The notation f( x) does that, to an extent, too, but leaves f a mystery that is unraveled somewhere else (on the right hand side of the = in the definition statement) but never seen again. Rather than write out or speak all the words “the Dilation (of x) about center point 0 by scale factor s,” we abbreviate the important parts with just single letters to encapsulate this long-winded phrase. While this notation may at first seem daunting, it actually may be less mysterious than the traditional f( x) language. In Figure 2, students use the Number Line, Point, and Dilate tools to create a point restricted to the number line and dilate it about the origin to obtain a point labeled D 0, s( x). They focus in particular on connecting the geometric behavior of dilation and translation to the observed numeric values of their variables on a number line. Having explored reflection and other geometric transformations in two-dimensional Flatland, students then restrict the domain of these transformations into the Lineland (one-dimensional) environment of a number line (Abbott, 1886). By dragging x, students observe that x and r m( x) always move at the same speed, but not always in the same direction, and they can investigate how to drag x so the variables move in the same direction or in opposite directions.
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