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Topic:
[mathlearn] Teaching Conics In High School Algebra 2
Replies:
4
Last Post:
Jan 30, 2011 1:44 AM




Re: [mathlearn] Teaching Conics In High School Algebra 2
Posted:
Jan 29, 2011 3:34 PM



On Sat, 29 Jan 2011, starcap50@aol.com wrote:
> Greetings: > > I am a private math tutor who is currently tutoring several students in > high school Algebra 2. I receive more requests to tutor Algebra 2 than > for any other subject. > > One of my students, at one particular school, is currently studying the > unit of Conic Sections. There is a test on this unit next week. > > The teacher is NOT allowing the students to use any "notes" for this > test. > > > NONE! > > This is also a private school. > > There is a tremendous amount of material to memorize or learn for this > test, including the following: __________: > > ***How to find the center and the radius of any circle whose equation is > given, located at the origin and not at the origin. > > Okay, that's fairly easy. But moving on............... > > ***How to find the length of the major axis, the length of the minor > axis, and the foci of any ellipse whose equation is given, including > those located at the origin and not at the origin.
This, and the demands below, would be quite difficult  and never have been taught in high school, besides  if by "whose equation is given" included those which are quadratic and include a nonzero "xy term". What you write below suggests that you are only considering conics whose axes are parallel with the x or y axes. If that it the case, I don't believe what is asked for is "a bit much", for in all cases what is needed is a "complete the square" process on both the quadratic in x and the one in y, and almost no "memorization" at all, provided your student understands what the completion of the square accomplishes. Working out a couple of examples with your students should get the idea across, so that the full abstraction in symbolism in not needed, and in fact would probably be confusion.
Unfortunately, not many textbooks do a good job in showing what is at stake when "completing the square" is required. The idea is this: If your student can answer the questions for an ellipse or hyperbola whose equation looks like 3x^2 + 12y^2 = 1, then if the equation is 3(x5)^2 + 12(y+7)^2 = 1 you will have the same curve but shifted so as to have its center at (5,7). Well, if you begin with something that looks really nasty, say 3x^2  8x + 11y^2 + 18x = 20, completing the squares will ultimately make the second equation look like the first one, recognizable as an ellipse or hyperbola of the same shape, recognizable as easily as that second equation, the one with the shifted center.
These manipulations are not difficult, but take time to sink in. I wouldn't expect a hurryup exam preparation to have any effect on a student who has never understood "completing the square" in connection with quadratic equations in one variable. Still, there might be some value in showing how to get the "reduced" versions for the equations of conics (with axes parallel with the x and y axes), worked out for the following examples, each one introducing just one new idea:
x^2 + y^2 = 1 x^2  y^2 = 1 x^2 + y^2 = 6 x^2  y^2 = 6 3x^2 + 3y^2  18 = 0 3x^2  3y^2  18 = 0 3x^2  12x + 3y^2  60y  18 = 0
and so on down the line, each equation introducing one extra complication, but exhibiting how the completing of the square can make the new equation resemble, in essence, its predecessor. I'm sorry I haven't time to write a whole chapter on the subject here, but a good list of graduated exercises should be possible for you to find, and add to the list of equations of conics I began above, until it becomes evident what has to be done with (say) 3x^+4x+5+6y^2+7y+8+9=0 to make it look like one of the earlier, simpler, examples. (The arithmetic can get tough here, though.)
I must say, however, that introducing "directrix" for ellipse and hyperbola is something I would not undertake in a high school algebra course, for while it is possible for someone who has time for it, it would hurry the whole subject too much for my taste. If you are tutoring students who are having difficulty, including such things would only confuse the whole story. Except that the parabola is most easily defined in terms of directrix and focus, so that I would expect a student who studies the parabola *at all* to be able to find those features of the horizontal or vertical conic when it is a parabola. Similarly for the latus rectum, an idea of value only if you intend to repeat some of the simpler work of Archimedes (which actually isn't at all simple).
As for the forbidden "notes", I fear the notes they would like to have on the desk would be formulas (things like "b/2a")that wouldn't teach them, or the teacher, anything worthwhile. Learning the relvence of "completing the square" is not only more important, it makes the whole subject easier.
For a parabola, knowing that the focus is on one side of the vertex and the directrix a line perpendicular to the axis and on the other side of the vertex (same distance away), tells you most of what you need to know to answer such questions (and you should be able to supply the rest without *memorizing* anything that looks like "b/2a"). See how the book does it; if it is explained at all, that explanation is worth a chapter full of formulas (which I never have managed to memorize, myself).

> ***How to find the axis of symmetry, the vertex, the focus, the directrix, > and the latus rectum for any parabola whose equation is given, including > those which are horizontal and vertical. > > ***How to find the center, the vertices, the equations of the asymptotes, > and the foci of any hyperbola whose equation is given, including those > located at the origin and not at the origin. > > ***How to change the equation of any of the above not given in standard > form to standard form and vice versa. > > ***How to graph any of the above on paper by hand only from both standard > form and nonstandard form. > > ***How to graph any of the above on a graphing calculator by solving for > "y" first. > __________ > > Once again, the students are allowed NO NOTES to assist them on this test. > > Do you think this is a "bit much", even for a private school? > > Every one of our local public high schools allow their Algebra 2 students > to use their notes on their test on Conic Sections. > > As for myself, I do not remember studying Conic Sections in this depth, > back when I had my own high school Algebra 2 course back in 1966. > > All comments will be welcome. > > Dennis > > > > > > > > [Nontext portions of this message have been removed] > > > >  > > Yahoo! Groups Links > > >
Ralph A. Raimi Tel. 585 275 4429 or (home) 585 244 9368 Dept. of Mathematics, Univ. of Rochester, Rochester, NY 14627 <http://www.math.rochester.edu/people/faculty/rarm/>



