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

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 Ralph A. Raimi Posts: 724 Registered: 12/3/04
Re: [math-learn] Teaching Conics In High School Algebra 2
Posted: Jan 29, 2011 3:34 PM
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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 non-zero "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(x-5)^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 hurry-up 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 non-standard 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
>
>
>
>
>
>
>
> [Non-text portions of this message have been removed]
>
>
>
> ------------------------------------
>
>
>
>

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/>

Date Subject Author
1/29/11 Dennis
1/29/11 Ralph A. Raimi
1/29/11 Robert Hansen
1/30/11 Ron Ferguson
1/30/11 Robert Hansen