Categories
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

pic1

Connecting Representations

Fenton's Mistake - Various Approaches

Which One Doesn’t Belong?

Fenton's Mistake - Various Approaches (6)

Commentary

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?

 

Categories
Quadratics Reasoning with Equations and Inequalities

How Teachers Would Respond to A Quadratics Mistake

Lots of responses to this great tweet. I wanted to understand the themes in what people were replying, so I went through everything and tried to summarize it here.

Response #1: Check Your Work, Start a Conversation

Response #2: Just Check Your Work (No Conversation Mentioned in Tweet)

Response #3: Explain the Zero Product Property

Response #4: Thinking About How to Teach the ZPP Unit

https://twitter.com/philliphsquared/status/741394601399967744

https://twitter.com/bowenkerins/status/741281577687240704

Response #5: Switch to a Graphical Context

https://twitter.com/EulersNephew/status/741315075223363584

Response #6: Ask for Explanations

https://twitter.com/ProfNoodlearms/status/741276100953706496

Response #7: Run a New Activity with the Whole Class

https://twitter.com/geometrywiz/status/741329839488000002

I’m sure I didn’t capture everyone’s response, and I don’t know what any of this means. But there you go.

Categories
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?

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

Completing the Square

photo (6)

 

What do you notice in this student’s work?

 

Thanks to Matt Owen for the mistake.

Categories
factoring Quadratics Reasoning with Equations and Inequalities

0 = -2

complex 4

 

How worried are you about this mistake?

Thanks again to Tina! Go Tina!

Categories
Graphing Linear, Quadratic, and Exponential Models* Quadratics

Axis of symmetry and vertex

What does this mistake reveal about the kid’s thinking? How would you take the kid’s thinking to the next level?

Categories
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?

Categories
Algebra 1 Quadratics

Quadratics Quiz

As we start a new week of Math Mistakes, if you have any pictures of student work that you find interesting in any way, please send it in. 

 

Today’s student has a couple of ideas worth drawing out. Do so in the comments.

Quadratic