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Complex Rotations

Stage: 5 Challenge Level:

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?

Thebault's Theorem

Stage: 5 Challenge Level:

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?

Footprints

Stage: 5 Challenge Level:

Make a footprint pattern using only reflections.

Napoleon's Theorem

Stage: 4 and 5 Challenge Level:

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?

A Roll of Patterned Paper

Stage: 4 Challenge Level:

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

Quaternions and Rotations

Stage: 5 Challenge Level:

Find out how the quaternion function G(v) = qvq^-1 gives a simple algebraic method for working with rotations in 3-space.

Rotations Are Not Single Round Here

Stage: 4 Challenge Level:

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. . . .

Tablecloth

Stage: 4 Challenge Level:

Colour the squares of the square tablecloth so that each square is the same colour as all the symmetrically placed squares and a different colour from the rest of the squares.

Frieze Patterns in Cast Iron

Stage: 4 and 5

A gallery of beautiful photos of cast ironwork friezes in Australia with a mathematical discussion of the classification of frieze patterns.

Arrow Arithmetic 2

Stage: 4 Challenge Level:

Introduces the idea of a twizzle to represent number and asks how one can use this representation to add and subtract geometrically.

A Resource to Support Work on Transformations

Stage: 4 Challenge Level:

This resources contains a series of interactivities designed to support work on transformations at Key Stage 4.

Symmetric Trace

Stage: 4 Challenge Level:

Points off a rolling wheel make traces. What makes those traces have symmetry?

Zodiac

Stage: 4 Challenge Level:

Cubic Rotations

Stage: 4 Challenge Level:

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?

Arrow Arithmetic 3

Stage: 4 Challenge Level:

How can you use twizzles to multiply and divide?

Arrow Arithmetic 1

Stage: 4 Challenge Level:

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.

The Matrix

Stage: 4 Challenge Level:

Investigate the transfomations of the plane given by the 2 by 2 matrices with entries taking all combinations of values 0. -1 and +1.

Rots and Refs

Stage: 5 Challenge Level:

Follow hints using a little coordinate geometry, plane geometry and trig to see how matrices are used to work on transformations of the plane.

Coke Machine

Stage: 4 Challenge Level:

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. . . .

Interpenetrating Solids

Stage: 5 Challenge Level:

This problem provides training in visualisation and representation of 3D shapes. You will need to imagine rotating cubes, squashing cubes and even superimposing cubes!

In a Spin

Stage: 4 Challenge Level:

What is the volume of the solid formed by rotating this right angled triangle about the hypotenuse?

Overlaid

Stage: 2, 3 and 4 Challenge Level:

Overlaying pentominoes can produce some effective patterns. Why not use LOGO to try out some of the ideas suggested here?

Stereoisomers

Stage: 5 Challenge Level:

Put your visualisation skills to the test by seeing which of these molecules can be rotated onto each other.

Garfield's Proof

Stage: 4 Challenge Level:

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.

Illusion

Stage: 3 and 4 Challenge Level:

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?

Triangles and Petals

Stage: 4 Challenge Level:

An equilateral triangle rotates polygons with equal length sides and produces an outline like a flower. What is its perimeter?

Hand Swap

Stage: 4 Challenge Level:

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. . . .

Cubic Spin

Stage: 5 Challenge Level:

Prove that the graph of f(x) = x^3 - 6x^2 +9x +1 has rotational symmetry. Do graphs of all cubics have rotational symmetry?

Cut Cube

Stage: 5 Challenge Level:

P, Q and R are points of trisection of 3 non-intersecting perpendicular edges of a cube. Where does the plane PQR cut the other edges of the cube? Describe the symmetries of the 'half cube' obtained.

Twizzle Arithmetic

Stage: 4 Challenge Level:

Arrow arithmetic, but with a twist.

Middle Man

Stage: 5 Challenge Level:

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?

Shuffles

Stage: 5 Challenge Level:

An environment for exploring the properties of small groups.

Plex

Stage: 2, 3, 4 and 5 Challenge Level:

Plex lets you specify a mapping between points and their images. Then you can draw and see the transformed image.

The Frieze Tree

Stage: 4

Patterns that repeat in a line are strangely interesting. How many types are there and how do you tell one type from another?

Get Cross

Stage: 4 Challenge Level:

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?