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# Simplifying Radicals

Lesson 11 of 16

## Objective: SWBAT simplify expressions involving square roots.

## Big Idea: In this lesson, students use previously acquired knowledge of square roots to simplify square roots completely.

*50 minutes*

#### LAUNCH

*10 min*

Project LaunchTask Simplifying Rad.docx on the board to start the lesson. Inform the students that the opposite sides in each figure are equal.

Allow the students to use their calculators. Students should be able to see that the yellow square in the middle is the actual perfect square because all four sides are equal. Call on a volunteer to verbally state how he/she found the answer.

Address the class and state ...

"YOU HAVE FOUND THAT √20 IS EQUAL TO 2√5 . CAN YOU DEMONSTRATE THAT THE EXPRESSIONS ARE EQUAL WITHOUT CALCULATING THEIR DECIMAL EQUIVALENCE?"

Tell the students to discuss their thoughts with their elbow partner. Depending on how much the students are struggling, I may or may not hint them by writing the property addressed in the previous lesson on the board: √a ∙ √b = √ab.

#### Resources

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#### NEW INFO

*20 min*

Students should recognize that √4 ∙√5 =√20. They should also be able to simplify further to √5 using the property of square roots learned in prior lessons.

Inform students that just like you can multiply square roots by using the Product of Square Roots Property, √a ∙√ b =√ab, you can rewrite a square root as a product by factoring the radicand.

Write this example on the board: √40 = √4 ∙ √10 which simplifies to 2√10

The key here is to find a **perfect square factor **of the radicand.

Ask student pairs to simplify the following two expressions. Write these on the board.

1. √27 2. √12

Most students will simplify these successfully. Now write √48. Ask them to simplify this expression.

Call on a volunteer to write their work on the board. Chances are that the student will rewrite this expression in one of two ways:

**√4 ∙ √12 **or** √6 ∙ √8 **

It is important to ask if anyone in the class did it differently.

Make sure at least three routes are on the board including the most efficient route (using 16 and 3). Say to the class:

”ALL THREE ROUTES ARE CORRECT BUT ONLY ONE OF THESE IS THE BEST ANSWER, CAN SOMEONE TELL ME WHICH ONE AND WHY? DISCUSS THIS FOR A MINUTE AMONG YOURSELVES.”

Allow time for discussion without calling on anyone who has their hand up trying to give an answer. Afterwards, call on a volunteer.

It is very important to ask students to demonstrate that the other two routes on the board lead to the same answer. Ask students to continue simplifying these until completely simplified.

Write the following on the board after this is done.

*A radical expression completely simplified when the value under the radical sign, (radicand), has no perfect square factors other than 1.*

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#### APPLICATION

*15 min*

Hand each student the APPLICATION Simplifying Square Roots.docx and ask students to work in pairs or small groups. Walk around to see how students are working. An ALTERNATIVE WORKSHEET.docx can be given to those students who are struggling.

In question 7 tell students that when expressions have the same square root, then they are like terms and their coefficients can be added or subtracted, just like 3x + 5x = 8x

In question 8, there are two routes. Make sure students see that they can either simplify the square root of 20 first, or multiply the radicands and simplify afterwards.

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#### CLOSURE

*5 min*

Ask students to complete the EXIT SLIP-simplifying square roots.docx.

Give the class 5 minutes to reflect upon the problems of the lesson. Use this to decide whether a part of the lesson needs to be retaught, or simply re-enforced or reviewed.

#### Resources

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- UNIT 1: Number Sense
- UNIT 2: Solving Linear Equations
- UNIT 3: Relationships between Quantities/Reasoning with Equations
- UNIT 4: Powers and Exponents
- UNIT 5: Congruence and Similarity
- UNIT 6: Systems of Linear Equations
- UNIT 7: Functions
- UNIT 8: Advanced Equations and Functions
- UNIT 9: The Pythagorean Theorem
- UNIT 10: Volumes of Cylinders, Cones, and Spheres
- UNIT 11: Bivariate Data

- LESSON 1: The Product of Powers Property
- LESSON 2: What if we take a power of a power?
- LESSON 3: Quotient of Powers
- LESSON 4: The Negative Exponent Property
- LESSON 5: Powers of Products and Quotients
- LESSON 6: Remember...the Properties of Powers
- LESSON 7: Square Root Solutions (Part 1 of 2)
- LESSON 8: Square Root Solutions (Part 2 of 2)
- LESSON 9: Cube Root Solutions
- LESSON 10: Multiply and Divide Square Roots
- LESSON 11: Simplifying Radicals
- LESSON 12: Scientific Conversions
- LESSON 13: Operations with Scientific Notations
- LESSON 14: Sun Facts (Part 1 of 2)
- LESSON 15: Sun Facts (Part 2 OF 2)
- LESSON 16: Round Robin Review (Unit 4/L1-6)