• Students can be evaluated by their work on the Square Root Equations.
• Student proficiency can be assessed based on their performance on the Cube Root Equations worksheet.
• To determine student concept mastery, use the Lesson 3 Exit Ticket.

Activity 1

Write the equation x² = 25 on the board. Ask students to find all the solutions and raise their hands when they are finished. When most of the class has finished, ask for all the solutions. If they don’t provide x = −5, tell them they’re missing one solution.

Illustrate the fact that it is easy to forget about the negative solution to an equation like x² = 25.

“How did you know that x = 5 was a solution?” Responses probably will include that
5 × 5 = 25. “How did you determine that x = −5 was a solution?” Use this as an opportunity to make sure students remember the rules for multiplying negative numbers, i.e., the product of two negative numbers is a positive number. Guide students toward recognizing that the square root of 25 is ±5. (Depending on the class, choose whether or not to introduce the ± symbol.)

“What are all the solutions to x² = 100?” (±10) “How did you know that x = 10 was a solution?” Here, encourage students, rather than noting that 10 × 10 = 100, to note that the square root of 100 is ±10. The goal is to get students thinking in terms of taking the square root of both sides.

“Remember that when solving equations, we typically want to get the variable, x in this case, all by itself. If the equation was x − 4 = 20, what operation would we perform on both sides of the equation to isolate x?” (Add 4.) “This works because addition and subtraction are inverse operations. What about if the equation was 5x = 80?” (Divide by 5.) “Again, this works because multiplication and division are inverse operations. In our problem, x² = 100, to solve it algebraically we need to know what the inverse operation is to squaring something. Any guesses?” (square root)

“When we take the square root of both sides, on the left-hand side we’re just left with x because taking the square root and squaring something are inverse operations. They cancel one another out. On the right-hand side, we have the square root of 100, which as we know is 10 and −10.”

Write x² = 31 on the board. “What are the solutions to this equation?” Students may struggle with this equation or even say there are no solutions. Remind students that their job is to isolate the x. In other words, get rid of the exponent. Guide students toward the realization that they should take the square root of both sides, resulting in x = ± the square root of 31. Make sure students include the negative square root. If the ± symbol wasn’t introduced before now, it should be introduced at this point as a convenient way to indicate both positive and negative solutions.

“Does 31 have a square root?” Here, it’s easy for students to say that it does not. Point out that it does have a square root, but it’s not a whole number. (Depending on the class, you can mention that it is an irrational number, if the class has been introduced to rational and irrational numbers.) “However, it’s a decimal that keeps going forever with no pattern to it, so we’ll just leave it as the square root of 31, thus we don’t have to round our answer. We know that our answer is precisely correct.”

Verbally give students several other equations to solve of the form x² = p where p is a perfect square (sometimes) and a non-perfect square (including fractions, decimals, etc.). Once the class seems to have mastered solving these types of equations, write the following equation on the board:

3x² = 90

“Remember, we want to get x by itself. But, here we have two problems – we need to get rid of the 3 and we also need to get rid of the exponent. What should we get rid of first?” Allow for some discussion, pointing out that if we take the square root of both sides of the equation, we have to take the square root of the 3. After some discussion, continue: “Think about the order of operations. When we are simplifying an expression, we know that exponents and roots come before division. However here, we are ‘undoing’ these operations to isolate the variable and solve the equation, and thus we need to use the reverse order of operations. This means that we must first divide both sides by 3, and then take the square root of either side of the equation.

“So, we’ll divide by 3 to get x² = 30. What are the solutions to our equation? ()

How about an equation like x² = −6? What are the solutions to this equation?” Allow for some discussion, exploring the idea of taking the square root of a negative number. Point out to students that you can take the square root of a negative number but you get what is called an imaginary number. “We don’t need to know any more about imaginary numbers right now. For our equation, though, we can say it does not have any real solutions because there’s no real number that is the square root of −6.”

Give each student a copy of the Square Root Equations worksheet (M-8-4-3_Square Root Equations and KEY.docx). Give them time to complete it before reviewing answers and collecting it.

Activity 2

Write x³ = 64 on the board and ask students to find all the real solutions to the equation. Here, responses will be mixed as students may assume it is a square root problem without looking closely, leading to responses of ±8. Other students may note it is a cube root but assume the solutions are ±4 (rather than just 4).

If students’ responses included ±8, point out that that variable is being cubed, not squared.

“How do we know that 4 is a solution to the equation?” (The cube root of 64 is 4, and
4 × 4 × 4 = 64.)

“What about −4? What is the value of −4 × −4 × −4?” (The value is −64.) If necessary, take a moment to explain how the sign of (−4)³ ends up being negative rather than positive.

“So, is −4 a solution?” (No.) “For this cube root problem, we really just have one solution: 4. So that’s one difference between solving problems with square roots and cube roots. With square roots, you typically have to remember to include the plus-or-minus sign. With cube roots, you don’t include it.”

“How about x³ = 8?” (x = 2)

“How about x³ = −8?” Students may respond that there is no solution. Remind them that when we did −4 × −4 × −4, we got a negative product. This is a hint that equations involving x³ can have negative solutions. Help students realize that the solution is x = −2.

Write x³ = 7 on the board. “What is the solution to this equation?” As before, students may say there isn’t one since 7 isn’t a perfect cube. “Remember, our goal is to isolate the x. When it was x squared, we took the square root. Now that it’s x cubed, what should we do?” By emphasizing the repeated word square, students should at least tentatively guess, even if they’ve never heard of it, to take the cube root of both sides.

“When we take the cube root of x³, the root and the exponent cancel each other out because they’re inverse operations. That just leaves us with x on the left-hand side. On the
right-hand side, we’re left with the cube root of 7.” Write  on the board.

“Notice that the cube root symbol is pretty much the same as the square root symbol. We have a radical sign, but there’s one small and important difference. We put a small 3 on the ‘shelf’ of the radical to show that it’s a cube root, which is the inverse of raising something to the third power. When we take a square root, we don’t write a little 2, although we could, because it’s just assumed since square roots are so common. That means if you see a radical sign without a small number, it means it is a square root (or ).

“So, our solution here is the cube root of 7. Again, we don’t put a ± sign here because when we cube a negative number, we don’t get a positive number.”

Write x³ = −4. “What is the solution for x?” Here, students have to combine both of the new concepts about cube roots: that a solution can be negative and that the solution needs to incorporate a cube root, rather than square root. Students should recognize that the solution here is .

Give students a copy of the Cube Root Equations worksheet (M-8-4-3_Cube Root Equations and KEY.docx) and give them time to complete it before reviewing answers and collecting it.

Before exiting, students should also complete the Lesson 3 Exit Ticket (M-8-4-3_Lesson 3 Exit Ticket and KEY.docx).

Extension:

Use the following strategies to tailor the lesson to meet the needs of students throughout the year.

• Routine: The ideas in this lesson can be repeated throughout the year as students learn to solve more complex equations involving square and cube roots, e.g.,, 4x³ −12 = x³ + 8.

Concepts in this lesson can also be introduced in a geometric sense when dealing with area and volume. For example, given that the volume of a cube is 125 in³, what is the length of one side of the cube? Questions of that type require setting up the equation x³ = 125 and solving.

• Small Group: To become more proficient with common squares and cubes (squares up to 144 and cubes up to 125), students can create flash cards and quiz one another. Students can also work in groups to see how many squares (or cubes) their group can memorize collectively and compare to other groups.
• Expansion: Students who are looking for a challenge beyond the requirements of the standard may explore the use of the ^x√ button on their calculators and use it to solve higher-degree equations such as x⁴ = 1,024. (Solving such equations without the ^x√ button may prove to be frustrating for students, although it can also help students get a sense of the rapid increase in the size of powers as the exponents increase.)

Likewise, students can use a calculator to begin to explore fractional and negative exponents. Having students evaluate, for example, n⁻¹ where n is every integer from 1 to 10 may allow students to determine what effect the negative sign is having on evaluating the exponent.