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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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Prove Pythagoras' Theorem using enlargements and scale factors.

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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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An environment that enables you to investigate tessellations of regular polygons

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A right-angled isosceles triangle is rotated about the centre point of a square. What can you say about the area of the part of the square covered by the triangle as it rotates?

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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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A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?

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Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

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The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?

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

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To avoid losing think of another very well known game where the patterns of play are similar.

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There are 27 small cubes in a 3 x 3 x 3 cube, 54 faces being visible at any one time. Is it possible to reorganise these cubes so that by dipping the large cube into a pot of paint three times you. . . .

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Use an Excel to investigate division. Explore the relationships between the process elements using an interactive spreadsheet.

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The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?

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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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Do you know how to find the area of a triangle? You can count the squares. What happens if we turn the triangle on end? Press the button and see. Try counting the number of units in the triangle now. . . .

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Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in them?

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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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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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How good are you at finding the formula for a number pattern ?

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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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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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This is an interactivity in which you have to sort the steps in the completion of the square into the correct order to prove the formula for the solutions of quadratic equations.

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The aim of the game is to slide the green square from the top right hand corner to the bottom left hand corner in the least number of moves.

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Find all the ways of placing the numbers 1 to 9 on a W shape, with 3 numbers on each leg, so that each set of 3 numbers has the same total.

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Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.

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Ask a friend to choose a number between 1 and 63. By identifying which of the six cards contains the number they are thinking of it is easy to tell them what the number is.

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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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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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Two engines, at opposite ends of a single track railway line, set off towards one another just as a fly, sitting on the front of one of the engines, sets off flying along the railway line...

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Place a red counter in the top left corner of a 4x4 array, which is covered by 14 other smaller counters, leaving a gap in the bottom right hand corner (HOME). What is the smallest number of moves. . . .

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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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A and B are two interlocking cogwheels having p teeth and q teeth respectively. One tooth on B is painted red. Find the values of p and q for which the red tooth on B contacts every gap on the. . . .

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Use Excel to explore multiplication of fractions.

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Can you make a right-angled triangle on this peg-board by joining up three points round the edge?

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Can you find a reliable strategy for choosing coordinates that will locate the treasure in the minimum number of guesses?

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A simple file for the Interactive whiteboard or PC screen, demonstrating equivalent fractions.

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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?

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Use an interactive Excel spreadsheet to explore number in this exciting game!

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Use Excel to investigate the effect of translations around a number grid.

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Use Excel to practise adding and subtracting fractions.

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An Excel spreadsheet with an investigation.

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Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?

This article gives you a few ideas for understanding the Got It! game and how you might find a winning strategy.

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Use an interactive Excel spreadsheet to investigate factors and multiples.

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Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.

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The interactive diagram has two labelled points, A and B. It is designed to be used with the problem "Cushion Ball"