Lesson 4 | Right Triangles and Trigonometry | 10th Grade Mathematics

Objective

Multiply and divide radicals. Rationalize the denominator.

Common Core Standards

Core Standards

The core standards covered in this lesson

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).

Seeing Structure in Expressions

A.SSE.A.2 — Use the structure of an expression to identify ways to rewrite it. For example, see x<sup>4</sup> — y<sup>4</sup> 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.

The Real Number System

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

Foundational Standards

The foundational standards covered in this lesson

8.EE.A.2

Expressions and Equations

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.

Criteria for Success

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

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

Tips for Teachers

Suggestions for teachers to help them teach this lesson

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.

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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

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 22 — Exercise 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 Fishtank Learning, 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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Target Task

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

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 22 — Exit Ticket, Question #1 and #2

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.

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.

EngageNY Mathematics Geometry > Module 2 > Topic D > Lesson 22
MARS Formative Assessment Lessons for High School Evaluating Statements about Radicals — Use the problems that focus on multiplication or division of radicals

Lesson 3

Lesson 5

Lesson Map

Topic A: Right Triangle Properties and Side-Length Relationships

Define the parts of a right triangle and describe the properties of an altitude of a right triangle.

G.CO.A.1 G.SRT.B.4

Define and prove the Pythagorean theorem. Use the Pythagorean theorem and its converse in the solution of problems.

G.SRT.B.4

Define the relationship between side lengths of special right triangles.

G.SRT.B.4 G.SRT.B.5

Multiply and divide radicals. Rationalize the denominator.

A.SSE.A.2 N.RN.A.2

Add and subtract radicals.

A.SSE.A.2 N.RN.A.2

Topic B: Right Triangle Trigonometry

Define and calculate the sine of angles in right triangles. Use similarity criteria to generalize the definition of sine to all angles of the same measure.

G.SRT.C.6

Define and calculate the cosine of angles in right triangles. Use similarity criteria to generalize the definition of cosine to all angles of the same measure.

G.SRT.C.6

Derive the relationship between sine and cosine of complementary angles in right triangles, and describe sine and cosine as angle measures approach 0°, 30°, 45°, 60°, and 90°.

G.SRT.C.7

Describe and calculate tangent in right triangles. Describe how the value of tangent changes as the angle measure approaches 0°, 45°, and 90°.

G.SRT.C.6 G.SRT.C.7

Solve for missing sides of a right triangle given the length of one side and measure of one angle.

G.SRT.C.8

Topic C: Applications of Right Triangle Trigonometry

Find the angle measure given two sides using inverse trigonometric functions.

G.SRT.C.8

Describe the relationship between slope and the tangent ratio of the angle of elevation/depression. Use the tangent ratio of the angle of elevation or depression to solve real-world problems.

G.SRT.C.8

Solve a modeling problem using trigonometry.

G.SRT.C.8

Topic D: The Unit Circle

Define angles in standard position and use them to build the first quadrant of the unit circle.

F.TF.A.2

Use the first quadrant of the unit circle to define sine, cosine, and tangent values outside the first quadrant.

F.TF.A.2

Topic E: Trigonometric Ratios in Non-Right Triangles

Derive the area formula for any triangle in terms of sine.

G.SRT.D.9

Verify algebraically and find missing measures using the Law of Sines.

G.SRT.D.10

Verify algebraically and find missing measures using the Law of Cosines.

G.SRT.D.10

Use side and angle relationships in right and non-right triangles to solve application problems.

G.SRT.D.11

Right Triangles and Trigonometry

Objective

Common Core Standards

Core Standards

Seeing Structure in Expressions

The Real Number System

Foundational Standards

Expressions and Equations

Criteria for Success

Tips for Teachers

Anchor Problems

Problem 1

Guiding Questions

References

Problem 2

Guiding Questions

Problem 3

Guiding Questions

Target Task

Problem 1

Problem 2

References

Additional Practice

Lesson Map

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