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?

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?

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.

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?

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?

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

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.

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

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.

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

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

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?

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

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

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

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?

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?

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

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?

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

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

These caterpillars have 16 parts. What different shapes do they make if each part lies in the small squares of a 4 by 4 square?

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

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?

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

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

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

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?

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?

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?

This task, written for the National Young Mathematicians' Award 2016, invites you to explore the different combinations of scores that you might get on these dart boards.

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?

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

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?

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?

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

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!

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?

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.

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?

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?

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?

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?

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?

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?