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Algebra II Module 1, Topic D: A Surprise From Geometry--Complex Numbers Overcome All Obstacles

  Classroom Resource Information  

Title:

 Algebra II Module 1, Topic D: A Surprise From Geometry--Complex Numbers Overcome All Obstacles

URL:

 https://www.engageny.org/resource/algebra-ii-module-1-topic-d-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 1, Topic D students extend their facility with solving polynomial equations to working with complex zeros. Complex numbers are introduced via their relationship with geometric transformations. The topic concludes with students realizing that every polynomial function can be written as a product of linear factors, which is not possible without complex numbers.  

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II
2 ) Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. [N-CN2]
Mathematics
MA2015 (2016)
Grade: 9-12
Algebra II
4 ) Solve quadratic equations with real coefficients that have complex solutions. [N-CN7]
Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
3. Define the imaginary number i such that i2 = -1.
Unpacked Content
Evidence Of Student Attainment:
Students:
Given an equation where x2 is less than zero,
  • Explain by repeated reasoning from square roots in the positive numbers what conditions a solution must satisfy, how defining a number i by the equation i2 = -1 would satisfy those conditions, and extend the real numbers to a set called the complex numbers.
  • Explain how adding and/or multiplying i by any real number results in a complex number and is real when the multiplier is zero.
Teacher Vocabulary:
  • Complex number
Knowledge:
Students know:
  • Which manipulations of radicals produce equivalent forms.
  • The extension of the real numbers which allows equations such as x2 = -1 to have solutions is known as the complex numbers and the defining feature of the complex numbers is a number i, such that i2 = -1.
Skills:
Students are able to:
  • Perform manipulations of radicals, including those involving square roots of negative numbers, to produce a variety of forms, for example, √(-8) = i√(8) = 2i√(2).
Understanding:
Students understand that:
  • When quadratic equations do not have real solutions, the number system must be extended so that solutions exist. and the extension must maintain properties of arithmetic in the real numbers.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI. 3.1: Define rational and irrational numbers.
ALGI. 3.2: Identify the product of a nonzero rational number and an irrational number as irrational.
ALGI. 3.3: Identify the sum of a rational number and an irrational number is irrational.
ALGI. 3.4: Discuss why the product of two rational numbers is rational.
ALGI. 3.5: Discuss why the sum of two rational numbers is rational.
ALGI. 3.6: Describe the properties of addition and multiplication.
ALGI. 3.7: Apply properties of fractions to add, subtract, multiply, and divide rational numbers.
ALGI. 3.8: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

Prior Knowledge Skills:
  • Combine like terms of a given expression.
  • Show on a number line numbers that are equal distance from 0 and on opposite sides of 0 have opposite signs.
  • Discover that the opposite of the opposite of a number is the number itself.
  • Give examples of positive and negative numbers to represent quantities having opposite directions in real-world contexts.
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.1 Determine the value of a quantity that is squared or cubed (limited to perfect squares and perfect cubes).

Mathematics
MA2019 (2019)
Grade: 9-12
Algebra I with Probability
9. Select an appropriate method to solve a quadratic equation in one variable.

a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2 = q that has the same solutions. Explain how the quadratic formula is derived from this form.

b. Solve quadratic equations by inspection (such as x2 = 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.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Solve quadratic equations where both sides of the equation have evident square roots by inspection.
  • Transform quadratic equations to a form where the square root of each side of the equation may be taken, including completing the square.
  • Use the method of completing the square on the equation in standard form ax2+bx+c=0 to derive the quadratic formula.
  • Identify quadratic equations which may be solved efficiently by factoring, and then use factoring to solve the equation.
  • Use the quadratic formula to solve quadratic equations.
  • Explain when the roots are real or complex for a given quadratic equation, and when complex write them as a ± bi.
  • Demonstrate that a proposed solution to a quadratic equation is truly a solution by making the original true.
Teacher 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).
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
ALGI.9.1: Define quadratic equation and zero product property.
ALGI.9.2: Solve one-step equations using addition and subtraction that are set equal to zero.
ALGI.9.3: Solve two-step equations using addition and subtraction that are set equal to zero.

a.
ALGI.9.4: Define completing the square.
ALGI.9.5: Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x - p)2= q that has the same solutions.
ALGI.9.6: Derive the quadratic formula from the form (x - p)= q.

b.
ALGI.9.7: Define quadratic formula, factoring, square root, complex number, and real number.
ALGI.9.8: Solve quadratic equations by completing the square.
ALGI.9.9: Solve quadratic equations by the quadratic formula.
ALGI.9.10: Solve quadratic equations by factoring.
ALGI.9.11: Solve quadratic equations by taking square roots.
ALGI.9.12: Recognize when the quadratic formula gives complex solutions.
ALGI.9.13: Write complex solutions as a ±bi for real numbers a and b.

Prior Knowledge Skills:
  • Identify perfect squares and square roots.
  • Define square root, expressions, and approximations.
  • Explain the distributive property.
  • Calculate an expression in the correct order (Ex. exponents, mult./div. from left to right, and add/sub. from left to right).
  • Recalving one-step equations.
  • List given information from the problem.
  • Identify the unknown, in a given situation, as the variable.
  • Test the found number for accuracy by substitution.
    Example: Is 5 an accurate solution of 2(x + 5)=12?
  • Calculate a solution to an equation by combining like terms, isolating the variable, and/or using inverse operations.
  • Define equation and variable.
  • Set up an equation to represent the given situation, using correct mathematical operations and variables.
  • Recognize the correct order to solve expressions with more than one operation.
  • Calculate a numerical expression (Ex. V=4x4x4).
  • Choose the correct value to replace each variable in the algebraic expression (Substitution).
Alabama Alternate Achievement Standards
AAS Standard:
M.A.AAS.11.9 Identify equivalent expressions given a linear expression using arithmetic operations.
Tags: associative, coefficients, commutative, complex numbers, distributive, equations, geometric transformations, linear factors, polynomial, quadratic
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Author: Hannah Bradley
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