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Golden Thoughts

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

From All Corners

Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.

Star Gazing

Find the ratio of the outer shaded area to the inner area for a six pointed star and an eight pointed star.

The Rescaled Map

Age 14 to 16 Challenge Level:

Hopefully students will enjoy the World Mapper images and that that will lead to discussion about the issues represented and about the process of producing images (maps) that communicate.

You may wish to draw out the different needs of map users. For example, a navigator may be concerned to have equal distances on a map matching equal distances on the real surface. Angles and bearings might be important, so that moving along a straight path on a map would correspond to taking a straight route when you travel.

Students may be encouraged to think about how maps may can be used to communicate characteristics like population density, rainfall, land under cultivation, import/export, availability of health care, etc.

That discussion would ideally include the ways in which a given method of representation plays down, or even misrepresents, certain features while giving emphasis to others.

Above all students should explore the idea that a representation is a compromise and that mathematics is used to understand the effect and scale of that compromise under different conditions.