Exploring and noticing

Suppose there is a train with 24 carriages which are going to be put together to make up some new trains. Can you find all the ways that this can be done?

Kate has eight multilink cubes. She has two red ones, two yellow, two green and two blue. She wants to fit them together to make a cube so that each colour shows on each face just once.

Have a go at this well-known challenge. Can you swap the frogs and toads in as few slides and jumps as possible?

How do you know whether you will reach these numbers when you count in steps of six from zero?

How many faces can you see when you arrange these three cubes in different ways?

This challenge asks you to investigate the total number of cards that would be sent if four children send one to all three others. How many would be sent if there were five children? Six?

Compare the numbers of particular tiles in one or all of these three designs, inspired by the floor tiles of a church in Cambridge.

Investigate how this pattern of squares continues. You could measure lengths, areas and angles.

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.