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

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An environment that enables you to investigate tessellations of regular polygons

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

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

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

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

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

Week 2

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

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

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

Challenge Level

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

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L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?

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

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

Challenge Level

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

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

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Can you find a way to turn a rectangle into a square?

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

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

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Can you work out the fraction of the original triangle that is covered by the inner triangle?

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Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?

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

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

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

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

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Draw some isosceles triangles with an area of $9$cm$^2$ and a vertex at (20,20). If all the vertices must have whole number coordinates, how many is it possible to draw?

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

Challenge Level

A tilted square is a square with no horizontal sides. Can you devise a general instruction for the construction of a square when you are given just one of its sides?