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with multiplication by a positive real number. We use a dual language system of both geometric and algebraic designations to communicate motions in the coordinate plane. We form a sequence of motions by follow- ing one motion with another. For instance a hori- zontal shift 2 units followed with a horizontal scale by factor 3 moves ( u , v ) to ( u + 2, v ) and ( u + 2, v ) to (3( u + 2), v ). These changes of ordered pairs show the order of the arithmetic operations addi- tion and multiplication which build up algebraic expressions of coordinates; this is algebraic lan- guage. We write programs for sequences of mo- tions (compositions) by listing their individual labels in the order they are applied; in other words, programs are ordered lists of motion labels. In this instance we write program Xshift 2| Xscale 3; this is geometric language. Programs may be of any length. We display programs horizontally, vertical- ly or in whatever arrangement is convenient; it’s the ordered sequencing that is important. We dis- cuss programs in more detail under the section entitled Programs . From Motions to Equations. To illustrate our approach we consider the following problem: Problem 1: Find an equation for the solid pa- rabola shown in Figure 2 . To find an equation for the solid parabola means we seek the shape of its ordered pairs. To this end take points with ordered pair ( u , v ) on the solid parabola. How are u and v related for exactly

those points making up the parabola’s geometric picture; that is, what is the shape of ordered pair ( u , v ) for points on the solid parabola? If the solid parabola had only been the parent parabola we would be done for then each second coordinate would be the square of its first coordinate. We are not so lucky. But all is not lost. We will move the solid parabola to the parent parabola where we know the shape of or- dered pairs (see dotted parent parabola in Figure 3). What program can we write to move the solid pa- rabola to the dotted parent parabola? In comparing the two parabolas there appear to be several se- quences of motions (actually there are infinitely many sequences but only a few practical ones). We discuss two such sequences and their programs. Program (1) : Suppose we move the vertex of the solid parabola to the origin. To do this we shift the solid parabola 2 units horizontally and −1 unit vertically (see dashed parabola in Figure 4). Then we flip the dashed parabola vertically (see dashed/ dotted parabola in Figure 4) and finally we scale that image vertically by a factor of 4 (see dotted parent parabola in Figure 4). We write the corresponding program Xshift 2| Yshift− 1| Yflip | Yscale 4 to move ( u , v ) to ( u + 2, −4( v − 1)) which lies on the dotted parent parabola. The point of recording this sequence of ordered pairs ( u , v ) → ( u + 2, v − 1) → ( u + 2, −4( v − 1)) is that we know the shape of ordered pairs on the parent parabola; namely, second coordinates are the square of first coordinates. Since

Figure 3

Figure2 igure 2.

Figure 3.

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