Discovery of the Conic Sections and their properties
a unit lesson plan designed by Rebecca A. Zeier

(Prerequisite knowledge: formula of circles and how to graph circles
Book Used: Intermediate Algebra for College Students (forth edition) by Allen R. Angel)

Day 1: Exploring the shape and formula of an ellipse.

Objective: Students will be able to:
* Describe the conic section, the ellipse, as the intersection of a plane at an angle to an imaginary axis of a cone
* Compare the ellipse to what they already know about the circle
* Relate simple parameter changes in the equation to corresponding changes in the graph
* Discover the formula of an ellipse
* Find the center, vertices, and foci of the ellipse from the equation

Activity (2 parts):

Part 1: construction of an ellipse using play-doh and dental floss
** Divide students into groups of four, with each student serving a particular role: recorder, materials manager, task keeper, and reporter.
** Ask the materials manager to come up and get enough play-doh and dental floss for the members of their group.
** Each student in these groups is to form a cone using play-doh. All the students then slice their individual cones by drawing the dental floss through the cone parallel to the base and perpendicular to an imaginary line or axis passing through the center of the cone.
** The recorder draws the results on the paper provided to the group. The group is to identify this shape and write down the general formula (note: since the prerequisite knowledge about circles is assumed, this task should simply be review for them). Ask the students to think of some real-world objects that look like the circle they have just formed.
** After they have done the task of making the circle, students are to reshape their cones and proceed to form their 1st conic section: the ellipse. This cut is done by slicing the cone at an angle to the imaginary axis, but not through the base of the cone. Again, the recorder draws the results on the group paper and list some real-world examples of the shape just formed.
** Students are to compare the circle (which they are already familiar with) to the ellipse and record what kind of similarities and differences there are between the two.
** When they are finished, all students are to raise their hand. I will then check that this task has been done by having the reporter "report" the groups findings. When I have checked their findings, they may now move onto part 2 of the activity.

Part2: Exploration of the formula of an ellipse
** The students are to go to the "Ellipse" activity at ExploreMath.com to discover more about ellipses.
** Each student in their small group is to fill out the attached worksheet of guided questions (worksheet 1).
** At the end of the class period, I will randomly select one group member’s worksheet to grade.

Checking for Comprehension:
* I will walk around as the students are working in their small groups periodically asking questions
* Random collection of worksheet 1 to be filled out during activity

Assignment:
* Homework on circles and ellipses (see attached HW1)
* # 1-10 on page 594 – 585: describe the each ellipse in words (i.e.: center, foci, etc). Do not graph.

Homework 1 – Circles and Ellipses

1. Write an explanation of the effects of changing (h,k) on both the circle and the ellipse.

2. Explain the effect of "a" and "b" on the ellipse. How does this differ from a circle?

3. Create 5 problems of your own that would be ellipses and circles. (be sure to indicate which are ellipses and which are circles)

4. Do problems 1-10 on page 584 – 585. Describe in words what each ellipse would look like (i.e.: where the center is, foci, major and minor axis, etc). You do not have to graph each ellipse.

5. Write a summary of the difficulties, if any, that your group experienced during the exploration. What valuable things did you learn from the various experiments?

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Day 2: Exploring the shape and formula of parabolas.

Objective: Students will be able to:
* Describe the conic section, the parabola, as the intersection of a plane parallel to an imaginary axis of a cone
* Relate simple parameter changes in the equation to corresponding changes in the graph.
* Discover the formula of a parabola
* Find vertex, directrix, and focus of a parabola using the formula.

Activity (2 parts):
Part 1. Construction of the Parabola
** Students will get into the same groups as the day before, with each student serving a particular role: recorder, materials manager, task keeper, and reporter (it may be a good idea to switch roles among the students as the day before).
** Ask the materials manager to come up and get enough play-doh and dental floss for the members of their group.
** Each student in these groups is to form a cone using play-doh. All the students then slice their individual cones by drawing the dental floss through the cone parallel to the imaginary line on the side of the cone and passing through the base of the cone.
** The recorder draws the results on the paper provided to the group. Ask the students to think of some real-world objects that look like the circle they have just formed.
** When they are finished, all students are to raise their hand. I will then check that this task has been done by having the reporter "report" the groups’ findings. When I have checked their findings, they may now move onto part 2 of the activity.

Part 2: Exploration of the formula of a parabola
* The students are to go to the "Parabola" activity at ExploreMath.com to discover more about parabolas.
* Each student in their small group is to fill out the attached worksheet of guided questions (worksheet 2).
* At the end of the class period, I will randomly select one group member’s worksheet to grade.

Checking for Comprehension:
* I will walk around as the students are working in their small groups periodically asking questions
* Random collection of worksheet 2 to be filled out during activity

Assignment:
* Homework on parabolas (see attached HW2)
* # 1 – 10 on pages 590 – 591: describe the each parabola in words (i.e.: vertex, opens upward or downward, etc). Do not graph.

Homework 2 – Parabolas

1. Do problems 1-10 on page 590 – 591. Describe in words what each parabola would look like (i.e.: where the vertex is, if it opens upward or downward, etc). You do not have to graph each ellipse.

2. Write the equation of the following parabolas

1. (h, k) = (2,3) p = 3 turns down
2. (h,k) = (-3,-1) p = 1 turns up

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Day 3: Exploring shape and formula of hyperbolas

Objective: Students will be able to:
* Describe the conic section, the hyperbola, as the intersection of a plane to two cones nose to nose.
* Relate simple parameter changes in the equation to corresponding changes in the graph.
* Discover the formula of the hyperbola
* Find the center, foci, vertices, and asymptotes of hyperbolas

Activity (2 parts):
Part 1. Construction of the Hyperbola
** Students will get into the same groups as the day before, with each student serving a particular role: recorder, materials manager, task keeper, and reporter (it may be a good idea to switch roles among the students as the day before).
** Ask the materials manager to come up and get enough play-doh and dental floss for the members of their group.
** Each student in these groups is to form two cones using play-doh and place the cones nose to nose. All the students then slice their individual cones by drawing the dental floss from the base of one cone parallel to the imaginary axis and through the cones to the base of the second cone.
** The recorder draws the results on the paper provided to the group. Ask the students to think of some real-world objects that look like the circle they have just formed.
** When they are finished, all students are to raise their hand. I will then check that this task has been done by having the reporter "report" the groups’ findings. When I have checked their findings, they may now move onto part 2 of the activity.

Part 2: Exploration of the formula of a hyperbola
* The students are to go to the "Hyperbola" activity at ExploreMath.com to discover more about hyperbolas.
* Each student in their small group is to fill out the attached worksheet of guided questions (worksheet 3).
* At the end of the class period, randomly select one group member’s worksheet to grade.

Checking for Comprehension:
* I will walk around as the students are working in their small groups periodically asking questions
* Random collection of worksheet 3 to be filled out during activity

Assignment:
* Homework on hyperbolas (see attached HW3)
* # 1-10 on page 599 – 600. Describe in words what each hyperbola will look like (i.e. where the center is, foci, asymptotes, etc). Do not graph.

Homework 3 – The Hyperbola

1. Write the equation for the following hyperbolas.

a. (h,k) = (2,3) a = 4 b = 9 lies in the x-direction

b. (h,k) = (-1,-2) a = 9 b = 4 lies in the y-direction

c. (h,k) = (4,5) a = 4 c = 16 lies in the y-direction

2. Do problems 1-10 on page 599 - 600. Describe in words what each hyperbola would look like (i.e.: where the center is, foci, asymptotes, etc). You do not have to graph each hyperbola.

3. What differences do you notice about the hyperbola and ellipse?

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Day 4: Graphing the conic sections

Objectives: Students will be able to:
* Recognize and distinguish between the formulas for ellipses, parabolas, and hyperbolas.
* Graph circles, ellipses, parabolas, and hyperbolas given the equations of each given appropriate information

Lesson:
** The first 15 minutes or so will be given to the students to finish up any unanswered questions or questions they got wrong from the three worksheets from the previous days.
** The rest of this day’s lesson will mostly consist of "direct instruction" where I will ask questions of the class about what they discovered in the past three days.
** As a whole class we will discuss the differences seen in the equations of the different conic sections and the students will learn how to distinguish between them.
** From the previous lessons, the students engaged in activities in which the graph was given and they were to discover the formulas of the conic sections. Now that they have explored and discovered the formulas and what certain properties of the formula represent, we will use that information to graph the conic sections from the formulas.
** I will show them how to graph the conic sections using the formula by asking them questions on what they know about the formulas (which they should have learned in the past three days).
** Students will be given the opportunity to try to graph some conic sections on their own.
** Students can check their graphs to see if they are accurate using the TI Graphing Calculators.

Checking for comprehension:
* Discussion in class with reciprocal questioning

Assignment:
* # 29 – 46 All on page 600. Decide which conic section each equation represents
* # 29 – 46 even on page 600. Graph the conic sections.

Homework 4 – ALL Conic Sections

1. Do problems 29 - 46 ALL on page 600.

2. Graph problems 29 – 46 EVEN on page 600

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Day 5 and Day 6 Conic Section Project:
(perhaps a 7th day if needed … in which case the last two days will be pushed back)

Objectives: The students will be able to:
* apply the graphing of the conic sections to real-world situations by creating a collage.

Activity:
* The students will cut pictures of conic sections from magazines or internet printouts and glue them to the poser board creating a collage.
* Students will follow the given guide lines (also see scoring rubric):
** There must be five different examples of each of the four conic sections: circle, ellipse, parabola, hyperbola.
** At lease five of the examples should be of nature and at least five should be of architecture
** The remaining examples can be the students choice.
** For each picture, the conic should be traced with a marker, and the poster should have a title.
** General equations should be written by each conic section (looking for whether they recognize the properties of the different forms of equations … for example whether the parabola opens upward or downward)
** Students are encouraged to be creative!

Checking for Comprehension:
* completion of project
* presentation of project

Assignment:
* no formal homework will be given … just that students should be working on their projects

Conic Sections Project

Goal: You will create a collage of pictures illustrating conic sections (circles, ellipses, parabolas, and hyperbolas) found in nature (leaves, flowers, body parts, etc.), architecture (bridges, doorways, etc.), and everyday items (appliances, logos, furniture, etc.).

Include:

1. Pictures of the entire objects where the conic section is found
2. Five different examples for each of the four conic sections: circles, ellipses, parabolas, and hyperbolas (no repeat pictures are allowed)
3. Trace, in marker, the conic in each picture
4. General equations for each conic picture
5. Title for the poster
6. CREATIVITY!!!

Illustration:
* white or colored poster board
* use scissors to cut out pictures (no tearing)
* use glue to paste pictures (no taping)

 Category Points Possible Earned Points 5 examples of circles 5 points (1pt for each) 5 examples of ellipses 5 points (1pt for each) 5 examples of parabolas 5 points (1pt for each) 5 examples of hyperbolas 5 points (1pt for each) Examples of nature (at least 5) 5 points (1pt for each) Examples of architecture (at least 5) 5 points (1pt for each) Examples of everyday items 15 points (1pt for each) Tracing of the conic sections 20 points (1pt for each) Title 5 points General equation given (1pt for right conic equation … 3pts for correct form of conic equation) 60 points Neat/Unique/Appropriate Up to 10 extra points Presentation of Project 10 points Total Points: 150

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Day 7: Presentation of projects and review for test

Objective: Students will be able to:
* Present their group project collage
* Feel confident about the test

Activity:
* Each group will be given approximately 5 minutes to present their projects to the class.
* The students will be given the opportunity to ask any last minute questions that they may have regarding the conic sections in preparation for the test.
* If students have no questions, I will give them some sample examples for them to work on independently.
* We will go over those examples as a class.

Checking for Comprehension:
* grading of the presentation of the projects

Assignment:
* Study for the test

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Day 8: Test Day

Objective:
* Students will take a test and hopefully get an A+!

Activity:
* Students will take the test on conic sections (see attached)

Checking for comprehension: