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

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

On my calculator I divided one whole number by another whole number and got the answer 3.125 If the numbers are both under 50, what are they?

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?

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?

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.

On Friday the magic plant was only 2 centimetres tall. Every day it doubled its height. How tall was it on Monday?

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.

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?

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

Four Go game for an adult and child. Will you be the first to have four numbers in a row on the number line?

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

This challenge is a game for two players. Choose two numbers from the grid and multiply or divide, then mark your answer on the number line. Can you get four in a row before your partner?

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?

Take the number 6 469 693 230 and divide it by the first ten prime numbers and you'll find the most beautiful, most magic of all numbers. What is it?

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

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?

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

At the beginning of May Tom put his tomato plant outside. On the same day he sowed a bean in another pot. When will the two be the same height?

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

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?

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

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.

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?

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?

This multiplication uses each of the digits 0 - 9 once and once only. Using the information given, can you replace the stars in the calculation with figures?

Can you complete this jigsaw of the multiplication square?

Can you see how these factor-multiple chains work? Find the chain which contains the smallest possible numbers. How about the largest possible numbers?

This article for teachers looks at how teachers can use problems from the NRICH site to help them teach division.

Can you work out the arrangement of the digits in the square so that the given products are correct? The numbers 1 - 9 may be used once and once only.

What is the lowest number which always leaves a remainder of 1 when divided by each of the numbers from 2 to 10?

The Man is much smaller than us. Can you use the picture of him next to a mug to estimate his height and how much tea he drinks?

Here is a picnic that Petros and Michael are going to share equally. Can you tell us what each of them will have?

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

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.

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

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

This article for teachers describes how modelling number properties involving multiplication using an array of objects not only allows children to represent their thinking with concrete materials,. . . .

In November, Liz was interviewed for an article on a parents' website about learning times tables. Read the article here.

Resources to support understanding of multiplication and division through playing with number.

Explore Alex's number plumber. What questions would you like to ask? What do you think is happening to the numbers?

This problem is designed to help children to learn, and to use, the two and three times tables.

After training hard, these two children have improved their results. Can you work out the length or height of their first jumps?

Look at what happens when you take a number, square it and subtract your answer. What kind of number do you get? Can you prove it?

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

This challenge encourages you to explore dividing a three-digit number by a single-digit number.