# Introducing quadric surfaces in high school

I am presenting an enrichment session on 3D geometry and quadric surfaces to able 15-year-old secondary school students. They know algebra but not calculus. They have learned about equations of parabolas and circles but not other conic sections such as ellipses and hyperbolas.

I was thinking of explaining what 3D geometry is ($$z$$-axis will be unfamiliar), showing a few examples such as the sphere and the cone, and then getting students to match the equations of surfaces with their graphs.

Does anyone have any nice ideas for how to introduce this topic or for anything else that could be covered in this session?

## 2 Answers

This may be a bit of a stretch, but once you introduce the sphere, you could at least mention how to shade a sphere for 3D graphics: if $$\theta$$ is the angle between the light source ray and the normal to the sphere at point $$p$$, then shade $$p$$ with a light intensity proportional to $$\cos \theta$$. This is known as Lambertian shading.

So when $$\theta =0$$, the light is head-on, and the intensity is maximized. When $$\theta \rightarrow 90^\circ$$, the intensity goes to zero:

Of course the same holds for any smooth surface, but the surface normals for a sphere are the easiest to visualize.

There are many sources for this calculation, e.g., Flat Shading.

I think it's too early to be showing this sort of topic and discussing the connection to equations. So resist that mathy temptation to only think of the topic analytically. These kids haven't even done the equation of an ellipse yet, let alone analytical geometry.

Given you have to do this topic, the way to approach it is descriptively, not analytically. Do something where you show the different shapes, give their name, perhaps indicate how they can be created by rotation (descriptively or with a movie, NOT via equations). Any applications or instances in nature, architecture, machinery, fiction, the arts would be good also.

I like the idea to do an activity, but I'm having a hard time thinking of one that is not hokey. But my initial poor brainstorm:

*recognition test at period end

*flash card recognition game

*clay modeling (of 3-D and 2-D shapes)

If you want to share the equations, I would do it in some en passent manner, where you are not requiring the kids to understand them. Like just a random detail included in the descriptions, but not emphasized or explained. Maybe even say "you'll learn these in college". But I'm not sure even of that.

If you can make it so they have fun and recognize the names of the different new animals, then it will have been a good trip to the zoo. Think of this more as an interest awakener than a skill builder.

Sorry I don't have the complete peeled-grape edible answer for you. It's a tough spot really. But I think if you go with too much of an algebraic approach (which is the natural instinct of people with sophistication, like yourself) it will be a mistake, given your audience's algebraic newbieness.

P.s. It would also be helpful if you told us more of the basics (and considered them yourself in terms of how the constraints affect the solution space):

*where exactly are the kids in development? (Will they be doing ellipse equations soon?) Algebra 1 or 2? (I realize that's US-centric.) You actually gave us a fair amount of info, but anything else on the audience would help. E.g. AGE.

*How long is your session? (A useful tip: try to end early, NOT late. Both students/teacher will appreciate it. Classic way to give good speeches!)

*How bright are the kids? Above average, average, below? Affects how difficult you can be and also some remarks about college prep, etc.

EDIT: Maybe tweak (i.e. change) the topic? Think a little outside the box. The topic you mentioned looks like it was picked as a CLASS of equations. This is a very normal way for you to think. But is not really appropriate for the students yet. Maybe do some general descriptive topic in solid geometry (e.g. weird shapes and examples in real life, see the excellent thread in this forum on that topic. Extra credit if you use your Google-fu to find it.) Another very fun (and useful) descriptive topic is 2-D and 3-D symmettry. No, don't do any discussion using matrices or abstract algebra. Just list the types of operations. Then students can classify objects (real or pictures), to list their elements at least. You can even do easy hands on toothpick/styrofoam ball (or clay) activities. Another fun topic is the platonic solids, including higher dimension types (DESCRIPTIVE treatment, remember).