At every level I’ve taught at, I’ve found a lot of students are confused about the syntax of algebraic notation. Here’s a handout I gave to my middle-school, high-school precalculus, and college calculus students, showing the coefficient, a single variable, and an exponent, and saying what the defaults are if anything is missing.

I think many of my students found this helpful, but it could be used as the basis for something more engaging, maybe even *perplexing*, to use Dan Meyer’s word (Ten Design Principles For Engaging Math Tasks). Let’s challenge students to think about what the defaults should be — no, what they *must* be — if parts of the term are missing, as they often are. If there’s no coefficient, you *must* assume it’s 1. Why? Because multiplying by 1 doesn’t change the value; multiplying by anything else does. And so on.

Feel free to use these however you like (though I’d appreciate it if you’d give me credit). Higher-resolution versions are available of both the “challenge” version and the “reference” version.

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