- Take a square or small piece of wax paper
about six inches long. Draw a line across the length about an 1
and 1/2 inch from the edge. This is the directrix of our
parabola.
- Now draw a free point above this directrix
somewhere near the center area of the paper. This is the focus of
your parabola.
- Now fold the paper so that the directrix
touches the focus. Make a good crease this is our fold line.
Continue from one end of the directrix to the other until you get
a good forest of fold lines. Hold the paper up to the
light.
What figure do these fold lines
make?
What relationship do the fold lines have with
the directrix and focus?
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We can also explore this paper folding construction using
Geometer's Sketchpad. Open a New Sketch
and get ready to go .
- Select the line tool (not the segment) hold
down the shift key and construct a line across the bottom of the
screen. What is this line called?
- Construct a free point somewhere above this
line. What is this point called?
- Select the line (directrix) and choose
CONSTRUCT from top menu the scroll down to POINT ON OBJECT. A new
point will appear on the directrix, notice that you can drag this
point around on the line and the line position (hence the slope
does not change) while if you choose any of the other points on
the line they change the slope.
- Select this point and hold down the shift
button while selecting the focus.
- Go to CONSTRUCT menu and choose SEGMENT. While
this segment is still selected Go to CONSTRUCT menu and choose
POINT AT MIDPOINT. Select the midpoint and the segment and the go
to CONSTRUCT PERPENDICULAR LINE.
What does this line represent?
What happens as you move the point on the
directrix?
What relation have we formed with our previous
activity?
- Select this line and go to DISPLAY menu choose
TRACE LINE. Go to the point on the directrix (the one joining the
focus with a segment) and move it back and forth across the
directrix. What does this appear as?
- Select the directrix and the point on it. Go
to EDIT menu and scroll down to ACTION BUTTON and choose
ANIMATE.
- Move the Animate button around the sketch to
desired location then Double Click What happens?
- To Make a Snazzier Sketch you might want to
incorporate some colors. Select your fold line and go to DISPLAY
go to COLOR and choose a color you want Animate it. COOL
HUH?
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- How can we prove this is a parabola or rather;
What is a good way we can perform a "Sketchpad Proof" that this is
a parabola.
- Select the directrix and the point on it. Go
to CONSTRUCT PERPENDICULAR LINE. Keep this line selected and then
select your fold line. CONSTRUCT POINT AT
INTERSECTION.
- Select this point and go to DISPLAY TRACE
POINT. Double Click on ANIMATE. What do you Notice?
- Select the fold line and DISPLAY TRACE LINE
notice it is Checked. Uncheck it by choosing it. Double Click
ANIMATE.
- Try moving the focus around and animating What
do you notice? How is the distance between the focus and directrix
related to the shape of the parabola? What is the relation to the
movement of the fold line? What does the fold line appear to be at
the vertex of the parabola?
- Now we can explore a proof of how the figure
we constructed is indeed a parabola. What should we start with?
Answer all questions above and describe explorations in the space
that follows.
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- Maybe we can start by constructing a segment
from the point that is traced to the point at the focus. How does
this segment relate to the segment from directrix point to the
focus? Choose MEASURE LENGTH to explore this. Also Choose SHOW
LABELS for all the points so measurements are visible. Now using
some congruence theorems from triangles we are well on our way to
establishing a "Sketchpad Proof" of a parabola. Use the Text
option to describe some of what is going on in the
sketch.