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

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

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

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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 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

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.

Challenge Level

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

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

Challenge Level

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

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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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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 red square and a blue square overlap. Does the red square always cover a quarter of the blue square?

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An environment that enables you to investigate tessellations of regular polygons

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

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

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

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

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

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

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

Challenge Level

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

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?

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

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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?

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

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

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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

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

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?

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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?

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

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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.

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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.

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

Challenge Level

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.