Conjecturing and generalising

It would be nice to have a strategy for disentangling any tangled ropes...

This challenge is a game for two players. Choose two of the numbers to multiply or divide, then mark your answer on the number line. Can you get four in a row?

Watch this film carefully. Can you find a general rule for explaining when the dot will be this same distance from the horizontal axis?

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

Can you predict when you'll be clapping and when you'll be clicking if you start this rhythm? How about when a friend begins a new rhythm at the same time?

If you count from 1 to 20 and clap more loudly on the numbers in the two times table, as well as saying those numbers loudly, which numbers will be loud?

Can you describe this route to infinity? Where will the arrows take you next?