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* *Reflection: Diverse Entry Points
Tailgating and Solving Radical Equations - Section 1: Warm Up

In this Warm up, I am modeling a radical equation that police officers can use to measure skid marks and estimate how fast a car was travelling when it began to brake at the scene of an accident. (The phenomena of Tailgating is common and relatively easy for my students to understand.)

During the lesson several of my students understood how to evaluate the radical expression in Number 1. It refers back to previous lessons on substitution, formulas, and evaluating. Evaluating the radical expression on one side of the equation to determine how fast the car was traveling was not too challenging for my students.

Several students did struggle with solving the equation in Number 2. With the length of the skid mark represented by d, a variable, my students had difficulty reasoning about the problem. Here are some reflections on two examples of student work:

- Student 1 understood how to evaluate the speed of the car given the distance of the skid marks in number one of the Warm Up, and had the correct answer of approximately 53.5 miles per hour. On Number 2, this student set up the equation correctly, but did not know how to solve for d, by squaring both sides.
- Student 2 also did number one correctly. (Some of the students left the radical around the answer and did not evaluate the square root.) However, this student substituted incorrectly for the variables in Number 2. I think that he had difficulty using the language of algebra problem. This student also did not know how to undo the square root to solve the problem.

*Diverse Entry Points: Providing Different Levels of Challenge at the Start of a Lesson*

# Tailgating and Solving Radical Equations

Lesson 5 of 11

## Objective: SWBAT evaluate radical expressions and solve radical equations using order of operations and inverse operations.

#### Warm Up

*10 min*

The purpose of this lesson is for students to identify the difference between evaluating a radical expression and solving a radical equation. I allow students about five minutes to complete the Tailgaiting-Warm up. I will have them hand it in when they are done.

After completing the Warm Up, students will take notes on how to solve a radical equation. Later in the lesson, they will be given an opportunity to correct their Warm Up in the Closure Activity. I provide two samples below of student work in the reflection where I talk about some of the challenges provided by today's Warmup tasks.

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#### Guided Notes

*15 min*

After the Warm Up, I will focus on teaching students how to solve radical equations conceptually, algebraically, and graphically. I begin today's Guided Notes session by questioning students about the domain of the graph of the square root of x function. I ask, "What can x not be in the domain?" Most of my students recognize that the expression under the radical may be equal to zero, but it cannot be less than zero.

I plan to graph the first problem with the class as we complete the Guided Notes. I want to make sure that all of my students can visually see the solutions for x. We enter each expression on each side of the equation as a function, and we graph both of them. As a class, we discuss the guiding points in solving a radical equation, and students write them down. The key points that I emphasize are listed below:

- Are there restrictions on x in the equation?
- Isolate the radical before solving?
- Are the solution(s) real or extraneous?

Next, I will ask my students to solve several radical equations algebraically. My students have previously learned that taking the square root, and squaring, are inverse operations. Therefore, to eliminate the radical, they will be aware that squaring is a possible solution. I will make sure that they idea of squaring both sides of the equation is discussed.

I model reviewing Examples 5 and 6 in this TI-Nspire Solving Radical Equations video in which we compare the algebraic method to the graphing method.

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#### Independent practice

*20 min*

The Independent Practice should take my students about 20 minutes to complete. Afterward, students should have a better understanding of how the inverse operations of squaring and taking the square root work to undo each other to solve equations.

After the Guided Practice, my students generally work diligently on the Independent Practice. I walk around the room to assist students, and to monitor their progress while they are working. Some students will need help with their Calculator. I am also relatively forward about checking their algebraic work. As usual, I plan to require my students to do a check either graphically or algebraically.

I will collect today's Independent Practice in order to check student understanding of solving radical equations.

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- UNIT 1: Introduction to Functions
- UNIT 2: Expressions, Equations, and Inequalities
- UNIT 3: Linear Functions
- UNIT 4: Systems of Equations
- UNIT 5: Radical Expressions, Equations, and Rational Exponents
- UNIT 6: Exponential Functions
- UNIT 7: Polynomial Operations and Applications
- UNIT 8: Quadratic Functions
- UNIT 9: Statistics

- LESSON 1: Introduction to Radicals
- LESSON 2: Apply the Pythagorean Theorem to a Broken Telephone Pole and an Isosceles Right Triangle.
- LESSON 3: The Pythagorean Theorem and the Distance Formula
- LESSON 4: Finding the Distance or the Midpoint of a Line Segment on the Coordinate Plane
- LESSON 5: Tailgating and Solving Radical Equations
- LESSON 6: Renovate a Park by Applying Radicals and Formulas
- LESSON 7: Add and Subtract Radical Expressions
- LESSON 8: Gallery Walk of Application Problems Involving Radicals
- LESSON 9: Multiplying Radical Expressions
- LESSON 10: Dividing Radicals Made Easy Through the History of Rationalizing
- LESSON 11: Simplify and Rewrite Radicals as Rational Exponents and Vice Versa.