exponents Expressions Expressions and Equations

Combining like terms…

john weisenfeld

…is probably looking like an unfortunate name for adding monomials when you see a mistake like this one. Right?

Or wrong? Speak up in the comments. And more thanks to John Weisenfeld for the mistake.


Arithmetic with Polynomials and Rational Expressions exponents

Simplifying exponents

pic 2 edited down


How would you help this kid out?

Thanks to Chris Shore for the submission!

exponents Linear, Quadratic, and Exponential Models* Seeing Structure in Expressions

Negative and Fractional Exponents



I want to share a theory on this mistake:

The student had an association between negative exponents and reciprocals and “half-powers” and square roots. As the student was parsing the problem he “fulfilled his obligation” to use that association on the number. I guess what I’m positing is that the mind works by making a connection, and then remaining in tension until that connection is used in a problem. (I’ve often had the experience of feeling as if there’s an insight that I haven’t used yet in solving a problem, and it’s like having a small weight on my back.) The mind comes to relief at the moment that the insight is used.

The student’s mind made the connection between negative powers and reciprocals and was in tension. He then used this insight at the first opportunity he saw, to relieve himself from the burden of his insight.

Some of you might disagree. For instance, you might think that the student had just memorized some rule poorly, had no conceptual understanding of powers, and gave the answer that he did.

But I think that the answer felt right because he used the fact that he knew. I’d predict that this student would be able to answer x^{1/2} correctly.

If you think that the student just memorized a rule, is there any reason to think that a student would get a question such as x^{1/2} correct?


Arithmetic with Polynomials and Rational Expressions exponents Expressions Seeing Structure in Expressions Simplifying expressions

Combining Like Terms + Exponents

Here are 7 mistakes. There represent all of the variety of mistakes from a selection of 36 students. The first two mistakes were repeated by several students, but the last 5 were unique in the sample.


IMG_2605IMG_2604IMG_2603IMG_2602 IMG_2600 IMG_2599

Which of these mistakes would you predict? Which ones surprise you? Can you make sense of them all?

exponents Expressions & Equations

“Prove that 5^0 = 1.”

exponents Expressions & Equations Middle School

Exponents – Positive Integer Powers


(via Chris Robinson)

exponents Rational Expressions Seeing Structure in Expressions


What’s going on in this (reconstructed) student work? Tell a story in the comments.

And then go thank Christopher Danielson for sharing this stuff.

exponents The Number System

Zero and Exponents

This post is brought to you by David Wees. Thanks, David!

Distributive Property exponents Expressions and Equations

The Distributive Property of Exponents

Why is this mistake so enticing, and how might you help students avoid it?


Thanks again to Anna Blinstein for the submission. Follow her! Virtually! Not literally!

Arithmetic with Polynomials and Rational Expressions exponents Rational Expressions Seeing Structure in Expressions


What’s the mistake? Diagnose the disease, and find the cure in the comments.

Thanks to Anna Blintsein for the submissions. Go follow her on twitter!