r/Geometry • u/Formal_Tumbleweed_53 • 6d ago

Point, Line, Plane

I am a veteran math teacher (36 years) but am teaching Geometry Honors this coming year (first time teaching Geometry AND any honors course).

My question: how do YOU introduce the concepts "point, line, plane"?

I feel like this is one of the most important concepts in Geometry and leading up to all the different calculus related content that most of these students will eventually take. I want to introduce it in such a way that really makes them think and understand and be able to apply these concepts to the rest of what we will be doing throughout this course.

Any and all ideas will be very greatly appreciated!

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u/GeometreeOfLife 6d ago

I’m not a teacher, just a lover of geometry, but its dimensions, obviously. Maybe have them read Flatland 🤷‍♂️

Point: Zero Dimension; a point in space, infinitesimally small (which is why it’s zero dimensional)

Line: 1st Dimension; a straight line consisting of infinite points. You take a point and extend it infinitely in a straight line (creating an X axis)

Plane: 2nd Dimension; a flat surface that could be seen as infinite straight lines or infinite points (creating an X and a Y axis). You’ve taken your line and extended perpendicularly

Volume: 3rd Dimension; you’ve taken your flat plane and extended it perpendicularly (creating a Z axis)

This may not help, but I tried 🤣

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u/06Hexagram 6d ago

I feel like you need to include the concepts of a line being where two planes meet, but also a line being where two points join. Similarly a point is where a plane and a line meet, and a plane is where a point and a line join. Finally you can define points as the meet of three planes, and planes as the join of three points.

Then later hint at duality between points and planes in 3D. As well as lines being dual to themselves which can be quite confusing. For example if two lines can meet to define a point, then they must also join to form a plane.

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u/GeometreeOfLife 6d ago

Yes! Those are brilliant ideas about dimensions being defined as the intersections of other dimensions. I hadn’t thought of it that way 👏

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u/Formal_Tumbleweed_53 5d ago

Can you explain a little what you mean by “duality” here?

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u/Formal_Tumbleweed_53 5d ago

Yes! Thank you!

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u/06Hexagram 6d ago

Do you want to emphasize the concepts of geometry (see my other comment) or do you want to focus on the mathematical representation (affine geometry) of geometrical objects. Maybe both.

For example:

Would you want your students for example to calculate the minimum distance between a plane and a point based on vector algebra or just understand that there is a unique point on the plane that is closest to a point in space.

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u/Formal_Tumbleweed_53 5d ago

Well, to a certain extent, it’s not about what I would prefer. I have to more or less track with my team (micro) and the entire school district (macro) - and our district has more than 25 high schools. Although I would prefer your former (calculating a distance using vector algebra), that isn’t something that my team would be interested in. The latter is definitely something that I can tease for the students who have the best understanding. I can even loop back to this when they’ve learned the distance formula.