factoring Quadratics Reasoning with Equations and Inequalities

Which Activity Would You Choose?

In an earlier post, I shared Michael Fenton’s scenario and categorized the responses he got on twitter.

There were at least seven distinct responses that teachers offered to Fenton’s prompt. Wow! This makes me think two things:

  1. Fenton’s scenario was so thought-provoking that it yielded an amazing variety of responses.
  2. How come there was so much disagreement about how to act in this scenario?

Part of the disagreement, I think, comes from what went unspoken in Fenton’s mistake. We didn’t know if this mistake was shouted on in a discussion or found on a piece of paper. We don’t know if this is one of those times when we can afford to have a one-on-one conversation with a kid in response to her mistake, or if our response will be scrawled on her paper and returned. Was this a common error, or an isolated mistake? Could our response be an activity for the class instead of a chat?

While one-on-one conversations are crucial in teaching, they are hard to talk about. By their nature, they’re improvisational and somewhat unstructured. I’d also argue that opportunities for one-on-one conversations can be rare, and they get rarer as the number of students in your class grows larger.

Revising the Scenario

So let’s add some details to Fenton’s scenario. This was a mistake in an Algebra 1 class. Smart kids, thoughtful teacher, but when she collects papers after an ungraded check-in she finds that about half her class made Fenton’s mistake. Oh no! She decides that she’s going to launch class the next day with a brief activity to help advance her kids’ thinking.

Her first idea is to try a string of equations. She has three different drafts. Which one would you choose, and why?

Equation String 1

Fenton's Mistake - Various Approaches (5)

Equation String 2

Fenton's Mistake - Various Approaches (1)


Equation String 3

Fenton's Mistake - Various Approaches (2)

Other Activities

Then, she has some other ideas. Maybe equation strings aren’t the right move? She comes up with three other activities: Working With Examples, Which One Doesn’t Belong and Connecting Representations.

Working With Examples


Connecting Representations

Fenton's Mistake - Various Approaches

Which One Doesn’t Belong?

Fenton's Mistake - Various Approaches (6)


The meta-question here is about the conversation. Can we have a conversation with so many options? I don’t know. I worry that maybe I should have just limited discussion to the equation strings.

What would do?

My first reaction is that I like the equation strings, because it most directly gets at the issue of overextending the zero-product property to other equations.

But what I really want to do is lay out a sequence of 3-4 activities that I could do in sequence to develop this idea for a class.

And do I know enough to answer that question? Wouldn’t that depend on the math that we’d already studied and the math that’s coming up next?

Do we learn anything from thinking about these questions?


factoring Quadratics Reasoning with Equations and Inequalities

0 = -2

complex 4


How worried are you about this mistake?

Thanks again to Tina! Go Tina!

Arithmetic with Polynomials and Rational Expressions factoring Feedback

Factoring and Canceling

kelli prine

Kelli Prine submits the above, and asks “This student keeps factoring the terms and canceling.  What can I do to clarify the process?”

I expect some excellent comments on this one. Don’t disappoint.

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.


A-REI.4 factoring Quadratics

Solving versus factoring

What does the student have right, and what does the student have wrong? How would you help?

Algebra 1 factoring horizontal and vertical lines linear functions

“Graph x = 3”


The mistake is pretty easy. But what’s the underlying misconception?

A-SSE.2 Algebra 1 factoring

What does he think “factoring” means?


What does this kid think “factoring” means? And, at least to me, the more interesting question is: where did that conception of “factoring” come from? Tease it out in the comments.