Use Stronger and Clearer Each Time to give students an opportunity to revise and refine their response to the last question. 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 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, invite students to share their responses. Discuss the connection between the graphs and the solutions to the equations. The important idea is that the operation of taking the square root results in a positive number, but there are two square roots of every non-zero number, one positive and one negative. This means that when Tyler takes the square root of each side of the equation, he does not get an equivalent equation. To find all possible solutions, he should remember that has two roots, and , and that 4 has two square roots, 2 and -2. This gives him the pair of equations and .

If students have trouble getting started consider saying:

• “Can you explain how you would solve .”

• “How could substituting values for help you find a number that, when squared, equals 4?”

Invite students to share how they solved the equation. Record student strategies and display them for all to see throughout the discussion.

Then display two alternate strategies for solving the equation. First, a graphic solution.

Show students this graph of the solutions to :

Ask, “How can we see the solutions to in this graph?” (The lines cross at and , so those are the solutions.)

Then a partial algebraic solution.

Tell students that this is a way of solving the problem that results in only one of the solutions. Ask, “Why doesn’t this method give us both solutions?” (Taking the square root of each side gives us only the positive square root.) Then invite students to share their advice for solving equations like this one. (Instead of taking the square root, set up two equations:  and .)

Invite previously selected groups to share their reasoning for the last question. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.
Connect the different responses to the learning goals by asking questions, such as:

• “Which strategy is quicker to recognize from Tyler’s equation?” (Knowing that the symbol represents only non-negative values lets me know right away that there are no solutions.)
• “Which method is better for explaining to someone who might not understand?” (Using the visual of a graph is helpful to show someone who doesn’t understand.)

Ask students to share their responses and reasoning. After each response, invite any students who thought of the problem in a different way to share their reasoning. Remind students that squaring each side of an equation sometimes results in a new equation that has solutions that the old equation doesn’t have.