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 offers students the opportunity to consider the underlying structure behind multiples and remainders, as well as leading to some very nice generalisations and justifications.
Display the image of the four bags (available as a PowerPoint slide).
Alternatively, you could start with this image, with 7s, 10s, 13s and 16s.
Take Three from Five and What Numbers Can We Make Now? are suitable follow-up problems.
Begin by asking students to explore what happens when they add two, three, four... numbers chosen from a set of bags containing $2$s, $4$s, $6$s and $8$s. Can they explain their findings?