Unit 4, Lesson 9
Solving Radical Equations (Optional)
Algebra 2
9.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-REI.A.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
Building Toward
Materials
NoneActivity Narrative
This Math Talk focuses on solving equations involving a radical. It encourages students to think about where the variable is and to rely on the structure of the equation to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students solve more complex equations.
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneSolve these equations mentally:
Student Response
Loading...
Building on Student Thinking
Activity Synthesis
The goal of this discussion is to review students’ strategies for solving equations involving roots.
To involve more students in the conversation, consider asking:
- “Who can restate ’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 ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. Examples:
Advances: Speaking, Representing
- “First, I because .”
- “I noticed , so I .”
Advances: Speaking, Representing
9.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
Students practice setting up equations involving squares, cubes, square roots, and cube roots from geometric contexts.
Launch
Arrange students in groups of 2.
Ask students to recall these geometry facts.
- “What is a formula for the area of a rectangle?” ()
- “What is a formula for the volume of a rectangular prism?” ()
- “What does it mean to scale an object with a scale factor of ?” (All the lengths in the object are multiplied by the same value, . This creates an object that looks similar, but is larger if or smaller if .)
Remind students that when an object is scaled so that lengths are multiplied by the scale factor , any areas in the scaled copy are scaled by a factor of and any volumes in the scaled copy are scaled by a factor of .
The last question refers to finding volume by water displacement. If any students ask about this process, ask the class if they have ever found the volume of an object this way in their science classes. If necessary, explain that the volume of an object can be found by pouring a known volume of water in a container and then submerging the object. The volume of the object is the combined volume of the object and the water minus the known volume of the water before the object was added. In this case, it can be difficult to find the volume of an airplane either by geometric or displacement methods, but a scale model is small enough to have its volume measured in this way.
Encourage students to discuss their thinking with their partner. If partners disagree, they should work to reach agreement.
9.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
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 discussion with a partner and whole-class discussion.
Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. After the first 2–3 minutes, check in with students to provide feedback and encouragement. Ensure that students are creating equations with the required number of solutions.
Supports accessibility for: Attention, Social-Emotional Functioning
Supports accessibility for: Attention, Social-Emotional Functioning
Activity
NoneLesson 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 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.)
Student Lesson Summary
Areas of figures are often related to two lengths multiplied together, and volumes of objects are often related to three lengths multiplied together. If the lengths are related, this can result in an equation with powers of 2 or 3. Here are some examples:
- The area of a circle with radius is .
- The area of an equilateral triangle with side length is .
- The volume of a sphere with radius is .
- The volume of a regular tetrahedron (a pyramid made from 4 equilateral triangles) with side length is .
We can solve each of these equations for or to give us an equation that would allow us to find the radius or side length from the area or volume. These equations would include either a square root or cube root.
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:
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 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 .
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 , we can rearrange it to get and then think, “If the positive square root of is 6, then must be 36, because the positive square root of 36 is 6. So must be , because .” If we check this result, we see that is a solution to the original equation because .