Children S Constructions of Computational Algorithms

Prepublication Draft

Fold, Plot, Simulate, Do Algebra:

Using Technology to Help Students

Understand the Parabola

Larry L. Hatfield

Have you seen this happen? Algebra students have completed their introductory study of the algebra of lines. The next chapter in the textbook is quadratics. As an introduction, students are asked to sketch a graph of y = x2. Of course, most students construct a table of pairs using simple integers and plot a few of the points. What occurs next is quite predictable (see Figure 1) and even reasonable. After all, what have they just been doing in the previous chapter? Yes, they were graphing lines by using two plotted points! Therefore, they produce a broken-line graph based on pairs of points. For most students, there is nothing in this approach to motivate or suggest the need for graphing a curve.

[Insert Figure 1]

I prefer to reverse the direction of this mathematical development— namely, start with parabolas that students can explore in order to construct ideas about this interesting curve and how we can “mathematize” it. I have used this approach with many groups of students over more than three decades, and have found many advantages and a great deal of benefit from the approach and results. First, let’s examine the sequence of activities and developments, and then we can consider the benefits of such experiences.

Making and Exploring

To begin, I provide each student with a piece of ordinary kitchen waxed paper (about 30 cm long). I model how to fold and crease to represent a line d, and marking point F (about in the middle of the sheet). These can be shown on an overhead projector quite clearly, as waxed paper is translucent. Students fold to make line d, and then I direct different groups of students to vary where they mark point F—there are several different setups, so point F is either about 2 cm or 4 cm or 6 cm, etc. (“1 thumb width, 2 thumb widths,” etc.) from line d.

[Insert Figure 2a, 2b]

Next, they are given the general direction for folding: “fold a point of the line d onto the point F and crease” (Figure 2a). This is to be repeated, using a different point of line d each time (Figure 2b). [To help see line d, students can color it with an overhead pen using a ruler; it helps to mark point F in color, also.]

As homework, I ask them to complete 50 or more such folds. If done in class, while they do the folds, I ask them to envision in their “mind’s eye” paths in several situations, such as when a ball is thrown from home plate to second base versus from to centerfield, or the ball’s path while playing catch with oneself versus the path of a towering homerun, or the path when a sack of flour is dropped from a low-flying airplane to hit a target on the ground, or the path of light in a flashlight beam. From their mental images, they might try to draw a trace of what they imagine, as the envelope of a curve is beginning to appear from their wax paper-folding actions.

Stage 1. Intuitive Descriptions When they have folded enough waxed paper “lines,” they are asked to examine, compare and discuss what they have made. Students almost always observe that they see a curved shape. I like to point out that what we see is not really a curve, and ask them to recognize that it is only a bunch of straight lines. “If there were a curve, how do you think each of these lines might be related to it?” Some student will suggest that these might be tangents to the curve.

Someone will quickly point out how it appears to be symmetric. We see that the line of symmetry seems to include point F, and to be perpendicular to line d. By folding line d onto itself through F, they can make the axis of symmetry. From comparing their wax paper folding with others, they rather quickly see that these “curves” seem to vary in “openness,” with those less “open” being made when the point F is closer to line d. These intuitions can be voiced quite quickly, and at this time we may list these as informal conjectures.

[Insert Figure 3]

Stage 2. From measures to informal locus definition I ask them to use a ruler (or compass) to explore how the apparent points of the curve might be related in any special ways with the “set-up”---that is, the particular positions of line d and point F. To help, they might mark and label some specific points of the curve. Recording their measures in a chart may lead to them making conjectures (Figure 4)

Many students will observe and be able to frame a statement about “curve points being just as far from point F as they are from the line d.” This can be shown nicely using a compass by placing its point on the curve (call it point A), setting the pencil at point F, and drawing a circle (see Figure 4). Of course, that circle will be tangent to line d, and we can mark the point of tangency and label it point A. By doing several of these, and asking them to look for any special connections, many will notice that line AA seems to be perpendicular to line d but intersect the curve. This can be tested directly by paper folding line d onto itself through point A, and creasing to make the perpendicular. Recognizing this property becomes important later as we develop the algebra of this “curve.”

[Insert Figure 4]

Stage 3. A computer re-presentation model At this stage, I focus on how we can model these actions and results, using technology. Using Geometer’s Sketchpad (Jackiw, 2001), students can reconstruct the sequence of steps they produced when they paper folded the first crease. To help, I guide them through an analysis of the wax paper folding of the first crease to recognize that each crease is the perpendicular bisector of segment FX. These construction steps are shown in Figure 2a, and include the following: construct and label line d and point F; construct and label X as “point on object;” construct segment FX, its midpoint M, and a perpendicular to FX at M; hide FX and M.

Then, by dragging point X along line d, their representation of a “folded crease” becomes a dynamic model. They can see how each position of X results in a different “folded” line. Then, by setting the line to trace and dragging point X, they can produce and record many such simulated “folds.” By animating point X on line d, they can produce and admire the more complete “curve” (being reminded that GSP makes the envelope of the curve).

By varying the relative position of point F, with respect to line d (including “above” and “below”), they can witness the variations in the shape of the curve. Sometimes, at this point, I encourage students to create interesting art forms using this construction with two or more setups (starting line and point), making very nice display items. I might also ask them to ponder what might be produced if the starting lines are, say, perpendicular or parallel or intersecting at special angles, such as 60. This sometimes leads to an individual investigation or project for those who become interested. Some student creations are shown in Figure 5.

[Insert Figure 5]

Investigating Algebraic Representations

Stage 4. Using coordinates with the construction In this stage, I ask them to place their waxed paper “curve” on an x-y coordinate grid. Most of them will place it as shown in Figure 6. Before agreeing to use that, we do consider other placements to show that the graph could occur in any orientation; later we will consider other positions. Based upon their “curve” and coordinate system, they identify and label various points of the “curve,” as well as the coordinates of point F. Some students make a chart that includes the x-value for each point X that seemed to produce a particular point on the “curve,” as well as the x- and y-values for that “curve” point. They observe various ideas about how the values seem to vary. Beyond reinforcing ideas we recognized earlier, their study of these values typically does not lead to much progress in trying to find an algebraic equation for a particular “curve.” If that is their problem, they are blocked from solving it.

[Insert Figure 6]

Stage 5. Algebraic reasoning to find the quadratic equation Typically, it is necessary to suggest that we try to simplify our approach, so that we can analyze the algebra to be found in the steps of the construction. For this, I ask that they reconstruct the basic “folded crease,” this time using the GSP coordinate system. We agree to let the x-axis be the starting line d, and they construct point F on the y-axis and point X on the x-axis; both are “moveable” points so relative positions can be varied. They use the same construction steps to find the perpendicular bisector of segment FX. Our goal is to be able to find (name) every point P on the curve. Each point X results in a point P, and we want to see how these are related. Our strategy is to find point P as the intersection of lines MP and XP.

We need to identify the coordinates of points F, X, and the midpoint M of segment FX. We do this first for a particular placement of point F (0,2) and point X (4,0) (see Figure 7). Here they are using their knowledge of the algebra of lines, first to find M (2,1). How to find the equation of MP? We only have one point, M. Many can suggest that if we knew the slope of MP, then we could write its equation using the point-slope form. But, the slope of MP is the negative reciprocal of the slope of FX, and this can be found. Thus, they can find the equation of the “folded line” MP to be y=2x. The equation of the perpendicular XP for this particular placement of X is x=4. Solving the system of linear equations, they can now find the specific point of the curve is (4,8). We can test our algebraic approach, by comparing that solution to what GSP graphs.

Next, we take the “big” (parametric) step of using the same analysis for any abscissa value (resulting from any position of point X; see Figure 8). For F (0,f) and X (x,0), we can find y = (1/2f)(x2 + f2). Students can “test” this general result, using the value of f from their folded envelope to compute several points of the curve. In Figure 8, where F (0,2), they can first trace the points P to show the parabola as x varies. Then, they can simplify and enter the equation y=.25*x^2+1 which GSP graphs. The two curves will coincide!

[Insert Figure 7 and Figure 8]

Stage 6. Reflections and extensions Through discussion, I encourage the students to describe what they have done, starting with the wax paper folding. In many cases, they are able to recognize that we started with something we made first by hand and then with GSP, and by studying the “curve” we realized several key ideas about it. We usually list all of the ideas we found. Some students recognize that we found an algebraic equation of a particular “curve.” By varying the location of point F (using a variable, f), we have an algebraic equation for any such “curve.” [Perhaps, at this point I will actually begin to use the standard name, parabola, and suggest that they try to find out more about this on the web. In particular, I ask that they try to find places where parabolas occur in the world.]

I like to stimulate extensions of this development. By placing a dinner plate face down, we can trace around it to make a circle on the wax paper. I ask that they choose a point, F (again, varying distance from the circle) and follow the same folding pattern: fold lots of points X of the circle onto the given point F. They quickly produce results, make comparisons, and form intuitive conjectures about possible properties. Quickly, they do the same steps, using GSP. The “wows!” when they animate point X on the circle are visible, and without much prompting they move on to vary the position of point F to produce various shapes of the same curve. Then, more exclamations occur when someone moves it outside the circle, and others do the same! This always sets the stage for an eager group of students who want to see if they can identify properties and use coordinates and algebra to produce a general result or explanation.

Benefits of this approach

For me, this sequence of activities embodies so many of the significant elements of the vision presented in the Principles and Standards for School Mathematics (NCTM 2000). My focus is on the experiential processes and outcomes for my students, including the following.

First, the fundamental quality of experience for both students and teacher is engagement in “sense making:” trying to figure out what they can about this “curve” they have folded, and how it can be described (both synthetically and algebraically). The aim is to develop conceptual understanding. The starting point involves “making one,” so it begins and unfolds as a constructive experience---manipulatively, mathematically and psychologically speaking. It does not begin with “finished” formalizations (such as a quadratic equation to be graphed), but rather sets a goal for them to “find” what might be mathematical about the object that they made. In such an approach, students may not even realize that they are “doing mathematics,” which too often is characterized by unpacking given formalizations of “finished” mathematics and by imitating textbook procedures for acting on them. Too often, school mathematical experiences are framed in students accommodating to given abstractions and methods in which proficiency, rather than deep meaning, becomes the goal. Thus, the very approach taken to this content fosters the qualities of the key principles.

As articulated in the Standards, across all grade levels students should “create and use representations to organize, record, and communicate mathematical ideas” (p. 360). In these activities, students experience a variety of representations of the basic object of study, the parabola. Shultz and Waters (2000) emphasize the importance of multiple representations, including concrete, tabular, graphical, and algebraic. On the nature of representations, Cuoco observes that “Representations don’t just match things; they preserve structure…Representations are ‘packages’ that assign objects and their transformations to other objects and their transformations” (2001, p. x).

Through constructive actions, students concretely build a representation of what to them is a new (and hopefully interesting) visual curve. This first occurs concretely with waxed paper and folding actions. They then re-construct what they have done, in a less concrete (graphics) world of GSP, which nonetheless embodies the same “actions.” This is a re-presentation for them. The power resulting from this step is manifold: they gain in accuracy, speed, completeness, and flexibility, plus it sets the stage for later entering the algebraic world via “coordinatizing.” Yet, to take that step, they return to the concrete embodiment as a context for sense-making, first placing their folded envelope onto a coordinate grid to identify points and interpret the algebra of the lines involved. Afterwards, they can do this on the GSP coordinate axes to confirm again their emerging theories. This interplay across representational systems appears to be important to building conceptual understandings. Central to all of this development is the direct involvement of the students through their own actions, guided and reinforced by the analytical conversation that I, as the teacher, stimulate and maintain.

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“Looking back” and “looking ahead”

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References

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