Rewriting Quadratic Expressions in Factored Form (Part 3)

10 min

Narrative

This Math Talk focuses on recalling strategies for multiplying. It encourages students to use the structure of the expressions to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students look at quadratic expressions that can be written as (a+b)(a−b)(a+b)(a-b).

To multiply the values, students need to look for and make use of structure (MP7).

Launch

Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:

  • Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
  • Invite students to share their strategies, and record and display their responses for all to see.
  • Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.

Keep all previous problems and work displayed throughout the talk.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Task

Evaluate mentally.

  • 9⋅119 \boldcdot 11
  • 19⋅2119 \boldcdot 21
  • 99⋅10199 \boldcdot 101
  • 119⋅121119\boldcdot121

Sample Response

  • 99. Sample reasoning: 9⋅10=909 \boldcdot 10 = 90 and 9⋅1=99 \boldcdot 1 = 9, so 9(10+1)=90+9=999(10 + 1) = 90 + 9 = 99.
  • 399. Sample reasoning: 20⋅21=42020 \boldcdot 21 = 420 and (20−1)⋅21=20⋅21−21=420−21=399(20-1)\boldcdot 21 = 20 \boldcdot 21 - 21 = 420 - 21 = 399.
  • 9,999. Sample reasoning: (100−1)(100+1)=1002−100+100−1=10,000−1=9,999(100 - 1)(100+1) = 100^2 - 100 + 100 - 1 = 10,000 - 1 = 9,999.
  • 14,399. Sample reasoning: (120−1)(120+1)=1202−120+120−1=14,400−1=14,399(120-1)(120+1) = 120^2 - 120 + 120 - 1 = 14,400 - 1 = 14,399.
Activity Synthesis (Teacher Notes)

To involve more students in the conversation, consider asking:

  • “Who can restate ‾\underline{\hspace{.5in}}’s reasoning in a different way?”
  • “Did anyone use 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?”
  • “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I ‾\underline{\hspace{.5in}} because . . . .” or “I noticed ‾\underline{\hspace{.5in}}, so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Standards
Building On
  • 6.EE.3·Apply the properties of operations to generate equivalent expressions. <em>For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.</em>
  • 6.EE.A.3·Apply the properties of operations to generate equivalent expressions. <span>For example, apply the distributive property to the expression <span class="math">\(3 (2 + x)\)</span> to produce the equivalent expression <span class="math">\(6 + 3x\)</span>; apply the distributive property to the expression <span class="math">\(24x + 18y\)</span> to produce the equivalent expression <span class="math">\(6 (4x + 3y)\)</span>; apply properties of operations to <span class="math">\(y + y + y\)</span> to produce the equivalent expression <span class="math">\(3y\)</span>.</span>

15 min

10 min