Applications of the Pythagorean Theorem

5 min

Narrative

This Math Talk focuses on estimating the decimal value of each square root expression. It encourages students to think about what the square root symbol means and to rely on what they know about perfect squares to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students apply the Pythagorean Theorem.

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

Mentally find the value of each expression to the nearest half.

  • 24\sqrt{24}
  • 7\sqrt{7}
  • 42\sqrt{42}
  • 10+97\sqrt{10}+\sqrt{97}

Sample Response

  • 5. Sample reasoning: The square root of 25 is 5, and 24 is very close to 25. 
  • 2.5. Sample reasoning: 7 is just a little over halfway between 222^2 and 323^2.
  • 6.5. Sample reasoning: 42 is almost halfway between 626^2 and 727^2.
  • 13. Sample reasoning: The square root of 10 is a little more than 3, and the square root of 97 is a little less than 10. 

Synthesis

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.
  • 8.NS.A·Know that there are numbers that are not rational, and approximate them by rational numbers.
  • 8.NS.A·Know that there are numbers that are not rational, and approximate them by rational numbers.

15 min

15 min