Lesson 2 | Pythagorean Theorem and Volume | 8th Grade Mathematics

Objective

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

Common Core Standards

Core Standards

The core standards covered in this lesson

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.

The Number System

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.

The Number System

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

The foundational standards covered in this lesson

7.NS.A.2.D

The Number System

7.NS.A.2.D — Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.
7.NS.A.3

The Number System

7.NS.A.3 — Solve real-world and mathematical problems involving the four operations with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions.

Criteria for Success

The essential concepts students need to demonstrate or understand to achieve the lesson objective

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.
Investigate the solution of $${x^2=2}$$ and informally conclude that $${\sqrt{2}}$$ is an irrational number.
Approximate the value of irrational square roots to given decimal values.

Tips for Teachers

Suggestions for teachers to help them teach this lesson

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

Problems designed to teach key points of the lesson and guiding questions to help draw out student understanding

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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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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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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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

A set of suggested resources or problem types that teachers can turn into a problem set

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Target Task

A task that represents the peak thinking of the lesson - mastery will indicate whether or not objective was achieved

Problem 1

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

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Problem 2

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

Student Response

An example response to the Target Task at the level of detail expected of the students.

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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.

Open Middle Decimal Approximations of Roots — Challenge
Smarter Balanced Assessment Consortium: Item and Task Specifications MAT.08.ER.3.000NS.A.136
Illustrative Mathematics Calculating the Square Root of 2
Open Middle Rational and Irrational Roots
Open Middle Rational and Irrational Numbers
EngageNY Mathematics Grade 8 Mathematics > Module 7 > Topic B > Lesson 11 — Problem Set
Illustrative Mathematics Estimating Square Roots

Lesson 1

Lesson 3

Lesson Map

Topic A: Irrational Numbers and Square Roots

Define, evaluate, and estimate square roots.

8.EE.A.2

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

8.NS.A.1 8.NS.A.2

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

8.NS.A.2

Represent rational numbers as decimal expansions.

8.NS.A.1

Represent decimal expansions as rational numbers in fraction form.

8.NS.A.1

Topic B: Understanding and Applying the Pythagorean Theorem

Understand the Pythagorean Theorem as a relationship between the side lengths in a right triangle.

8.G.B.6

Understand a proof of the Pythagorean Theorem.

8.G.B.6

Use the converse of the Pythagorean Theorem to determine if a triangle is a right triangle.

8.G.B.6

Find missing side lengths involving right triangles and apply to area and perimeter problems.

8.G.B.7

Solve real-world and mathematical problems using the Pythagorean Theorem (Part I).

8.G.B.7

Solve real-world and mathematical problems using the Pythagorean Theorem (Part II).

8.G.B.7

Find the distance between points in the coordinate plane using the Pythagorean Theorem.

8.G.B.8

Topic C: Volume and Cube Roots

Define and evaluate cube roots. Solve equations in the form $${x^2=p}$$ and $${x^3=p}$$.

8.EE.A.2 8.NS.A.2

Solve real-world and mathematical problems involving the volume of cylinders and cones.

8.G.C.9

Solve real-world and mathematical problems involving the volume of spheres.

8.G.C.9

Solve real-world problems involving multiple three-dimensional shapes, in particular, cylinders, cones, and spheres.

8.G.C.9

Pythagorean Theorem and Volume

Objective

Common Core Standards

Core Standards

The Number System

The Number System

Foundational Standards

The Number System

The Number System

Criteria for Success

Tips for Teachers

Lesson Materials

Anchor Problems

Problem 1

Guiding Questions

Problem 2

Guiding Questions

Problem 3

Guiding Questions

References

Problem Set

Target Task

Problem 1

Problem 2

Student Response

Additional Practice

Lesson Map

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