I recently taught several of my classes about the three major properties of relations: reflexive, symmetric and transitive (which I'm going to assume you are familiar with...if not, you probably don't care much about this post anyway). In my math class experience, they were taught as:

Reflexive

for all x: x = x

Symmetric

for all x,y: if x = y then y = x

Transitive

for all x,y,z: if x = y and y = z then x = z

To which I had always responded with an internally muttered, "ummm, duh". But what they didn't tell me was that it's just not a big deal that 7=7. Of course it does. That's not what the reflexive property is really all about. Relations can be any kind of true/false comparison between any kinds of objects.

6 = 9 (false, compares numbers with "equals")

8 < 12 (true, compares numbers with "less than")

Angle X is complementary to angle Y (compares angles with "complementary)

p implies q (compares facts with "implies)

and more.

And the three properties might apply to all of them. So as the sets get more fun and the comparisons get more interesting, the kids actually get excited about deciding which properties apply. Here are the examples I use (and the properties they have):

**comparisons on numbers:**

equals (R, S, T)

greater than (T)

not-equal-to (S)

*[note: not-equal-to isn't transitive since the first and third numbers could be the same]*

divides-evenly (R, T)

**comparisons on angles:**

congruent (R, S, T)

complementary-to (S)

**comparisons on facts:**

implies (R, T)

*[note: the fact that it isn't symmetric is a good review of how the truth of a conditional says nothing about the truth of its converse]*

**comparisons on students in school:**

has-a-class-with (R, S)

has-English-class-with (R, S, T)

*[note: because you can only be taking one English class at a time]*

has-tied-the-shoes-of (R)

*[note: gets laughs]*

has-kissed-on-the-mouth (S)

*[note: i tell them this is PG-13 rated. gets big laughs and giggles and "eeew"s (when considering transitivity especially)]*

**comparisons on all people:**

sibling-of (S)

*[note: strictly defined as full biological or adoptive sibling...no half- or step-. not transitive for the same reason not-equal-to isn't]*

brother-of (none!)

I tend to think that a lot of people don't understand that math vocabulary and symbols are supposed to reflect the patterns of thought that go on in our heads. Math is not about an arbitrary set of rules to apply to bizarre symbols. The students think this is barely math, but I tell them it's very much math--it's just that they never realized what's really math.

And the next day, they all know *exactly* why every segment is congruent to itself: "Because congruence is reflexive!"