Students may distribute the exponent across the addition inside the parentheses in the first part of the task. Ask them what it means to “square” something. Suggest that they write out the two factors, for example,  , and then apply the distributive property.

If students aren’t sure of the definition of a trinomial, remind them that a binomial is an expression that has 2 terms. How might that relate to a trinomial?

The purpose of this discussion is for students to clarify the characteristics of a perfect square trinomial and how to rewrite such an expression as a binomial squared. 

Use Critique, Correct, Clarify to give students an opportunity to improve a sample written response to how to square a binomial by correcting errors, clarifying meaning, and adding details. 

• Display this first draft: “If I apply the distributive property to the expression , then I get , because each term inside the parentheses is squared.”
  Ask, “What parts of this response 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. Listen for students who clarify the meaning of the distributive property and amplify the language that students use to explain why expands to . This will help students evaluate and improve on the written mathematical arguments of others.

• 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.

Ask students to share their strategies for determining which trinomials were perfect squares, and for rewriting those expressions. For example, students may say that if the constant term is the square of half the coefficient of , then the expression is a perfect square trinomial. For the expression , ask students what could be changed in order for it to be a perfect square trinomial (for example, if the final term were 5² or 25, it would be a perfect square trinomial).

Invite students to imagine they were looking at the work of a peer who incorrectly rewrote as . How could this student check the work? (One method is to distribute the expression to see if it matches the original expression.)

Students may struggle to decide whether the coordinates of the circles’ centers are positive or negative. Encourage them to rewrite the equation in the form . Remind them that we subtract the coordinates of the center from the given point to get the distance between the center and the point.

Ask students to rearrange the circle equation from the second problem so that there is a 0 on one side of the equation: . Display these three forms of this equation, emphasizing that these are all equivalent equations and, therefore, represent the same circle:

The purpose of the discussion is to make connections between different forms of the equation in preparation for completing the square. Ask students:

• “In which form is it easiest to find the center and radius of the circle?” (The first one.)
• “Compare and contrast the second and third forms.” (Each form contains the terms , , , and , but the constant terms are different.)
• “How can you go from the first form to the second one?” (Distribute the two sets of squared binomials.)
• “How can you go from the second form to the third one?” (Combine like terms.)
• “How can you go from the second form to the first one?” (Rewrite the perfect square trinomials as squared binomials, and rewrite 64 as 8².)