fall-2017-final

second coordinates v are related for exactly those points on this parabola: “to get second coordinate v take first coordinate u and square.” We describe this shape by writing equation Y = X 2 , where X represents first coordinates and Y represents sec- ond coordinates. We call this particular parabola the parent parabola because of its basic structure. In the paragraph above we made a subtle distinction between the equation, Y = X 2 , and the algebraic shape v = u 2 . To heighten the role repre- sentations play in our article, we illustrate three representations in Figure 1: equations in the top row, order pairs in the middle row, and graphs in the bottom row.

tions and ordered pairs: “ X ” stands for “take the first coordinate” and u is the coordinate taken. Please note that almost our entire article takes place in the middle row representation of ordered pairs (Figure 1). That is most unusual and will take some adjustment on your part (our readers). In secondary mathematics textbooks, in general, the equation of the parent parabola is writ- ten with small letters as y = x 2 . Substitution or replacement rules are applied to equations at the same representation level and almost always origi- nate with the parent. These authors do not pick a generic point on the parent parabola with say or- dered pair ( x , y ) and move it about the coordinate plane. Consequently they have no need to distin- guish between equation variables and ordered pair variables. We use small letters “ u ” and “ v ” instead of small letters “ x ” and “ y ” to avoid confusion with our use of capital letters “ X ” and “ Y ”, as well as textbooks’ use of small letters “ x ” and “ y ” for writ- ing equations. Motions. Before we illustrate our approach with a problem, we go over the actions and labels associ- ated with the motions we apply in this article. We apply three types of motions with each having horizontal and vertical components. We call these three types: shifts , flips and scales. (We define and label these motions specific to the coordinate plane to avoid confusion with transformations such as translations, reflections and dilations that are de- fined in the Euclidean plane.)

Figure 1.

We digress for a moment to discuss differ- ences between our approach and the approach of secondary mathematics textbooks in general. To emphasize the distinction between equation repre- sentation and ordered pair representation we use capital letters for equations and small letters for ordered pairs. For us “ X ” stands for “take the first coordinate” and “ Y ” stands for “take the second coordinate.” To say that Y = X 2 describes the equa- tion of the parent parabola is to say that a point is on the parabola if and only if the square of its first coordinate is equal to its second coordinate. So if we take a point with ordered pair ( u , v ) that lies on the parent parabola then the square of first coordi- nate u is equal to the second coordinate v . In sym- bols we write v = u 2 . Here is another way to look at the distinction between representations of equa-

Types of Motions

Xshift b

Horizontal shift ( u , v ) → ( u + b , v )

Yshift d

( u , v ) → ( u , v + d )

Vertical shift

Xflip

Horizontal flip ( u , v ) → (− u , v )

Yflip

( u , v ) → ( u , − v )

Vertical flip

Xscale a

Horizontal scale ( u , v ) → ( au , v )

Yscale c

( u , v ) → ( u , cv )

Vertical scale

where a, b, c, d are real numbers, a > 0 and c > 0

Notice that each of these six basic motions is associated with an arithmetic operation, either addition or multiplication: shifts are associated with addition by a real number, flips are associated with multiplication by –1 and scales are associated

Virginia Mathematics Teacher vol. 44, no. 1

45