"Just Enough" Sets of Word Problems

Students learn what they think about

Word problems are hard to teach. Some students seem to figure out word problems without any support while others are constantly confused, and it often feels like I can't make any progress.

Here's the big idea of this post: when I encounter a tough word problem skill, I want to design an activity where students do "just enough" thinking to develop their word problem skills. If the question asks students to do too much, the subtleties of word problems get lost in the shuffle. If it asks too little, students will whiz right past all the words and ignore them entirely. I want students to think, "what math is this situation connected to, and why?"

I want to start with a few pitfalls that teachers fall into when teaching word problems.

Too Much Focus on the Process

One common strategy for word problems is annotation, though there are some others that are very structured and not annotation-based. I'm not categorically against annotation, but sometimes structured annotation creates students who are great at circling numbers and underlining questions, but who don't actually think about the meaning of the problem they are trying to solve. Process becomes window dressing, not learning. Again, I’m not opposed to a structured process, but if that's your only strategy it's probably going to come up short.

Too Few Problems

Another common approach is to have lengthy discussions about a single carefully-chosen word problem. This can be helpful, but many students need more practice than that. The lengthy discussion takes valuable time away from chances to practice that learning and consolidate understanding. A word problem strategy needs to be more than going deep on a single problem.

Too Much Going On

Word problems are complicated. There's the context, but there's also the math. The key I want students learning in a word problem is the context. There will be chances later to solve problems from start to finish, but when students are learning how to solve a type of word problem, solving the entire problem can put too much on students and distract from the learning goal.

Too Much Repetition

Here are a few word problems:

• A rectangular garden has a length of 12 meters and a width of 8 meters. What is the area of the garden?

• A rectangular rug measures 9 feet in length and 6 feet in width. What is the area of the rug?

• A rectangular field is 25 kilometers long and 15 kilometers wide. What is the area of the field?

These problems aren't likely to get students thinking about area. In each problem, students can multiply the two numbers together and ignore the context entirely. Too many word problems are predictable and repetitive, creating shortcuts for students to avoid thinking about the context.

Just Enough

My approach is what I call "just enough" sets of problems. "Just enough" means the problems do just enough to get students thinking about the meaning of the word problem and how it connects to a mathematical idea, without asking so much that students get lost in the details.

Students learn what they think about. In this routine I ask myself, "what are students thinking about, and how can I get them thinking about the right stuff?"

There are two components of "just enough" problems:

• Strip down what students are doing to focus on the key idea. Don't solve the whole thing; categorize the problem as this type or that type, or write an equation but don't worry about solving it.

• Include just enough variety to keep students thinking about the context.

A Few Examples

Here is a set of word problems that lead to addition or subtraction equations. The task for students is to write an equation for each problem (though not to solve it). If I did all addition equations, there wouldn't be much thinking to do. If I added in other types of word problems or worried about having students solve all of the equations, that would dilute the goal. Instead, students are narrowly focused: what parts of the word problem signal addition vs subtraction?

• Keisha had some marbles. She gave 4 to her cousin. Now she has 5.

• Liam saved $12 over the weekend. Now he has $35 total.

• A bird flew some distance, then another 18 miles. Altogether, it traveled 45 miles.

Here is a set of word problems that require either differentiation or integration. The task isn't to solve anything, it's only to identify whether the problem is a differentiation or integration problem.

• A car is moving along a straight road. You know the function that gives its position over time. You want to find its velocity at a particular moment.

• The rate at which water is leaking from a tank is known. You want to determine how much water has leaked over a 3-hour period.

• A population of bacteria grows according to a given function over time. You want to know how fast the population is growing at time t=5 hours.

Here is a set of word problems that lead to px + q = r equations. Here the equation structure is the same each time, but students need to think about which value is p, which is q, and which is r. Again, the goal isn't to solve anything, it's only to write the equation.

• A movie ticket costs the same for each person. Four friends buy tickets and spend a total of $44. They also spend $4 on parking.

• A plumber charges $60 for a service call, plus $45 per hour. The total cost of a job was $195.

• A class orders pizzas that cost the same amount each. They also pay a $6 delivery fee. If they spent $72 for 6 pizzas and the delivery, what did each pizza cost?

A few more ideas of topics for “just enough” sets of word problems:

• Circumference vs area of circles

• Percent increase vs decrease

• Writing exponential functions

• Find the slope in a linear context

• Permutation vs combination

• Write the equations for a systems-of-equations problem

Writing Word Problems

If you're like me you absolutely fucking hate writing word problems. Writing word problems is probably the most cognitively draining thing I do as a teacher. Not only that, I'm not very good at it. My contexts end up repetitive, I use the same key words over and over again, and I struggle to get enough variety.

I've been using AI to write these sets of word problems. I don't use AI to generate very many of my classroom materials. I wrote a while back about a nice worksheet. I wrote that myself. It doesn't take a ton of time to write a worksheet for something like combining like terms. I'm particular about these things, so if I have AI do it I'll often spend a bunch of time fixing things and tinkering with the formatting, and it would've been faster just to do it myself. Word problems are the one big exception for me. I still read through the word problems and make tweaks. I usually ask AI for more problems than I need and pick out or modify the ones I like best. Even doing that it takes way less time than writing all the problems myself.

I have a sample prompt that you can feel free to use. It's in this footnote¹ because it's a bit long.

Facilitation

I use a few different strategies for these sets of problems. One is to drop them all on a handout. Maybe model the first one, then have students solve a bunch on their own, then come together to discuss. Another is to drop them on a series of slides. I can do a series of think-pair-shares, or if it's something simple like area vs circumference I can use choral response or mini whiteboards to keep things moving quickly and get more practice in. There are lots of other ways to facilitate this that will depend on your norms and routines. The most important goal is to get all students thinking about as many word problems as possible. In general I aim for around 8 problems at a time. That's enough to get some good practice in without going too crazy on the skill. If it's a mess, I step back and try again the next day.

Closing

The broad principle here is that students learn what they think about. The two common mistakes teachers make with word problems are that students are thinking about too many different things, which means they aren't learning the word-problem-specific skills we want them to learn; or word problems are too repetitive so students aren't thinking about the words at all. The goal of "just enough" word problems is to focus student thinking on the essential ideas I want them to learn.

The other piece that’s important is to increase the amount of practice I give, and to focus the goals so that practice is manageable. Too many curricula don’t have enough word problems, or stick them all at the end, or don’t sustain practice on a specific type of word problem until students get the hang of it. The goal here is to get a bit of sustained practice, but also to narrow down what students are doing so that practice doesn’t take forever.

1. I want to emphasize that prompting generative AI is more art than science. Part of this prompt is just getting the model to format its response in a way that saves me work. No matter how I prompt it, some of the word problems often aren’t what I asked for or are written in confusing ways. The best way to use this prompt is to ask for more questions than you need and cut out the ones you like least. Prompt:

I am a math teacher. I want you to write word problems for me. The word problems should be written in the following style: Each problem should be numbered. The problems should not have any blank lines between them. Instructions should be written in between problems when possible. For instance, if the problems all ask students to write an equation, the instruction "Write an equation for each problem" should be at the top before the numbered problems. As much as possible instructions should be limited. I will frame the purpose of the worksheet for students when I give it to them, so I can narrate areas to focus on or conceptual leaps to take. Problems should gradually increase in difficulty, starting with very simple and predictable problems, and gradually getting harder. The contexts should vary, so the mathematical ideas stay the same but connect to as many different contexts as possible. The structure of the word problems should avoid repetitive "tells," for instance repeated key words that give away the answer to the problem.

Give me 20 word problems that require students to write an addition or subtraction equation of the form x + p = q or x - p = q.

(Change that last sentence to match your goal.)