[Makeover] Systems of Equations

Here is the oldest kind of math problem that exists:

Some of you knew what kind of problem this was before you had finished the first sentence. You could blur your eyes and without reading the words you saw that there were two unknown quantities and two facts about them and you knew this was a problem about solving a system of equations.

Whoever wrote this problem knows that students struggle to learn how to solve systems and struggle to remain awake while solving systems. I presume that’s why they added a context to the system and it’s why they scaffolded the problem all the way to the finish line.

How could we improve this problem — and other problems like this problem?

I asked that question on Twitter and I received responses from, roughly speaking, two camps.

One group recommended we change the adjectives and nouns. That we make the problem more real or more relevant by changing the objects in the problem. For example, instead of analyzing an animated movie, we could first survey our classes for the movie genres they like most and use those in the problem.

This makeover is common, in my experience. I don’t doubt it’s effective for some students, particularly those students already adept at the formal, operational work of solving a system of equations through elimination. The work is already easy for those students, so they’re happy to see a more familiar context. But I question how much that strategy interests students who aren’t already adept at that work.

Another strategy is to ignore the adjectives and nouns and change the verbs, to change the work students do, to ask students to do informal, relational work first, and use it as a resource for the formal, operational work later.

This makeover is hard, in my experience. It’s especially hard if you long ago became adept at the formal, operational work of solving a system of equations through elimination. This makeover requires asking yourself, “What is the core concept here and what are early ways of understanding it?”

No adjectives or nouns were harmed during this makeover. Only verbs.

The theater you run charges $4 for child tickets and $12 for adult tickets.

  1. What’s a large amount of money you could make?
  2. What’s a small amount of money you could make?
  3. Okay, your no-good kid brother is working the cash register. He told you he made:
    • $2,550 on Friday
    • $2,126 on Saturday
    • $1,968 on Sunday

    He’s lying about at least one of those. Which ones? How do you know?

This makeover claims that the core concept of systems is that they’re about relationships between quantities. Sometimes we know so many relationships between those quantities that they’re only satisfied and solved by one set of those quantities. Other times, lots of sets solve those relationships and other times those relationships are so constrained that they’re never solved.

So we’ve deleted one of the relationships here. Then we’ve ask students to find solutions to the remaining relationship by asking them for a small and large amount of money. There are lots of possible solutions. Then we’ve asked students to encounter the fact that not every amount of money can be a solution to the relationship. (See: Kristin Gray, Kevin Hall, and Julie Reulbach for more on this approach.)

From there, I’m inclined to take Sunday’s sum (one he wasn’t lying about) and ask students how they know it might be legitimate. They’ll offer different pairs of child and adult tickets. “My no-good kid brother says he sold 342 tickets. Can you tell me if that’s possible?”

Slowly they’ll systematize their guessing-and-checking. It might be appropriate here to visualize their guessing-and-checking on a graph, and later to help students understand how they could have used algebraic notation to form that visualization quickly, at which point the relationships start to make even more sense.

If we only understand math as formal, operational work, then our only hope for helping a student learn that work is lots and lots of scaffolding and our only hope for helping her remain awake through that work is a desperate search for a context that will send a strong enough jolt of familiarity through her cerebral cortex.

That path is wide. The narrow path asks us to understand that formal, operational ideas exists first as informal, relational ideas in the mind of the student, that our job is devise experiences that help students access those ideas and build on them.

BTW. Shout out to Marian Small and other elementary educators for helping me see the value in questions that ask about “big” and “small” answers. The question is purposefully imprecise and invites students to start poking at the edges of the relationship.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. The math that is “easy” (it’s easy when it makes sense) for the student is harder to teach. The hard math (doesn’t require making sense, a lot of procedures to memorize and apply) is easier to teach and is much more prevalent.

  2. I like the idea, but it’s hard for me to determine what your first goal for the students is. “What’s a small number….?” 4. “What’s a large number…?” I don’t know, depends on the size of the theater.

    • Fair question. With “small” and “large,” I’m just trying to get students working with the relationship in ways that are relatively unconstrained. I need them to multiply $4 and $12 by numbers of tickets, and to realize that the numbers of child and adult tickets will vary, that they can be any whole number.

    • It’s in the paragraph that starts, “This makeover claims that the core concept of systems is ….”

  3. I like the idea of asking students to identify the option that isn’t possible, as it can serve an additional purpose.

    There is reason to believe that large numbers of students are in the habit of not really reading the problem, and just doing something, anything, with the numbers in the statement of the problem. Of course there are lots of strategies that students are taught that help them to get answers without understanding the problem (e.g., is/of = %/100). So if we could get more students to make sense of what is being asked, that might be a step forward.

    If students routinely see, as the first piece of a problem, a question that asks which option is not reasonable, and it is easy (so in this case I’d create an option that is an odd number, e.g., $2,297 on Thursday). This requires reading and making sense of the set-up for the problem, and allows the student a chance to feel a little clever — when they say to themselves, “I see what you did there,” they can feel some success. Identifying the unreasonable option is also a good topic for a quick conversation among students.

    I also like following that with the task of identifying the other option Dan provided that is not possible, that requires more thought.

    There are variations on this general strategy (assisting students with making sense of problems before they start solving them) and I believe that they are valuable for students working with all sorts of problems, standard and non-standard.

  4. What a happy coincidence that I was testing on systems of equations today in my 9th grade Algebra I, and I needed a post-test extra credit/filler. I used this blog post as inspiration. Students worked through a similar situation at their seats silently while others finished. (I started with “What is your favorite movie and why?”) I then asked for how much it would cost for 10 adults to go to the movies, 10 kids to go, and the the maximum and minimum that the theater could have made if they sold 342 tickets that day (and then continued onto the multiple totals, which ended up being too much for my kids without a chance to talk about it). More than half of the students sailed through, but others did surprising, “math-y” things with the numbers. (I was shocked to see my “top” student tell me it would cost $17 for 10 adults to attend. [A kid ticket was $7 and an adult $11 in my example, because that’s what it costs at our local theater.] Her work shown: 10+7, both numbers in the problem. I got to have a good conversation with her about drawing out a situation, which she then did (10 stick figures, each with an 11 above their heads), and got the “right” answer.) My favorite part of the exercise was to hear kids talk to each other (even though they were not supposed to be talking during the test!). I will use this next year as a small-group activity (BEFORE the test!). Others have questioned the “small” and “large” part, but I got a glimpse of the value of that thinking with my maximum and minimum questions. What is a large amount of money for a theater? What would a “slow night” look like in terms of sales?

  5. Perhaps another leitmotif in pseudocontextual questions is starting with knowledge you could not possibly know to arrive at knowledge you should already know, like the lovely (though fictitious) anecdote about Niels Bohr and the barometer.

    In the original question, some third party runs a theatre. They have told us (we’re not the people who run the theater, per the wording of the question) that they sold 342 tickets and made $2550, I guess out of the kindness of their hearts. For some unknown reason, they did not tell us the breakdown of adult/children’s tickets, even though they should have easy access to that information, because computers and databases.

    In the revised question, there is a plausible explanation for how you have come to know what you know and why it is not more comprehensive.

  6. What a stimulating article to read. I’m feeling stuck in a rut with my Alg 1 classes but I’m ready to create and innovate now!