Explore the meaning behind the algebra and geometry of matrices
with these 10 individual problems.
Follow hints using a little coordinate geometry, plane geometry and
trig to see how matrices are used to work on transformations of the
Investigate the transfomations of the plane given by the 2 by 2
matrices with entries taking all combinations of values 0. -1 and
Choose some complex numbers and mark them by points on a graph.
Multiply your numbers by i once, twice, three times, four times,
..., n times? What happens?
Put your visualisation skills to the test by seeing which of these
molecules can be rotated onto each other.
What groups of transformations map a regular pentagon to itself?
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?
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
Arrow arithmetic, but with a twist.
What is the volume of the solid formed by rotating this right
angled triangle about the hypotenuse?
Make a footprint pattern using only reflections.
This problem provides training in visualisation and representation
of 3D shapes. You will need to imagine rotating cubes, squashing
cubes and even superimposing cubes!
A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.
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?
Take any parallelogram and draw squares on the sides of the
parallelogram. What can you prove about the quadrilateral formed by
joining the centres of these squares?
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. . . .
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?
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.
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.
Overlaying pentominoes can produce some effective patterns. Why not
use LOGO to try out some of the ideas suggested here?
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?
Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.
Find out how the quaternion function G(v) = qvq^-1 gives a simple
algebraic method for working with rotations in 3-space.
Plex lets you specify a mapping between points and their images.
Then you can draw and see the transformed image.
Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational
symmetry. Do graphs of all cubics have rotational symmetry?
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?
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. . . .
Find the shape and symmetries of the two pieces of this cut cube.
How can you use twizzles to multiply and divide?
Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?
Rotate a copy of the trapezium about the centre of the longest side
of the blue triangle to make a square. Find the area of the square
and then derive a formula for the area of the trapezium.
An environment for exploring the properties of small groups.
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. . . .
Mark a point P inside a closed curve. Is it always possible to find
two points that lie on the curve, such that P is the mid point of
the line joining these two points?