Algebra 1 Reasoning with Equations and Inequalities Seeing Structure in Expressions Simplifying expressions

A Class Set of Identities Work

Predict: What responses to this prompt would you expect from my Algebra 1 students? (Prior to this problem my kids had mostly worked with integer arithmetic, solving linear equations in one-variable and graphing scenarios and equations.)


Study: What do you notice in this (small) class set of responses? Note anything that surprises you.

Kid 1:


Kid 2


Kid 3


Kid 4


Kid 5




Kid 6


Kid 7


Kid 8


Kid 9


Wrap Up

How did your predictions hold up? What surprised you the most? What’s something you wish you knew more about?

logarithms Seeing Structure in Expressions

ln(5) – ln(4) = ln

ln(5) – ln(4) = ln

Thanks, Jack!

Expressions Rational Expressions Seeing Structure in Expressions

Squaring doesn’t make equivalent fractions



Squaring doesn’t make equivalent fractions.

Thanks again to Gregory Taylor for the submission.


Absolute Value Seeing Structure in Expressions

Value of Absolute Value


Kids just can’t seem to figure out absolute value. Pat writes,

“I’ve posted a mistake I see ALL THE TIME from my students when working on Absolute Value Equations. Looking for any advice on how other teachers are handling this issue. Thanks!”

Incidentally, I think that there’s a very good argument for this sort of thing being struck from the curriculum in its entirety. It’s entirely isolated in the curriculum, unconnected to anything else.

Thanks to Pat John for the submission!

Exponents Seeing Structure in Expressions

-3X as 3 Negative X’s


Check out this kid’s explanation for why you end up with “+3x” from “-(-3x)”:

“You have -3x, so that’s three negatives, and then you have this other negative and that makes four…”

You can find this explanation at around 3:33 in the video below:

Exponents strike again!

(Or, maybe I misunderstood the kid’s explanation in the video? Lemme know if I did, please!)

Thanks to Jonathan for the submission.

factoring Seeing Structure in Expressions


john weisenfeld 2

You grade this on Sunday. What do you do on Monday? Go over the procedure that this student almost flawlessly executed?  Again?  What do you emphasize?  Checking your answer?

Thanks to John Weisenfeld 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?

Expressions Seeing Structure in Expressions

Number Tricks and Expressions


Comment on anything, but note the parentheses.


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.