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

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

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

Three dice are placed in a row. Find a way to turn each one so that the three numbers on top of the dice total the same as the three numbers on the front of the dice. Can you find all the ways to do. . . .

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?

Find out about Magic Squares in this article written for students. Why are they magic?!

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

Fancy a game of cricket? Here is a mathematical version you can play indoors without breaking any windows.

For this challenge, you'll need to play Got It! Can you explain the strategy for winning this game with any target?

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?

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?

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.

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

Use your logical reasoning to work out how many cows and how many sheep there are in each field.

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

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

What happens when you add three numbers together? Will your answer be odd or even? How do you know?

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

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?

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

The idea of this game is to add or subtract the two numbers on the dice and cover the result on the grid, trying to get a line of three. Are there some numbers that are good to aim for?

If each of these three shapes has a value, can you find the totals of the combinations? Perhaps you can use the shapes to make the given totals?

Use the 'double-3 down' dominoes to make a square so that each side has eight dots.

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?

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

Use these four dominoes to make a square that has the same number of dots on each side.

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.

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

This problem is based on the story of the Pied Piper of Hamelin. Investigate the different numbers of people and rats there could have been if you know how many legs there are altogether!

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

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?

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

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

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?

Mrs Morgan, the class's teacher, pinned numbers onto the backs of three children. Use the information to find out what the three numbers were.

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

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?

Have a go at this game which involves throwing two dice and adding their totals. Where should you place your counters to be more likely to win?

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?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

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?

Where can you draw a line on a clock face so that the numbers on both sides have the same total?

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?

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