Which pairs of cogs let the coloured tooth touch every tooth on the other cog? Which pairs do not let this happen? Why?

What happens to the area and volume of 2D and 3D shapes when you enlarge them?

Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

These clocks have only one hand, but can you work out what time they are showing from the information?

Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.

Imagine a large cube made from small red cubes being dropped into a pot of yellow paint. How many of the small cubes will have yellow paint on their faces?

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?

Can you lay out the pictures of the drinks in the way described by the clue cards?

Although almost any NRICH activity could be tackled in a group, these lower primary activities have been created specifically with group work in mind.

This article for teachers suggests a range of activities to help children get better at working in groups.

Two video clips of classes organised into groups to work on Counting Cogs.

A video clip of Jo Boaler talking about Complex Instruction.

Can you lay out the pictures of the drinks in the way described by the clue cards?

You are organising a school trip and you need to write a letter to parents to let them know about the day. Use the cards to gather all the information you need.

Complete these two jigsaws then put one on top of the other. What happens when you add the 'touching' numbers? What happens when you change the position of the jigsaws?

Group work depends on effective team work. This article describes attributes of effective team work and links to "Team Building" problems that can be used to develop learners' team working skills.

In this challenge, you will work in a group to investigate circular fences enclosing trees that are planted in square or triangular arrangements.

Drawing a triangle is not always as easy as you might think!

Can you find a quadratic equation which passes close to these points?