Conjecturing and generalising

Take any four digit number. Move the first digit to the end and move the rest along. Now add your two numbers. Did you get a multiple of 11?

Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: ×2 and -5. What do you think?

Liam's house has a staircase with 12 steps. He can go down the steps one at a time or two at time. In how many different ways can Liam go down the 12 steps?

The Egyptians expressed all fractions as the sum of different unit fractions. Here is a chance to explore how they could have written different fractions.

There are lots of ideas to explore in these sequences of ordered fractions.

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

Charlie and Alison have been drawing patterns on coordinate grids. Can you picture where the patterns lead?

Can you work out what step size to take to ensure you visit all the dots on the circle?