Square Root Solutions (Part 2 of 2)
Lesson 8 of 16
Objective: SWBAT find and understand square root solutions to simple equations
1. Hand each student a SqRt2 Entrance Slip.docx
2. Allow students to use a calculator and give them a few minutes to answer the questions. Students will find that the expressions in questions 1 and 2 are equivalent and that the the two radical expressions when multiplied result in the radicand.
3. Ask students to try other square roots, multiplying them so they can see that squaring and square roots are inverse operations.
4. Ask the class to come up with a rule for all cases. Students usually conclude with a rule (√x)(√x) = x
5. Write on the board: (√b)2 = b and (√b2 )= b and provide some examples using perfect squares so students can clearly see the pattern without have to use a calculator.
Ex: (√4)2 = (2)2 = 4
(√9)2 = (3)2 = 9
Ask the students to narrate what happens when they take the square root of a number squared. (They usually say..."they cancel" or "you get the same answer" Yes! Squaring a number and "Square Rooting" are inverse operations, just like multiplication and division, addition and subtraction. This face can be used to simplify, evaluate expressions, and solve simple equations.
Hand out the SqRt2 ACTIVITY SHEET.docx sheet to each student. Students work well in pairs, here. Motivate students to choose a partner they don't usually work with for the sake of change. The worksheet is broken up into activities to identify moments in the class to raise important issues.
Once Activity 1 is complete, make sure that students are aware that there are two answers for Question 3. Most students will have found a single answer, x = 0.8. Ask the class to identify another value for x that will also yield a result of 0.64. Students should realize that -0.8 is the other solution. If they take long just ask, "Are you just thinking of numbers greater than zero?"
At this point state that every positive number except 0 has two square roots, one positive and one negative. The square roots of x are √x, the positive root, and -√x, the negative root. The symbol √x , stands for only its positive square root. This root is called the principle square root.
Students should be able to conclude that x1/2 = √x Students may use a calculator to verify this with numerical values.
Ask students to enter (-4)1/2 in their calculator. Ask "What is the result? Can you understand what the calculator reports this result?" (They should get error on their calculator.) Work the class to produce a statement about the square root of any negative number. For example, the square root of a negative number is undefined because no value squared gives a negative result.
Technology Note: Students may have different calculators, so make sure they use parenthesis for the exponent (or 0.5).
Use ACTIVITY 3 in the SqRt2 ACTIVITY SHEET.docx sheet to close the lesson and assess students.
Question 1: Some students will immediately change x1/2 to the √x . Ask students to use the Power of a Power Property to solve for x as well.
Question 2: Despite the two routes students can take, students will usually square root both sides first. Since they have not learned the Square Root of Products Property, tell students to clear the coefficients first and that they will be able to go that route in coming lessons.
Question 3: Remind students of the Power of a Quotient Rule (if necessary). I will let the class struggle with it first and ask out loud, "Can we use one of the properties of powers here?" Try to gear students to using power or a power and square both sides.
Question 4: If students encounter difficulty here, ask them to analyze question 3. They should be able to reason their way toward a solution. The students should be able to figure that they take the square root of both 1 and 9, to get the result 1/3.