Separate the Radicals

The launch section of this lesson has basically two parts; the first part is to help students identify radical equations. The second part is to access student’s prior knowledge (via a video clip) on the multiplication property of radicals and recall that when you square a square-root expression, or cube a cube-root, etc., your result is always the radicand.

1. Project the resource, id_RadicalEquations on the board. Ask students to analyze both columns, then come up with one sentence that defines a radical equation. Allow students to discuss this with their immediate partner. Then call on students to give you a formal definition.

Students will usually say that a radical equation is one that has the variable under the square root sign.

• Remind students that the radical sign can be of a square root, cube root, etc. and that the number representing the root is the “index”. (show where it is written, when not a square root)
• Indicate that the “stuff” in the radical sign is called the Radicand.

Accept an answer like “An equation where the variable is found inside the square root sign, or cube root sign etc.”

Once appropriate answers are given, you should write on the board for all to see. “An equation with at least one variable within the Radicand”

Before continuing, tell students that in future lessons, they will encounter equations with an x in the radicand and one outside, like x + 2 = 3√x. Solving this type of equation may lead to solutions that do not satisfy the original equation, called Extraneous Solutions.

2.  Show the following video:

Most students should already know what’s going on here because they’ve mastered using the Pythagorean Theorem in 8^th grade, yet it’s a good refresher and helps them understand what to do with other nth roots when solving radical equations.  

After the video, ask the following questions:

• With (√b)², b must be ≥ 0…..why? (students should respond with understanding of why the square root of a negative value is undefined. Milk the problem a bit by asking if this is true of cube roots, 4^th roots, ect. ( see misconception below)
• How would you treat ³√x  if I wish to obtain x as a  result? (students should say, cube it)
• So, in general, (ⁿ√anything)ⁿ = ?   (students should answer “anything”)    

• Simplify:  (³√3x + 5)³       note: The radicand is 3x + 5

Misconception: Students may think that √-3x cannot be simplified because there is a negative in the radicand.  Tell them that if x < 0, making the radicand positive, the whole expression is defined. Students sometimes think that only the variable is the radicand and not the entire expression under the radical.

Ask students: "What is the main goal one has when asked to solve an equation?"

        (desired answer is “isolate the variable”)

Followup Question: "Ok, so we need to isolate the variable….and how do we do that?”

        (desired answers are “we use inverse operations” or “we use the properties of equality to justify the steps in the process”)

Task Prompt: “Right! So let’s look at the following radical equation” (write ³√x  + 2 = 7, on the board)

Tell students to solve we need to isolate the variable and since the variable is the radicand, we must isolate the radical part of the equation.  This we will do in all the equations seen in this lesson.

Have them do it at their desks and then call someone to the board.  They should realize that they must  cube both sides once they get to ³√x = 5.

Common mistake: Watch for students that leave the final answer in the form  √x = 125. This is a misconception of what is happening when we take an nth root expression to the n power. These students are trying to memorize patterns or procedures with little or no underlying mathematics.

Activity Format: The activity should be done independently. Student's will practice solving simple radical equations.  Handout  RadicalEquations_Activity1;  Allow discussion among students.

Common mistake: In question 3, students may not realize that there is no solution to the problem. Indicate to them that there is no value in the real number system whose square root is negative.  In the third step of question 6, for that matter, √-3x cannot equal -9. 

CLOSURE     2-1 task

In the 2-1 task, each student tells an “elbow” partner 2 MAIN POINTS of the lesson and 1 QUESTION that they have about anything covered that day. The teacher then calls on any student at random and asks what main points and question his/her elbow partner gave him/her.

Proceed to write a few main points and questions students say, on the whiteboard for all to see.

Discuss some of the main points and questions students gave.

Homework: SeparateRadicals_HW