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

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?

Annie and Ben are playing a game with a calculator. What was Annie's secret number?

This project challenges you to work out the number of cubes hidden under a cloth. What questions would you like to ask?

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?

Write the numbers up to 64 in an interesting way so that the shape they make at the end is interesting, different, more exciting ... than just a square.

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?

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?

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?

Explore Alex's number plumber. What questions would you like to ask? Don't forget to keep visiting NRICH projects site for the latest developments and questions.

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.

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

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

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?

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

Arrange eight of the numbers between 1 and 9 in the Polo Square below so that each side adds to the same total.

Use these head, body and leg pieces to make Robot Monsters which are different heights.

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?

Move from the START to the FINISH by moving across or down to the next square. Can you find a route to make these totals?

Choose four of the numbers from 1 to 9 to put in the squares so that the differences between joined squares are odd.

Benâ€™s class were cutting up number tracks. First they cut them into twos and added up the numbers on each piece. What patterns could they see?

Look carefully at the numbers. What do you notice? Can you make another square using the numbers 1 to 16, that displays the same properties?

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?

A game for 2 people using a pack of cards Turn over 2 cards and try to make an odd number or a multiple of 3.

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

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

This challenge combines addition, multiplication, perseverance and even proof.

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?

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?

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

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

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

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

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

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?

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?

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

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

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?

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?

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?

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?

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 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?

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

Arrange the numbers 1 to 6 in each set of circles below. The sum of each side of the triangle should equal the number in its centre.

Lolla bought a balloon at the circus. She gave the clown six coins to pay for it. What could Lolla have paid for the balloon?