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Broad Topics > Transformations and their Properties > Reflections

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Surprising Transformations

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

I took the graph y=4x+7 and performed four transformations. Can you find the order in which I could have carried out the transformations?

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Reflecting Lines

Stage: 3 Challenge Level: Challenge Level:1

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.

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Can You Explain Why?

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Can you explain why it is impossible to construct this triangle?

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...on the Wall

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Explore the effect of reflecting in two intersecting mirror lines.

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Mirror, Mirror...

Stage: 3 Challenge Level: Challenge Level:1

Explore the effect of reflecting in two parallel mirror lines.

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Transformation Game

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

Why not challenge a friend to play this transformation game?

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Orbiting Billiard Balls

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

What angle is needed for a ball to do a circuit of the billiard table and then pass through its original position?

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Let Us Reflect

Stage: 2 Challenge Level: Challenge Level:1

Where can you put the mirror across the square so that you can still "see" the whole square? How many different positions are possible?

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Shady Symmetry

Stage: 3 Challenge Level: Challenge Level:1

How many different symmetrical shapes can you make by shading triangles or squares?

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Reflecting Squarely

Stage: 3 Challenge Level: Challenge Level:1

In how many ways can you fit all three pieces together to make shapes with line symmetry?

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Clocks

Stage: 2 Challenge Level: Challenge Level:1

These clocks have been reflected in a mirror. What times do they say?

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Attractive Tablecloths

Stage: 4 Challenge Level: Challenge Level:1

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?

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What Am I?

Stage: 2 Challenge Level: Challenge Level:1

Can you draw the shape that is being described by these cards?

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It's Times Again

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Numbers arranged in a square but some exceptional spatial awareness probably needed.

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Building with Longer Rods

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

A challenging activity focusing on finding all possible ways of stacking rods.

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National Flags

Stage: 2 Challenge Level: Challenge Level:1

This problem explores the shapes and symmetries in some national flags.

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Building with Rods

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

In how many ways can you stack these rods, following the rules?

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So It's Times!

Stage: 2 and 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

How will you decide which way of flipping over and/or turning the grid will give you the highest total?

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Paint Rollers for Frieze Patterns.

Stage: 3 and 4

Proofs that there are only seven frieze patterns involve complicated group theory. The symmetries of a cylinder provide an easier approach.

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Exploded Squares

Stage: 1 Challenge Level: Challenge Level:1

This practical activity challenges you to create symmetrical designs by cutting a square into strips.

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Rose

Stage: 5 Challenge Level: Challenge Level:1

What groups of transformations map a regular pentagon to itself?

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2010: A Year of Investigations

Stage: 1, 2 and 3

This article for teachers suggests ideas for activities built around 10 and 2010.

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Matrix Meaning

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Explore the meaning behind the algebra and geometry of matrices with these 10 individual problems.

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Mirror, Mirror

Stage: 2 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Can you place the blocks so that you see the relection in the picture?

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Hexpentas

Stage: 1 and 2 Challenge Level: Challenge Level:1

How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?

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A Resource to Support Work on Transformations

Stage: 4 Challenge Level: Challenge Level:1

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

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The Fire-fighter's Car Keys

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A fire-fighter needs to fill a bucket of water from the river and take it to a fire. What is the best point on the river bank for the fire-fighter to fill the bucket ?.

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Quaternions and Reflections

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

See how 4 dimensional quaternions involve vectors in 3-space and how the quaternion function F(v) = nvn gives a simple algebraic method of working with reflections in planes in 3-space.

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Plex

Stage: 2, 3 and 4 Challenge Level: Challenge Level:1

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

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Making Maths: Indian Window Screen

Stage: 2 Challenge Level: Challenge Level:1

Can you recreate this Indian screen pattern? Can you make up similar patterns of your own?

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Simplifying Transformations

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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?

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Combining Transformations

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Decoding Transformations

Stage: 3 Challenge Level: Challenge Level:1

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

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Rotations Are Not Single Round Here

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

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. . . .

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One Reflection Implies Another

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

When a strip has vertical symmetry there always seems to be a second place where a mirror line could go. Perhaps you can find a design that has only one mirror line across it. Or, if you thought that. . . .

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A Roll of Patterned Paper

Stage: 4 Challenge Level: Challenge Level:1

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

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Footprints

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Make a footprint pattern using only reflections.

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Screen Shot

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

A moveable screen slides along a mirrored corridor towards a centrally placed light source. A ray of light from that source is directed towards a wall of the corridor, which it strikes at 45 degrees. . . .

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Shuffles

Stage: 5 Challenge Level: Challenge Level:1

An environment for exploring the properties of small groups.

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Coordinating Classroom Coordinates

Stage: 2

This article describes a practical approach to enhance the teaching and learning of coordinates.

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Shaping up with Tessellations

Stage: 2 and 3

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. . . .

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Hidden Meaning

Stage: 2 Challenge Level: Challenge Level:1

What is the missing symbol? Can you decode this in a similar way?

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The Matrix

Stage: 5 Challenge Level: Challenge Level:1

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

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Matching Frieze Patterns

Stage: 3 Challenge Level: Challenge Level:2 Challenge Level:2

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

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Triangles in a Square

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.

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The Eyeball Theorem

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

Two tangents are drawn to the other circle from the centres of a pair of circles. What can you say about the chords cut off by these tangents. Be patient - this problem may be slow to load.

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Transforming the Letters

Stage: 2 Challenge Level: Challenge Level:1

What happens to these capital letters when they are rotated through one half turn, or flipped sideways and from top to bottom?

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Reflect Again

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Follow hints to investigate the matrix which gives a reflection of the plane in the line y=tanx. Show that the combination of two reflections in intersecting lines is a rotation.

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Rots and Refs

Stage: 5 Challenge Level: Challenge Level:1

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

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Complex Rotations

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

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?