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Square Subtraction

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Square Subtraction

Choose any number. This is going to be your particular number for this proof.
Square your chosen number.
Subtract your starting number.
Is the number you're left with odd or even?
Create a model or a picture of your calculation, using your chosen number, and examine this model carefully.
Can you use this one model to prove that your result is always true and not just true for the particular number that you chose to start with?



Why do this problem?

This problem captures the essence of generic proof. This is a tricky concept to grasp but it draws attention to mathematical structures that are not often addressed at primary school level. It is possible that only very few children in the class may grasp the idea but this is still a worthwhile activity which provides opportunities for children to explore odd and even numbers and the relationship between them. Generic proof involves examining one example in detail to identify structures that will prove the general result. Proof is a fundamental idea in mathematics and in encouraging them to do this problem you will be helping them to behave like mathematicians.

The article entitled Take One Example will help you understand how this problem supports the development of the idea of generic proof with the children. Reading it will help you to see what is involved.

This problem builds on the ideas explored in Two Numbers Under the Microscope, Take Three Numbers and Odd Times Even. You may find it helpful to tackle these before this one as we offer less support in the posing of the question in this case.

Possible approach

Introduce the idea using numbers that the children are comfortable to work with and can represent easily either on paper or using apparatus such as Multilink cubes, counters or Dienes blocks. It will be helpful to use apparatus or drawings that support a model of multiplication based on arrays. The children might all be working with different numbers but should all arrive at the same conclusion. This result is the focus of the generic proof. The task is to examine the example for features that will be true in every case and so establish an argument to support their conjecture. This argument is the generic proof.

A different approach to proving the same result can be found in the problem Odd Squares.

Key questions

How would you like to represent these numbers?

Is the number you're left with odd or even?

Can you see anything in your example that would work in exactly the same way if you used a different starting number?

Can you say what will happen every time you choose a number, square it and subtract the number you chose?

Can you convince your friend that this is true?


Possible extension

Look at the relationship between successive square numbers. For example, what is the difference between 5 squared and 6 squared? Can you find a general rule? Can you prove it by looking at the structure of a specific case?

Possible support

It may be helpful for children who are struggling to look at Odd times even. Rehearsing the nature of generic proof may also be helpful and the article Take one example may help  you do do this with the children.