Unit 6, Lesson 14
The Quadratic Formula and Complex Solutions
Integrated Math 2
14.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSN-CN.C.7
Solve quadratic equations with real coefficients that have complex solutions.
Materials
NoneActivity Narrative
This Math Talk focuses on the types of solutions to quadratic equations. It encourages students to think about whether the solutions are real or non-real and to rely on the structure of the equations to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students examine the discriminant of a quadratic equation.
To classify the solutions, students need to look for and make use of structure (MP7).
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
NoneMentally decide whether the solutions to each equation are real numbers or not.
Activity Synthesis
The goal of this discussion is to review students’ strategies for determining the type of solutions to some quadratic equations.
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
14.2
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
Over several lessons, students have developed a concept of complex numbers. In this activity, students use that understanding to consider a situation where the quadratic formula leads to a negative number inside the symbol. The focus of the activity is on how complex numbers come into play rather than performing the computation, which is the focus of a later activity.
Launch
Arrange students in groups of 2. Give a few minutes of quiet think time before asking students to share their ideas with their partner. If there is disagreement, encourage students to work to reach agreement. Follow with a whole-class discussion.
14.3
Activity
Instructional Routines
MLR7: Compare and ConnectMaterials
NoneActivity Narrative
In this row-game activity, students solve equivalent quadratic equations given in different forms and then compare their results. The structure of a row game gives students an opportunity to construct viable arguments and critique the reasoning of others (MP3).
Monitor for students who use these strategies to solve the quadratic equations:- Reason about the solution using the form .
- Complete the square.
- Use the quadratic formula.
This activity uses the Compare and Connect math language routine to advance representing and conversing as students use mathematically precise language in discussion.
Lesson Synthesis
Display these equations one at a time, and ask students, “Would you solve this with the quadratic formula, by completing the square, or another method? Why?”
Student Lesson Summary
Sometimes when we use the quadratic formula to solve a quadratic equation, we get a negative number inside the square root symbol. This means that the solutions to the equation must involve imaginary numbers. For example, consider the following equation:
Using the quadratic formula, we know that:
or
This means that the two solutions are:
and