Jacob Disston asks his students to focus on a category of symbol strings (expression, equation, inequality), separate them, and discuss how they know how to categorize them. Students are asked to make statements like "I know this is an equation because..." or "I think ... what do you think?"

7th Grade Math - Algebraic Equations, Inequalities, & Properties*Jacob Disston, Willard Middle School, Berkeley Unified School District, Berkeley, California*

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JACOB DISSTON: Do you guys have them sorted?

STUDENT: Yes

JACOB DISSTON: Can I ask you guys a favor and go sort the ones on the board, under the posters...Is this an equation?

STUDENT: Oh yeah, this is an equation

JACOB DISSTON: Good, put all the equations up here

STUDENT: Mr. Disston, what if I could solve this whole side?

JACOB DISSTON: I will, I'll give you...I'll think about what I'll give you. What I want you to do right now is get all the equations in that pile. I see other equations that are in here. I want to see all the equations right here.

STUDENT: This one?

JACOB DISSTON: Is that an equation?

STUDENT: It is.

JACOB DISSTON: How do you know?

STUDENT: Yes it is.

JACOB DISSTON: Why?

STUDENT: 5 times...wait

JACOB DISSTON: Why is this an equation? Look at it you guys, look, look, look.

STUDENT: It's not an equation cause it doesn't have a variable

STUDENT: These are equations

STUDENT: It is an equation cause it has a variable...I can't think of it

JACOB DISSTON: What's the word that's in an equation?

STUDENT: Equal

JACOB DISSTON: Equal. Does it have an equal sign?

STUDENT: Yes

JACOB DISSTON: Ok, so boom. Find all the ones that have equal signs. If I hold this up and say is this an expression, an inequality or an equation?

STUDENT: No, it looks just like an answer.

STUDENT: I think it's an equation.

JACOB DISSTON: Why?

STUDENT: Because it has an equal sign

JACOB DISSTON: But you are unsure? So we can put this one in a separate pile and say we don't know what it is and we'll come back to it. How about this one?

STUDENT: That one I think might be an equation.

JACOB DISSTON: Why?

STUDENT: I think is an equation because it has...all the other ones have an equal sign right, and I think...

JACOB DISSTON: Does this one?

STUDENT: I don't know.

JACOB DISSTON: Is it an equation?

STUDENT: I don't know

JACOB DISSTON: Does it have by her definition?

STUDENT: By my definition?

JACOB DISSTON: You say it has an equal sign

STUDENT: I say it equals something...

JACOB DISSTON: Does this one have an equal sign?

STUDENT: It could be an expression though.

JACOB DISSTON: Why?

STUDENT: It just run by numbers and signs

JACOB DISSTON: Ok, but with addition doesn't it seem like it would always equal something? So does this have an equal sign?

STUDENT: No

JACOB DISSTON: Do we call it an equation?

STUDENT: No, we call it an expression.

JACOB DISSTON: Expression...does this? Ok, so what I want you to really quick is get them into your pile. So why don't you put equations here, expressions there, inequalities there.

COMMENTARY BY COACH LINDA FISHER: These clips bring up the importance of creating opportunities for students to use mathematical vocabulary for a purpose. While everyone is using the same language of equations, students seem to have a variety of definitions for what that means. Some students name it an equation because there’s multiplication. For others the variable signifies equations. Students seem to puzzle over words for describing signs: there’s addition, subtraction, and other signs like equal. They understand that the difference is significant, but don’t know how to classify it. How do you think about things you can’t name? As adults, we sometimes take meanings for granted. Equation seems like an obvious or given definition, but as I listen to students in the clips I hear many meanings or unintentional learning that they have attached to the word. Notice that even the writing promotes further discussion and negotiation of the definitions.