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Watch this animation. What do you notice? What happens when you try more or fewer cubes in a bundle?

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Suppose we allow ourselves to use three numbers less than 10 and multiply them together. How many different products can you find? How do you know you've got them all?

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Can you each work out the number on your card? What do you notice? How could you sort the cards?

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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?

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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 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,. . . .

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

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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.

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

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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.

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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?

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Can you put these four calculations into order of difficulty? How did you decide?

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

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

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Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

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

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

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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?

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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?

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Your vessel, the Starship Diophantus, has become damaged in deep space. Can you use your knowledge of times tables and some lightning reflexes to survive?

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This task offers an opportunity to explore all sorts of number relationships, but particularly multiplication.

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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!

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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?

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If you had any number of ordinary dice, what are the possible ways of making their totals 6? What would the product of the dice be each time?

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

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

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

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

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

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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?

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

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

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

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

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In this simulation of a balance, you can drag numbers and parts of number sentences on to the trays. Have a play!

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Investigate the different ways that fifteen schools could have given money in a charity fundraiser.

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This task combines spatial awareness with addition and multiplication.

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

In this article for primary teachers, Lynne McClure outlines what is meant by fluency in the context of number and explains how our selection of NRICH tasks can help.

This article for primary teachers encourages exploration of two fundamental ideas, exchange and 'unitising', which will help children become more fluent when calculating.

In this article, Alf outlines six activities using the Gattegno chart, which help to develop understanding of place value, multiplication and division.

Alf describes how the Gattegno chart helped a class of 7-9 year olds gain an awareness of place value and of the inverse relationship between multiplication and division.

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Does this 'trick' for calculating multiples of 11 always work? Why or why not?

Related resources supporting pupils' understanding of multiplication and division through playing with numbers.

More resources to support understanding multiplication and division through playing with numbers

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

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Watch the video of this game being played. Can you work out the rules? Which dice totals are good to get, and why?