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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.
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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
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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
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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
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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
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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.
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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
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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.
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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.
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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
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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.
(incomplete)
“Looking back” and “looking ahead”
(incomplete)
References
© Larry L. Hatfield
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© Larry L. Hatfield
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© Larry L. Hatfield
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© Larry L. Hatfield
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© Larry L. Hatfield
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© Larry L. Hatfield
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© Larry L. Hatfield
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