On Word Problems

Introduction

I was recently engaged in a discussion about word problems with my sister, who observed that some people seem to be innate word-problem solvers, while others, who are perfectly intelligent otherwise, can't seem to grasp it for anything. Those of us who are innate word-problem solvers assume that everyone else should be able to see the solutions just as obviously as we can. But some people need to be taught how to interpret a word problem before ever they can solve it. And then, they need lots of practice learning to apply a particular technique to different kinds of word problems. I have often told people that although I am not a creative mathematician (I don't make up new methods for solving problems), I am quite adept at applying techniques I've learned to situations I've never seen before. It is this characteristic that makes me good at math.

So, I'd like to give some pointers for how to teach others to solve word problems. We'll start with the very basics -- the four basic operations: addition, subtraction, multiplication and division. For each of these operations, there is a set of words that should trigger awareness of which operation we're dealing with. Knowing this vocabulary can help us to write solutions.

EXAMPLES
Here are four similar problems that use the four different operations:

• Jonny has 21 pencils. He gives 8 to his sister Megan. How many does he have left?
• Brenna has 21 pencils. She wants to give them away to her three sisters Delaney, Megan and Rosie so that each has the same number of pencils. How many should she give to each sister?
• Delaney has 12 pencils and Rosie has 2 pencils. How many do they have altogether?
• Brenna, Delaney and Jonny each have 8 pencils. How many do they have altogether?

You probably figured each problem out pretty quickly. But how did you know what operation to perform? What words clued you into the operation?

First, the operations that reduce a quantity.

• In the first problem, the words "gives" and "have left" tell us that we're looking for a difference -- subtraction. This "real life" situation allows us to instruct our students to look at the real meaning of the sentences. What happens when you give away pencils? Do you have more or less than what you started with? Are you adding more pencils to your collection or are you taking some away? Words that indicate "taking away" lead us to subtraction.
• In the second problem, we also see "give away". But this time, the reduction of our pencils happens in a slightly different way. Now, the word "each" indicates that we want to split up the pencils evenly. Antytime you are splitting up a set of items evenly, you want to divide.

Now the operations that augment a quantity.

• The third problem seems fairly simple and straightforward. The two key words here are "and" and "altogether". "Altogether" often indicates that we are increasing a quantity somehow. The word "and" often indicates addition (although not always).
• The fourth problem also uses the word "altogether", but now we see the word "each" as well. Here, the word "each" indicates multiplication, since we are increasing our quantity.

Of course, not all word problems are so straight-forward as these. There are often twists and turns and odd vocabulary that make it hard to determine what we should do sometimes. But we start with the basics so that we can build our toolbox of techniques. Later, we can use our toolbox to solve any problem.

Word Problem Vocabulary

Here are some additional words that translate into the various operations.

• Subtraction: less, minus, difference, how much more (and variations)
• Addition: more, plus, sum, altogether, how much in total
• Multiplication: per, each, for every, altogether, of
• Division: each, split up, divided evenly, per, out of

The words each and per can appear in both multplication and division word problems. The key to determining which operation to use is to decide if any of the numbers you've been given represents a "total" number of items. If so, then you're dealing with division. If you are being asked to decide on a total, then you're dealing with multiplication.