Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
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?
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
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?
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?
Can you make sense of these three proofs of Pythagoras' Theorem?
Draw some quadrilaterals on a 9-point circle and work out the angles. Is there a theorem?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
An environment that enables you to investigate tessellations of regular polygons
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.
Never used GeoGebra before? This article for complete beginners will help you to get started with this free dynamic geometry software.
How can visual patterns be used to prove sums of series?
The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Draw some angles inside a rectangle. What do you notice? Can you prove it?
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. . . .
Can you work out how to produce different shades of pink paint?
Can you find an efficent way to mix paints in any ratio?
There are lots of ideas to explore in these sequences of ordered fractions.
Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects its speed at each stage.
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
It would be nice to have a strategy for disentangling any tangled ropes...
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
The opposite vertices of a square have coordinates (a,b) and (c,d). What are the coordinates of the other vertices?
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?
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?
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?
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?
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.
P is a point on the circumference of a circle radius r which rolls, without slipping, inside a circle of radius 2r. What is the locus of P?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
We can show that (x + 1)² = x² + 2x + 1 by considering the area of an (x + 1) by (x + 1) square. Show in a similar way that (x + 2)² = x² + 4x + 4
What happens when we multiply a complex number by a real or an imaginary number?
Join pentagons together edge to edge. Will they form a ring?
Complex numbers can be represented graphically using an Argand diagram. This problem explains more...
Draw a square. A second square of the same size slides around the first always maintaining contact and keeping the same orientation. How far does the dot travel?
Cartesian Coordinates are not the only way!
What happens when we add together two complex numbers?
Let's go further and see what happens when we multiply two complex numbers together!
Can you devise a system for making sense of complex multiplication?
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
Take any triangle, and construct squares on each of its sides. What do you notice about the areas of the new triangles formed?
Kyle and his teacher disagree about his test score - who is right?
Use these four dominoes to make a square that has the same number of dots on each side.
How well can you estimate angles? Playing this game could improve your skills.
Move the point P to see how P' moves. Then use your insights to calculate a missing length.
How does the position of the line affect the equation of the line? What can you say about the equations of parallel lines?