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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?

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Are these statements always true, sometimes true or never true?

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Each child in Class 3 took four numbers out of the bag. Who had made the highest even number?

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Are these statements relating to odd and even numbers always true, sometimes true or never true?

This article introduces the idea of generic proof for younger children and illustrates how one example can offer a proof of a general result through unpacking its underlying structure.

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What happens when you add three numbers together? Will your answer be odd or even? How do you know?

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Explore the relationship between simple linear functions and their graphs.

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In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

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Four of these clues are needed to find the chosen number on this grid and four are true but do nothing to help in finding the number. Can you sort out the clues and find the number?

This article for teachers describes how number arrays can be a useful representation for many number concepts.

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There are exactly 3 ways to add 4 odd numbers to get 10. Find all the ways of adding 8 odd numbers to get 20. To be sure of getting all the solutions you will need to be systematic. What about. . . .

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Try to stop your opponent from being able to split the piles of counters into unequal numbers. Can you find a strategy?

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In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

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I added together some of my neighbours' house numbers. Can you explain the patterns I noticed?

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Charlie has made a Magic V. Can you use his example to make some more? And how about Magic Ls, Ns and Ws?

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Can you order the digits from 1-3 to make a number which is divisible by 3 so when the last digit is removed it becomes a 2-figure number divisible by 2, and so on?

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Can you put the numbers 1-5 in the V shape so that both 'arms' have the same total?

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These red, yellow and blue spinners were each spun 45 times in total. Can you work out which numbers are on each spinner?

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Great Granddad is very proud of his telegram from the Queen congratulating him on his hundredth birthday and he has friends who are even older than he is... When was he born?

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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?

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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?

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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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Ben’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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Investigate the smallest number of moves it takes to turn these mats upside-down if you can only turn exactly three at a time.

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Think of a number, square it and subtract your starting number. Is the number you’re left with odd or even? How do the images help to explain this?

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Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

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How many different sets of numbers with at least four members can you find in the numbers in this box?

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A game for 2 or more people. Starting with 100, subratct a number from 1 to 9 from the total. You score for making an odd number, a number ending in 0 or a multiple of 6.

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There are ten children in Becky's group. Can you find a set of numbers for each of them? Are there any other sets?

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You can trace over all of the diagonals of a pentagon without lifting your pencil and without going over any more than once. Can the same thing be done with a hexagon or with a heptagon?

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In how many different ways can you break up a stick of 7 interlocking cubes? Now try with a stick of 8 cubes and a stick of 6 cubes.

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

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Is it possible to place 2 counters on the 3 by 3 grid so that there is an even number of counters in every row and every column? How about if you have 3 counters or 4 counters or....?

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Place the numbers 1, 2, 3,..., 9 one on each square of a 3 by 3 grid so that all the rows and columns add up to a prime number. How many different solutions can you find?

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Square numbers can be represented as the sum of consecutive odd numbers. What is the sum of 1 + 3 + ..... + 149 + 151 + 153?

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Can you find examples of magic crosses? Can you find all the possibilities?

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Alison, Bernard and Charlie have been exploring sequences of odd and even numbers, which raise some intriguing questions...

In this 7-sandwich: 7 1 3 1 6 4 3 5 7 2 4 6 2 5 there are 7 numbers between the 7s, 6 between the 6s etc. The article shows which values of n can make n-sandwiches and which cannot.

Read all about Pythagoras' mathematical discoveries in this article written for students.