Pythagorean Theorem and Volume

Lesson 1

Math

Unit 7

8th Grade

Lesson 1 of 16

Objective


Define, evaluate, and estimate square roots.

Common Core Standards


Core Standards

  • 8.EE.A.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.

Foundational Standards

  • 6.EE.B.5
  • 7.NS.A.3

Criteria for Success


  1. Understand the square root of a positive number, $$p$$, is written as $$\sqrt{p}$$ and represents the number that when squared is equal to $$p$$, such that $$\left( \sqrt{p} \right)^2=p$$.
  2. Understand the side length of a square with an area $$p$$ is equal to $$\sqrt{p}$$.
  3. Evaluate square roots of perfect squares.
  4. Estimate square roots of non-perfect squares to whole number values. 

Tips for Teachers


  • It will be helpful for students to know the square values of integers 1 through 20 in order to quickly identify and evaluate square roots of perfect squares. Students could create a list of these values and use it as a reference throughout the lesson and unit. 
  • Throughout the unit students may use scientific calculators as needed.
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Anchor Problems


Problem 1

Three squares are shown below with their area in square units. 

a.   What is the side length of each square?

b.   Describe the relationship between the side length of a square and its area.

c.   Another square has an area of $${{{25}}}$$ square units. Write an equation to show the relationship between the side length, $$s$$, of this square and its area.

d.   This equation can be solved by asking, “What number, when squared, equals $${{{25}}}$$?” Another way of saying this is, “What is the square root of $${{{25}}}$$?” Use this information to write two ways to represent the solution to the equation.

Guiding Questions

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Problem 2

The square below has an area of $$8$$ square units. 

a.   Estimate the side length of the square. Between what two whole numbers will the side length be? Explain your reasoning.

b.   Write an equation to represent the relationship between the side length, $$s$$, of this square and the area.

c.   What is the side length of the square?

Guiding Questions

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Problem 3

a.   Find the side length of a square with area:

  1. $${196 \space \mathrm{u}^2}$$
  2. $${81 \space \mathrm{u}^2}$$
  3. $${169 \space \mathrm{u}^2}$$
  4. $${{25\over{49}} \space \mathrm{u}^2}$$
  5. $${0.64 \space \mathrm{u}^2}$$

b.   What is the side length of a square with area $${50 \space \mathrm{u}^2}$$? Between what two whole numbers is the side length of the square?

Guiding Questions

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Problem Set

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Give your students more opportunities to practice the skills in this lesson with a downloadable problem set aligned to the daily objective.

Target Task


The square below has an area of $${20}$$ square units. 

Taylor writes the equation $$s^2={20}$$  to find the measure of the side length of the square. She reasons that the solution to the equation is $$\sqrt{20}$$ and concludes that the side length of the square is $${10}$$ units.

Do you agree with Taylor? Explain why or why not. If you disagree, include the correct side length of the square.

Student Response

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Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

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Lesson 2

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Irrational Numbers and Square Roots

Topic B: Understanding and Applying the Pythagorean Theorem

Topic C: Volume and Cube Roots

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