Right Triangles and Trigonometry
Lesson 4
Objective
Multiply and divide radicals. Rationalize the denominator.
Common Core Standards
Core Standards
?
-
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
?
-
8.EE.A.2
Criteria for Success
?
- 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
?
- 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
?
Problem 1
Simplify the following:
$${\sqrt{32}}$$
$${\sqrt{5^{10}}}$$
$${\sqrt{4x^4}}$$
Guiding Questions
References
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
Problem 3
Find the area of the triangle below.
Guiding Questions
Problem Set
?
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.
-
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
Target Task
?
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
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..