this might look like in the classroom

A conjecture is a little like a guess where someone might say, ‘I believe the following statement is true’. The main thing about one is that it has not been proved or disproved. Once a conjecture is proved beyond doubt then it becomes a theorem.

Moving from a conjecture to a convincing proof is hard for most Key Stage 2 children. Children are more likely to give examples of the conjecture working than come up with a more general reason.

But it is possible. For example, when thinking about the conjecture ‘an odd number plus an odd number always equals an even number’ a child might reason:

Well an odd number is always an even number plus one. So if you add two odd numbers together the even bits make an even number and the two extra ones make 2, which is even, so the answer must be even.

## Taking this mathematics further

#### The power of counter-examples

To disprove a conjecture you only need one example, a counter-example. For example you can disprove the conjecture ‘;all numbers have an even number of factors’ by noticing that 9 has three factors: 1, 3 and 9. Although this could in turn lead to another conjecture related to numbers that have exactly three factors, or an odd number of factors!

#### An interesting fact

One of the most famous conjectures was the misnamed Fermat’s Last Theorem. It was misnamed because although Fermat claimed to have found a clever proof of his conjecture none could be found among his notes after his death. The conjecture taunted mathematicians for over three centuries before a English mathematician called Andrew Wiles finally proved it in 1994. Now it can properly be called a theorem. Because it has been proved it is therefore no longer a conjecture.

If you want to find out more about Fermat’s Last Theorem follow this link.

## Making connections

In order to develop the skills of making or trying to prove/disprove conjectures the child needs to be given the opportunity to develop their thinking skills (which include information processing, creative thinking, reasoning, evaluating and enquiry). You can do much to help this by giving open-ended tasks to the children in your class, particularly those that involve identifying patterns in number or shape.

They also need to be in an environment where they feel at ease to take risks and get things wrong. Developing the thinking that they can find out from mistakes is important. Saying to the child something like ‘that’s interesting, what can we learn from it’, rather than ‘that’s wrong’ is also important.

Children should initially find examples that match given conjectures. Once they are able to do this you could ask them to make up their own conjectures with examples to try to prove them.

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