Right Triangles and Trigonometry

Lesson 4

Objective

Multiply and divide radicals. Rationalize the denominator.

Common Core Standards

Core Standards

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  • A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x4 — y4 as (x²)² — (y²)², thus recognizing it as a difference of squares that can be factored as (x² — y²)(x² + y²).

  • N.RN.A.2 — Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Foundational Standards

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  • 8.EE.A.2

Criteria for Success

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  1. Multiply and divide radicals by following properties of radicals. For example: $${3\sqrt{7}\cdot2\sqrt{5}=\left(2\cdot3\sqrt{(7\cdot5)}\right)}$$
  2. Describe that radicals follow the same rules as exponents with power of a power and power of a quotient. For example: $${\sqrt{\left(\frac{2}{3}\right)}=\frac{\sqrt{2}}{\sqrt{3}}}$$
  3. Rationalize the denominator in a radical expression when there is a radical term in the denominator in algebraic expressions. 
  4. Identify when it is proper to "rationalize the denominator."
  5. Rationalize the denominator in a radical expression when there is a radical term in the denominator in geometric problems with special right triangles. 
  6. Calculate the area of right triangles.

Tips for Teachers

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  • This lesson extends work done in Algebra 1. The properties of radicals should be familiar to students but will need some review. 
  • The focus of this lesson is on working with numeric radical expressions, but students should practice with algebraic radical expressions as well.

Anchor Problems

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Problem 1

Simplify the following:

$${\sqrt{32}}$$

$${\sqrt{5^{10}}}$$

$${\sqrt{4x^4}}$$

Guiding Questions

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References

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 22Exercise 2 and 3

Geometry > Module 2 > Topic D > Lesson 22 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..

Modified by The Match Foundation, Inc.

Problem 2

Each of these statements are TRUE for some values. Identify the excluded values, then describe what the statement says about the property. Feel free to use an example. 

Statement 1:   $${\sqrt{ab}=\sqrt{a}\cdot\sqrt{b}}$$

Statement 2:   $${\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}=\frac{\sqrt{ab}}{b}}$$

Statement 3:   $${c\sqrt{a}\cdot d\sqrt{b}=cd\sqrt{ab}}$$

 

Guiding Questions

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Problem 3

Find the area of the triangle below.

Guiding Questions

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Problem Set

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

  • Include problems where students need to identify the form of expression that is most useful given the goal of the problem. 
  • Include error analysis problems, such as “What’s the mistake? $${\frac{\sqrt{2a}}{2}=a}$$” or with a specific example “What is the mistake? $${\frac{\sqrt{6}}{2}=\sqrt{3}}$$.”
  • Include problems where there are variable expressions in the radicand. 
  • Include problems where students need to find a missing measurement of a right triangle, including using special right triangles. 
  • Include problems where one of the sides of a right triangle is given in radical form and students need to find the area of the triangle, including using special right triangles, similar to Anchor Problem #3. 

Target Task

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Problem 1

What is the value of $$x$$ that will make the following equation true? Explain your reasoning.

$$\sqrt{5}\cdot3\sqrt{x}=15\sqrt{6}$$

Problem 2

Write each expression in its simplest radical form.

$${\sqrt{243}=}$$

 

$${\sqrt\frac{7}{5}=}$$

References

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 22Exit Ticket, Question #1 and #2

Geometry > Module 2 > Topic D > Lesson 22 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..