In this activity students make use of structure (MP7) as they make connections between polynomial long division and the reasoning used in completing a diagram. They also critique a statement that is intentionally unclear and incorrect and improve it by clarifying meaning, correcting errors, and adding details (MP3).

The remainders for division in this activity continue to be 0 since the focus is on dividing by known linear factors. In future lessons, students will focus on what a non-zero remainder means both for polynomials and for rewriting expressions of rational functions.

This is the first time Math Language Routine 3: Critique, Correct, Clarify is suggested in this course. In this routine, students are given a “first draft” statement or response to a question that is intentionally unclear, incorrect, or incomplete. Students analyze and improve the written work by first identifying what parts of the writing need clarification, correction, or details, and then writing a second draft (individually or with a partner). Finally, the teacher scribes as a selected second draft is read aloud by its author(s), and the whole class is invited to help edit this “third draft” by clarifying meaning and adding details to make the writing as convincing as possible to everyone in the room. Typical prompts are: “Is anything unclear?” and “Are there any reasoning errors?” The purpose of this routine is to engage students in analyzing mathematical writing and reasoning that is not their own, and to solidify their knowledge and use of language.

Display this image of Jada's partially completed long division for all to see:

Use Critique, Correct, Clarify to give students an opportunity to improve Jada's explanation by correcting errors, clarifying meaning, and adding details. 

• Display this first draft:
  "I placed the term ‘²’ under ‘²’ after I multiplied.”
  Ask, “What parts of this explanation are unclear, incorrect, or incomplete?” As students respond, annotate the display with 2–3 ideas to indicate the parts of the writing that could use improvement.
• Give students 2–4 minutes to work with a partner to revise the first draft. 
• Display and review these criteria:
  • A correct explanation of what is happening in each step
  • An explanation of the reasoning behind each step
  • Precise use of language, including words such as “dividend,” “divisor,” “quotient,” “terms,” and “factors.”
• Select 1–2 students or groups to slowly read aloud their draft. Record for all to see as each draft is shared. Then invite the whole class to contribute additional language and edits to make the final draft even more clear and more convincing.

Students have now seen two ways of representing division: using a diagram to work backward to determine what to multiply the divisor by, and using long division to find the quotient directly. Students should understand that these are two strategies for doing the same thing, and that they each have advantages.

Students practice using long division to factor when one factor is already known. Both polynomials in this activity were factored by students in a previous lesson using a diagram, and students will compare the two methods during the whole-class discussion.

The first question has the division started, to give students an example to study.  Students are working through the logic of the polynomial division to identify other factors, so graphing technology is not an appropriate tool.

Invite 1–2 students per question to share their long division work and how they calculated the other linear factors of the original expression. If not pointed out by students, highlight how, as with diagrams, the order of division does not matter for the final result.

Conclude the discussion by displaying the completed diagram for from a previous lesson next to the way they factored it using long division in this activity.

Give students brief quiet think time to consider what is similar and what is different about the two methods, and then invite students to share their observations.

Invite 1–2 students per question to share their reasoning. Select previously identified students to share unique strategies where possible.