Complex Numbers Solutions

Learning Resource Type

Lesson Plan

Subject Area

Mathematics

Grade(s)

8, 9, 10, 11, 12

Overview

This lesson is designed to teach the students that some quadratic equations will have imaginary solutions. The lesson will examine the concept of complex numbers in terms i. The student will use the quadratic formula to solve the equations and write the the solutions in the form a +bi.

This lesson results from the ALEX Resource Gap Project.

Mathematics (2019) Grade(s): 8 - Grade 8 Accelerated

MA19.8A.11

Select an appropriate method to solve a quadratic equation in one variable.

UP:MA19.8A.11

Vocabulary

  • quadratic equation
  • Square root
  • Factoring
  • Completing the square
  • quadratic formula
  • Derive
  • Real numbers
  • Imaginary numbers
  • Complex numbers

Knowledge

Students know:
  • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
  • The method for completing the square.
  • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.

Skills

Students are able to:
  • Take the square root of both sides of an equation.
  • Factor quadratic expressions in the form x2+bx+c where the leading coefficient is one.
  • Use the factored form to find zeros of the function.
  • Complete the square.
  • Use the quadratic formula to find solutions to quadratic equations.
  • Manipulate equations to rewrite them into other forms.

Understanding

Students understand that:
  • Solutions to a quadratic equation must make the original equation true and this should be verified.
  • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
  • Different procedures for solving quadratic equations are necessary under different conditions.
  • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
  • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).
Mathematics (2019) Grade(s): 8 - Grade 8 Accelerated

MA19.8A.11b

Solve quadratic equations by inspection (such as $x^2 = 49$), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation, and recognize that some solutions may not be real. [Algebra I with Probability, 9]
Mathematics (2019) Grade(s): 09-12 - Algebra I with Probability

MA19.A1.9

Select an appropriate method to solve a quadratic equation in one variable.

UP:MA19.A1.9

Vocabulary

  • Completing the square
  • Quadratic equations
  • Quadratic formula
  • Inspection
  • Imaginary numbers
  • Binomials
  • Trinomials

Knowledge

Students know:
  • Any real number has two square roots, that is, if a is the square root of a real number then so is -a.
  • The method for completing the square.
  • Notational methods for expressing complex numbers.
  • A quadratic equation in standard form (ax2+bx+c=0) has real roots when b2-4ac is greater than or equal to zero and complex roots when b2-4ac is less than zero.

Skills

Students are able to:
  • Accurately use properties of equality and other algebraic manipulations including taking square roots of both sides of an equation.
  • Accurately complete the square on a quadratic polynomial as a strategy for finding solutions to quadratic equations.
  • Factor quadratic polynomials as a strategy for finding solutions to quadratic equations.
  • Rewrite solutions to quadratic equations in useful forms including a ± bi and simplified radical expressions.
  • Make strategic choices about which procedures (inspection, completing the square, factoring, and quadratic formula) to use to reach a solution to a quadratic equation.

Understanding

Students understand that:
  • Solutions to a quadratic equation must make the original equation true and this should be verified.
  • When the quadratic equation is derived from a contextual situation, proposed solutions to the quadratic equation should be verified within the context given, as well as mathematically.
  • Different procedures for solving quadratic equations are necessary under different conditions.
  • If ab=0, then at least one of a or b must be zero (a=0 or b=0) and this is then used to produce the two solutions to the quadratic equation.
  • Whether the roots of a quadratic equation are real or complex is determined by the coefficients of the quadratic equation in standard form (ax2+bx+c=0).
Mathematics (2019) Grade(s): 09-12 - Algebra II with Statistics

MA19.A2.11

Solve quadratic equations with real coefficients that have complex solutions.

UP:MA19.A2.11

Vocabulary

  • Complex solution
  • Quadratic equation
  • Real coefficients

Knowledge

Students know:
  • strategies for solving quadratic equations

Skills

Students are able to:
  • apply the quadratic equation.
  • provide solutions in complex form.

Understanding

Students understand that:
  • all quadratic equations have two solutions: real or imaginary.
  • Some contextual situations are better suited to quadratic solutions.

Primary Learning Objectives

The student will be able to solve quadratic equations using the quadratic formula.

The student will be able to write complex solutions in the form a + bi.

Procedures/Activities

Before

As the students enter the classroom, the teacher will have the bell ringer, Bell Ringer with Complex Numbers, displayed on the interactive whiteboard. The directions are on the sheet. The students can use their technology devices or regular graph paper. The answers are hidden in the color, white. The teacher will highlight under the word, answers, and change the color to black. The teacher will click on the website link to open the graphs. The teacher will ask students to comment on the second part of the bell ringer.

During

  1. The teacher will use the bell ringer graphs to show the students that when the graph does not cross the x-axis, then the equation does not have real solutions. The teacher will say, “Solutions to the quadratic equation can be either real or imaginary/complex.” The teacher will show the students that when the graph crosses or touches the x-axis, then the solutions are real solutions.
  2. The teacher will write on the interactive whiteboard, “The square root of -1 is ____”. The students will turn and talk. The teacher will give one minute for the students to discuss their answers. The teacher will call on two or three students to fill in the blank. The answer is i, which means imaginary. The teacher can do an informal assessment.
  3. The teacher will introduce the video from YouTube. The teacher will discuss the idea that the quadratic formula may have a possible negative square root. The equations will have imaginary roots or complex solutions.
  4. The teacher will show the video, https://www.youtube.com/watch?v=jU_aLT2YMjA.
  5. The teacher will ask for questions or comments.
  6. The teacher will show the next video, https://www.youtube.com/watch?v=H5AM1bzqCQw . Stop the video when the presenter has the equation on the screen. The teacher will ask the students to work with a partner and solve the equation. The teacher will walk around the room and monitor student behavior and engagement. The teacher will give one-on-one instruction if needed.
  7. After three or four minutes, the teacher will continue playing the video. The teacher will do an informal assessment while the students are checking their work.
  8. As the video is ending, the teacher will pass out the worksheet, Quadratics with Complex Solutions. The teacher will place three students in each group. The intervention will be to allow the students to use the website in the materials section. (https://www.mathpapa.com/quadratic-formula)
  9. The teacher will call on students to write their work and answers on the interactive whiteboard.
  10. For the accelerated students, the teacher will give them the worksheet, Accelerated Complex Numbers.

After

The teacher will pass out the exit slip from the attachments, Exit Slip Complex Solutions. The teacher will use the exit slip as the formal assessment. The students will turn in the exit slip as they leave the classroom. 

Assessment Strategies

Informal

The teacher will ask students questions during the lesson as well as monitor the students work during the group assignment.

Formal

The teacher will use the exit slip as the formal assessment.

 

Acceleration

The accelerated students will have a worksheet to complete called Accelerated Complex Numbers.

Intervention

The students will work in groups. A peer-tutor will be assigned by the teacher. The webiste, https://www.mathpapa.com/quadratic-formula, can be used with the devices. The teacher will do one-on-one with students that are still struggling with the formula or simplifying the radicals.

Total Duration

31 to 60 Minutes

Background/Preparation

Teacher

The teacher will need to preview the website, https://www.desmos.com/calculator. The teacher will use the website to show the difference between real and non-real solutions to quadratic equations. If the parabola crosses or touches the x-axis, then the equation has real solution(s). If the parabola does not cross or touch the x-axis, then the equation has imaginary or complex solutions.

Student

The student should remember that the square root of -1 is i. Therefore, the student will be asked to simplify square roots of negative numbers. The student will need to be able to use the quadratic formula with quadratic equations. The student will need to be able to simplify square roots. The student needs to be able to graph quadratic equations.

Materials and Resources

Technology Resources Needed

Approved Date

2017-07-18

Owner (Author)

morganboyd
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