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The Shape of Ordered Pairs Harold Mick and Benjamin Bazak

The purpose of this article is to present an alternative approach that applies transformations (motions) to problems of the type given an equa- tion find its graph and given a graph find its equa- tion that are dealt with in the secondary school mathematics curricula. By an alternative approach we mean a decidedly different way to deal with these problems than is found in current textbooks and reflected in Virginia’s Standards of Learning. Most of this article is spent laying the foundation of our approach. In the beginning, we focus on problems of the type given a graph find its equation and then continue to problems of the type given an equation find its graph but stop short of actually sketching the graph. Instead we challenge you, the reader, to complete the “last step” of our approach. Our audience includes secondary mathe- matics teachers, mathematics educators, mathemat- ics supervisors, secondary mathematics curriculum specialists and curious readers. However we direct our attention primarily to high school mathematics teachers. Our intent is to help teachers develop a deeper level of understanding of the mathematics associated with applying motions as a way to ap- proach equations and graphs . In short, we address teachers as learners. We invite you (the reader) to put on your “math magic” hat and stretch your imagination as you accompany us on our exploration. We intro- duce our approach by exploring “shapes” of or- dered pairs. We are going to put a geometric spin on what is usually considered an algebraic concept. We restrict our discussion to parabolas. Our inspi- ration will come from a parabola’s location, orien- tation (up or down) and geometric shape (narrow or wide). However, we will concentrate on what a parabola “looks like” algebraically. What can we say about the “shape” of ordered pairs? How are complicated “shapes” of ordered pairs understood by comparing them to simpler “parent shapes”?

The mathematical playground for this arti- cle is the coordinate plane. The layout of the coor- dinate plane with its two intersecting perpendicular axes is such that each point has an address giving its location in the plane. We write these addresses in the form of ordered pairs . For instance, the or- dered pair ( u , v ) consists of two coordinates u and v . The first coordinate u is a real number repre- senting the directed distance from the Y -axis. The second coordinate v is a real number representing the directed distance from the X -axis. The intersec- tion of these two directed distances locates a point A parabola has a geometric shape; its ver- tex may be anywhere in the coordinate plane, it may open up or down, it may be narrow or wide. With all this variance of geometric shapes the cor- responding ordered pairs vary too. Suppose you are “standing” on a parabola. Now look down and follow ordered pairs of points on the parabola. What do you see? Clearly the individual first and second coordinates of each ordered pair change as points move along the path of the curve. You can become dizzy watching these changes. But as you look at these changing ordered pairs you may no- tice something that remains the same. What is it you may ask? As we closely examine these ordered pairs we see a relationship between first and second coordinates that remains the same as the ordered pairs move along the curvature of the parabola. We call this relationship the shape of ordered pairs . For instance, points with ordered pair ( u , v ), where the second coordinate is the square of the first coordinate, form a parabola with its vertex at the origin. From this geometric shape we extract an algebraic shape of ordered pair ( u , v ) in the form of statement v = u 2 . This algebraic shape or rela- tionship remains the same as points with ordered pair ( u , v ) move along this special parabola. This shape, v = u 2 , shows how first coordinates u and with ordered pair ( u , v ). Shape of Ordered Pairs.

Virginia Mathematics Teacher vol. 44, no. 1

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