In this activity, students practice connecting quadratic expressions in the form 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.

In this partner activity, students take turns matching equivalent quadratic expressions in different forms. As students trade roles explaining their thinking and listening, they have opportunities to explain their reasoning and critique the reasoning of others (MP3). Students will refer to these matches in the next activity.

The purpose of the discussion is to find ways to recognize quadratic expressions that can be written as a perfect square in factored form.

Once all groups have completed the matching, discuss the following:

• “Which matches were tricky? Explain why.”
• “Did you need to make adjustments in your matches? What might have caused an error? What adjustments were made?”

Select students to share their solutions and methods for determining the matches. Here are some additional questions for discussion:

• “Is checking 1 of the terms enough to figure out the match?” (No, some of the 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.)

In this activity, students use the matches from the previous activity to identify parts of different forms of equivalent expressions. They then notice patterns and draw conclusions connecting the forms of quadratic expressions. By using each of the pairs students express regularity in repeated reasoning (MP8).

When students articulate their reasoning, they have an opportunity to attend to precision in the language they use to describe their thinking (MP6). They might first propose less formal or imprecise language, and after sharing with a partner, revise their explanation to be clearer and stronger.

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 responses in the table. Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to what they noticed about the terms in each 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?”

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

After Stronger and Clearer Each Time, select students to share the things they notice. Ask students,

• “Using the patterns you noticed, can you write the standard form for the quadratic expression quickly?” (Yes, the standard form is .)
• “Do the patterns you noticed work for other factored forms like ?” (No, the patterns work only for factored forms that can be written as perfect squares.)