Materials:
Activity Sheet: 1 per group
Sno-cone Cups (or paper plates): 2 per group
Construction paper: 1 sheet per group
Stapler: 1 per group
Scissors: 1 pair per group
Discussion of Lesson:
The teacher will begin the lesson
by placing the list of knowledge from the previous lesson on the overhead
and (very briefly) reviewing it. Discussion of the homework will follow.
The teacher will provide feedback on the solutions before moving into a
discussion on parabolas.
Students will be encouraged to develop
their own definitions of a parabola. The teacher will demonstrate the construction of a parabola emphasizing the properties of the parabola and
its focus, directrix and axis of symmetry. This discussion will be used
to guide students toward discovering how a parabola is constructed by the
intersection of a double right circular cone and a plane. Before beginning
activity one, use a flashlight (a cone) to demonstrate how a parabola is
formed by the intersection of a cone and a plane (the wall).
In activity one, the students will use a model similar to the
one used in the previous lesson to discover that a parabola is formed when
a plane intersects only one nappe ("half") of a double right circular
cone and that plane is parallel to an "edge" of the cone. The
teacher will provide guidance to the groups as they construct their parabolas
and will clear up any misconceptions or misunderstandings the students have
concerning the construction process. Once the constructions are complete,
the teacher will facilitate a discussion concerning these constructions
and their relation to the definition of a parabola. The students will then
move on to exploring the equation of a parabola.
This exploration will begin with the teacher explaining that the standard
equation for a parabola like that for the circle, ellipse and hyperbola
is derived from the distance formula. The teacher will present to the students
a given parabola with a given focus, directrix and axis of symmetry. In
activity two the students will use their definitions of a parabola
along with the distance formula to derive the standard equation for this
given parabola. The teacher will circulate checking for correctness and
providing guidance. The students will then generalize the equation to that
of an arbitrary parabola with arbitrary focus and directrix and a horizontal
or vertical axis of symmetry.
In activity three, the students will work on several examples in
which they are either asked to determine the standard equation for a parabola
with given focus, directrix and axis of symmetry or to determine the focus,
directrix and axis of symmetry of the parabola given the standard equation.
In activity four, the students will work on several examples in which they are either asked to determine the general equation for a parabola with given focus, directrix and axis of symmetry or to determine the focus, directrix and axis of symmetry of the parabola given the general equation.
To end the discussion, the teacher will place the list of knowledge back on the overhead this time having the students add to that list the knowledge of parabolas they gained from this lesson. To assess the students progress, they will be asked to write in their journals about their knowledge of the parabola. As a minimum they should give their own definition of a parabola, describe how to obtain the equation of an parabola given its focus, directrix and axis of symmetry and explain why the general equation of a parabola is a special case of the general second degree (quadratic) equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0. Also, students will be given a couple of exercises in which they will further explore parabolas.
To construct a parabola you will need two boards of equal width, two tacks, a piece of string equal to the width of each board and a pencil.
1. Drive one tack into the board near the bottom (point C).
2. Drive the other tack into the top corner of the other board (point A).
3. Tie the string to the two tacks.
4. Use a pencil to keep the string taut against the top board.
5. Slide the top board back and forth (keeping the string taut) to trace the parabola.
Activities:
Activity
1: Constructing A Parabola By "Slicing"
a Double Right Circular Cone
1. Flatten one nappe of your double right circular cone.
2. Construct a line parallel to one of the edges of your flattened cone.
3. Construct an arc tangent to this line.
Make sure this line intersects one edge and the base of your flattened cone.
4. Cut out the marked arc.
5. Open your cone back up and place the sheet of construction paper between
the two sections of the cone. Mark the intersection between the plane (i.
e. the construction paper) and the cone. What figure do you get? Will the
plane intersect the other half of the double right circular cone? Why or
why not? Is this the only parabola that can be created by intersecting the
cone with a plane?
6. What do you think would happen if the above plane passed through the vertex of the cone? This figure is called a degenerate parabola.
Activity
2: Deriving the Standard Equation
for a Parabola using the Distance Formula
Rotate roles before beginning this activity.
1. Let FP be the distance between the focus and a point on the given parabola
and AP be the distance between the directrix and a point on the given parabola.
Compare these two distances. Are they the same or different? What will happen
to this comparison if you choose another point on the parabola? Use the
formula for the sum of the distances to determine the standard equation
of the given parabola below.
2. Generalize the standard equation for an arbitrary parabola such as the one below. What would happen if the parabola opens down? Right? Left? What would happen if the vertex was not located at the origin? Describe how the focus, directrix and vertex relates to this equation.
Activity
3: Determining the standard equation
of an parabola/ Determining the focus, vertex, directrix and axis of symmetry
of a parabola
Rotate roles before beginning this activity.
1. Determine the standard equation for the parabola with focus (0, 3) and
directrix y=-3. What is the vertex and axis of symmetry?
2. Determine the standard equation for the parabola with focus (3, -2) and
directrix y=4. What is the vertex and axis of symmetry?
3. Determine the standard equation for the parabola with focus (4,3) and
vertex (2, 3).What is the directrix and axis of symmetry?
4. Determine the standard equation for
the parabola with focus (-1, -5) and directrix x=2. What is the vertex and
axis of symmetry?
5. Determine the focus, vertex, directrix,
axis of symmetry and direction of opening for the parabola with standard
equation (y+2)^2=8(x-3).
6. Determine the focus, vertex, directrix, axis of symmetry and direction
of opening for the parabola with standard equation (x+5)^2=-4(y-2).
Activity
4: Determining the general equation
of a parabola/ Determining the focus, directrix, vertex, axis of symmetry
and direction of opening of an parabola with a general equation
Rotate roles before beginning this activity.
1. Determine the general equation for each of the parabolas in activity
3.
2. Determine the general equation for a parabola with focus (2, -3) and directrix x=5. What is the vertex, axis of symmetry and direction of opening?
3. Determine the general equation for a parabola with focus (-4, 6) and a point on the parabola (-5, 8). What is the vertex, axis of symmetry and direction of opening?
3. Determine the general equation, vertex,
axis of symmetry and direction of opening for a parabola with the general
equation 2x^2-4x+y+4=0.
4. Find the focus, vertex, directrix and axis of symmetry of the parabola
with the general equation 5y^2+10y-7x-2=0.
Assessment:
Journal Assignment: In your journal write about what you have learned
about parabolas. Give your own definition of a parabola, its focus, directrix,
vertex, axis of symmetry and direction of opening. Describe the construction
of a parabola using a plane and a double right circular cone (i. e. what
is the mathematics behind the construction we did in class?). Describe how
to obtain the equation of a parabola given its focus and vertex, directrix
or a point. Explain how to determine the direction of opening of a parabola
given its equation. Explain why the general equation of a parabola is a
special case of the general second degree quadratic equation Ax^2+Bxy+Cy^2+Dx+Ey+F=0.
Exercises: Complete the following exercises using your knowledge of parabolas.
1. If a ball is thrown vertically upward from the ground with initial velocity 20 ft/s, its distance above the ground at the end of t seconds is given by the formula y=20t-16t^2. At what time will the ball be 5 feet above the ground? Once the ball is thrown how many seconds will it take for it to begin to fall? How long will it take for it to reach the ground?
2. A bridge has a consists of a pair of parabolic arches that are 50 feet high in the center and 150 feet wide at the bottom. Find the height of the arch 30 feet from the center.
3. The length of fencing available to enclose a rectangular lot is 200 feet. Express the enclosed area as a function of the width of the lot. When will the area be at a maximum?
Did you know?
When a baseball is thrown from the outfield to the infield, it traces a near parabolic path.
Most satellite dish antennas are in the shape of a paraboloid, a three dimensional parabola.
The span of a suspension bridge (the Golden Gate Bridge) is in the shape of a parabola because that is the shape that most evenly distributes the weight of the bridge that must be supported.
The reflexive properties of the parabola are utilized by both automotive and audio engineers. If a light source is located at the the focus of a parabola, the light rays will reflect off the parabola parallel to the axis of symmetry of the parabola. Automobile headlights are often designed with parabolic reflectors. Audio engineers use the reverse concept to collect sound waves at the focus. A parabolic microphone operates in this way.
