In this activity, students encounter quadratic expressions in factored form where at least one of the factors is a difference.

Monitor for students who:

• Find it useful to reason with a diagram.
• Transfer what they learned when multiplying two sums in the preceding lesson—that is equivalent to —and use negative numbers for and .

Either approach is welcome, but it is important to transition students from seeing rectangular diagrams as a way to organize the terms of the factors to directly applying the distributive property.

Students see that can be expressed as , so the sides of the diagram can be labeled accordingly. The negative values in the expressions are not problematic when the expressions do not represent side lengths of a rectangle and the partial products don’t represent areas.

Some students will write the expression in the last question as . Remind them that just as means , the expression means .

Invite previously selected students to share how they reasoned about expanding the factors. Sequence the discussion of the strategies by the order listed in the Activity Narrative. If possible, record and display their work.

Connect the different responses to the learning goals by asking questions, such as:

• “How do the different strategies deal with the second term of each factor being negative?” (Rewrite the factor as adding on the opposite, then proceeding as before.)
• “Use one of these strategies to write an equivalent expression for . Does it matter if either or both of or are negative?” (An equivalent expression is . This is true regardless of the signs of and .)

Students have seen quadratic expressions in both standard form and factored form since the beginning of the unit. In this activity, they learn to distinguish the expressions by their forms and to refer to each form by its formal name. Refining their language about the different forms prepares students to be more precise in their thinking about the graphs of quadratic functions in later lessons (MP6).

Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the question, “Why do you think that form is called factored form?” In this structured pairing strategy, students bring their first-draft response into conversations with 2–3 different partners. They take turns being the speaker and the listener. As the speaker, students share their initial ideas and read their first draft. As the listener, students ask questions and give feedback that will help clarify and strengthen their partner's ideas and writing.

If time allows, display these prompts for feedback: 

• “_____ makes sense, but what do you mean when you say. . . ?”
• “Can you describe that another way?”
• “How do you know . . . ? What else do you know is true?”

Close the partner conversations, and give students 3–5 minutes to revise their first draft. Encourage students to incorporate any good ideas and words that they got from their partners, to make their next draft stronger and clearer.

As time allows, invite students to compare their first and final drafts. Select 2–3 students to share how their drafts changed and why they made the changes that they did.

Define a quadratic expression in standard form explicitly as . Explain that we refer to as the coefficient of the squared term , as the coefficient of the linear term , and as the constant term.

Ask students:

• “How would you write in standard form?” ( )
• “The expression  has only two terms. Is it still in standard form?” (Yes, there is no constant term, which means is 0.)
• “How would you write in standard form?” ( )
• “What are the values of and ?” (-4, -1, and 7)

Then, clarify that a quadratic expression in factored form is a product of two factors that are each a linear expression. For example, , , and all have two linear expressions for their factors. An expression with two factors that are linear expressions and a third factor that is a constant, for example: , is also in factored form.