Square subtraction
Problem
For this activity, you just need to follow three simple steps:
- Choose any number.
- Square your chosen number.
- Subtract your starting number.
Once you've done that, answer the question: 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 model to prove that your result is always true, not just true for the particular number that you started with?
Getting Started
How about using a pattern of dots to show your square number?
Student Solutions
Nearly $80$ solutions were sent in! These had many examples that had even answers. We did suggest that you went a bit further and here we have some ideas.
First from OwenӬ from the Montessori School of Wooster, OhioӬ, USAӬ
A square number subtracted by its self (X squared - X) always has to be even. It will always be a even number because an even number times a odd number (3 x 3 = 3 x 2) is always even
(eg. 4 x 4 - 4 = 4 x 3 = 12, 5 x 5 - 5 = 4 x 5 = 20)
AmyӬ from Claremont PrimaryӬ School in the UKӬ had the following thoughts;
2 x 2 = 4
4 - 2 = 2 (even)
Answer is even with an even number starting point
3 x 3 = 9
9 - 3 = 6 (even)
Answer is even with an odd number starting point
Therefore answer is always even. Why?
The answer will always be even because an odd number squared makes another odd number. An even number squared makes an even number. And an even number take away another even number makes another even number. But quite strangely an odd number take away another odd number also makes an even number.
From Jasmine, Arran, Rebecca and Thomas Ӭat St. Mary's School ShawburyӬ in England , we had a similar thought;
The answer is even. We know this because, when you square an even number you always get an even number, but when you take an even number away from the number that you now have, you still remain an even number, no matter what number you started with.
When you square an odd number, it should result as another odd number, but when you subtract an odd number, from the number you have now, it will result as an even number.
e.g. Even squared = even - even = even.
e.g. Odd squared = odd - odd = even.
We tried this with $1,2, 3$ digit numbers and they all ended even, we also tried different odd and even numbers and found the same result, all the answers were even.
OllyӬ from Bourton Meadow AcademyӬ in EnglandӬ, wrote;
It is always even because the multiples of an even number are even
$(2a)(2a+1) = 6a$
If you use an odd number it is simply this formula backwards
$(2a+1)(2a) = 6a$
Finally Victor and ElliottӬ from Kenakena SchoolӬ in New ZealandӬ;
$X ² - X =$ even. If $X$ is odd. $X ²$ is odd. so $X ² - X = $ even.
If $X$ is even. $X ²$ is even. so $X ² - X = $even.
Side proof : how even - even = even and odd - odd = even:
If n and e are arbitrary whole numbers, then even = $2n$ (I think by this they mean that you can write an even number as $2n$), so even - even $= 2n - 2e = 2(n - e)$. Thereby, even - even = even.
So odd$ = 2n + 1$. So odd - odd = $2n + 1 - (2e + 1) = 2n - 2e + 1 - 1 = 2(n- e)$.
Thereby, odd minus odd = even.
We had a few very late in the month suggestions from Jack, Daniel, Sophiie, Isobel and Ryan at Keidmarch Primary School
Thank you all for your hard work and the emails.
Teachers' Resources
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.