What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
Show that for any triangle it is always possible to construct 3
touching circles with centres at the vertices. Is it possible to
construct touching circles centred at the vertices of any polygon?
This problem encourages students to think about the properties of numbers, including divisibility and remainders. It may help students gain a deeper understanding of linear sequences (and perhaps quadratic sequences if they explore Level 2 and 3). The interactivity encourages students to begin the problem experimentally before working more theoretically as they engage with the main ideas.
What is true about any pair of rules where it is not possible to light up both lights?
If two sequences are described by the rules $an+b$ and $cn+d$, can you explain the conditions for determining whether the lights will ever switch on together?
Some students may wish to use ideas of modular arithmetic to prove their findings; reading this article first may help.
The problem Shifting Times Tables offers an introductory challenge for exploring linear sequences.
Students could use a 100 square as a visual way to record sequences and see where (if) they coincide.