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

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

There are over sixty different ways of making 24 by adding, subtracting, multiplying and dividing all four numbers 4, 6, 6 and 8 (using each number only once). How many can you find?

How can we help students make sense of addition and subtraction of negative numbers?

Can you draw a continuous line through 16 numbers on this grid so that the total of the numbers you pass through is as high as possible?

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.

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

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

How many solutions can you find to this sum? Each of the different letters stands for a different number.

There are nasty versions of this dice game but we'll start with the nice ones...

What do the digits in the number fifteen add up to? How many other numbers have digits with the same total but no zeros?

In this 100 square, look at the green square which contains the numbers 2, 3, 12 and 13. What is the sum of the numbers that are diagonally opposite each other? What do you notice?

Start by putting one million (1 000 000) into the display of your calculator. Can you reduce this to 7 using just the 7 key and add, subtract, multiply, divide and equals as many times as you like?

The clockmaker's wife cut up his birthday cake to look like a clock face. Can you work out who received each piece?

Use 4 four times with simple operations so that you get the answer 12. Can you make 15, 16 and 17 too?

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

Find out what a Deca Tree is and then work out how many leaves there will be after the woodcutter has cut off a trunk, a branch, a twig and a leaf.

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

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

Find a great variety of ways of asking questions which make 8.

Some Games That May Be Nice or Nasty for an adult and child. Use your knowledge of place value to beat your oponent.

Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?

The Scot, John Napier, invented these strips about 400 years ago to help calculate multiplication and division. Can you work out how to use Napier's bones to find the answer to these multiplications?

Choose four different digits from 1-9 and put one in each box so that the resulting four two-digit numbers add to a total of 100.

Make one big triangle so the numbers that touch on the small triangles add to 10. You could use the interactivity to help you.

These sixteen children are standing in four lines of four, one behind the other. They are each holding a card with a number on it. Can you work out the missing numbers?

How have the numbers been placed in this Carroll diagram? Which labels would you put on each row and column?

In sheep talk the only letters used are B and A. A sequence of words is formed by following certain rules. What do you notice when you count the letters in each word?

In your bank, you have three types of coins. The number of spots shows how much they are worth. Can you choose coins to exchange with the groups given to make the same total?

Tim had nine cards each with a different number from 1 to 9 on it. How could he have put them into three piles so that the total in each pile was 15?

There were chews for 2p, mini eggs for 3p, Chocko bars for 5p and lollypops for 7p in the sweet shop. What could each of the children buy with their money?

Use the interactivities to fill in these Carroll diagrams. How do you know where to place the numbers?

Twizzle, a female giraffe, needs transporting to another zoo. Which route will give the fastest journey?

Winifred Wytsh bought a box each of jelly babies, milk jelly bears, yellow jelly bees and jelly belly beans. In how many different ways could she make a jolly jelly feast with 32 legs?

Go through the maze, collecting and losing your money as you go. Which route gives you the highest return? And the lowest?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Using the statements, can you work out how many of each type of rabbit there are in these pens?

Investigate the different distances of these car journeys and find out how long they take.

You have 5 darts and your target score is 44. How many different ways could you score 44?

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

A group of children are using measuring cylinders but they lose the labels. Can you help relabel them?

In this investigation, you are challenged to make mobile phone numbers which are easy to remember. What happens if you make a sequence adding 2 each time?

Can you arrange fifteen dominoes so that all the touching domino pieces add to 6 and the ends join up? Can you make all the joins add to 7?

If the numbers 5, 7 and 4 go into this function machine, what numbers will come out?

A lady has a steel rod and a wooden pole and she knows the length of each. How can she measure out an 8 unit piece of pole?

I was looking at the number plate of a car parked outside. Using my special code S208VBJ adds to 65. Can you crack my code and use it to find out what both of these number plates add up to?