Choose any three by three square of dates on a calendar page...
Make a set of numbers that use all the digits from 1 to 9, once and
once only. Add them up. The result is divisible by 9. Add each of
the digits in the new number. What is their sum? Now try some other
possibilities for yourself!
What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
This problem looks like a number task, possibly revision about multiples, but it becomes a question about establishing why something can never happen, and creating a real proof of this. When it comes, the proof often feels powerful, satisfying and complete, and students leave feeling they have achieved something.
Allow negative numbers, as long as they will allow you negative multiples of $3$ (and zero).
At some stage there may be mutterings that it's impossible. A possible response might be:
"Well if you think it's impossible, there must be a reason. If you can find a reason then we'll be sure."
Once they have had sufficient thinking time, bring the class together to share ideas.
If it hasn't emerged, share with students Charlie's representation from What Numbers Can We Make?
All numbers fall into one of these 3 categories:
Type A (multiple of $3$)
Type B (of the form $3n+1$)
Type C (of the form $3n+2$)
We have found that trying to use algebraic expressions as above, is tricky, students often end up with n having two or more values at once. Students are unlikely to know the notation of modular arithmetic, but the crosses notation above is sufficient for the context, and it suggests a geometrical image that students can use in explaining their ideas.
"Which combinations of A, B and C give a multiple of three?"
"Can you find examples in our list on the board where you gave me one of those combinations?"
A few minutes later...
"Great, then all you have to do is find a combination of As, Bs and Cs that doesn't include AAA, BBB, CCC or ABC!"
"It's impossible! All the combinations will include AAA, BBB, CCC or ABC!"
"OK, but can you prove it? Can you convince me that it's impossible?"