It is common for algebra students to be confused about the difference between unknowns and variables. After all, they are both often represented by the letter x. But they are not at all the same things.
An unknown in algebra is simply an amount whose value is not stated or that you have yet to figure out.
A variable is an amount that changes.
Sometimes unknowns are variables. Sometimes they are amounts that do not vary, also called constants. Sometimes variables are unknown, and sometimes they are known.
The relationship between the two types of amounts looks like this:
Why does it matter? Because the techniques of algebra can be used to achieve two quite different objectives: 1) determine the value of an unknown, whether it is constant or a variable, and 2) determine the mathematical relationship of two or more variables, whether or not they are known or unknown.
And that means that the solution of a problem can take very different forms. Some problems call for finding the value of an unknown amount, and the answer is of this form:
x = ______
Other problems call for determining how two or more variables are related, and the answer might be of this form:
an expression involving x = an expression involving y
If students don’t understand the difference between unknowns and variables, they will be unable to understand which kind of a solution a problem calls for.
In the academic world of manipulating algebraic expressions as modeled by a textbook or teacher in order to get a specified result, this understanding may not be essential to getting a right answer and passing an exam. But in the real world of solving actual problems in engineering, business and finance, graphic arts, health and fitness, and a host of other fields of activity, knowing what you’re trying to do is essential to success.
This confusion may or may not be one of the reasons many students have so much trouble learning algebra. But it certainly doesn’t help.
By the way: When I asked ChatGPT to give me an example of a problem involving unknown variables as well as unknown non-variable amounts, it failed to understand the difference five times in a row. I then explained the difference and it got one right, then failed the next six times in a row even with the modified prompt. It’s important to recognize that AI is simply a highly sophisticated search engine and pattern recognizer with a natural language interface, there is no actual intelligence there. See the Real Algebra blog “AI and Mathematics Success”.