Syntax of Math Notation: the Anatomy of a Term

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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