Today in my Algebra class we were doing word problems, and I gave my students a problem involving someone's age. One of my students commented, "I wish there was an age problem that the answer was my age, so when people ask me how old I am, I can just give them the problem."

So I asked how old she was, and then wrote the following on the board:

"When I was half as old as I am now, I was 21 years younger than I will be when I am twice as old as I am now. How old am I?"

This problem turned out to be a really good problem for my students, as it provided a launching-point to discuss some grammatical constructions. In previous classes, we'd talked about the fact that when you're trying to convert an English language statement into an equation, you should look for a linking verb to serve as the "equals" in the equation. In most - but not all - cases, this is the verb "to be." The students get very used to seeing the word "is" as the link between two halves of an equation.

We've also discussed the fact that "is" isn't always a linking verb; sometimes it's a helping verb, as in the following expression:

"Five is multiplied by a number"

In this case, "is" is not linking anything - it is a helping verb that goes with "multiplied".

We've also discussed the fact that the verb "to be" has a lot of forms - in addition to "is", other forms are: "are", "am", "was", "will be", and several others. 

What made this word problem different was the fact that there were so many forms of the verb "to be", so identifying the linking verb was more challenging. Below, I've written the statement of the problem with all the "to be" forms highlighted.

"When I was half as old as I am now, I was 21 years younger than I will be when I am twice as old as I am now."

As you can see, there are six "to be" words highlighted. That's a lot of possibilities to consider!

This is where we talked about "subordinating conjunctions." That's a terrifying sounding grammatical term, but I assured them that subordinating conjunctions aren't that difficult to understand. I told them - tongue in cheek -  "It's really simple: a subordinating conjunction is a word that introduces a subordinating clause."

They were not amused by this.

I didn't go into much depth into what a subordinating clause is (after all, I'm not an English teacher!). I told them that a subordinating clause is a phrase that contains a noun and a verb, but isn't a complete sentence. The important thing, for our purposes, is to know the types of words that begin subordinating clauses (the conjunctions). The following list is very helpful, as it contains many of the subordinating conjunctions students will see in a math problem:

after, before, unless, although, if, until, as, since, when, because, than, while.

I asked my students to look at the word problem again, find all the subordinating conjuctions, and then underline the phrases that follow the conjunctions. The subordinating conjunctions are shown in bold (with the "to be" forms bold and green).

"When I was half as old as I am now, I was 21 years younger than I will be when I am twice as old as I am now."

There are five subordinating conjunctions, and for each one of them there is a verb "to be". Now ask the students to identify the one "to be" form which does not fall within a subordinating clause. It's the word "was" in the phrase "I was 21 years younger". That instance of the verb "to be" is the linking verb connecting the two sides of the equation.

Once the students understand that, the whole problem becomes easier to work out:

If x is my age now, the problem is: