To introduce points, lines, planes and just the general notion of undefined terms in Geometry, I like to start with a game of Socratic questioning. Usually students give the background on Socrates, and we talk a little about his life and how he operated.
Then I divide the class into pairs, one as “the explainer” and one as “the questioner” and write a question on the board such as: “What is a rectangle?”
Here are the rules:
1) the explainer must make a good-faith attempt to explain the concept at hand
2) the questioner must ask a question designed to expose a flaw in the definition or to frustrate the explainer
3) Neither player can give up. No matter how good the explanation, the questioner must find something to question, and no matter how annoying the question, the explainer must try to answer it.
After a few minutes, I have the class vote on which role they thought did a better job, then we share some of the more frustrating questions that were asked. This is usually awesome, as the questions will range from the completely off-the-wall to the remarkably-apposite-to-the-lesson. “What is space?” “What is a point?” “What is a thing?” “What is a shape?”
We usually play four rounds, switching roles each time. The first two rounds I use actual defined objects (rectangle, circle, square, etc.), while the last two rounds I go with “What is a line?” and “What is a point?” The proportion of the class who thinks the explainer “won” goes sharply down in rounds 3 and 4.
I use this to talk about the four basic types of mathematical statement: Undefined Term, Definition, Axiom/Postulate, and Theorem, and show how definitions are based on undefined terms.
I played a game with a student where I asked: “What is a point?” and he said, “Don’t make me answer that, give me something else.” I let him pick what he wanted to explain, and showed how in a few questions, I could get him right back to “What is a point?”
Try this out with your classes — loads of fun, starts the year off well, and really gets students thinking about definitions in a more precise way.