Lesson 5

The purpose of this Math Talk is to elicit strategies and understandings students have for distributing pairs of binomials. These understandings help students develop fluency and will be helpful later in this lesson when students will need to be able to complete the square to find the center and radius of a circle.

In this activity, students have an opportunity to notice and make use of structure (MP7) as they recall area models or other ways to represent the distributive property.

Remind students that in a previous activity, they found that a circle can be defined by an equation in the form \((x-h)^2+(y-k)^2=r^2\) where \((h,k)\) is the center of the circle and \(r\) is the length of the circle’s radius. Tell them that in the next several activities, they’ll use algebra to work with circles in the coordinate plane.

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?”

Students practice squaring binomials. Then, they look for patterns in their results to help identify and rewrite perfect square trinomials.

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, \((x+4)(x+4)\), 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?

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 \(x\), then the expression is a perfect square trinomial. For the expression \(x^2+10x+20\), 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 \(x^2-8x+16\) as \((x-8)^2\). How could this student check their work? (One method is to distribute the expression \((x-8)^2\) to see if it matches the original expression.)

This activity builds toward completing the square. Students rewrite an equation of the form \((x-h)^2 + (y-k)^2 = r^2\), putting it in the form \(x^2 + y^2 + ax +by + c = 0\). Then, they’re presented with an equation for a circle in which the squared binomials have been expanded. Students rewrite 2 perfect square trinomials in factored form, then identify the center and radius of the circle.

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 \((x-h)^2+(y-k)^2=r^2\). Remind them that we subtract the coordinates of the center from the given point \((x,y)\) 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: \(x^2+6x+y^2-10y-30=0\). Display these 3 forms of this equation for all to see, emphasizing that these are all equivalent equations and therefore represent the same circle:

\((x+3)^2+(y-5)^2=8^2\)

\(x^2+6x+9+y^2-10y+25=64\)

\(x^2+6x+y^2-10y-30=0\)

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 \(x^2\), \(6x\), \(y^2\), and \(\text-10y\), 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.)