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?
Move the point P to see how P' moves. Then use your insights to calculate a missing length.
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?
Can you explain what is happening and account for the values being displayed?
Can you find an efficent way to mix paints in any ratio?
Can you work out how to produce different shades of pink paint?
Can you make sense of these three proofs of Pythagoras' Theorem?
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.
Never used GeoGebra before? This article for complete beginners will help you to get started with this free dynamic geometry software.
Can you work out the fraction of the original triangle that is covered by the inner triangle?
How well can you estimate angles? Playing this game could improve your skills.
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
Draw some angles inside a rectangle. What do you notice? Can you prove it?
An environment that enables you to investigate tessellations of regular polygons
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?
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?
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.
How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?
Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?
Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?
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?
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
Move the corner of the rectangle. Can you work out what the purple number represents?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
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. . . .
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?
Some of the numbers have fallen off Becky's number line. Can you figure out what they were?
Under what circumstances can you rearrange a big square to make three smaller squares?
Complex numbers can be represented graphically using an Argand diagram. This problem explains more...
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
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?
Cartesian Coordinates are not the only way!
Take any triangle, and construct squares on each of its sides. What do you notice about the areas of the new triangles formed?
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
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.
Can you devise a system for making sense of complex multiplication?
Let's go further and see what happens when we multiply two complex numbers together!
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.
Use these four dominoes to make a square that has the same number of dots on each side.
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
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?
Can you find triangles on a 9-point circle? Can you work out their angles?
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.
There are lots of ideas to explore in these sequences of ordered fractions.
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?
Join pentagons together edge to edge. Will they form a ring?
Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings.