Teaching Conic Sections: Tips & Strategies

In summary, the conversation was about teaching conic sections in a precalc class and how to approach it without relying too heavily on memorizing formulas. Suggestions were given to use a Pre-Calculus book, demonstrate with a 3D model, derive equations from definitions and the distance formula, and discuss real life applications. It was recommended to show the similarities and differences between conic sections and to use resources like online demonstrations and old geometry books. An interesting experiment involving ice cream was also mentioned. Finally, it was suggested to use animations and software like geogebra to help with visualization.
  • #1
kathrynag
598
0
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?
 
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  • #2
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.
 
  • #3
symbolipoint said:
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.
 
  • #4
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.
 
  • #5
As someone who not long ago learned 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.
 
  • #8
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.
 
  • #9
http://en.wikipedia.org/wiki/Dandelin_spheres" [Broken] :wink:

fun! :tongue2:​
 
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  • #10
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.
 
  • #11
tiny-tim said:
http://en.wikipedia.org/wiki/Dandelin_spheres" [Broken] :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!
 
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  • #12
I wish I got ice cream when we did conics years ago!

That's rather cool.
 
  • #13
hi micromass! :smile:
micromass said:
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:
 
  • #14
Pity they didn't realize they could cut the ice cream into two identical spheres of ice cream!
 
  • #15
tiny-tim said:
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...
 
  • #16
jhae2.718 said:
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:
 
  • #17
Banach and Tarski only ever bought one scoop of ice cream in their lives...
 
  • #18
jhae2.718 said:
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
 
  • #19
Show this animation in class!
http://schools.spsd.sk.ca/mountroyal/Hoffman/MathC30/Conics.MOV [Broken]

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.
 
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  • #20
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...
 
  • #21
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
 
  • #22
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.
 
  • #23
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!)
 
  • #24
physics girl phd said:
(My physics class does make ice-cream when studying Thermo!)

Ice cream made with LN is fun!
 

1. What are the main types of conic sections?

The main types of conic sections are the circle, ellipse, parabola, and hyperbola. These are formed by the intersection of a plane with a cone at different angles.

2. How are conic sections used in real life?

Conic sections have many practical applications in areas such as architecture, engineering, astronomy, and physics. For example, parabolas are used in satellite dishes and headlights, while ellipses are used in the design of bridges and sports stadiums.

3. What are some strategies for teaching conic sections?

Some effective strategies for teaching conic sections include using visual aids such as diagrams and interactive activities, incorporating real-life examples, breaking down the material into smaller chunks, and providing plenty of practice problems for students.

4. How can students remember the formulas for conic sections?

A helpful tip for remembering the formulas for conic sections is to use mnemonics or visual aids. For example, the acronym "SOHCAHTOA" can be used to remember the trigonometric ratios for conic sections, or students can use a diagram to visualize the relationship between the different parts of a conic section.

5. How can I make conic sections more engaging for students?

One way to make conic sections more engaging for students is to relate the material to their interests and real-life situations. For example, students can explore the trajectory of a basketball shot using the concept of a parabola, or they can design their own roller coaster using ellipses and hyperbolas. Additionally, incorporating hands-on activities and group work can also make the material more engaging for students.

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