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

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?

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.

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

Can you use the numbers on the dice to reach your end of the number line before your partner beats you?

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

The picture shows a lighthouse and many underwater creatures. If you know the markings on the lighthouse are 1m apart, can you work out the distances between some of the different creatures?

In this game, you can add, subtract, multiply or divide the numbers on the dice. Which will you do so that you get to the end of the number line first?

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

A game for 2 or more players. Practise your addition and subtraction with the aid of a game board and some dried peas!

Can you make a train the same length as Laura's but using three differently coloured rods? Is there only one way of doing it?

How would you count the number of fingers in these pictures?

A game for 2 or more players with a pack of cards. Practise your skills of addition, subtraction, multiplication and division to hit the target score.

Can you each work out the number on your card? What do you notice? How could you sort the cards?

Can you hang weights in the right place to make the equaliser balance?

Well now, what would happen if we lost all the nines in our number system? Have a go at writing the numbers out in this way and have a look at the multiplications table.

A game for 2 players. Practises subtraction or other maths operations knowledge.

We start with one yellow cube and build around it to make a 3x3x3 cube with red cubes. Then we build around that red cube with blue cubes and so on. How many cubes of each colour have we used?

What happens when you add the digits of a number then multiply the result by 2 and you keep doing this? You could try for different numbers and different rules.

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

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?

If you have ten counters numbered 1 to 10, how many can you put into pairs that add to 10? Which ones do you have to leave out? Why?

On the table there is a pile of oranges and lemons that weighs exactly one kilogram. Using the information, can you work out how many lemons there are?

Amy has a box containing domino pieces but she does not think it is a complete set. She has 24 dominoes in her box and there are 125 spots on them altogether. Which of her domino pieces are missing?

Rocco ran in a 200 m race for his class. Use the information to find out how many runners there were in the race and what Rocco's finishing position was.

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.

Annie cut this numbered cake into 3 pieces with 3 cuts so that the numbers on each piece added to the same total. Where were the cuts and what fraction of the whole cake was each piece?

Can you make a cycle of pairs that add to make a square number using all the numbers in the box below, once and once only?

Tell your friends that you have a strange calculator that turns numbers backwards. What secret number do you have to enter to make 141 414 turn around?

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

This group activity will encourage you to share calculation strategies and to think about which strategy might be the most efficient.

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

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?

Arrange three 1s, three 2s and three 3s in this square so that every row, column and diagonal adds to the same total.

This problem is based on a code using two different prime numbers less than 10. You'll need to multiply them together and shift the alphabet forwards by the result. Can you decipher the code?

Investigate what happens when you add house numbers along a street in different ways.

Place the numbers 1 to 6 in the circles so that each number is the difference between the two numbers just below it.

Using 3 rods of integer lengths, none longer than 10 units and not using any rod more than once, you can measure all the lengths in whole units from 1 to 10 units. How many ways can you do this?

An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.

Here are the prices for 1st and 2nd class mail within the UK. You have an unlimited number of each of these stamps. Which stamps would you need to post a parcel weighing 825g?

Can you design a new shape for the twenty-eight squares and arrange the numbers in a logical way? What patterns do you notice?

If you have only four weights, where could you place them in order to balance this equaliser?

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?

Number problems at primary level that may require determination.