 Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum                On the surface, this activity is a simple game that gives students practice with coordinates. But it is also intended as an introduction to taxicab geometry, which is a gateway to non-Euclidean geometry. If that sounds too formidable, fear not. Here's the difference: in regular, everyday (Euclidean) geometry, the shortest distance between two points is a straight line. But in a taxicab, a straight line is not always possible. You have to follow the streets. So the distance is the number of blocks you travel along the streets. The picture below shows a Euclidean, straight-line (red) distance of 5 (remember the Pythagorean theorem and the 3-4-5 triangle?), while the same distance in taxicab geometry (blue) is 7. The clues you get in this activity are taxicab distances, given in blocks. Very young students can find the treasure by guess and check. Older elementary students can develop and articulate good strategies. Underlying this activity, however, is the idea of the definition of a circle. In regular geometry, a circle is the set of points equidistant from a center. If this were a Euclidean treasure hunt, when you got a clue (the treasure is 30 meters away) you would know that the treasure was on a circle centered at your location with a radius of 30 meters. In taxicab geometry, you can make a similar statement—but the circle has a different shape. This picture shows a "taxicab circle" with a radius of three blocks. When you get your second clue, you would then look for the intersection of two circles. In Euclidean space, these are usually two points. In taxicab space, the result is often different. This perspective may give you good ideas for discussion questions. You may not want to define "taxicab circles," but you could ask students, after each turn, What are the possible locations for the treasure? The NCTM Standards don't refer to taxicab geometry directly, but they do specify learning about the properties of plane figures. The first item under each grade-level range, for example, asks students to "analyze characteristics and properties of two- and three-dimensional geometric objects." The characteristics and properties of the circle (discussed above) are some of the most important of these, but circles can be tricky because you can't "count" the distance—but in taxicab geometry you can. Thus, this activity helps develop good background in informal spatial reasoning for very young children, and offers good practice in formal reasoning for more experienced students. An Internet search for "taxicab geometry" will yield interesting results. For example, a project by Pascal Tesson at McGill University (Montreal, Canada) lists these references: Krause, Eugene F. 1975. Taxicab geometry. Menlo Park: Addison-Wesley. Gardner, Martin. 1980. Mathematical games. Scientific American 243 (November): 18–30.