Time for a post about geometry, which I tutor in addition to algebra and many other subjects.
I especially enjoy helping students learn how to do proofs, which I find is the hardest area of geometry for most kids.
Recently I came up with an analogy to help students understand the special usefulness of definitions in geometric proofs.
The analogy is: Definitions are like reversible coats.
What? … you say.
Coats. Reversible coats. As in two for the price of one.
Similarly with definitions: you get two IF-THEN statements for the price of one when you work with a definition.
Here’s what I mean.
First consider a “standard theorem” in geometry, viewed in the IF-THEN format.
Theorem: IF two angles are complements of the same angle, THEN they are congruent.
Notice that the converse of this statement doesn’t make much sense:
IF two angles are congruent, THEN they are complements of the same angle. (What other angle? We haven’t even mentioned another angle!)
But when it comes to definitions, you can:
a) First, turn the definition into and IF-THEN statement, and
b) Secondly, you can flip that IF-THEN statement around, and this new statement, called the “converse,” will always be true. You can bank on it!
Example of a definition: A right angle is an angle that measures 90 degrees.
And here’s one IF-THEN statement that flows out of this definition:
1) IF an angle is a right angle, THEN it measures 90 degrees.
But notice that the converse is also true:
2) IF an angle measures 90 degrees, THEN it is a right angle.
Let’s try this again, for the definition of perpendicular lines.
Definition: Two lines are perpendicular if they form four right angles.
First IF-THEN statement:
1) IF two lines are perpendicular, THEN they form four right angles.
Second IF-THEN statement, the converse.
2) IF two lines form four right angles, THEN the lines are perpendicular.
I am wondering if you are wondering why this is true. Why is it that, for definitions, both the statement and its converse are always true? The reason, I believe, has to do with the nature of a definition. With a definition, we are giving a name to some geometrical object, and stating what we consider to be the defining characteristic of that object.
To take a nonsensical example, suppose that you live in a world that has objects called “Snurfs,” which are measured in units called “Goobles.” Now imagine that some of the Snurfs are special because they have a measure of 100 Goobles. This fact makes these Snurfs so special that you wind up talking about them a lot. And because you talk about them a lot, it is helpful to give them a name. So you do give them a name; you decide to call them “Wombats.” What this means is that anytime a Snurf has a measure of 100 Goobles, you will call it a Wombat. And anytime you see the thing you call a Wombat, you can be sure that it will have a measure of 100 Goobles. For that is just what you have decided the word Wombat will mean. Based on this, you put forth the formal definition:
A Wombat is a Snurf with a measure of 100 Goobles.
Given this definition, notice that you can create two IF-THEN statements:
1) IF a Snurf is a Wombat, THEN its measure is 100 Goobles.
And you can also state the converse, and it will be true:
2) IF a Snurf has a measure of 100 Goobles, THEN it is a Wombat.
To me, this is how definitions work. They involve people noticing something they are talking about, and they decide to give it a name so they can talk about it more easily. When they define what the word means, they attach the word to the primary characteristic of this thing, and through this act, the word is born, and along with it, its definition.
Anyhow, in terms of doing geometry, the important thing to keep in mind is that all definitions can be used reversibly. So, going back to the example of the right angle, here’s what this means.
If, in the course of a proof, you establish that a particular angle is a right angle, you can conclude that the measure of this angle is 90 degrees. Reason: Definition of a right angle.
And similarly, if in a proof you establish that a particular angle has a measure of 90 degrees, then you can conclude that this angle is a right angle. Reason: Definition of a right angle.
This reversibility factor is why, when you read through geometric proofs, you will notice that “Definition of … ” is used quite often as a reason for steps. Because they are logically reversible, definitions are TWICE as useful as standard theorems.
Comments on: "How to Use Definitions in Geometry" (5)
Of course, first you have to convince students that they CAN’T do this with regular if-then statements! Most of my students will switch if-thens around willy-nilly: converse, inverse, contrapositive–who cares?
It’s only after they learn to be careful about the differences that they appreciate this flexibility with definitions.
Denise, that’s interesting that students believe that they can turn logical IF-THEN statements around “willy-nilly.” I wonder if color-coding the IF and the THEN clauses, with the IF part in one color and the THEN part in another color, would help them see that these two clauses are significantly different. And I wonder if that awareness would inhibit students from turning the statements around. I have a system for doing this that works quite well. Have you ever seen this sort of color-coding done?
I think the main problem is just that they are not used to thinking logically, and their intuition isn’t always a reliable guide. After a few lessons, they sort of get it — but then we have to go over it again frequently as we meet new topics: “Yes, you have to prove it BOTH directions.”
I don’t think color coding would matter. They can tell which part is the “if” and which is the “then.” They just don’t understand why they can’t switch them as needed. After all, it works sometimes (for definitions, as you explain above), so why not whenever they want it to?
Interesting comment. It sounds like some of your students simply assume that the logic should flow in both directions. And perhaps the more that they need/want it to flow in both directions (for proofs, for finding measures of angles & segments, perhaps), the more they think that it “should” flow both ways.
Hmmm … I wonder if this would help. Present a lesson on nothing more than logic, a lesson that directly explores the question of when a statement flows both ways, and when it flows just one way. One possibility would be to give students worksheets with a variety of true IF-THEN statements that relate, not to geometry, but just to life. Students would examine these statements and write out the converse in a space provided. Then they would determine whether or not the converse is also true. After doing this for perhaps ten or more statements, students could be prompted to write about whether or not the statements that do flow both ways have something in common, and the same for those that flow in only one direction. This could lead to students develop ideas as to how you can tell if a statement is reversible or not. Example of a statement that flows both ways: If today is Tuesday, then tomorrow is Wednesday. Example of a statement that flows just one way: If Joe lives in Santa Fe, then Joe lives in New Mexico. In a follow-up discussion, you could bring up the Venn-diagram way of analyzing such statements, too, to help students gain a visual approach to understanding the flow of logic. Do you think this sort of approach might work with your students? My thinking in making this proposal is that by addressing the question directly, you would at least make students “think twice” before assuming that any given statement will flow both ways … and you would also get them to start asking interesting questions about logic.
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