Geometry and Algebra - October 2006, All Stages

Problems

problem icon

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.

problem icon

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?

problem icon

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.

problem icon

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?

problem icon

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?

problem icon

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.

problem icon

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?

problem icon

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?

problem icon

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?

problem icon

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' ?

problem icon

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 ?

problem icon

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?

problem icon

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?

problem icon

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?

problem icon

Pick's Quadratics

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

Find a quadratic formula which generalises Pick's Theorem.

problem icon

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.

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
Copyright © 1997 - 2018. University of Cambridge. All rights reserved.
NRICH is part of the family of activities in the Millennium Mathematics Project.