Teaching Conic Sections

[kathrynag]

So I currently teach a precalc class and new this year we are required to teach conic section.
We cover parabolas, circles, ellipses, and hyperbolas. Since I haven't taught this before, I was wondering if anyone has suggestions on how to teach it? The book we use has a bunch of formulas, but I'm looking for a way to teach it to my students without using all the formulas so they don't have to memorize a bunch of formulas before their exam. What has worked for others?

[symbolipoint]

You should be able to design good lessons directly based on the book sections. As long as you use a Pre-Calculus book you will have rich enough information available.

Be sure to demonstrate the conic sections using a realistic three-dimensional model. Also use the definitions of each conic section and the distance formula to derive the equation for each conic section, and include the analytical cartesian graph for each.

You are right on-target about not just giving a bunch of formulas. The demonstration and the derivations are important for learning and understanding.

[Ryker]

You are right on-target about not just giving a bunch of formulas.My opinion is that the formulas should still be given, though. It sounded to me from the original post as if they were to be scratched altogether.

[armolinasf]

As a student who struggled through conic sections, I found that by exploring how they were really just variations of the of the same things cemented my understanding of the topic. So if I were in your shoes I would try to show the similarities and differences of the different sections. Specifically between the hyperbola and parabola and the circle and ellipse.

[Eldar]

As someone who not long ago learnt Conic Sections, I found the derivations of the formula much easier than remembering them. It was good to see the formulas at first but I much preferred the derivations.

As above said, use a 3D model as well. The 2D drawing didn't really do it justice for me.

[jhae2.718]

These may be of some use: http://demonstrations.wolfram.com/topic.html?limit=20&topic=Conic+Sections

[micromass]

I used to love learning about conic sections! Certainly when the teacher explained why conic sections are important in real life. You should really mention the applications of conics. See: http://britton.disted.camosun.bc.ca/jbconics.htm

[kathrynag]

I was definitely going to derive the formulas using the distance formula and talk about applications. I wouldn't scratch formulas altogether, but our book has like 8 different formulas, which isn't fair to give all of them to my students if I don't give them on an exam.

[tiny-tim]

Dandelin spheres (http://en.wikipedia.org/wiki/Dandelin_spheres) :wink: …

fun! :tongue2:

[qspeechc]

Something one of my physics professors said to our class is recalled to me by this thread. He said students of today are so used to tv games, comics, etc. rather than playing with things with their hands, that they can't visualize 3d objects anymore. He was of course exaggerating. I think it quite odd if a student can't visualize what's going on with conic sections, so yes a model would be quite good. Maybe you could get someone to cut it at all the right angles.

Also, the old books on geometry, particularly solid geometry, should be good with conic sections, so maybe go down to the library and have a look at them.

[micromass]

Dandelin spheres (http://en.wikipedia.org/wiki/Dandelin_spheres) :wink: …

fun! :tongue2:

Ah yes, I have a fun experiment using that! You should certainly teach that!

Here it goes: Dandelin was a Belgian, and some people decided to celebrate him. So what they did was the following. They made an ice-cream cone, they put a small biscuit in there (they made it like an ellipse so it would fit inside the cone). And they they put a ball of ice-cream in the cone. Then they would sell it to people.

After the people eat the ice-cream, they would take out the biscuit and observe that the only place where the ice-cream touched the elliptical biscuit was in the focus of the ellipse. This would give an intuitive demonstration of Dandelin's theorem.

I always tought that it was very clever, and it was quite the financial succes too!

[jhae2.718]

I wish I got ice cream when we did conics years ago!

That's rather cool.

[tiny-tim]

hi micromass! :smile:
After the people eat the ice-cream, they would take out the biscuit and observe that the only place where the ice-cream touched the elliptical biscuit was in the focus of the ellipse. This would give an intuitive demonstration of Dandelin's theorem.

did they then discover another, smaller, ice-cream (just like discovering another layer of chocolates when you finish the first layer!), which touched the other focus on the other side of the biscuit? :wink:

[jhae2.718]

Pity they didn't realize they could cut the ice cream into two identical spheres of ice cream!

[micromass]

hi micromass! :smile:


did they then discover another, smaller, ice-cream (just like discovering another layer of chocolates when you finish the first layer!), which touched the other focus on the other side of the biscuit? :wink:

Haha :biggrin: In fact, I think they they did do this, yes, but I'm not sure...

[micromass]

Pity they didn't realize they could cut the ice cream into two identical spheres of ice cream!

Us Belgians don't like choice :rofl:

[jhae2.718]

Banach and Tarski only ever bought one scoop of ice cream in their lives...

[micromass]

Banach and Tarski only ever bought one scoop of ice cream in their lives...

Fermat on the other hand, had millions of paper notebooks, but it was never enough... Tragic

[Caramon]

Show this animation in class!
http://schools.spsd.sk.ca/mountroyal/Hoffman/MathC30/Conics.MOV

It will be very helpful for the students who are not as good at mathematics and have a hard time visualizing the double-napped cone and intersections.

[micromass]

Nice animation Caramon!

I would also suggest you let them play with software like geogebra. I really does help to gain familiarity with the conics...

[kathrynag]

Well,
my book has all of these equations for the parabola:
x^2=4py focus (0,p)
y^2=4px focus (p,0)
(x-h)^2=4p(y-k) focus (h,k+p), directrix y=k-p
(y-k)^2=4p(x-h) focus (h+p,k) directrix x=h-p
Circles
(x-h)^2+(y-k)^2=r^2
Ellipse:
if major axis horizontal:(x-h)^2/a^2+(y-k)^2/b^2=1
vertices (h-a,k), (h+a,k)
foci (h-c,k), (h+c,k) where c^2a^2-b^2
if major axis vertical:(x-h)^2/b^2+(y-k)^2/a^2=1
vertices (h,k-a), (h,k+a)
foci (h,k-c), (h,k+c) where c^2a^2-b^2

My problem with all of these formulas is that the actual formulas aren't too bad, but all the additional information like foci, directrix, and vertices will be difficult for them to remember

[symbolipoint]

Kathrynag complained in this way:


that the actual formulas aren't too bad, but all the additional information like foci, directrix, and vertices will be difficult for them to remember

That is the reason for teaching the derivations of the conic sections equations. If the students do not know the meaning and use of foci, directrix, then they do not yet understand conic sections.

[physics girl phd]

This thread is making me want ice-cream... and cones... and biscuits. Gee... why can't I teach math? (My physics class does make ice-cream when studying Thermo!)

[jhae2.718]

(My physics class does make ice-cream when studying Thermo!)

Ice cream made with LN is fun!