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

Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?

This challenge extends the Plants investigation so now four or more children are involved.

This challenging activity involves finding different ways to distribute fifteen items among four sets, when the sets must include three, four, five and six items.

How could you put these three beads into bags? How many different ways can you do it? How could you record what you've done?

What two-digit numbers can you make with these two dice? What can't you make?

What happens when you round these three-digit numbers to the nearest 100?

Have a go at balancing this equation. Can you find different ways of doing it?

Can you complete this calculation by filling in the missing numbers? In how many different ways can you do it?

In this calculation, the box represents a missing digit. What could the digit be? What would the solution be in each case?

Can you work out some different ways to balance this equation?

How could you arrange at least two dice in a stack so that the total of the visible spots is 18?

Three children are going to buy some plants for their birthdays. They will plant them within circular paths. How could they do this?

This activity involves rounding four-digit numbers to the nearest thousand.

In this town, houses are built with one room for each person. There are some families of seven people living in the town. In how many different ways can they build their houses?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

What happens when you round these numbers to the nearest whole number?

Bernard Bagnall discusses the importance of valuing young children's mathematical representations in this article for teachers.

As you come down the ladders of the Tall Tower you collect useful spells. Which way should you go to collect the most spells?

In this article for teachers, Elizabeth Carruthers and Maulfry Worthington explore the differences between 'recording mathematics' and 'representing mathematical thinking'.

This article, written for teachers, looks at the different kinds of recordings encountered in Primary Mathematics lessons and the importance of not jumping to conclusions!

Here are some short problems for you to try. Talk to your friends about how you work them out.