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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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ABC and DEF are equilateral triangles of side 3 and 4 respectively. Construct an equilateral triangle whose area is the sum of the area of ABC and DEF.

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

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If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.

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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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With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.

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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?

Dave Hewitt suggests that there might be more to mathematics than looking at numerical results, finding patterns and generalising.

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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?

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

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Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

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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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Make some loops out of regular hexagons. What rules can you discover?

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Many numbers can be expressed as the difference of two perfect squares. What do you notice about the numbers you CANNOT make?

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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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Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

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

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

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

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

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What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

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First of all, pick the number of times a week that you would like to eat chocolate. Multiply this number by 2...

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

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Euler discussed whether or not it was possible to stroll around Koenigsberg crossing each of its seven bridges exactly once. Experiment with different numbers of islands and bridges.

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Rectangles are considered different if they vary in size or have different locations. How many different rectangles can be drawn on a chessboard?

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Can you find sets of sloping lines that enclose a square?

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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.

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Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?

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

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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?

Alf Coles writes about how he tries to create 'spaces for exploration' for the students in his classrooms.

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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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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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Janine noticed, while studying some cube numbers, that if you take three consecutive whole numbers and multiply them together and then add the middle number of the three, you get the middle number. . . .

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Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

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Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?

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A counter is placed in the bottom right hand corner of a grid. You toss a coin and move the star according to the following rules: ... What is the probability that you end up in the top left-hand. . . .

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Can you describe this route to infinity? Where will the arrows take you next?

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This task encourages you to investigate the number of edging pieces and panes in different sized windows.

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A red square and a blue square overlap so that the corner of the red square rests on the centre of the blue square. Show that, whatever the orientation of the red square, it covers a quarter of the. . . .

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The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?