Question

Dear Mitch,

Do you have any ideas on how I can teach geometric terms such as point, line, space geometry, all of which are undefined things in geometry?

Is there any way I can use analogy, demonstration, or a questioning technique?

Thank you,

Miss Elsie 

Answer

Dear Miss Elsie,

Sure!

To answer your question, there are so many ways to do this that I find it hard to limit my response to a manageable size and scope. But, what follows is one attempt, with my personal favorite part of it, which is self-contained, toward the bottom and emphasized with bold lettering; so, if the first portion of the response doesn't seem to be the kind of thing you were looking for, by all means scroll down to the bold fonted type and start again there. 

First, for "point", the best analogy I know of to bring the idea from the world of the abstract to the world of the "real" is to think of a "point" as an ADDRESS.  In one way, a person's address (the thing which tells you where a person's home is LOCATED) is not "real", at least in the sense that it is an arbitrary name made up by someone, and, unless it is located in a new development or in a newly formed town, it was probably named a long time ago, when most streets were named.  On the other hand, it IS "real" in the sense that it indicates a unique location in the world.  For example, if someone lives at 24 Elm Street, Mount Highpoint, New York, 11955, USA, although the name "Elm Street" and the name "Mount Highpoint" and the name "New York" were all "made up", or chosen by people a long time ago, and although those people a long time ago could just as easily have named the town "Mount Big Hill," and named the state "East Bigtown" and named the country "Joined Together Places of Sun and Snow" ("JTPSS" instead of "USA"), the words still indicate one particular, unique place in the world, just as the point (4,3) on our (x,y) coordinate graphing system represents one and only one unique location in the first quadrant exactly four units toi the right and three units up from a predetermined place (the "origin").  Long ago, the person who devised the (x,y) coordinate graphing system could have chosen to use the first variable (the x) to indicate the point's placement along the vertical axis (the vertical locater), and the second position (the y) as the locater of the position along the horizontal axis, i.e. an inversion of our x, y system that would look like our current depiction of a y, x system, but he/she did not, and so the x, y address location is our current-day, perfectly defined system of locating points and lines, etc. on our standard coordinate graphing system,  even though its name and naming system was invented.  After all, in the real world, made up or not, three simple, short lines of text is enough of an indicator of location ("address") to enable the postal system and its mail carriers to get a letter from a rural farmhouse on one side of the world to a small studio apartment otherwise hidden in the middle of an urban highrise apartment building on the other side of the world.

From this idea you can derive and develop the idea of a line, which is just a particular set of points, and three-dimensional geometry (or "space geometry"), which is characterized by addresses or "coordinates" defined by three terms: x, y, and z.

Jumping ahead a bit to a higher and more interesting level,  most people who have studied basic geometry know that three non-colinear points form a plane.  This, of course, means that the three points are not in a single line.  So, ask your class why they think it is that the 3 such points that define a particular plane cannot be lined up.  After the group shares ideas and discusses them, step over to the classroom door, which -- HOPEFULLY -- will have three hinges, and dramatically slam it closed.  SAY:  "See those 3 hinges? Think of them as points, and notice they are lined up.  Notice how many different planes the door swung through along its journey?"

ANSWER: "An infinite number of planes."

Therefore, when in a line, any three points do not define -- or give the "address" of -- a single, unique plane.  Therefore, as an "address" of a plane they are useless, and mail sent to a point on one of those planes could end up in any one of a zillion or more of the planes the door swung through along its journey from the open to the closed position.

Then ask the class what the advantage is of a 3-legged stool over a 4-legged stool?  Have them make a few of each with toothpicks stuck into little circles of corrugated cardboard... and the answer they will discover is this:  3-legged stools can never wobble.  Why?  Becuse 3 points define a unique plane on the floor, whereas a set of 4 points (each of the endpoints of 4 legs) makes one of them an "extra", and the chance of that leg's endpoint landing on the exact same plane as the other three is 1-in-a-million. The best we can hope for is to get the 4th leg's endpoint to be close enough to the plane of the floor to land it close enough for the stool's wobbling to be too small to notice. 

I hope this helps.  If so, feel free to let us know.  If you'd like, we will gladly post some more tips!

-- Mitch