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What is the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle?

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What is the total number of squares that can be made on a 5 by 5 geoboard?

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Sets of integers like 3, 4, 5 are called Pythagorean Triples, because they could be the lengths of the sides of a right-angled triangle. Can you find any more?

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Pick a square within a multiplication square and add the numbers on each diagonal. What do you notice?

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If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

Spotting patterns can be an important first step - explaining why it is appropriate to generalise is the next step, and often the most interesting and important.

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Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel. . . .

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Polygons drawn on square dotty paper have dots on their perimeter (p) and often internal (i) ones as well. Find a relationship between p, i and the area of the polygons.

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Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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Can you find a general rule for finding the areas of equilateral triangles drawn on an isometric grid?

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A little bit of algebra explains this 'magic'. Ask a friend to pick 3 consecutive numbers and to tell you a multiple of 3. Then ask them to add the four numbers and multiply by 67, and to tell you. . . .

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Can you show that you can share a square pizza equally between two people by cutting it four times using vertical, horizontal and diagonal cuts through any point inside the square?

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

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It would be nice to have a strategy for disentangling any tangled ropes...

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It's easy to work out the areas of most squares that we meet, but what if they were tilted?

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Choose a couple of the sequences. Try to picture how to make the next, and the next, and the next... Can you describe your reasoning?

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Caroline and James pick sets of five numbers. Charlie chooses three of them that add together to make a multiple of three. Can they stop him?

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Jo has three numbers which she adds together in pairs. When she does this she has three different totals: 11, 17 and 22 What are the three numbers Jo had to start with?”

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Start with any number of counters in any number of piles. 2 players take it in turns to remove any number of counters from a single pile. The loser is the player who takes the last counter.

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A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.

An article for teachers and pupils that encourages you to look at the mathematical properties of similar games.

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Try entering different sets of numbers in the number pyramids. How does the total at the top change?

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When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

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A 2 by 3 rectangle contains 8 squares and a 3 by 4 rectangle contains 20 squares. What size rectangle(s) contain(s) exactly 100 squares? Can you find them all?

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How many moves does it take to swap over some red and blue frogs? Do you have a method?

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Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?

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Choose four consecutive whole numbers. Multiply the first and last numbers together. Multiply the middle pair together. What do you notice?

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Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

This article for teachers describes several games, found on the site, all of which have a related structure that can be used to develop the skills of strategic planning.

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Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

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Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?

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Consider all two digit numbers (10, 11, . . . ,99). In writing down all these numbers, which digits occur least often, and which occur most often ? What about three digit numbers, four digit numbers. . . .

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Can you find the values at the vertices when you know the values on the edges?

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We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4

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Take any two positive numbers. Calculate the arithmetic and geometric means. Repeat the calculations to generate a sequence of arithmetic means and geometric means. Make a note of what happens to the. . . .

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Jo made a cube from some smaller cubes, painted some of the faces of the large cube, and then took it apart again. 45 small cubes had no paint on them at all. How many small cubes did Jo use?

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

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Is there an efficient way to work out how many factors a large number has?

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Nim-7 game for an adult and child. Who will be the one to take the last counter?

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Got It game for an adult and child. How can you play so that you know you will always win?

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Imagine we have four bags containing numbers from a sequence. What numbers can we make now?

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It starts quite simple but great opportunities for number discoveries and patterns!

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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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Some students have been working out the number of strands needed for different sizes of cable. Can you make sense of their solutions?

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Can you find the values at the vertices when you know the values on the edges of these multiplication arithmagons?

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Imagine we have four bags containing a large number of 1s, 4s, 7s and 10s. What numbers can we make?

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How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?