logarithms Seeing Structure in Expressions

ln(5) – ln(4) = ln

ln(5) – ln(4) = ln

Thanks, Jack!

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.

Functions High School: Functions Interpreting Functions Uncategorized


tina infinity


The question asked for the range of the function SQRT(x).  What are your thoughts about the way the student answered the question. What does it show about what this particular student knows.*

In the past, I’ve publicly kvetched about how the only questions that we grapple with on this blog are about the particularities, rather than the generalities, of student work. This is a time when I think that the most interesting questions are accessed through thinking about what this particular kid was thinking. I’d also be interested in hearing how you think this represents a trend in students’ thinking, in general.

Thanks to Tina for the submission!

Exponents Feedback Linear, Quadratic, and Exponential Models* logarithms

Constants and Exponents

pam patterson 3


Open thread. Go wild!

(Thanks to Pam for the submission.)


Four approaches to adding up all integers from 1 to 203.

We’re looking at #4 here, guys: “Use a non-calculator shortcut to add up all the integers from 1 to 203.”






Are these differences significant? Which is most appropriate to introduce to students first? Do you introduce multiple techniques explicitly to your classes? What does this student work indicate about the way this class was taught?

Or, talk about whatever you like.



Odd Pairing

“Add up all of the numbers between 1 and 100.”

Easy, once you realize that (1, 100) make a pair whose sum is 101, (2, 99) make another such pair, etc.

“Add up all of the numbers between 1 and 101.”

Harder to make sense of, because there’s an odd number of integers that you’re summing, so they can’t be paired off evenly.

I’ve got (at least) pieces of student work that show (at least) 3 different ways of handling an odd number of integers, as in the problem above.

In the comments below, try to catch ’em all. I’ll update this post with the actual student work tomorrow.

Update: For pictures of student work on this, see the next day’s post.

Complex Numbers The Complex Number System

Negative Square Roots

complex 3


Thanks to Tina, we’ve got this great example of a tiny little error that crops up during complex numbers. Here’s my take on it: there’s no way that this kid would make this mistake if their problem was just “Simplify the square root of negative 4.”  When the skill is laid forth is such a direct way, it’s very clear what the student is supposed to do. But when the skill is embedded in a much more complex problem, the student “handled” the negative root by realizing that this was a context that deserved a complex number. Happy and satisfied that they noticed and “handled” every aspect of the problem, the student moved on.

I like calling these sorts of mistakes “local maxima” mistakes, and I think they’re fairly common. To me, the importance of these sorts of mistakes is that they reveal the problem with testing any skill in isolation of others. I’m <i>absolutely sure</i> that this student could simplify the square root of negative four if plainly asked to. But  that didn’t mean that this student was able to use that skill in this context, when there are many more things to juggle.

To me, this means that you can’t really assess any individual skill in that sort of isolation. Instead, I’d prefer an assessment system that gives students a bunch of chances to use a skill — unprimed — in the context of a fairly difficult problem. If the student can simplify negative radicals in 3-4 more involved problems, then I’m pretty confident that this kid has that skill down.

Arithmetic with Polynomials and Rational Expressions Feedback Rational Expressions Rational Expressions

Factoring + Rational Expressions

IMG_20130331_171842 IMG_20130331_172201

What’s the fastest way of helping these students?

Thanks to Anna for the submissions.


Linear, Quadratic, and Exponential Models* Quadratic Functions Quadratic Functions Quadratics Radicals

Completing the Square, II

matt owen


Matt submits the above, and Matt writes, “I think it’s especially interesting that this student left the mistake on the board even though she had found the correct solutions by graphing in Desmos.  I’m not really sure if she did half of forty, or sqrt 4 and then stuck a zero on it (she wasn’t sure either).”

I vote for “half of 40.” You?