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