Lesson 12

The purpose of this Math Talk is to elicit strategies and understandings students have for multiplying binomials to get a quadratic in standard form. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to factor and complete the square in the associated Algebra 1 lesson.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

In this activity, students practice connecting quadratic expressions in the form \((x+a)^2\) with associated functions in standard form. In the associated Algebra 1 lesson, students are beginning the process of completing the square. Fluency with recognizing quadratic expressions in standard form that are equivalent to perfect squares in factored form is important for students to begin completing the square.

Students will refer to these matches in the next activity.

Arrange students in groups of 2. Tell students that for each expression in column A, one partner finds an associated function in column B and explains why they think it is equivalent. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next expression in column A, the students swap roles. If necessary, demonstrate this protocol before students start working.

The purpose of the discussion is to find ways to recognize quadratic expressions that can be written as a perfect square in factored form. Select students to share their solutions and methods for determining the matches. Ask students,

• “Is checking 1 of the terms enough to determine the match?” (No, the last two expressions have functions that are very similar. By process of elimination, sometimes 1 term is enough, but is not enough in general.)
• “What patterns do you notice in the standard form of perfect squares?” (Things students may notice, but are not required to internalize at this point: The quadratic term in the standard form is the linear term in the factored form squared. The constant term in the standard form is the constant term in the factored form squared. The linear term in the standard form is twice the product of the linear and constant term from the factored form.)

Display the first pair of equivalent expressions for all to see.

\((x+9)^2\) and \(x^2 + 18x + 81\)

Ask,

• “Which expression is in factored form?” (\((x+9)^2\))
• “Which expression is in standard form?” (\(x^2 + 18x + 81\))
• “What does it mean to say that a term is linear?” (The term consists of a coefficient and \(x^1\) or just \(x\).)
• “What is a coefficient?” (A number multiplied by the variables in a term. For example, 18 is the coefficient of \(x\) in the linear term for this example.) Use the paired cards from the previous activity.

The purpose of the discussion is to identify some connections between the standard form of a quadratic expression and the factored form when the expression can be written as a perfect square. Select students to share their solutions and things they notice. Ask students,

• “Using the patterns you noticed, can you write the standard form for the quadratic expression \((x+5)^2\) quickly?” (Yes, the standard form is \(x^2 + 10x + 25\).)
• “Do the patterns you noticed work for other factored forms like \((x+2)(x-3)\)?” (No, the patterns only work for factored forms that can be written as perfect squares.)