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Rolling Triangle

Stage: 3 Challenge Level:

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.

Turning Triangles

Stage: 3 Challenge Level:

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.

Isometrically

Stage: 3 Challenge Level:

How many unique symmetrical shapes can you make by shading four small triangles?

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?

...on the Wall

Stage: 3 Challenge Level:

Explore the effect of reflecting in two intersecting mirror lines.

Notes on a Triangle

Stage: 3 Challenge Level:

Can you describe what happens in this film?

John's Train Is on Time

Stage: 3 Challenge Level:

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?

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.

Weighty Problem

Stage: 3 Challenge Level:

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. . . .

Matching Frieze Patterns

Stage: 3 Challenge Level:

Sort the frieze patterns into seven pairs according to the way in which the motif is repeated.

Rollin' Rollin' Rollin'

Stage: 3 Challenge Level:

Two circles of equal radius kiss 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?

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?

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. . . .

Decoding Transformations

Stage: 3 Challenge Level:

See the effects of some combined transformations on a shape. Can you describe what the individual transformations do?

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?

Combining Transformations

Stage: 3 Challenge Level:

Does changing the order of transformations always/sometimes/never produce the same transformation?

Simplifying Transformations

Stage: 3 Challenge Level:

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?

Turning Tangles

Stage: 3 Challenge Level:

Look carefully at the video of a tangle and explain what's happening.

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?

In a Spin

Stage: 4 Challenge Level:

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

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.

Overlap

Stage: 3 Challenge Level:

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. . . .

Transformation Game

Stage: 3 Challenge Level:

Why not challenge a friend to play this transformation game?

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.

Zodiac

Stage: 4 Challenge Level:

Symmetric Trace

Stage: 4 Challenge Level:

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

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.

Arrow Arithmetic 3

Stage: 4 Challenge Level:

How can you use twizzles to multiply and divide?

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.

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. . . .

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. . . .

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

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?

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.

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.

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?

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.

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?

Twizzle Arithmetic

Stage: 4 Challenge Level:

Arrow arithmetic, but with a twist.