1. T AXICAB G EOMETRY An exploration of Area and Perimeter Within city blocks.

2. E UCLIDEAN G EOMETRY Type of geometry generally taught in High School Named after mathematician Euclid, circa 300 BC Created under several assumptions 2 dimensional plane containing points, lines, circles, angles, measures and congruence 5 Postulates Defining Euclidean Geometry

3. 5 P OSTULATES We can draw a unique line segment between any two points. Any line segment can be continued indefinitely. A circle of any radius and any center can be drawn. Any two right angles are congruent. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended infinitely, meet on that side on which are the angles less than the two right angles. ( The Parallel Postulate ).

4. T HERE ARE OTHER TYPES OF GEOMETRY !!! Here are some examples: SPHERICAL GEOMETRY Based on a sphere, instead of a coordinate plane Think of Geometry based on the Globe How is it possible for a person to walk 10 miles south, then 10 miles west, and then 10 miles north, and return to their home? Consider if their home is on the North Pole! TAXICAB GEOMETRY

5. W HAT IS TAXICAB GEOMETRY ? Introduced by Eugene F. Krause Redefines Euclidean Distance Euclidean Distance – A line segment represents the shortest distance from one point to the other. “As the Crow Flies” Formula comes from Pythagorean Theorem

6. D EFINING T AXICAB D ISTANCE Consider the coordinate plane as city blocks. Taxicab may only travel along the city streets. Distance is found by counting how many units one must travel to get from one point to another, moving only horizontally and vertically.

7. W HY IS T AXICAB G EOMETRY B ENEFICIAL ? Students can relate. Taxicab Treasure Hunt Address alternatives to Euclidean Geometry. Often people do not even know they exist. Focuses on need for set rules, postulates and definitions. Explore common ideas in a new parameter. Conic Shapes, Perpendicular Bisectors, etc. Circles:

8. A CTIVITY 1 Sketch Triangle with Vertices: A (-3, 2), B (1, 4), and C (2, 2) Find Euclidean Distance of all 3 sides Find Taxicab Distance of all 3 sides Sketch the path you took from each vertex to find the distance

9. A CTIVITY 2 – P ERIMETER AND AREA Find Euclidean Perimeter of the triangle. Find Taxicab Perimeter of the Triangle. Find the Euclidean Area of the Triangle. Can we find Taxicab Area? Explain what this means and how you would find it.

10. 1. Describe a situation when it would make most sense to use Taxicab Geometry. Explain. 2. Describe a situation when it would make most sense to use Euclidean Geometry. Explain. Ticket out the Door