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

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

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

Use this grid to shade the numbers in the way described. Which numbers do you have left? Do you know what they are called?

The triangles in these sets are similar - can you work out the lengths of the sides which have question marks?

Which is quicker, counting up to 30 in ones or counting up to 300 in tens? Why?

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

Can you replace the letters with numbers? Is there only one solution in each case?

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

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?

There are four equal weights on one side of the scale and an apple on the other side. What can you say that is true about the apple and the weights from the picture?

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?

Grandma found her pie balanced on the scale with two weights and a quarter of a pie. So how heavy was each pie?

This big box multiplies anything that goes inside it by the same number. If you know the numbers that come out, what multiplication might be going on in the box?

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

Chandrika was practising a long distance run. Can you work out how long the race was from the information?

Work out Tom's number from the answers he gives his friend. He will only answer 'yes' or 'no'.

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.

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

Use this information to work out whether the front or back wheel of this bicycle gets more wear and tear.

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

All the girls would like a puzzle each for Christmas and all the boys would like a book each. Solve the riddle to find out how many puzzles and books Santa left.

56 406 is the product of two consecutive numbers. What are these two numbers?

Can you work out some different ways to balance this equation?

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

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

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?

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?

Using the numbers 1, 2, 3, 4 and 5 once and only once, and the operations x and ÷ once and only once, what is the smallest whole number you can make?

Have a go at balancing this equation. Can you find different ways of doing it?

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 calculation by filling in the missing numbers? In how many different ways can you do it?

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?

In the multiplication calculation, some of the digits have been replaced by letters and others by asterisks. Can you reconstruct the original multiplication?

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

Find a great variety of ways of asking questions which make 8.

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

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

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?

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.

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.

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

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,. . . .

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?