Lesson Plan -- Math 097 Grade Level:

9-10 Title: Section 6.3

Author: Kyle Linford

Enduring Understanding:

Students will gain an understanding of how to apply the multiplicative property of radicals to simplify radical expressions.

Objectives:

SWBAT:

Apply the multiplicative property of radicals to simplify expressions

Rewrite terms as the product of factors

Draw connections between the multiplicative property of radicals and the other properties of exponents

Apply the order of operations to simplify radical expressions

Content Standard(s): CCSS.MATH.CONTENT.HSN.RN.A.1—Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notion for radicals in terms of rational exponents. CCSS.MATH.CONTENT.8.EE.A.1—Know and apply the properties of integer exponents to generate

equivalent numerical expressions. For example, 32 ∙ 3−5 = 3−3 = (1

3)3

=1

27.

Resources: Miller, Julie, O’Neil, and Hyde. MTH 097 and MTH 169. McGraw Hill, n.d. Print.

Vocabulary: Multiplicative Property of Radicals

Materials: N/A

Assessment Strategies FORMATIVE: Solving in class problems SUMMATIVE: Completing the exit ticket and the independent practice

Differentiation: Ability:

Reviewing the calculations of the nth roots helps struggling learners refresh their memories, draw connections to prior knowledge, and limit the amount of prerequisite knowledge.

Allowing students to work with others on the independent practice to answer questions allows for struggling learners to have better support with the content.

Having students work at their own pace helps struggling learners take their time with problems.

Providing an answer key helps all students check their work and analyze where they made mistakes.

Making connections helps all learners build their connected thinking about the topic and refer back to prior knowledge.

Providing two methods for viewing the process for simplifying radical expressions by the multiplication property helps all learners better understand how to simplify the expressions and why the properties work.

Intelligences:

The class discussions and teacher commentary in the Build section and the Guided Practice help

auditory learners.

The exploration of the connections between the multiplicative property and the other properties of exponents helps visual learners better interpret how the exponential process works by writing out the work in expanded form.

The use of the board and expanded form to show the work helps visual learners.

Allowing students to work with others on the independent practice helps interpersonal learners share and develop their ideas for how to solve problems.

Allowing students the opportunity to work alone on the independent practice helps intrapersonal learners focus individually on their work.

Instructional Activities & Strategies

ANTICIPATORY SET: (5 minutes)

Bell Work: When students walk into the classroom, they will be instructed to follow the task designated

on the board. The task will be for them to simplify exponential expressions and exponents in expanded

form.

The instructor will launch the lesson by having students recall their understanding of exponents in expanded form and how to utilize the rules for the exponents.

BUILD: (10 minutes)

Input:

The instructor will first present students with the problem √4. Students are expected to recall that this problem requires them to find the square root of the number 4. Ask: What is this problem asking you to do? The instructor will then define the notation and components of radicals for students. Ask: How are square roots related to exponents? Students will be guided to observing that the square root operation is essentially an inverse of exponents: they are to find what numbers when multiplied together get you 4. The instructor will then present students with the exponent notation for roots and explain how to interpret them. Ask: Can I take different roots, like cube roots? What if I wanted to find 4

th roots where a numbered is

multiplied it by itself 4 times? If I can do it with exponents, does it make sense that I can do the reverse process? Students will be guided to observing that they can, and the instructor will define this as the index of the radical expression and show how the notation fits in. The instructor will then have the class produce the answer of 2 and -2. If students are unable to produce the negative answer, the instructor will guide them to observing that any number multiplied by itself is a positive number. The instructor will then have students observe that these square root operations have 2 solutions. Ask: What does this remind us of when we have a positive and a negative solution for the problem? Students will be guided to connecting absolute value equations with this concept.

Ask: Will we ever only have 1 answer? What situation does not have both a positive and a negative solution? Students will be guided to observing that the square root of 0 has only one solution.

Ask: What about √−4? The class will be guided to articulating that there is no real number solution for a negative radicand. The instructor will make sure to state this under the real number system to be mathematically accurate. The instructor will then segue into exploring the values of the radicands. The instructor will inform students that these values are perfect squares. The instructor will define perfect squares for students as natural numbers that have integer square roots. The class will then find some perfect squares.

The instructor will first present students with the problem √32. Ask: Is this number a perfect square? What value to you get if you type it into your calculator? The class will observe that the answer produces a long decimal. Ask: This number will go on forever and never repeat a pattern. What do we call these numbers? Students will be guided to recalling that these values are irrational numbers.

Ask: Will we ever find the exact value of √32? The class will be guided to observing that they cannot; however, the instructor will inform them that they can simplify it to get a better understanding of its value. The instructor will inform the class that a property allows them to simplify radical expressions—the multiplication property of radicals. The instructor will define this property and have the class explore why it works. After having written the property, the instructor will rewrite the property using exponents for the square roots. The class will then be asked to analyze what property this relates to. Ask: Does this property look familiar to us? Students will be guided to observing that it is essentially the power rule. The instructor will utilize a concrete example of the power rule to remind students if they are not sure. Ask: What would be our goal then for working with a number we cannot take the square root of? How can I utilize this property to help me simplify my answer? Students will be guided to observing that they want to break the number down into factors. Ask: How do we break 32 down into factors, and what numbers do we break it down into? Modeling: The instructor will present the class with exploring how numbers can be written as the product of their factors. Using 4 as an example, the instructor will produce the factor tree breakdown for the number. The students will then be shown that 4 is the product of all the prime factors for the number, and the instructor will substitute them into the radical sign to replace the radicand. The instructor will then recall students’ understanding of square roots.

Ask: What do we want to do when we take the square root of something? What does it ask us to find? Students will be guided to observing that they want to find what two numbers when multiplied together get 4. The instructor will show that since there is a pair of 2’s, meaning 2 times 2, they can say that the value is 2. The instructor will then utilize 32 as an example. Ask: What is the prime factor decomposition of 32? Utilizing a tree diagram, the instructor will show the breakdown of 32 and substitute it into the radical sign as the radicand. The students will then take pairs of numbers and pull them outside of the radical. When all pairs have been removed, the class will then multiply everything out to show the simplified form.

The instructor will then have the students practice this idea with √8, √48, and √50. The instructor will then show the property with variables by writing them in expanded form. The students can then make pairs to be simplified. Students will then be asked if they can make their work any easier to simplify. Students will be guided to

observing that the expanded form produces values that are perfect squares. The problem of √32 will be

shown again but with the expanded form showing √4 ∙ 4 ∙ 2 or √16 ∙ 2. Students will then be asked to apply the multiplication property of radicals to simplify this work. Check for Understanding: While students are working on simplifying the radical terms, the instructor will monitor their work and check for any misconceptions. To check for student understanding, the instructor will utilize quick, formative feedback by questioning students about their comfort level with this topic by indicating with a thumb up, down, or in the middle when the topic has been explained.

GUIDED PRACTICE: (15 minutes)

To help students develop their understanding of simplifying radical expressions and the properties of

exponents, the instructor will have the class work through simplifying an expression that combines both

numerical and variable terms: √18𝑥3. The instructor will first substitute the prime factor decomposition

and the expanded form of the terms into the radicand. Students will be guided to identifying what

perfect square they can break 18 down into to simplify the radical. When students observe that 9 is a

perfect square they can use, the instructor will rewrite this expression as 9 and 2. They will then be

guided to simplifying the expression.

Students will then be presented with radicals that allow for utilizing the order of operations to further

simplify:

Ask: What properties am I using to simplify this expression?

INDEPENDENT PRACTICE: (20 minutes)

The class will spend the rest of the class session working on practice problems and asking the instructor

questions. The class will then come back together to after 10 minutes to go over how some selected

problems are solved. Students will share their answers on the document camera, and the instructor will

see if students have any questions on how these problems were solved.

CLOSING: (15 minutes)

The instructor will close the lesson by having students record connections with the work they have done

in previous sections and how this topic is a continuation of their work with exponents. Students will first

record their answers in their notes and then share a few with the class. The instructor will remind

students of past topics.

The class will then complete an exit ticket where they are asked to simplify radical expressions using the

multiplicative property of radicals.

Note:

Name: _________________________

Unit 3, Section 6.1 & 6.3 – Simplifying Radical Expressions

Thinking Column: Can a ever be a negative number? Why? How many solutions do square roots have?

Notes: Vocabulary

nth Root

Square Root

Perfect Squares/Cubes

I. Square Root Overview

a. Definition:

b is a square root of a if 𝑏2 = 𝑎.

b. Components of a Square Root

√4

II. nth Root Overview

a. Definition of nth Root: b is an nth root of a if 𝑏𝑛 = 𝑎.

Does this property appear familiar? Where have we seen it?

b. Components of an nth Root

c. “Perfect” Indexes

Perfect Squares Perfect Cubes Perfect Fourth 12 = 13 = 14 = 22 = 23 = 24 = 32 = 33 = 34 = 42 = 43 = 44 = 52 = 53 = 54 =

III. Simplifying Expressions

√32, √𝑥5, √50𝑥3 a. Multiplication Property of Radicals”

b. Simplifying Constants

√32 =

Steps: 1. Test 2. Test

3. Test Practice Problems:

1. √8 =

2. √48 =

3. √50 =

c. Simplifying Variable Terms

√𝑥5 =

Steps: 1. Test

2. Test

3. Test

d. Simplifying Expressions

√18𝑥3

2√300

30

√𝑞11

𝑞5=

Practice Problems

1. 6√50 =

2. √98

2=

3. √125𝑝3𝑞2 =

4. −8√8 =

5. √4ℎ

100ℎ5=

6. 10√63

12=

Main Idea/Purpose: Connections:

Name: _________________________

Unit 3, Section 6.3 Exit Ticket

Task: On your own, simplify the following radical expressions completely. When finished, turn in your exit ticket.

1. √80𝑥3𝑦4 =

2. 7√24𝑘5

2𝑘=

Name: _________________________

Unit 3, Section 6.3 Exit Ticket

Task: On your own, simplify the following radical expressions completely. When finished, turn in your exit ticket.

1. √80𝑥3𝑦4 =

2. 7√24𝑘5

2𝑘=