Geometry and Algebra - October 2006, All Stages

Problems

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Tessellating Capitals

Age 5 to 7 Challenge Level:

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

Age 5 to 7 Challenge Level:

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

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Coordinate Challenge

Age 7 to 11 Challenge Level:

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

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Tubular Path

Age 7 to 11 Challenge Level:

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

Age 7 to 11 Challenge Level:

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

Age 7 to 11 Challenge Level:

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

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Decoding Transformations

Age 11 to 14 Challenge Level:

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

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Combining Transformations

Age 11 to 14 Challenge Level:

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

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Simplifying Transformations

Age 11 to 14 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?

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Points in Pairs

Age 14 to 16 Challenge Level:

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

Age 14 to 16 Challenge Level:

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

Age 14 to 16 Challenge Level:

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

Age 14 to 18 Challenge Level:

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

Age 16 to 18 Challenge Level:

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

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Pick's Quadratics

Age 16 to 18 Challenge Level:

Find a quadratic formula which generalises Pick's Theorem.

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Proof of Pick's Theorem

Age 16 to 18 Challenge Level:

Follow the hints and prove Pick's Theorem.