Geometry and Algebra - October 2006, All Stages

Problems

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Tessellating Capitals

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

Have you ever tried tessellating capital letters? Have a look at these examples and then try some for yourself.

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Four-triangle Arrangements

Stage: 1 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

How many different shapes can you make by putting four right- angled isosceles triangles together?

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Coordinate Challenge

Stage: 2 Challenge Level: Challenge Level:1

Use the clues about the symmetrical properties of these letters to place them on the grid.

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Tubular Path

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

Can you make the green spot travel through the tube by moving the yellow spot? Could you draw a tube that both spots would follow?

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A Maze of Directions

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

Use the blue spot to help you move the yellow spot from one star to the other. How are the trails of the blue and yellow spots related?

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Escher Tessellations

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

This practical investigation invites you to make tessellating shapes in a similar way to the artist Escher.

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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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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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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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Points in Pairs

Stage: 4 Challenge Level: Challenge Level:1

In the diagram the radius length is 10 units, OP is 8 units and OQ is 6 units. If the distance PQ is 5 units what is the distance P'Q' ?

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The Line and Its Strange Pair

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

In the diagram the point P' can move to different places along the dotted line. Each position P' takes will fix a corresponding position for P. If P' moves along a straight line what does P do ?

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Mapping the Wandering Circle

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

In the diagram the point P can move to different places around the dotted circle. Each position P takes will fix a corresponding position for P'. As P moves around on that circle what will P' do?

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Intersections

Stage: 4 and 5 Challenge Level: Challenge Level:1

Change one equation in this pair of simultaneous equations very slightly and there is a big change in the solution. Why?

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Lattice Points

Stage: 5 Challenge Level: Challenge Level:1

Why are there only a few lattice points on a hyperbola and infinitely many on a parabola?

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Pick's Quadratics

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

Find a quadratic formula which generalises Pick's Theorem.

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Proof of Pick's Theorem

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

Follow the hints and prove Pick's Theorem.