# = for equals, ≡ for identity?

November 2, 2011 am30 12:47 am

Do any of you out there use ≡ for “identity”? In your own work? With little kids?

I am considering introducing it to 9th grade algebra students.

They think symbols are cool.

I know they get confused when solving 4(1-2b) = 2(3-b) – 2 , maybe the special symbol would help?

What do you think?

6 Comments
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I had a couple teachers in High School who would use it…none of them Math teachers. My chemistry teacher was the one that taught it to me.

That being said, when I finally got to college, I saw “” used far more frequently particularly in proofs classes and in situations of equivalence for other classes…and so out of habit, I tend to use that one. The only time I saw the one you are describing was in reference to equivalence relations modulo n in group theory.

I’m not sure if my experience is normal; but, since it is short hand, it will vary slightly from professor to professor and teacher to teacher and book to book.

Giving them exposure sure couldn’t hurt, though. I say do it.

Sorry…it seems that the comments tried to turn it into an html tag…

The trend I noticed was to use double tailed arrows like this “==>” (single tailed arrows are used exclusively for logical implications).

The equivalence would just be a double tailed arrow with two arrow heads on each side.

Also, now that I’m thinking about it further…I believe I remember hearing that the three barred equal sign meaning “is defined as” where as “= =” like you might see in a programming language for a logical test is “equivalent to.”

FWIW, I’ve successfully used == in this context. 9th graders seem to like to think they are using stuff computer programmers use. For the rare sophisticated class I do a side bar about logic testing in if() statements and add other standard symbols such as >=, <=, .

If := is assignment, how about :=: for “having been assigned as”?

The only time I would use the identity symbol is with trig. identities to differentiate between identities and equations.

As a sometimes geometry teacher and college math teacher, I have seen the 3-line-equals symbol used in some books for congruent (or isomorphic, or homeomorphic, or…).

As a teacher of future teachers, I can tell you that part of the problem with early problem solving in algebra–problems like:

4(1-2b) = 2(3-b) – 2

is that students have learned through long experience in elementary school that = means “the answer comes next”, and have not learned that = means “the same as”. As late as 5th or 6th grade (maybe later) students will tell you that you can’t write 7=4+3. The symbol = is used with both meanings–certainly by students, possibly by you too when you’re not teaching. Think about it–how often do you see (in, say, a surface area calulation) 4×5=20×4=80+32=112? So, one of the things you have to teach in pre-algebra and algebra 1 is the equivalence meaning of the = sign.

The word “equal” isn’t strongly enough defined for them as equivalence. One thing that’s suggested for slightly lower grades teachers (but you might give it a try anyway) it to change the language you use when you read the equation out loud, so that instead of saying “equals” you say “is the same as” or “has the same value as”, so that you’re contributing verbally to students gaining the equivalence meaning for =. The problems you give where students are solving equations also automatically teaches the equivalence meaning for the equals sign.