New For Teachers! Fun Ways to Teach Math

By Alfred S. Posamentier, Ph.D.

Where in the World Are You?

This is a popular riddle
that has some very interesting extensions, yet seldom considered.
It requires some “out of the box” thinking
that can have some favorable lasting effects on students. Let’s
consider the question:

Where on earth can you be so that you can walk one mile south,
then one mile east, and then one mile north and end up at the
starting point?

Mostly through guess and test a clever student will stumble
on the right answer: the North Pole. To test this answer, try
starting from the North Pole and travel south one mile and
then east one mile. This takes you along a latitudinal line
which remains equidistant from the North Pole, one mile from
it. Then travel one mile north to get you back to where you
began, the North Pole.

Most people familiar
with this problem feel a sense of completion. Yet we can
ask: Are there other such starting points, where we can take
the same three “walks” and end up at
the starting point? The answer, surprising enough for most
people, is yes.

One set of starting points is found by locating the latitudinal
circle, which has a circumference of one mile and is nearest
the South Pole. From
this circle walk one mile north (along a great circle, naturally),
and form another latitudinal circle. Any point along this second
latitudinal circle will qualify. Let’s try it.

Begin on this second latitudinal circle (the one farther north).
Walk one mile south (takes you to the first latitudinal circle),
then one mile east (takes you exactly once around the circle),
and then one mile north (takes you back to the starting point).

Suppose the first latitudinal
circle, the one we would walk along, would have a circumference
of ½ mile. We could
still satisfy the given instructions, yet this time walking
around the circle twice, and get back to our original starting
point. If the
first latitudinal circle had a circumference of ¼ mile,
then we would merely have to walk around this circle four times
to get back to the starting point on this circle and then go
north one mile to the original starting point.

At this point, we can take a giant leap to a generalization
that will lead us to many more points that satisfy the original
stipulations, actually an infinite number of points! This set
of points can be located by beginning with the latitudinal
circle, located nearest the south pole, which has a -mile circumference,
so that the 1- mile walk east (which is comprised of n circumnavigations)
will take you back to the point on this latitudinal circle
at which you began your walk. The rest is the same as before,
that is, walking one mile south and then later one mile north.
Is this possible with latitude circle routes near the North
Pole? Yes, of
course!

This unit will provide
your students with some very valuable “mental
stretches,” not normally found in the school curriculum.
You will not only entertain them, but you will be providing
them with some excellent training in thinking logically.#

Editor’s Note:
This is a new column by Dr. Alfred S. Posamentier, Dean of
the School of Education at City College of NY, author of
over 35 books on math, member of the NYS Standards Committee on Math. This was taken from Math Wonders: To Inspire Teachers and
Students, by Alfred S. Posamentier (Alexandria, VA: Association
for Supervision and Curriculum Development, 2003)