Stringing together equations

Related essay: The importance of precise language

In module 12 (page 2-4), Balance Beams, Student A wrote 3+4 = 7+5 = 12.   Paying attention to the context of the statement, it is clear that Student A meant if you add 3 and 4, you get 7 and when you add 5 to that, you get 12.  This is an example of a fairly common misuse of notation, but as discussed in “The importance of precise language,” these innocent uses of imprecise language can cause problems.

The equality symbol has a very precise meaning.  If two quantities are “connected” by an equals sign, it means the two quantities are the same (or congruent or equivalent, depending on the context).  More to the point, “=” does not mean “and next, we do.” I tend to go a little overboard with this issue with my own students, saying that “equality is sacred – don’t abuse it!”  While students might feel that Student A’s mistake is just an abbreviation, but this can cause trouble.  Since a string of equalities, like

A = B = C = D = E = F

literally means that A=B, B=C, C=D, D=E, and E=F, a correct conclusion is that all of the quantities A, B, C, D, E, and F are equal to one another.  That is clearly not the case for Student A, since 3+4 does not equal 12.

There are some mathematicians and educators who prefer that strings of equalities or inequalities never be used in order to avoid such a misunderstanding.  However, a string of inequalities can be a very useful time-saving method of demonstrating how one complicated expression (for example) is actual equivalent to another.  As long as we are careful and acknowledge the “sanctity” of the equals sign, there will be no confusion!