A white cross is placed symmetrically in a red disc with the central square of side length sqrt 2 and the arms of the cross of length 1 unit. What is the area of the disc still showing?
A security camera, taking pictures each half a second, films a
cyclist going by. In the film, the cyclist appears to go forward
while the wheels appear to go backwards. Why?
The first part of an investigation into how to represent numbers
using geometric transformations that ultimately leads us to
discover numbers not on the number line.
Introduces the idea of a twizzle to represent number and asks how
one can use this representation to add and subtract geometrically.
A design is repeated endlessly along a line - rather like a stream
of paper coming off a roll. Make a strip that matches itself after
rotation, or after reflection
I noticed this about streamers that have rotation symmetry : if
there was one centre of rotation there always seems to be a second
centre that also worked. Can you find a design that has only. . . .
Arrow arithmetic, but with a twist.
Charlie likes tablecloths that use as many colours as possible, but insists that his tablecloths have some symmetry. Can you work out how many colours he needs for different tablecloth designs?
There are thirteen axes of rotational symmetry of a unit cube. Describe them all. What is the average length of the parts of the axes of symmetry which lie inside the cube?
Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.
Points off a rolling wheel make traces. What makes those traces
This resources contains a series of interactivities designed to
support work on transformations at Key Stage 4.
How can you use twizzles to multiply and divide?
What is the volume of the solid formed by rotating this right
angled triangle about the hypotenuse?
A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.
Plex lets you specify a mapping between points and their images.
Then you can draw and see the transformed image.
Triangle ABC has equilateral triangles drawn on its edges. Points
P, Q and R are the centres of the equilateral triangles. What can
you prove about the triangle PQR?
My train left London between 6 a.m. and 7 a.m. and arrived in Paris
between 9 a.m. and 10 a.m. At the start and end of the journey the
hands on my watch were in exactly the same positions but the. . . .
Overlaying pentominoes can produce some effective patterns. Why not
use LOGO to try out some of the ideas suggested here?
See the effects of some combined transformations on a shape. Can
you describe what the individual transformations do?
Does changing the order of transformations always/sometimes/never
produce the same transformation?
Look carefully at the video of a tangle and explain what's
This article describes the scope for practical exploration of tessellations both in and out of the classroom. It seems a golden opportunity to link art with maths, allowing the creative side of your. . . .
How many different transformations can you find made up from
combinations of R, S and their inverses? Can you be sure that you
have found them all?
Here is a chance to create some attractive images by rotating
shapes through multiples of 90 degrees, or 30 degrees, or 72
How many different symmetrical shapes can you make by shading triangles or squares?
Can you describe what happens in this film?
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. . . .
The coke machine in college takes 50 pence pieces. It also takes a certain foreign coin of traditional design. Coins inserted into the machine slide down a chute into the machine and a drink is duly. . . .
The diagram shows a very heavy kitchen cabinet. It cannot be lifted but it can be pivoted around a corner. The task is to move it, without sliding, in a series of turns about the corners so that it. . . .
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Sort the frieze patterns into seven pairs according to the way in
which the motif is repeated.
This article for teachers suggests ideas for activities built around 10 and 2010.
Why not challenge a friend to play this transformation game?
Two circles of equal radius touch at P. One circle is fixed whilst the other moves, rolling without slipping, all the way round. How many times does the moving coin revolve before returning to P?
Explore the effect of reflecting in two intersecting mirror lines.
A triangle ABC resting on a horizontal line is "rolled" along the
line. Describe the paths of each of the vertices and the
relationships between them and the original triangle.
A train leaves on time. After it has gone 8 miles (at 33mph) the driver looks at his watch and sees that the hour hand is exactly over the minute hand. When did the train leave the station?
The triangle ABC is equilateral. The arc AB has centre C, the arc
BC has centre A and the arc CA has centre B. Explain how and why
this shape can roll along between two parallel tracks.