August 2003, All Stages

Problems

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Transformations Tables

Stage: 2 Challenge Level: Challenge Level:1

These grids are filled according to some rules - can you complete them?

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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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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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Transformations on a Pegboard

Stage: 2 Challenge Level: Challenge Level:1

How would you move the bands on the pegboard to alter these shapes?

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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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The Number Crunching Machine

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

Put a number at the top of the machine and collect a number at the bottom. What do you get? Which numbers get back to themselves?

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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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Threesomes

Stage: 3 Challenge Level: Challenge Level:1

Imagine an infinitely large sheet of square dotty paper on which you can draw triangles of any size you wish (providing each vertex is on a dot). What areas is it/is it not possible to draw?

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Bow Tie

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

Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.

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Just Repeat

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

Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?

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Stringing it Out

Stage: 4 Challenge Level: Challenge Level:1

Explore the transformations and comment on what you find.

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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?

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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.