Curriculum  /  Math  /  8th Grade  /  Unit 7: Pythagorean Theorem and Volume  /  Lesson 2

Pythagorean Theorem and Volume

Lesson 2

Math

Unit 7

8th Grade

Lesson 2 of 16

Objective

Understand that some numbers, including $${\sqrt{2}}$$, are irrational. Approximate the value of irrational numbers.

Common Core Standards

Core Standards

  • 8.NS.A.1 — Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.
  • 8.NS.A.2 — Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Foundational Standards

  • 7.NS.A.2.D
  • 7.NS.A.3

Criteria for Success

  1. Understand that a rational number is any number that can be represented as a fraction with a non-zero denominator, and an irrational number is any number that is not rational. Rational numbers can be written as decimals that terminate or eventually repeat; irrational numbers are represented as decimals that neither terminate nor repeat. 
  2. Investigate the solution of $${x^2=2}$$  and informally conclude that $${\sqrt{2}}$$ is an irrational number. 
  3. Approximate the value of irrational square roots to given decimal values. 

Tips for Teachers

  • In this lesson, students reason quantitatively about the value of $$\sqrt{2}$$, leading them to understand that some values do not have an exact decimal representation (MP.2). They consider other values around the value of an irrational number to make sense of its numerical value and quantity.
  • Note, if students are using calculators in eighth grade, they should get practice with using scientific calculators.

Lesson Materials

  • Scientific calculator (1 per student)
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Anchor Problems

25-30 minutes

Problem 1

Investigate the value of $${{\sqrt{2}}}$$.

a.   Between what two whole numbers is $${{\sqrt{2}}}$$?

b.   Can you find a value for $$x$$, in fraction or decimal form, that satisfies the equation $$x^2=2$$? If you cannot find an exact value, what is the closest value you can find?

Guiding Questions

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Student Response

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Problem 2

Approximate the value of $${\sqrt{32}}$$ to the nearest hundredth. 

Use the table below as a possible strategy to help you organize your work.

Low value (Low value)$$^2$$ (High value)$$^2$$ High value
       
       
       

Guiding Questions

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Student Response

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Problem 3

Decide whether each of the following numbers is rational or irrational. Explain your reasoning for each one. 

a.   $${0.33\bar{3}}$$

b.   $${\sqrt{4}}$$

c.   $${\sqrt{2}={1.414213}…}$$

d.   $${1.414213}$$

e.   $${\pi=3.141592…}$$

f.   $${11}$$

g.   $${{1\over7}=0.\overline{142857}}$$

h.   $${12.34565656\overline{56}}$$

Guiding Questions

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Student Response

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References

Illustrative Mathematics Identifying Rational Numbers

Identifying Rational Numbers, accessed on July 20, 2016, 3:34 p.m., is licensed by Illustrative Mathematics under either the CC BY 4.0 or CC BY-NC-SA 4.0. For further information, contact Illustrative Mathematics.

Problem Set

15-20 minutes

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

Target Task

5-10 minutes

Problem 1

Which of the following are irrational numbers? Select all that apply.

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Student Response

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Problem 2

Approximate the value of $${\sqrt{11}}$$ to the nearest tenth. 

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.

Next

Locate irrational values approximately on a number line. Compare values of irrational numbers.

Lesson 3
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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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