Sketchpad Worksheet CI336 Spring 1997

Travers/ Hendrix/ Dildine

Feb. 22 1997

Parabolas and Paper Folding

You can explore the marvels and mysteries of the conic sections using a sheet of paper. Any sheet of paper can do the job however, wax paper or "paddy paper" allows for easy viewing of the constructions. Let's take a look at how we can fold a parabola using wax paper.

  1. Take a square or small piece of wax paper about six inches long. Draw a line aross the length about an 1 and 1/2 inch from the edge. This is the directrix of our parabola.
  2. Now draw a free point above this directrix somewhere near the center area of the paper. This is the focus of your parabola.
  3. 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?

    -------------------------------------------------------------------------------------- We can also explore this paper folding construction using Geometer's Sketchpad. Open a New Sketch and get ready to go .

  4. 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?
  5. Construct a free point somewhere above this line. What is this point called?
  6. 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.
  7. Select this point and hold down the shift button while selecting the focus.
  8. Go to CONSTRUCT menu and choose SEGMENT. While this segment is still selected Go to CONSTRUCT meny 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 previos acitivity?

  9. 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?
  10. Select the directrix and the point on it. Go to EDIT menu and scroll down to ACTION BUTTON and choose ANIMATE.
  11. Move the Animate botton around the sketch to desired location then Double Click What happens?
  12. 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?

    Answer the Questions in the above steps and explain what happened

    in this space and save it for handin.

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  13. 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.
  14. 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.
  15. Select this point and go to DISPLAY TRACE POINT. Double Click on ANIMATE. WHat do you Notice?
  16. Select the fold line and DISPLAY TRACE LINE notice it is Checked. Uncheck it by choosing it. Double Click ANIMATE.
  17. 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?
  18. 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. -------------------------------------------------------------------------------
  19. Maybe we can start by constructing asegment 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.