Reasoning about Square Roots

5 min

Teacher Prep
Setup
Display one problem at a time. Students give a signal when they have an answer and a strategy. After each problem, give students 1 minute of quiet think time and follow with a whole-class discussion.

Narrative

This Math Talk focuses on analyzing symbolic statements about square roots. It encourages students to think about the meaning of the square root symbol and to rely on what they know about the relationship between squares and square roots to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students plot solutions to equations of the form x2=nx^2=n on the number line (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.

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

Decide mentally whether or not each statement is true.

  • (5)2=5\left( \sqrt{5} \right)^2=5
  • (9)2=3\left(\sqrt{9}\right)^2 = 3
  • (10)2=100\left(\sqrt{10}\right)^2 = 100
  • (16)=22\left(\sqrt{16}\right)= 2^2

Sample Response

  • True. Sample reasoning: 5⋅5=5\sqrt{5}\boldcdot\sqrt5=5
  • False. Sample reasoning: 9⋅9=9\sqrt{9}\boldcdot\sqrt9=9, not 3
  • False. Sample reasoning: 10⋅10=10\sqrt{10}\boldcdot\sqrt{10}=10, not 100
  • True. Sample reasoning: 16=4\sqrt{16}=4 and 22=42^2=4
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
  • 8.EE.2·Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.
  • 8.EE.A.2·Use square root and cube root symbols to represent solutions to equations of the form <span class="math">\(x^2 = p\)</span> and <span class="math">\(x^3 = p\)</span>, where <span class="math">\(p\)</span> is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that <span class="math">\(\sqrt{2}\)</span> is irrational.

15 min

10 min