Lesson 9

9.1: Math Talk: Radical Equations (5 minutes)

Warm-up

This Math Talk encourages students to think about the structure of the equations to mentally solve problems (MP7). For example, where is the variable: inside or outside of the radical? Is it apparent from the structure of the equation that the last question does not have a solution? The goal is to encourage students to think about the meaning of the symbols in an equation before applying procedures to solve, which will be helpful later in the lesson when students solve radical equations strategically.

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization

Student Facing

Solve these equations mentally:

$$\sqrt[3]{x} = 1$$

$$\sqrt{7} = \sqrt{x-1}$$

$$\sqrt{100} = 2x$$

$$\sqrt{x + 1} = \text{-}5$$

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate $$\underline{\hspace{0.3 in}}$$’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to $$\underline{\hspace{0.3 in}}$$’s strategy?”
• “Do you agree or disagree? Why?”
Speaking: MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because…” or "I noticed _____ so I….” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)

9.2: Getting to the Root of the Problem (15 minutes)

Optional activity

This activity is optional because it includes additional practice students may not need. Students practice solving radical equations.

Launch

Arrange students in groups of 2. Encourage students to discuss their thinking with their partner. If partners disagree, they should work to reach agreement.

Engagement: Provide Access by Recruiting Interest. Leverage choice around perceived challenge. Invite students to select 4–5 of the equations to solve, but to make sure they select at least one equation with no solution. Chunking this task into more manageable parts may also support students who benefit from additional processing time.
Supports accessibility for: Organization; Attention; Social-emotional skills

Student Facing

Find the solution(s) to each of these equations, or explain why there is no solution.

1. $$\sqrt{a - 5} = 5$$
2. $$\sqrt[3]{a - 5} = 5$$
3. $$\sqrt[3]{b} = \text{-}2$$
4. $$\sqrt{c} + 2 = 0$$
5. $$\sqrt[3]{3 - d} + 4 = 0$$
6. $$\sqrt{7} = \sqrt{x-1}$$
7. $$\sqrt{36} = 3y$$
8. $$22z = \sqrt[3]{11}$$

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

Select students to share their responses. Be sure to include the equation $$\sqrt{c} + 2 = 0$$ in the discussion since it is the only equation given that has no solution.

Reading, Writing, Speaking: MLR3 Clarify, Critique, Correct. Before students share their responses for the equation $$\sqrt{c}+2=0$$, present an incorrect answer and explanation. For example, “The solution to the equation $$\sqrt{c}+2=0$$ is 4. First subtract 2 from each side of the equation to get $$\sqrt{c}=-2$$. Then square each side of the equation to get $$c=4$$.” Ask students to identify the error, critique the reasoning, and write a correct explanation. As students discuss with a partner, listen for students who clarify when an equation involving square roots has no solution. Invite students to share their critiques and corrected explanations with the class. This will help students evaluate, and improve upon, the written mathematical arguments of others.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness

9.3: Write Your Own Equation (15 minutes)

Optional activity

This activity is optional because it includes additional practice students may not need. Students create their own equations involving radicals that have different numbers of solutions. Generating new questions requires more innovative thinking than solving given equations, allowing students to better make sense of radical equations (MP1).

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

Launch

Arrange students in groups of 2. Give students 8–10 minutes of quiet work time followed by partner- and whole-class discussion.

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students are creating equations with the required number of solutions and solving them correctly.
Supports accessibility for: Memory; Organization

Student Facing

1. Write an equation that includes a radical symbol with:
1. one solution
2. no solutions
3. two solutions
2. Switch with a partner and solve their equations.

Student Response

For access, consult one of our IM Certified Partners.

Student Facing

Find all solutions to the equation $$\sqrt{x} = \sqrt[3]{x}$$. Explain how you know those are all of the solutions.

Student Response

For access, consult one of our IM Certified Partners.

Anticipated Misconceptions

If students struggle longer than is productive, suggest thinking of graphs of functions that have different numbers of intersection points. Also suggest using $$x^2$$ somehow, since they know that $$x^2=k$$ has 2 solutions for positive $$k$$.

Activity Synthesis

Ask students to check their solutions with their partners. Ask students to share any equations they felt helped them learn the most and discuss those equations with the class. If any students used graphs to help figure out their equations, invite them to share and display their visuals for all to see.

Lesson Synthesis

Lesson Synthesis

In this lesson, students practiced solving various radical equations. Here are some questions for discussion:

• “What kinds of mistakes did you used to make with radical equations that you no longer make? What do you understand now that you didn’t before?” (I used to square each side before checking whether a positive square root was set equal to a negative number. Now I know that the $$\sqrt{}$$ symbol means the positive square root, so I am aware to check that the equation makes sense before trying to solve it.)
• “What are some common mistakes that are easy to make with these types of equations, and what would you tell someone so that they will avoid those mistakes?” (It might be tempting to look at an equation with a cube root and cube each side right away, but it is easier to isolate the radical before cubing.)

9.4: Cool-down - A Radical Notion (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Whenever we have an equation with a radical symbol that contains a variable, we can solve it by isolating the radical and then raising each side of the equation to a power in order to get a new equation without radicals. Here is an example:

\displaystyle \begin{align}\text- 4 &= \sqrt[3]{5p+1} \\ (\text- 4)^3 &= \left(\sqrt[3]{5p+1}\right)^3 \\ \text- 64 &= 5p +1 \\ \text- 65 &= 5p \\ \text- 13 &= p \end{align}

Sometimes this results in an equation with solutions that do not make the original equation true. If we use this strategy, it is good to check the solutions to the new equation we got after raising each side to a power, to be sure they make the original equation true. In this example, we did find a solution to the original equation because $$\sqrt[3]{5(\text- 13)+1} = \text-4$$.

Another way to solve these equations is to reason about what the answer is, instead of raising each side to a power. For example, if we are solving $$\sqrt{1-x}+5 = 11$$, we can rearrange it to get $$\sqrt{1-x}= 6$$ and then think, “If the positive square root of $$1-x$$ is 6, then $$1-x$$ must be 36, since the positive square root of 36 is 6. So $$x$$ must be $$\text-35$$, since $$1-(\text-35)=36$$.” If we check this result, we see that $$\text-35$$ is a solution to the original equation because $$\sqrt{1- (\text-35)} + 5 = 11$$.