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

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?

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?

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!

Claire thinks she has the most sports cards in her album. "I have 12 pages with 2 cards on each page", says Claire. Ross counts his cards. "No! I have 3 cards on each of my pages and there are. . . .

Can you find different ways of creating paths using these paving slabs?

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

A game for 2 people. Use your skills of addition, subtraction, multiplication and division to blast the asteroids.

Use your logical-thinking skills to deduce how much Dan's crisps and ice-cream cost altogether.

Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?

Zumf makes spectacles for the residents of the planet Zargon, who have either 3 eyes or 4 eyes. How many lenses will Zumf need to make all the different orders for 9 families?

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?

There is a clock-face where the numbers have become all mixed up. Can you find out where all the numbers have got to from these ten statements?

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

Use the information to work out how many gifts there are in each pile.

There were 22 legs creeping across the web. How many flies? How many spiders?

EWWNP means Exploring Wild and Wonderful Number Patterns Created by Yourself! Investigate what happens if we create number patterns using some simple rules.

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

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

What do you notice about the date 03.06.09? Or 08.01.09? This challenge invites you to investigate some interesting dates yourself.

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

Can you find which shapes you need to put into the grid to make the totals at the end of each row and the bottom of each column?

Shut the Box game for an adult and child. Can you turn over the cards which match the numbers on the dice?

Skippy and Anna are locked in a room in a large castle. The key to that room, and all the other rooms, is a number. The numbers are locked away in a problem. Can you help them to get out?

During the third hour after midnight the hands on a clock point in the same direction (so one hand is over the top of the other). At what time, to the nearest second, does this happen?

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

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

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

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?

This number has 903 digits. What is the sum of all 903 digits?

Number problems at primary level that may require resilience.

Number problems at primary level that require careful consideration.

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.

Can you arrange 5 different digits (from 0 - 9) in the cross in the way described?

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?

Bernard Bagnall recommends some primary school problems which use numbers from the environment around us, from clocks to house numbers.

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?

Peter, Melanie, Amil and Jack received a total of 38 chocolate eggs. Use the information to work out how many eggs each person had.

On a calculator, make 15 by using only the 2 key and any of the four operations keys. How many ways can you find to do it?

There are 44 people coming to a dinner party. There are 15 square tables that seat 4 people. Find a way to seat the 44 people using all 15 tables, with no empty places.

Find another number that is one short of a square number and when you double it and add 1, the result is also a square number.

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

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?

In a Magic Square all the rows, columns and diagonals add to the 'Magic Constant'. How would you change the magic constant of this square?

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.