## Chronic Conics

### Steve Friedlander

#### Description

The students will use two activities to be able to draw four different conic sections. One of the activities is of a physical nature while the second activity is a more traditional pencil and paper activity.

#### Objectives

Using a rectangular coordinate system (graph), applies and algebraically verifies properties of two-and three- dimensional figures, including distance, midpoint, slope, parallelism, and perpendicularity.

Describes, analyzes and generalizes relationships, patterns, and functions using words, symbols, variables, tables and graphs.

Determines the impacts when changing parameters of given functions.

#### Materials

-String
-Orange cones
-Tape measure
-Cartesian Coordinate System that can be projected from an overhead.
-Dry erase markers
-Graphing calculator (optional)
-Graphing paper

#### Preparations

1) Find a place to hold the first activity. Decide where you want the students to stand for each of the conic sections. You may also want to decide who will be your foci (Focus 1 and Focus 2) for each class. Being the foci does require a little common sense.
2) Gather materials. Copy worksheet.
3) On the day of the first activity, watch the Weather Channel if you are going outside!
4) Perform the first activity as outlined above.
5) Before your class arrives the next day, project a Cartesian Coordinate System onto a white board. Outline the x- and y-axes so that the coordinate system is now quartered, then draw a coordinate system in each quadrant. Write the equations from above in the quadrants.
6) Have the students graph their figures.
7) The procedure for obtaining points is largely your choice. In the beginning, you might have students use t-charts, select their own values for ' x', and then find the resulting values for 'y'. Later, you could use the equations for the graphs to determine critical points for each figure in order to constuct a graph.
8) Assign the evaluation instrument (worksheet).

#### Procedures

Prior Knowledge : Since this is an Algebra II class, students are expected to know how to graph lines and have a working knowledge of parabolas.

First Activity : You (and the students) get to go outside! At the very least, get out of your classroom and go to a big area, like the gym or cafeteria. There will be a series of instructions you will give the students. As the activity progresses, give as little information as possible, allowing the students to try to figure the constructions of the conic sections. Only prompt when necessary.

1) Have all of the students stand the same distance from a straight line drawn on the ground or from the orange cones set up in a straight line or from a wall. The result is that the students will make a straight line.

2) Next have one student stand alone in the middle of the field or floor. Have the remaining students all stand the same distance from that person. You should now have a circle. The student in the middle is the center of the circle.

3) Now, the fun(?) part. Cut a piece of string; approximately 40 to 50 feet long. Place two students (Focus #1 and Focus #2 at locations determined from step 1 of Teacher Preparations.), about 12 feet apart in the middle of the field or floor, each holding the ends of the string. Mark where each 'focus' student is standing in case one loses his/her place. A third student takes the string somewhere near the middle and walks away so that the string is taut about 4 feet off the ground. Continue this process with several students, though each student should initially grab the string at a different point. Each student should walk out to a different location from the other students so that there are many 'points' on this stringed curve. Accomplished correctly, when finished, you will have an ellipse; that is, each student stands the sum of the distances from the two students in the middle (the foci).

4) Now, the real fun (note the sarcasm here). Use the same set-up as in #3 above. The process starts off the same way with the foci students holding the ends of the string. A third student, student A, grabs the string and walks out while another student, student B, uses a tape measure to find the difference between the two pieces of string. We’ll call this distance X. The first student stands distance X away from the focus he/she is nearest. This process continues until you get a hyperbola. If Student C has the same distance as Student A from the same focus, place Student C on the opposing -side- of the same branch of the hyperbola. Note that this process is far from perfect and, quite frankly, tedious. Stop this part when you feel the students get the idea how a hyperbola is constructed, the difference of the distances from the foci.

5) Finally, have one student stand several feet away from a wall or line in the ground or the cones. Students are to stand equidistant from the wall (line, cones) and the student. You should get a parabola. The wall represents the directrix while the student is the focus.

Second Activity : (usually the day after the first activity). Ok, you’re back in the classroom. Project a Cartesian Coordinate System onto your board. In each quadrant, draw a separate coordinate system and write one of these equations in each quadrant : 1) x
2 + y
2 = 36,
2) x
2 + 4y
2 = 64, 3) 4x
2 - y
2 = 36, and 4) x = 4y
2 . Students, working in groups, are assigned one of the equations to graph on the board. The students may use any prior knowledge they have in order to find points to graph. Some students solve for 'y' (they’ll forget the plus-or-minus sign every time!), use the guess and check method, use t-charts, or, as the more enlightened students might, use their graphing calculators. After each equation is correctly graphed, conduct an observation session. That is, what characteristics of the equation let the students know the conic section they must graph. These characteristics include, but are not limited to :
Equation 1 : a circle - have both x
2 and y
2, same signs in front of x
2 and y
2, same coefficients for x
2 and y
2.
Equation 2 : an ellipse - have both x
2 and y
2, same signs in front of x
2 and y
2, different coefficients for x
2 and y
2.
Equation 3 : a hyperbola - have both x
2 and y
2, different signs in front of x
2 and y
2, coefficients for x
2 and y
2 do not matter.
Equation 4 : a parabola - either x
2 or y
2.
Emphasize the shapes of the figures. For instance, a parabola looks more like a -U- than a -V-. Following this activity, assign the worksheet, which is an attachment to this plan.

#### Assessments

In general, look for the students’ graphs to be the correct shapes and that there are enough points graphed to ensure accuracy. The last two questions on the worksheet could be considered optional for grading purposes and for discussion purposes only, though you might not tell the students that ahead of time.

In particular, you may use the rubric at the following URL for grading purposes.

http://www.indep.k12.mo.us/pdc/MAPS/Math8/math_rubric.htm%20copy

Or, you may want to score each graph as follows :
9-11 points : Excellent production. Few errors, the shape of the graph
is near flawless, sufficient number of points to generate
a superior product, excellent knowledge of the task.
6-8 points : Good production. Some errors due to insufficient number
of points chosen or points graphed incorrectly, some
inconsistency in the shapes of the graphs, good
0-5 points : Poor production. Numerous errors which lead you to
the conclusion that the student does not understand the
task, numerous errors in all areas.

The last two problems may be scored on a completion basis since some of the knowledge to complete the problems may not have been acquired yet. These two problems present a -preview of coming attractions- and can lead to some great discussion.

You may want to give some sort of a participation grade for the first activity. Since this is a 'FUN' learning activity, it might not require a participation grade nor is it necessary to assign related homework.

Web supplement for Chronic Conics
Conic Sections History

Web supplement for Chronic Conics
Foci of a Conic Secion

Web supplement for Chronic Conics
Conic Section Movie

Web supplement for Chronic Conics
Conic Section Problems

#### Attached Files

A worksheet to help with assessment for this activity.     File Extension: pdf