## When Not to Use an Equal Sign

I keep thinking about equal signs – when to use them and what do they mean.  Here is an example of when not to use an equal sign.

p = plane

I require that students define their variables when they do so-called word problems.  This attempt is wrong in several ways.  First this supposed sentence is missing a period.  Second it is ungrammatical. Using an equal sign in an English sentence in this way is not allowed.   Third this supposed sentence is not precise enough.  Here is an “improvement” typical of the text we use.

Let p = the plane’s speed in miles per hour.

Better precision but still wrong.  Equal signs can not be substituted for the words like is or in this case be.  Here is what’s typically found in algebra texts.

Let p be the plane’s speed in miles per hour.

Now we have a well-formed sentence but I don’t think it makes a lot of sense if you don’t know algebra-problem-speak.  The verb be(is) has around thirteen different uses as I counted here. I write the sentence in class this way.

Let p stand for  the plane’s speed in miles per hour.

I take this to mean.

Let the symbol p stand for  the plane’s speed as measured in miles per hour.

Now we are clear that we are using a symbol and that every time we see it we can substitute the words: the plane’s speed as measured in miles per hour.

Noticed that we are using the symbol p not x or y.  I teach my students to use variable symbols that have meaning in the context of the problem.  p reminds them that the variable has something to do with the plane.  With some classes I have had buy-in using the symbol $p_s$ for the plane’s speed.  I learned this useful habit when I was a professional programmer.  That sounds a little highfalutin but I was actually paid to program computers three separate times in my life.  The choice of variable is very important for reading and debugging programs. My favorite programming language – the very elegant FORTH places particular emphasis on naming variables and I took this to heart.

Addendum: I am grading papers this morning.  This sentence is also incorrect:  The tolerance limits are $9.39\le x\le 9.41$.