Completing the Square (Part 2)

5 min

Narrative

This Math Talk focuses on operations with fractions to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students solve quadratic equations involving fractions.

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

Launch

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.

Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

Student Task

Solve each equation mentally.

  • x+x=14x+x=\frac14
  • (32)2=x(\frac32) ^2 = x
  • 35+x=95\frac35 + x = \frac95
  • 112+x=14\frac{1}{12}+x=\frac14

Sample Response

  • 18\frac{1}{8}. Sample reasoning: 2x=142x = \frac{1}{4}, so x=12⋅14=18x = \frac{1}{2} \boldcdot \frac{1}{4} = \frac{1}{8}.
  • 94\frac{9}{4}. Sample reasoning: Squaring a fraction is equivalent to squaring the numerator and denominator separately.
  • 65\frac{6}{5}. Sample reasoning: x=95−35x = \frac{9}{5} - \frac{3}{5}
  • 212\frac{2}{12} (or equivalent). Sample reasoning: x=14−112=312−112x = \frac{1}{4} - \frac{1}{12} = \frac{3}{12} - \frac{1}{12}
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
Addressing
  • A-REI.A·Understand solving equations as a process of reasoning and explain the reasoning
  • HSA-REI.A·Understand solving equations as a process of reasoning and explain the reasoning.

10 min

20 min