Objectives:
1. Students will learn how conics sections were first 'discovered' and how
our knowledge of them has developed through time. In doing so students will
better appreciate the material they are learning and the history behind
it.
2. Students will investigate the relationship between a circle and the intersection
of a double right circular cone with a plane.
3. Students will investigate the standard and general equation of a circle
and will discover the relationship between the equation and the circles
center and radius.
Previous Knowledge:
For this lesson, students will have worked with the distance and midpoint
formulas. Students should have a general knowledge of the properties of
a circle. Though it is not essential, a knowledge of completing the square
would also be helpful.
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 classroom will be divided into groups as usual with 3 to 4 students
per group. In activity one each group will create a list of knowledge in
which they list everything that comes to mind that they know concerning
circles. Once the students have had ample time to create their list of knowledge,
the teacher will lead the students in a discussion on circles by calling
on a member from each group to share points from their list. The teacher
will list these points on the overhead and provide feedback to the students
on these points clearing up any areas which need further discussion. In
this discussion, students should be encouraged to develop a definition for
circle and radius. Using the students definitions, the teacher will create
a model of a circle using a thumbtack, a piece of string and a pencil. The
pencil will be tied to the string and the string mounted on a piece of paper
using a thumbtack. The teacher will use this apparatus to draw a circle
stressing the properties of the center, the radius and the circumference
points. In the general discussion of circles, if none of the students point
out that a circle is a type of conic section the teacher will do so in order
to move into a discussion on conic sections.
After pointing out that a circle is a type of conic section, the teacher
will begin a discussion on conic sections. This discussion should include
the different types of conics and some history behind
the study of conics. This discussion should lead into group exploration
of how the different types of conics can be formed by "slicing"
a double right circular cone with a plane. The creation of the circle can
be created by shining a flashlight onto a wall where the flashlight is perpendicular
to the wall. The light coming from the flashlight is in the shape of a cone
when it strikes the wall (a plane) a circle is formed. In this lesson students
will create in their groups a model of a circle using the "slicing" method. In later lessons, students will explore the other
conics using this same method. The teacher will provide guidance to the
students as they construct their circles 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 the constructions and their relation to the definition of a circle.
The students will then move on to exploring the equation of a circle.
Activity three will begin with the teacher explaining that the
standard equation for a circle is derived from the distance formula. The
teacher will present to the students a given circle (using the overhead)
with a given center and a given point on its circumference. The students
will be asked to recall their definitions of a circle (particularly that
each point on the circumference is equidistant from the center. The students
will then work in their groups to derive the standard equation for this
given circle using the distance formula. Once this is done students will
generalize the equation to that of an arbitrary circle.
In activities four and
five the students will then
work on several examples in which they are either asked to determine the
equation for a circle with given center and radius or to determine the center
and radius given the standard equation of a given circle.
The teacher will move forward by reminding students that the (standard)
equation of a circle can also be expressed in quadratic (or general) form
and that circles (as well as all conics) are quadratic relations. The teacher
will demonstrate this by taking the standard equation of a circle from the
previous examples and putting it in quadratic form. The teacher will then
ask the students how to determine the center and radius of a circle whose
equation is in quadratic form. The teacher will answer this question by
demonstrating completing the square. So that students are not misled into
believing all quadratics of the form, Ax^2+Cy^2+Dx+Ey+F=0, are equations
of circles, the teacher will ask the students if this is true. If the students
answer "yes", the teacher will give counter examples. If the students
answer "no", the teacher will ask the students for counter examples.
Students will work a couple of examples using completing the square to convert
the quadratic form of a circle to the standard form.
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
they gained from this lesson. To assess the students progress, they will
be asked to write in their journals about their knowledge of circles and conics. As
a minimum they should give their own definition of a circle, describe how
to obtain the equation of a circle given the center and a circumference
point and explain why the general equation of a circle 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
which help them further explore circles.
Activities:
Activity 1: Getting Ready to Learn About Circles
In your groups, construct a list of knowledge concerning
circles. Write down everything that comes to mind that you know about circles.
Activity
2: Constructing A Circle By "Slicing"
a Double Right Circular Cone
Menaechmus (370 B. C.) is credited as being the first person to use the
idea of "slicing" a cone with a plane to create the conics. He
used a process similar to the steps below to create the circle. See if you
can do the same.
1. Think of the two sno cone cups as the two nappes (halves) of a double
right circular cone. Flatten one of these two cones that make up the double
right circular cone. Unfold it. Flatten it in a different place. Repeat
several times.
2. With your cone still flattened, use
your compass to construct an arc centered at the endpoint of the cone.
3. Cut out the marked arc.
4. Open your cone back up and place the sheet of construction paper between
the two sections of this cone. Mark the intersection between the plane (i.
e. the construction paper) and the cone. What figure do you get? What do
you notice about the relationship between the plane and the axis of the
cone? Does this plane intersect the other nappe of the double right circular
cone? The base of the cone? Is this the only circle that can be created
by intersecting the cone with a plane?
Activity
3: Deriving the Standard Equation for a Circle using the Distance
Formula
Rotate roles before beginning this activity.
1. Determine the distance between the center, point A, and the circumference
point, point B of the circle below. What is this distance called in relation
to the circle? Use this formula to determine the standard equation of the
circle below.
2. Generalize the standard equation for an arbitrary circle such as the one below with center C and a circumference point P. (Hint: What do you do when you do not know a quantity?)
Activity
4: Determining the standard equation
of a circle/ Determining the center and radius of a circle
Rotate roles before beginning this activity.
1. Determine the standard equation for the circle with center at (1,-7)
and radius 6.
2. Determine the standard equation for the circle with center at (1/3, -1/2)
and radius 3/2.
3. Determine the center and radius for the circle with standard equation
(x+1)^2+(y-3)^2=25.
4. Determine the center and radius for the circle with standard equation
(x-3)^2+(y-1)^2=34.
Activity 5: Determining the center and radius of a circle with a general equation
Rotate roles before beginning this activity.
During the 16th century, Descartes discovered that the coordinate system
could be applied to the conics. He determined that all conics can be described
by equations of the form Ax^2+Bxy+Cy^2+Dx+Ey+F=0 and consequently all circles
can be described by an equation of the (general) form Ax^2+Cy^2+Dx+Ey+F=0.
Use the general equation of the circles given below to find their center
and radius.
1. Find the center and radius of the circle with the general equation x^2+y^2+4x+6y-3=0.
2. Find the center and radius of the circle with the general equation x^2+y^2+2x+6y-71=0.
3. Does the graph of every equation of the form Ax^2+Cy^2+Dx+Ey+F=0 determine a circle. If yes,
why? If no, give a counter example.
Assessment: Each student will get a copy of the following assignment
A History
of Conics
Conics sections were first explored by the Greeks over 2000 years ago. Menaechmus
is credited with creating the conic sections around 370 B. C. in his attempt
to solve the duplication of the cube problem. Apollonius built on the work
of Eudoxus and is credited with naming the conics- the hyperbola,
the parabola, and the ellipse in his On Conic Sections
in the third century B. C. and showed that they could be produced by
slicing a right circular cone. Around 300 A. D. Pappus compiled the works
of Apollonius and others in his Mathematical Analysis. It is here
that the first recorded statement of the focus-directrix property of the
three conic sections can be found. Hypatia also built on the works of Apollonius
in her book, On the Conics of Apollonius around 400 A. D. Arab Tabit
ibn Qorra (826-901) also completed a (translated) compilation of Apollonius'
work (and others). It is only through his versions of Books V, VI, and VII
of Apollonius' On Conic Sections that these books come to us today.
In the early sixteenth century, Copernicus modified the Ptolemaic point
of view that the planets revolved around the earth on a great sphere in
circular orbits by considering the sun, not the earth, as the center of
the universe. Later Johannes Kepler (1571-1630) refuted both of these theories
by proving that the planets actually revolve around the sun and do so in
elliptical patterns. At about the same time Descartes applied his coordinates
to the conics and found that all conics could be described by equations
of the form Ax^2+Bxy+Cy^2+Dx+Ey+F=0. It was not until this same period that
the broad applicability of conics became apparent and played a big role
in the development of calculus. Also during this time period, Edmund Halley
(1656-1742) restored the lost Book VIII of Apollonius' Conic Sections
translating them from Arabic even though he did not know a single word of
the language. It was not until 1832 when Jacob Steiner published his Systematische
Entwicklungen that the first actual definition of a conic appeared in
literature. He defined a conic as "the locus of the points of intersection
of corresponding lines of two homographic pencils with distinct vertices.
The French mathematician Germinal Pierre Dandelin (1794-1847) gave a method
involving spheres nested in cones to obtain a two-dimensional algebraic
characterization for the conics. Charlotte Agnus Scott (1858-1931) contributed
to study of conics through her book on plane analytical geometry, An
Introductory Account of Certain Modern Ideas and Methods in Plane Analytical
Geometry.
Journal
Assignment: In your journal write about what you have learned
about circles. Give your own definition of a circle. Describe the construction
of a circle 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 circle given its center and radius. Explain
why the general equation of a circle 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 circles.
1. A landscape architect is planning a circular garden to be planted on
a square plot of land. He must decide between the two configurations displayed
below. One layout is one large circle tangent to the sides of the square.
The other is four smaller circles of equal size that are tangent to each
other and the sides of the square. The landscape architect wants to choose
the configuration which will give him the largest amount of planting space.
Help him decide which configuration has a larger area. Will the equations
of the circles influence your answer. (Hint: You may first want to assign
values to length of the sides of the square. Solve the problem for that
case. Then generalize.
2. An interior designer is planning a tile pattern for the floor of a foyer. Part of the pattern consists of concentric circles of different colors as shown below. The designer knows the formula for the inner (red) circle is x^2+y^2+8x-12y-14=0 and for the outer (yellow) circle is x^2+y^2+8x-12y-12=0. The designer knows that 25 blue tiles will cover 1 square foot. If she can figure out the area between these two circles, she can determine how many blue tiles to order. Help her determine the area between the red and yellow circles. How many blue tiles should she order to cover this area?
Let's say the homeowner rejects the tile design. Instead she wants a pattern consisting of three nonconcentric circles such as the examples below. Assuming that the circles maintain the same equations as above, determine how this affects the number of tiles needed to complete each of the patterns.
3. You are competing in an archery contest. The target consists of three concentric circles. The outer circle is worth 25 points, the middle circle is worth 50 points and the inner circle is worth 100 points. It is your last shot in the final round of competition and with a score of 500 you need 50 points to capture 1st place. Using the figure below, what is the probability you win? What is the probability if you need 25 points? 100 points?
Did you know?
When an earthquake occurs, energy waves radiate from the the epicenter in concentric circles.