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An environment that enables you to investigate tessellations of regular polygons

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Make five different quadrilaterals on a nine-point pegboard, without using the centre peg. Work out the angles in each quadrilateral you make. Now, what other relationships you can see?

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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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Draw some angles inside a rectangle. What do you notice? Can you prove it?

Never used GeoGebra before? This article for complete beginners will help you to get started with this free dynamic geometry software.

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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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Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?

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Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.

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Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?

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Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?

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Join pentagons together edge to edge. Will they form a ring?

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A dot starts at the point (1,0) and turns anticlockwise. Can you estimate the height of the dot after it has turned through 45 degrees? Can you calculate its height?

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Under what circumstances can you rearrange a big square to make three smaller squares?

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Move the point P to see how P' moves. Then use your insights to calculate a missing length.

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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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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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What is the relationship between the angle at the centre and the angles at the circumference, for angles which stand on the same arc? Can you prove it?

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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?

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Can you find triangles on a 9-point circle? Can you work out their angles?

Week 2

How well can you estimate angles? Playing this game could improve your skills.

Move the corner of the rectangle. Can you work out what the purple number represents?

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The points P, Q, R and S are the midpoints of the edges of a non-convex quadrilateral.What do you notice about the quadrilateral PQRS and its area?

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P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?

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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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Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?

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Using a ruler, pencil and compasses only, it is possible to construct a square inside any triangle so that all four vertices touch the sides of the triangle.

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The length AM can be calculated using trigonometry in two different ways. Create this pair of equivalent calculations for different peg boards, notice a general result, and account for it.

Use the applet to explore the area of a parallelogram and how it relates to vectors.

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Can you make sense of these three proofs of Pythagoras' Theorem?

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Can you work out how to produce different shades of pink paint?

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Can you find an efficent way to mix paints in any ratio?

Use the applet to make some squares. What patterns do you notice in the coordinates?

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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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Complex numbers can be represented graphically using an Argand diagram. This problem explains more...

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The sine of an angle is equal to the cosine of its complement. Can you explain why and does this rule extend beyond angles of 90 degrees?

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What happens when we multiply a complex number by a real or an imaginary number?

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Let's go further and see what happens when we multiply two complex numbers together!

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Can you devise a system for making sense of complex multiplication?

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How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?

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Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

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Take any triangle, and construct squares on each of its sides. What do you notice about the areas of the new triangles formed?

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Can you explain what is happening and account for the values being displayed?

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Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?

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

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Investigate what happens to the equations of different lines when you reflect them in one of the axes. Try to predict what will happen. Explain your findings.