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 FEBRUARY 2005

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)