How can visual patterns be used to prove sums of series?
Semi-regular tessellations combine two or more different regular polygons to fill the plane. Can you find all the semi-regular tessellations?
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?
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?
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?
Join pentagons together edge to edge. Will they form a ring?
Can you make a right-angled triangle on this peg-board by joining up three points round the edge?
Can you find triangles on a 9-point circle? Can you work out their angles?
If you continue the pattern, can you predict what each of the following areas will be? Try to explain your prediction.
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
Join the midpoints of a quadrilateral to get a new quadrilateral. What is special about it?
Kyle and his teacher disagree about his test score - who is right?
Medieval stonemasons used a method to construct octagons using ruler and compasses... Is the octagon regular? Proof please.
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?
L triominoes can fit together to make larger versions of themselves. Is every size possible to make in this way?
Can you put the 25 coloured tiles into the 5 x 5 square so that no column, no row and no diagonal line have tiles of the same colour in 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. . . .
Draw some angles inside a rectangle. What do you notice? Can you prove it?
Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?
Is there a relationship between the coordinates of the endpoints of a line and the number of grid squares it crosses?
It would be nice to have a strategy for disentangling any tangled ropes...
The farmers want to redraw their field boundary but keep the area the same. Can you advise them?
Explore the area of families of parallelograms and triangles. Can you find rules to work out the areas?
Can you use small coloured cubes to make a 3 by 3 by 3 cube so that each face of the bigger cube contains one of each colour?
Construct two equilateral triangles on a straight line. There are two lengths that look the same - can you prove it?
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?
How good are you at estimating angles?
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?
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?
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.
In how many ways can you fit all three pieces together to make shapes with line symmetry?
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?
An environment that enables you to investigate tessellations of regular polygons
A tool for generating random integers.
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?
How much of the square is coloured blue? How will the pattern continue?
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?
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?
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 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?
Arrange the four number cards on the grid, according to the rules, to make a diagonal, vertical or horizontal line.
Investigate what happens to the equation of different lines when you translate them. Try to predict what will happen. Explain your findings.
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.
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.
Move the corner of the rectangle. Can you work out what the purple number represents?
Can you work out the fraction of the original triangle that is covered by the inner triangle?