How to Work a Math Problem
Copyright © 2003–2022 by Stan Brown, BrownMath.com
Copyright © 2003–2022 by Stan Brown, BrownMath.com
Summary: Many math students are terrified of “story problems”. This page shows you some techniques to tame the dread beast.
How to Succeed in Math
Also, the classic How to Solve It by G. Polya is available in most libraries. It’s chock full of complete examples that help you form good habits for solving word problems.
See also: This page is also available in Dutch.
When you face a word problem, do you tend to freeze up, believing that it will be hard? Can you say “self-fulfilling prophecy”? That MEGO factor (MEGO = My Eyes Glaze Over) might be your biggest problem all by itself.
When you find yourself panicking, try saying, “I can solve this!” And you know what? Using the techniques on this page, you can.
It should go without saying that you need to read the textbook before working the problems. If you skip that essential step the problems will be a lot harder than they need to be, maybe even impossible. Reading a textbook means more than just passing your eyes over the words; a separate page gives some tips.
Don’t immediately pick up your pencil and try to solve the problem. Read it once all the way through just to get a general sense of what it’s about. Is this a mixture problem? Am I finding dimensions of some shape? Are there several related quantities (ages, distances, etc.) that I must find? Am I looking for a probability?
At some point in the problem you’ll be told what you’re looking for. Usually (not always) this comes at the end or has a question mark, or both. “How old is John?” “Find the probability of ...” After you’ve read the problem, look back at the goal and write it down with an equal sign and a question mark:
Amount of vinegar added = ?
P(two oatmeal cookies) = ?
As a general rule, since you want the goal of the problem to be your variable (or one of your variables), you assign it a symbol. The thought process is “I don’t know the amount of vinegar added; I’ll call it A”. You should write this down:
Let A = amount of vinegar added
(In statistics, most of the variables already have standard symbols.)
If it’s not obvious from the first reading what you’re looking for, read the problem again to get that specific information. Many students seem to feel there’s something wrong if they read a problem more than once. In fact, there’s usually something wrong if you read it only once!
Now that you have a general idea of the problem, read it again with pencil in hand and start writing down the details.
If this is a physical problem, draw a quick sketch. If it’s a mixture problem or distance-rate-time, set up an empty table. Then you’ll be able to write things in the proper places as you go along.
As you come to each unknown quantity, give it a name (letter) and write down what it means, like this:
Let P = Pete’s age
No law says unknowns have to be x and y. If you use meaningful initials you will find the problem clearer in your mind.
When you come to any fact, write it down in symbols. If the area of the corral is 182 sq.ft., write
A = 182 sq.ft.
If you’ve drawn a picture or set up a table, write each new fact in the place that makes sense. (For instance, write the area in the middle of the shape that it’s the area of.)
When you come to any relationship, write it down in symbols. You may have to break it down into several steps: write all the steps. If Pete will be twice as old in four years as John was ten years ago, you’ll have something like this:
Let P = Pete’s age now
P+4 = Pete’s age in four years
Let J = John’s age now
J–10 = John’s age ten years ago
P+4 = 2(J–10)
Where many students go wrong is that they try to skip those preliminary steps and jump right to an equation. If you’ve carefully written down the first four steps, it should be easy to get the equation right.
Look over what you have so far, and remind yourself of your goal. You may have already developed the equation when writing down the given relationships. If not, ask what other facts you need: some formula that you’ve learned, the number of feet in a mile, etc. What do you need to make the problem complete?
Usually your plan will be in the form of an equation. Write it down.
If you’re stuck, ask yourself, “Have I used all the data?” Math problems sometimes give irrelevant facts, but usually to solve the problem you need every piece of information given.
Another good question: “Can I draw a picture?” If you drew one earlier, look back at it and ask if it’s reasonable for the quantities given in the problem. If the length is supposed to be four times the width, redraw your picture that way. If it’s a trig problem, make sure that the sides and angles are roughly in proportion to the measurements in the problem.
Sometimes the problem must be solved in steps. Ask yourself “Can I solve part of this problem?” A variation is to think backwards: “If I knew _____, that would show me the way to the solution. Is there any way I can find _____ from the given information?”
You can also ask yourself Have I previously solved a similar problem? If you have, probably the solution to this one will be similar as well.
“Can I rearrange the data?” Detectives work by juggling around their facts until they fall into a pattern. You can do the same thing. Try taking what you have and rearranging it. If you can make a chart or table, do it. Often that slight shift in perspective will get you started.
Polya’s book has lots more things to do when you get stuck.
If you get frustrated, you’ll be tempted to look at the back of the book, find the answer, and then figure out how to work backward from it to the facts. Resist that temptation! Never mind the philosophical reasons; there’s a very practical one: you won’t be able to do that on tests. The way to practice working problems forward is to work problems forward.
If you’ve really made a genuine effort and you’re just stuck, looking at the answer and working backward is better than nothing. But don’t fool yourself: if you can’t work a problem forward you haven’t mastered that part of the material. Once you’ve used the backward technique to figure out a problem, look in your book for other problems of the same general type (not too similar), and work them out forward, until you can do so correctly and with confidence.
Now carry out your plan. Do the computation, solve the equation, whatever.
Write down all the steps you follow. As you go along, make sure each step follows from the previous one. It’s tempting to kip steps, but that’s also a good way to make mistakes.
Students tend to skip over this final step, but it’s pretty important to your learning. (Skipping it is also a fertile source of lost points on tests.)
“This isn’t right. This isn’t even wrong!” — attributed to Wolfgang Pauli
Many students are so relieved to get to “the end” of a problem that they immediately leave it and go on. Then teachers see discounts greater than 100%, negative lengths, parents 4 years old, concrete dams 0.2 ft high, and so on. These are fun to share with other teachers, but they don’t do you much good.
Look at your answer in the context of the problem and see if it makes sense. If not, go back and find your mistake.
Look back at the problem and make sure that your answer is what was asked for. If you were asked for the point where two lines intersect, make sure you give both the x and y coordinates. If you were asked for the length of a box, make sure you give it (not the width).
Make sure your answer is in the proper form, and has the proper units. If you’re asked for a percent discount, it should be a percent (probably converted from decimal), and it should have a percent sign. Physical quantities like length and weight and liquid measure should have their units; money amounts should have a dollar sign or cent sign as appropriate.
Check your answer in the original problem, and see if it makes all the statements true. If it does, you can be pretty confident.
Suppose your answer doesn’t work in the original problem? Then you have to find your mistake. The first question is whether you wrote down a correct equation. If your answer fits in the equation but not the problem, you misunderstood the problem somehow and you need to go back and correct your understanding. If your answer also doesn’t work in the equation, find where you made a mistake in solving. Fix the mistake, finish the solution, and check your answer again in the original problem.
After you’ve checked your answer on your own, check the one in the back of the book (if available).
There should be no surprises at this point, but sometimes there are. Maybe there were two possible answers and you found only one of them, for instance. Go back and see what you missed.
Be alert to the possibility that the book’s answer is the same as yours but in a different form. Check whether your answer is equivalent. If so, and if the problem didn’t require a particular form of answer, you’re fine.
Don’t be tempted to look at the back of the book before you’ve checked your answer on your own. Remember that you will have to check your own answers on a test, and you need to practice doing that and finding mistakes.
Sometimes you just can’t get a problem. You’ve tried all the techniques to get yourself unstuck, but nothing is coming.
In this case, the best thing to do is probably to leave the problem for a while. If you’re very tired, maybe you’ve studied enough for that day. Or perhaps you just need a fresh perspective. Very often if you work another problem and then come back to this one you’ll find you can solve it.