Pythagorean Theorem and Volume

Lesson 3

Math

Unit 7

8th Grade

Lesson 3 of 16

Objective


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

Common Core Standards


Core Standards

  • 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

  • 6.NS.C.6

Criteria for Success


  1. Understand that irrational numbers have approximate locations on the number line.
  2. Approximate locations of irrational and rational numbers on the number line.
  3. Estimate the values of irrational expressions.
  4. Compare values of irrational numbers. 
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Anchor Problems


Problem 1

Ten points, A through J, are shown on the number line below. 

Which point on the number line best approximates the location of each value below?

a.   $$\pi$$

b.   $$\sqrt{3}$$

c.   $$\sqrt{9}$$

d.   $$\sqrt{50}$$

e.   $$-\sqrt{10}$$

f.   $$-\sqrt{5}$$

Guiding Questions

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Problem 2

Estimate the value of each irrational expression below. Then plot a point to approximate each location on the number line. 

a.   $${3\pi}$$

b.   $${\sqrt{50}+1}$$

c.   $${\sqrt{140}\over2}$$

d.   $$2\sqrt{10}$$

Guiding Questions

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Problem 3

For each pair of numbers, decide which number is greater without using a calculator. Explain your choices. 

a.   $${\pi^2 }$$ or $$9$$

b.   $${{\sqrt{50}}}$$  or $${\sqrt{51}}$$

c.   $${{\sqrt{50}}}$$ or $$8$$

d.   $${-2\pi}$$ or $${-6}$$

Guiding Questions

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References

Illustrative Mathematics Comparing Rational and Irrational Numbers

Comparing Rational and Irrational Numbers, accessed on July 20, 2016, 8:46 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

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Target Task


Problem 1

Are the two expressions below equivalent? Explain why or why not. 

$${\sqrt{10}\over5}$$    and   $${\sqrt{20}\over10}$$

Problem 2

Place the numbers in their approximate locations on the number line below: $$\sqrt 9$$$$\sqrt{15}$$$${\sqrt {49}}\over 2$$$$\sqrt 6$$

References

EngageNY Mathematics Grade 8 Mathematics > Module 7 > Topic A > Lesson 2Exit Ticket, Question #4

Grade 8 Mathematics > Module 7 > Topic A > Lesson 2 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

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

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Lesson 4

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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