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?

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

Challenge Level

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

Challenge Level

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?

Challenge Level

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

Challenge Level

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

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

Challenge Level

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

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

Challenge Level

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

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

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

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

Challenge Level

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

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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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An environment that enables you to investigate tessellations of regular polygons

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

Challenge Level

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

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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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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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

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?

Challenge Level

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

Challenge Level

There are lots of ideas to explore in these sequences of ordered fractions.

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

Challenge Level

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

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

Challenge Level

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?

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

Challenge Level

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

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

Week 2

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

Challenge Level

Some of the numbers have fallen off Becky's number line. Can you figure out what they were?

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

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

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?

Challenge Level

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

Challenge Level

Mr McGregor has a magic potting shed. Overnight, the number of plants in it doubles. He'd like to put the same number of plants in each of three gardens, planting one garden each day. Can he do it?

Challenge Level

Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.

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

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

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

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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.

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?