Challenge Level

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

Challenge Level

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?

Challenge Level

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.

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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?

Challenge Level

Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings.

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

Challenge Level

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

Challenge Level

In how many ways can you fit all three pieces together to make shapes with line symmetry?

Challenge Level

Triangle ABC has equilateral triangles drawn on its edges. Points P, Q and R are the centres of the equilateral triangles. What can you prove about the triangle PQR?

Challenge Level

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

Challenge Level

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. . . .

Challenge Level

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

Challenge Level

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

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

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

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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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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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An environment that enables you to investigate tessellations of regular polygons

Challenge Level

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

Challenge Level

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?

Challenge Level

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?

Challenge Level

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.

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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!

Challenge Level

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

Challenge Level

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

Challenge Level

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

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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.

Challenge Level

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

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

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Where will the point stop after it has turned through 30 000 degrees? I took out my calculator and typed 30 000 ÷ 360. How did this help?

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

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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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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.

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

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

Challenge Level

The farmers want to redraw their field boundary but keep the area the same. Can you advise them?

Week 2

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

Challenge Level

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

Challenge Level

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?

Challenge Level

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

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

Challenge Level

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