Math Story Problems: Encouragement and Advice for Students and Teachers

When story problems come your way, don't panic. In this post, I have some suggestions that I have used to help students get through story problems. Some of these suggestions involve specific, concrete steps that work in any type of story problem. But some just involve a shift in mindset -- an adjustment in what we think it should feel like when we do math. This attitude change is just as important – and practical – as are the more concrete suggestions.

The cartoonist Gary Larson has a cartoon in his "Far Side" series that comes to mind here. The cartoon’s caption is “Hell’s library”. The unfortunate resident of hell in the cartoon is apparently looking to get his mind off of his situation by visiting the very special library they have down there. But what does he find? Just shelves of books with titles like “Story Problems”, “More Story Problems”, “The Big Book of Story Problems”, and “Story Problems Galore”. The librarian on duty is you know who, and he’s not likely to help.

So often, math work consists of just solving equations that are handed to you. “Find x”.  Equation solving can be hard, but at least you know that if you remember the mechanics, you’ll probably get what you need. But with story problems, you have to come up with the equations yourself. You have to do the translation between the real world and the math world, and then back to the real world. That is, you must produce expressions and equations that capture or reflect the meaning of the words in the story problem. My students would often say that this is the tough part. It requires really knowing the meaning of math symbols and expressions in order to do the translation from English (or other natural language) into math-ese.

Let’s work on our first example. This is taken from this post’s image above:

Two trains left their stations heading toward each other at constant speeds on straight parallel tracks. The slower train travels 20 mph slower than the other. The slower train left its station 20 minutes before the faster train left its station. Three hours after the slower train departs, the trains pass each other halfway between the two stations. How far apart are the stations?

Step 1: Just read through the problem. At this point, don’t try to understand everything about the problem. Don’t try to memorize it. You are just familiarizing yourself with the things you will be thinking about later. In this example, there are different train speeds, different departure times, and the notion of distances between trains and train stations. At this point, don’t even think about any math expressions or equations.

Step 2: Focus now on the question that is asked. This is almost always the last sentence of the story problem.  It will ask for some numerical quantity. Pick a variable name for that quantity. This is called the solution variable. Then define the solution variable carefully, in writing. In our example, the question is in the last sentence:

How far apart are the stations?

The solution variable will represent the distance between the train stations.  We can give it the variable name “x” (because we are so used to “solving for x”), or we could give it a more meaningful name like “D”:

D = distance between the stations

Notice we didn’t just write “D = distance”. That wouldn’t have been good enough. In some story problems, we may get a long list of variables. In our current story problem, there are three different “distances” that are considered. We want to be able to quickly scan our list of variables (possibly for ideas about forming expressions) and know exactly what they mean. We don’t want any ambiguities.  Another good idea is to include units of measurement, so let’s do that:

D = distance between the stations in miles

We have chosen the distance unit “miles”. The question doesn’t specify units for the distance between the stations, but the question does reference “20 miles per hour” (mph) as the speed difference between the trains, so we choose miles as the distance unit.

Step 3: Start writing simple algebraic expressions or equations that convey the information in the problem. As you do this, make up whatever variables you need, writing them down in your list of variables. If you end up creating more variables than you need, it doesn’t hurt anything.  Go through the problem in any order you want. You can start at the beginning of the written story problem and go forward, or you can just tackle first whatever seems easiest.

Key point at this step: Don’t think about writing one master equation that has all the information in it. That’s working too hard. If you see that master equation immediately, good for you, but it’s just not necessary. We don’t want to work hard; we want to work patiently and carefully. You don’t have to be able to think about everything in the problem at once. Consider things piece by piece. Mathematics isn’t about being brilliant; it’s about not needing to be brilliant, because the rules of math will take you along in small steps. Math allows you to do more – in the long run –  by using straightforward, smaller steps. Break the problem down into simple, easily understood pieces. Later (in Step 4) you’ll combine these pieces – these smaller mathematical expressions – as needed.

Let’s do Step 3 for our problem. The first sentence is

Two trains left their stations heading toward each other on straight parallel tracks.

This doesn’t contain much actionable information that we can use to make an expression or equation. It just tells us that we’re dealing with train motions along a straight line between trains stations. This gives us context for other information in the story problem. We can keep that image of straight parallel tracks in mind as we go through the problem.

 Let’s go to the next sentence.

The slower train travels 20 mph slower than the other.

Does this equation really say what we want it to say? Let’s double-check. It says that if you start with the speed of the faster train and deduct 20 mph, then the result is the speed of the slower train. I’ll have a little more to say about this simple equation in a later discussion after we finish this story problem.

For some students, the custom of using the subscripts (in our example, “s” and “f”) doesn’t seem natural. Subscripts can look “complicated” to high school algebra or college intermediate algebra students. But they are not mathematically complicated; they’re just names of things. Our problem has two rates of motion, but we can’t use r as a variable twice in the same problem. We can call one of them “r” if we choose; then the other one might be “s” (maybe because it’s next in the alphabet). But it might be harder to remember which rate is which, using "r" and "s" than it is to

The next sentence says

The slower train left its station 20 minutes before the faster train left its station.

Double checking: yes, this equation says that the faster train has a smaller travel duration (which happens because it leaves later).

Notice that we converted 20 minutes to 1/3 of an hour. This is because our speeds here are measured in miles per hour. We want to be consistent in our units.

Next we have

Three hours after the slower train departs, the trains pass each other halfway between the two stations.

How can we express this in mathematical symbols? We have to do some interpretation here. We really have two pieces of information. The first part of the sentence tells us that the time duration for the slower train is 3 hours, so we have

Again, in Step 3, we’re not trying to “solve the problem”. We’re just writing down what we know. “What we know” consists of things expressly stated in the story problem, together with other basic principles that relate to the different variables, such as the principle that distance equals rate multiplied by time.

Step 4: At this point, you have all the math expressions you need. You don’t have to think about “the story problem” anymore (much) because you have converted all of the verbal information into mathematical expressions in Step 3. The next step is to combine these expressions into a solvable equation, containing the solution variable, and then solve for that solution variable. You’ll do this by following the basic algebraic laws. Focus on eventually arriving at an equation that has the solution variable as the only variable. One major tool will be variable substitution. This will allow you to merge the information in the problem into one main equation. Depending on the complexity of the problem, there may be some twists and turns. But you can always look at your variables list and your collection of expressions from Step 3 to provide clues.

A good place to start is any equation that contains the solution variable. For us, this is “D”.

We have

Well, we have this down to two variables instead of the five variables we started with. That’s good. But without knowing the rate of the faster train, we can’t solve for D. If you get stuck at a spot like this, start looking back through the list of variables and equations you developed in Step 3 to see if we have more information that we can use. Because you were thorough about translating the language from the story problem into math expressions and equations, the information you need should be found there.

What information haven’t we used yet? It’s the fact that the two trains travel the same distance.

That is,

D = 480 + 480

D = 960

So, the faster train travels a distance of 480 miles. 480 miles is also the distance that the slower train travels (until the trains pass each other halfway between the two stations). The two stations are 960 miles apart.

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We solved the story problem. I just want to recap some of the principles we applied. I have seen so many math students sit frozen in front of story problems. Often, they feel like they just don’t know where to start. The answer lies in just focusing on one thing at a time. Start with a simple statement about the solution variable. In our problem, the distance between the stations, D, is the total distance travelled between the two trains. You can just start by writing

D = distance travelled by faster train + distance travelled by slower train

That is to say, sometimes you can just write out something in English, if that helps. Turn English expressions into mathematical variables, later, if you choose. That’s fine. Start writing pieces of information down. In our story problem above, I count six or seven separate pieces of information that we itemized in Step 3. We’re not supposed to have to think about that many things at once. Math and Algebra rules make it so that we don’t have to think about them all at once! Gather the little pieces of information, assemble them using substitutions, and simplify expressions as it becomes possible.

I don’t know about you, but if my wife sends me to the store to get a list of items any longer than one item, I have to write the items down. I don't like to think about too many things at one time. I’m a mathematician, and when I do story problems, the process I described above is the way I have to do it.

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There are a couple of things to be careful about in Step 3, when we translate English to math. Earlier, I commented about the equation

A similar problem was encountered at a conference of math educators over 20 years ago. The educators were asked to write an equation that expresses the following:

A bag contains twice as many dimes as it contains nickels.

There was no complete “story problem”. The conference attendees were just asked to translate one piece of information into an equation.

When we see that phrase “twice as many dimes”, we may be itching to write down the expression

2d

and then use it in the equation

2d = n.

Do you see the error? This equation actually says that the number of nickels is twice as great as the number of dimes.  But that’s backwards from what the sentence says. The correct equation is

d = 2n.

At the math education conference I mentioned, a surprising number of teachers got this wrong (something like 20 or 30%). Now, one of the themes that will come up in these blog posts is that we can’t always trust experts, and we should try to think for ourselves. People are human and they can make mistakes.

The conference attendees may have all been excellent math teachers, but every one of them was a member of homo sapiens sapiens –  that subspecies of great thinkers, innovators, creators, and mistake-makers.

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I could talk now about the various ways of “checking our final answers” after we do the algebra and so on. But this post is getting a bit long, so I need to wrap it up. I don’t want to say too much and risk extinguishing any new passion I may have instilled in you for tackling math story problems.