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Will This Be on the Test? (December 2025)

by Aren Lew

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Welcome to the latest installment of our series, “Will This Be on the Test?” Each WTBotT features a new question similar to something adult learners might see on a high school equivalency test and a discussion of how one might go about tackling the problem conceptually.

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Welcome back to our continuing exploration of how to bring real conceptual reasoning to questions students might encounter on a standardized test.

I have argued throughout this series that many standardized test math questions can be approached visually and conceptually and that students who may not have studied the specific content the question is targeting may still have a solution path if they are disposed to make sense and think flexibly. In general, the more concrete and contextualized a question is, the more likely it is that multiple creative and flexible solution paths are available. I encourage students to see story problems as a gift because the context opens up more avenues to sense-making. However, questions that are more abstract or appear to involve higher level concepts are also sometimes accessible through a sense-making approach even when students have not learned the relevant procedures.

Here’s one such question for this month:

If you look in an algebra textbook or test prep book or website, it is likely to tell you that the way to solve an equation like this is by using inverse operations and isolating x. That works if it is an approach you understand and are comfortable with. But what if you haven’t learned all those algebra rules or they get muddled during a math panic and can’t remember what you are supposed to do?

This question is abstract and not accessible to every learner, but the understandings needed to solve it may be fewer than you think. My challenge to you is to solve this relying only on these basic understandings:

• x stands for a number. There is a number that can be put in the equation where x is and the equation will be true. That is the number you are trying to find.
• The equation is true if the values on both sides of the equal sign are the same.
• When a number and letter are written next to each other, that is a way of writing multiplication.
• Multiplication is performed before addition or subtraction.

How can you make sense of the equation using these understandings and find a value for x that makes it true?

Here are some approaches:

1. Make sense of the story. Chunk it up. Equations and mathematical notation in general are shorthand. They are ways of recording relationships or processes in symbols instead of words. But the same stories can be told in words. Here’s a way of telling the story of this equation:

There’s some number that if I multiply it by 6 and then subtract 3 from the answer, I’ll get 21.

A useful follow-up question to this reframing is:

What number gives me 21 if I subtract 3 from it?

One way to answer that question is to make use of the inverse relationship of addition and subtraction and add 3 to 21, but there are other ways, too. A student might picture a number line (more on that later) or pick a number that’s a little bigger than 21 and try counting down by 3 to see if they get to 21, or they might think, “I know 4 – 3 is 1, so 24 – 3 is 21.” 

Having answered that question, a student can then simplify the original story like this:

There’s some number that, if I multiply it by 6, I will get 24.

You might be thinking that this approach is not that different from a traditional approach of using inverse operations to isolate x. The difference is in its being more concrete. Using words and the idea of “some number” is more accessible to students whose algebraic thinking has not yet reached the level of abstraction that allows them to read and make sense of symbols as easily as they do words.

2. Use a story table. Another approach that makes the story of the equation visible is a story table that separates out the action into one step per column. A story table for this equation could look like this:

Start with a number ►	Multiply it by 6 ►	Subtract 3 ►	Did I get 21?
3	18	15	NO.
4	24	21	YES!

A story table is a great tool for using with guess-and-check. Because the question is multiple choice, there are only 5 numbers to try in the story table, but if this were a constructed response question, the story table would still be useful. The organization of the table helps students revise guesses thoughtfully and discern structure. Even without choices supplied, it likely won’t take students very many guesses before they start to think, “The only number that’s going to give me 21 when I subtract 3 is 24, so I need to make a guess that’s going to get a 24 in that column.” A story table is useful for approaching lots of types of algebra and, just like other concrete and visual approaches, students will discard it when their thinking reaches a level of abstraction where they no longer need it. You can read more about story tables in this writeup of a meeting of the Community of Adult Math Instructors (CAMI).

3. Use a bar model. Bar models are a great way to visualize quantitative relationships, and an equation is a statement about a relationship. Showing subtraction with bar models can be a little tricky. Here’s one possible approach:

In this model, the top bar shows 6x and the bottom bar shows that 21 is not as much as 6x. The difference between them is 3. How might this model help a student find the value of x? What other ways of representing this equation as a bar model make sense to you?

4. Illustrate the action on a number line. It might feel more accessible to show subtraction on a number line than with a bar model. In this case, an open number line that doesn’t have every tick mark shown and labeled is a useful choice because we don’t know the value of x. We can show the multiplication and subtraction by taking six jumps of the same size to the right from zero to represent the 6x and then taking three steps to the left to show subtracting 3. The point we land on is 21.

How might this number line help a student figure out the value of x? (You might notice that the arrow labeled -3 is shorter than the arrows labeled x, and there’s no way of knowing it should be until you know what x is. Try drawing the number line with that arrow being greater than x. How does it affect your reasoning? This drawing also assumes x is positive. These assumptions are also in the bar model approach. Depending on the numbers and relationships in the equation, a bar model or number line could be a useful tool or a confusing one. This is true of most tools. One reason why it’s good to have a big toolbox is so you can try something else if the first strategy you try isn’t helpful.)

As you read through these strategies, you might have seen inverse operations and isolating x at play behind the various visuals or ways of organizing thinking. It’s tempting to decide that teaching inverse operations is faster and that it’s simpler if everyone does it the same way. But the inverse operations strategy is the most abstract, and abstract thinking doesn’t stick if it doesn’t have a concrete foundation. You can’t tell students how to think abstractly—they have to build that themselves. Students who use tools that make sense to them to build a solid understanding of the structure of equations and the meanings of equality and operations will naturally progress to more abstract thinking in their own time.

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