Sarah and Hamilton
EMAT 6550
Lesson Plan
10/31/06
Telescope Design
Conic section: parabola
Goals:
1)
2)
3)
to use properties of parabolas and of reflection to determine the best telescope
design to maximize the amount of incoming light that is collected at the eye
piece
to employ problem solving skills and creativity to work through the many
things that are variable in this problem: position, length, and angle of flat
mirror; telescope width
to do all this through careful GSP construction
Level: Algebra II or higher
Prerequisite knowledge: familiarity with GSP, geometry knowledge (congruent triangles,
congruent angles), properties of parabolas, reflection properties
* This lesson will build on that knowledge by using GSP to examine how properties of
parabolas and reflection are integrated in telescope design.
Materials: computers equipped with GSP (at least one computer for every two students—one for
each student is better); projector and screen (or large blank space) to display teacher’s computer
screen
LESSON PLAN:
Day One:
Review of parabola definition: Sketch (on whiteboard/chalkboard/overhead) several
points that are equidistant from focus and directrix. Sketch curve through these points.
Explain reflection properties: angle of incidence equals angle of reflection. Sketch a
few examples of light reflecting off a straight mirror.
Discussion of how reflection properties relate to parabolas: Display (on projector) a
parabola (already constructed by the teacher on GSP) including all lines used in its construction.
Point out congruent triangles, congruent angles, and how incoming light (perpendicular to
directrix) reflects off parabola toward focus.
Students will go to computers and each student will construct a parabola using GSP,
following the directions provided. This will include a GSP sketch simulating light reflections.
(See Worksheet 1: “Constructing a parabola with GSP.”)
End class with a brief introduction of telescope design: light reflecting off parabolic
mirror and flat mirror.
Day Two:
Present/display GSP telescope (already constructed by the teacher), pointing out its
features (directrix, focus, parabolic mirror, flat mirror), and how each feature can be
adjusted/moved.
Discuss limitations introduced by various factors (ex. a mirror too long blocks too much
incoming light, while a mirror too short limits light reflected to eye piece).
Explain the goal: to use properties of parabolas and of reflection to determine the best
telescope design to maximize the amount of incoming light that is collected at the eye piece.
Exploration: Students, working in pairs, will go to computers and manipulate the GSP
telescope (the same one constructed and displayed by teacher—already provided on the
computers) to demonstrate how changes of each variable affect the path and amount of light
collected at the eye piece. Students will complete a worksheet with directed questions and make
conjectures about the best telescope design. (See Worksheet 2: Exploration.)
Day Three:
Wrap up: Students explain their conjectures about how to construct the best telescope,
sharing their ideas and rationale with the class.
Telescope Project
Worksheet 1
Constructing a parabola with GSP:
1. Begin with a line: label it “directrix.”
2. Plot a point not on the line, and label it “F” for focus.
3. Construct a point on the directrix: label it “Drag me.”
4. Construct a segment with endpoints F and “Drag me.”
5. Construct the perpendicular bisector of this segment (start with the midpoint, M).
6. Construct a line perpendicular to the directrix through “Drag me.”
7. Mark the intersection of these two perpendicular lines and label it P.
8. Trace P and move “Drag me” back and forth. This trace should look like a parabola.
9. Select P and “Drag me” and choose “Locus” under the “Construct” menu.
Simulating light reflection:
1. Is there a part of the construction that could represent light coming in parallel to the axis
of symmetry of the parabola? Construct a ray and color it to better define this beam of
light. Carefully place the endpoint of the ray so that it does not go past our parabolic
mirror. Label this ray “incoming light.”
2. A beam of light reflecting off of a curved surface behaves the same as a beam of light
reflecting off the line tangent to the curve. After reflecting off the parabola, where does
the light travel? Construct a segment to represent this part of the light’s journey. (Hint:
you could try constructing the perpendicular to the tangent line at the point of tangency
and reflecting the “incoming light beam” across this perpendicular.) Color the segment
the same color as your light beam in #1.
3. Move “Drag me” back and forth to see how beams of light reflect off the parabola.
Telescope Project
Worksheet 2
Exploration
Use the GSP Telescope Sketch provided to answer the following questions.
1. Given the position of the Focus (Fp), why is the flat mirror necessary to make this
telescope work?
2. Should the Light Focus (Flight) be inside, outside, or on the side of the telescope? Why?
(Think about where the viewer’s eye is.) What adjustments can you make to the sketch in
order to position this Focus in the right place?
3. Explore various angles for the flat mirror.
a) What are some pros and cons of a “steep” angle?
b) What are some pros and cons of a “less steep” angle?
4. Examine what happens when you change the length of the flat mirror. List the pros and
cons of lengthening and shortening the mirror.
5. Move the mirror closer to, and farther from, the focus. What are the pros and cons of
various positions of the flat mirror?
6. Explore various telescope widths. What happens if the telescope is too wide? Too
narrow?
7. If the telescope has width W, then let’s agree that light beams come in across a space that
is W wide. If the flat mirror has length M, then how big is the space that is blocked from
light beams? (You may need to consider an angle here and some trig)
8. So if the beam is W wide, and you know a beam ______________ wide gets blocked,
then there are two beams each ____________________ wide that get through and hit the
parabolic mirror. (You remember subtraction, right?)
9. What is the minimum distance away from the focus of the parabola that the center of the
mirror must be to ensure that the focus of the light satisfies the requirement in 1.?
10. Is there a maximum distance away from the focus that the mirror can be and successfully
collect light? Explain.
11. If we make the flat mirror angle such that the focus of the light will always be on top side
of the telescope, which light beam, lower or higher, will have a harder time getting back
to the flat mirror? Can you give a verbal explanation for this?
12. Since one beam will always have issues making it back to the flat mirror, let’s focus on
making all of the light from the trouble free beam hit the flat mirror. Where should one
end of the mirror lie to ensure that all the light from the top beam hits the mirror without
having any wasted mirror length? (drawing a clever segment here may help)
13. Make adjustments to all of the above variables (angle, length, and position of the flat
mirror, and telescope width), and make a conjecture about the best design for a
telescope—that is, the design that would maximize the amount of light that is collected
for the viewer to see. Draw a sketch of your telescope here: