This article invites you to get familiar with a strategic game called "sprouts". The game is simple enough for younger children to understand, and has also provided experienced mathematicians with significant food for thought.
A paradox is a statement that seems to be both untrue and true at the same time. This article looks at a few examples and challenges you to investigate them for yourself.
Weekly problem 48 - 2006
Choose any two odd numbers and one even number, such as $3, 5$ and $2$.
How would you like to represent these numbers?
Try adding them together and draw/make the representation of their sum.
What do you notice about the answer?
Look closely at your model.
Would it work in exactly the same way if you used different numbers but still two odds and one even?
Can you use your example to prove what will happen every time you add two odd numbers and one even number?
See if you can explain this to someone else. Are they convinced by your argument?
Once you can convince someone else, see if you can find a way to show the argument on paper. You might draw something or take a photo of things you have used to prove that your result is always true from your example.
Tell us about it by submitting your solution.
This problem supports the development of the idea of generic proof with the children. This is a tricky concept to grasp but it draws attention to mathematical structures that are not often addressed at primary school level. Generic proof involves examining one example in detail to identify structures that will prove the general result. 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. Proof is a fundamental idea in mathematics and in helping them to do this problem you will be encouraging them to behave like mathematicians.
By addressing the case of adding two odd numbers and an even number, a generic proof that adding two odd numbers and an even number always gives an even answer is developed based on the structure of odd and even numbers. 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.
Ask the children to choose two odd numbers and an even number and add them together. It is probably easiest if they choose ones that are easy to model and numbers that they are secure with.
Suggest that they make a model of their numbers using apparatus that is widely available in the classroom. Resist pointing them in specific directions unless they become stuck. If they are stuck then resources such as Multilink cubes, Numicon or squared paper will be helpful.
The idea is that they take a particular example and then see if they can see the general structure within that one example.
Can you see anything in your example that would work in exactly the same way if you used three different numbers but still one even and two odds?
Can you say what will happen every time you add any two odd numbers and one even numbers?
Can you convince your friend that this is true?
When adding three numbers there are a number of different combinations of odds and evens that are possible. Ask the children to explore what they are. Get them to identify the possible combinations and the features of those combinations that matter.
Does it matter whether the numbers are odd or even?
How many different cases can you find?
To work on the generic proofs for each case the children will need to consider them separately. Can you create a proof for each case using one example?
A possible extension would be to look at Three Neighbours.
Some other problems at higher stages that may also be worth exploring are Always a Multiple, Power Mad! or Seven Squares.
It may be helpful to return to Two Numbers Under the Microscope if the children are struggling with adding three numbers. This might help them to feel more comfortable with the rules they have proved in that problem and so build the foundations for this one.
The children may find it helpful to use representations of numbers such as these to support their thinking.