Download Quadratic Functions: Complex Numbers

Algebra II, Quarter 1, Unit 1.3
Quadratic Functions: Complex Numbers
Overview
Number of instruction days: 12-14
(1 day = 53 minutes)
Content to Be Learned
Mathematical Practices to Be Integrated

Develop a conceptual understanding of the
meaning of i.
2 Reason abstractly and quantitatively

Perform arithmetic operations (addition,
subtraction, and multiplication only) on the set
of complex numbers.

Factor quadratic expressions.

Solve quadratic equations with real coefficients
that have complex solutions.



Solve a system of equations consisting of a
linear equation and a quadratic equation in
two variables both algebraically and
graphically.
Use the process of factoring and completing
the square to analyze quadratic functions with
real and complex solutions.
Create quadratic equations and inequalities in
one variable and use them to solve problems.
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
Use the quadratic formula to solve for
complex solutions.

Compare functions represented in different
ways
5 Use appropriate tools strategically.

Use a graphing calculator to verify algebraic
solutions to systems of equations.
6 Attend to precision

Solve quadratic equations using several
methods.
7 Look for and make use of structure

Look for patterns of quadratics to help with
factoring.

Understand and make use of the patterns in
the powers of i in order to perform arithmetic
operations involving complex numbers.
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Algebra II, Quarter 1, Unit 1.3
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Essential Questions

What do the zeros of a quadratic function
represent?

How can quadratic equations with real root
coefficients have complex solutions

What are the various methods for solving
quadratic equations?

How do you determine the best method for
solving a system of equations?

How do you interpret the intersection of two
graphs in the context of a problem?

What do complex numbers represent?

What are the similarities and differences
between the real number system and the
complex number system?
Standards
Common Core State Standards for Mathematical Content
Number and Quantity
Quantities★
N-Q
Reason quantitatively and use units to solve problems.
N-Q.2
Define appropriate quantities for the purpose of descriptive modeling.★
The Complex Number System
N-CN
Perform arithmetic operations with complex numbers.
N-CN.1
Know there is a complex number i such that i2 = –1, and every complex number has the form
a + bi with a and b real.
N-CN.2
Use the relation i2 = –1 and the commutative, associative, and distributive properties to add,
subtract, and multiply complex numbers.
Use complex numbers in polynomial identities and equations. [Polynomials with real coefficients]
N-CN.7
Solve quadratic equations with real coefficients that have complex solutions.
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Algebra II, Quarter 1, Unit 1.3
Quadratic Function: Complex Numbers (12-14 days)
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Algebra
Creating Equations*
A-CED
Create equations that describe numbers or relationships [Equations using all available types of
expressions, including simple root functions]
A-CED.1
Create equations and inequalities in one variable and use them to solve problems.
Include equations arising from linear and quadratic functions, and simple rational and exponential
functions.*
Reasoning with Equations and Inequalities
A-REI
Understand solving equations as a process of reasoning and explain the reasoning
A-REI.1
Explain each step in solving a simple equation as following from the equality of numbers
asserted at the previous step, starting from the assumption that the original equation has a
solution. Construct a viable argument to justify a solution method.
Solve equations and inequalities in one variable [Quadratics with real and complex solutions]
A-REI.4
Solve quadratic equations in one variable.
b.
Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots,
completing the square, the quadratic formula and factoring, as appropriate to the initial
form of the equation. Recognize when the quadratic formula gives complex solutions
and write them as a ± bi for real numbers a and b.
Solve systems of equations [Linear-quadratic]
A-REI.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables
algebraically and graphically.[ For example, find the points of intersection between the line
y = –3x and the circle x2 + y2 = 3.]
Functions
Interpreting Functions
F-IF
Analyze functions using different representations [Linear, exponential, quadratic, absolute value, step,
piecewise-defined]
F-IF.9
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Compare properties of two functions each represented in a different way (algebraically,
graphically, numerically in tables, or by verbal descriptions). For example, given a graph of
both a quadratic and an algebraic function, say which has the larger maximum.
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Algebra II, Quarter 1, Unit 1.3
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Common Core State Standards for Mathematical Practice
2
Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically
and manipulate the representing symbols as if they have a life of their own, without necessarily
attending to their referents—and the ability to contextualize, to pause as needed during the
manipulation process in order to probe into the referents for the symbols involved. Quantitative
reasoning entails habits of creating a coherent representation of the problem at hand; considering the
units involved; attending to the meaning of quantities, not just how to compute them; and knowing and
flexibly using different properties of operations and objects.
5
Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a
spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient
students are sufficiently familiar with tools appropriate for their grade or course to make sound
decisions about when each of these tools might be helpful, recognizing both the insight to be gained and
their limitations. For example, mathematically proficient high school students analyze graphs of
functions and solutions generated using a graphing calculator. They detect possible errors by
strategically using estimation and other mathematical knowledge. When making mathematical models,
they know that technology can enable them to visualize the results of varying assumptions, explore
consequences, and compare predictions with data. Mathematically proficient students at various grade
levels are able to identify relevant external mathematical resources, such as digital content located on a
website, and use them to pose or solve problems. They are able to use technological tools to explore
and deepen their understanding of concepts.
6
Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the symbols
they choose, including using the equal sign consistently and appropriately. They are careful about
specifying units of measure, and labeling axes to clarify the correspondence with quantities in a
problem. They calculate accurately and efficiently, express numerical answers with a degree of precision
appropriate for the problem context. In the elementary grades, students give carefully formulated
explanations to each other. By the time they reach high school they have learned to examine claims and
make explicit use of definitions.
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7
Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students, for
example, might notice that three and seven more is the same amount as seven and three more, or they
may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7
× 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive
property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They
recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an
auxiliary line for solving problems. They also can step back for an overview and shift perspective. They
can see complicated things, such as some algebraic expressions, as single objects or as being composed
of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square
and use that to realize that its value cannot be more than 5 for any real numbers x and y.
Clarifying the Standards
Prior Learning
Starting in Grade 3, students applied properties of operations to multiply and divide. In Grade 8,
students learned to solve an equation of the form x2 = p, and they learned that some square roots of
positive numbers are irrational. In Algebra I, students learned to factor quadratic expressions to reveal
zeros, and they solved quadratic equations with real roots. In grade 6, students wrote, interpreted, and
used expressions and equations. In grade 7, students used properties of operations to generate
equivalent expressions. They also solved real-life problems using numerical and algebraic expressions
and equations and used properties of operations to generate equivalent expressions. Grade 8 students
worked with functions and learned that functions are a set of ordered pairs. Students in Algebra I used
technology to graph quadratic functions. They identified the key features of these functions from tables
and graphs. Students analyzed graphs of quadratic functions that modeled real-world problems, and
they identified appropriate domains of functions in relation to their graphs. Algebra 1 was the
introduction to factoring quadratics and completing the square. Quadratic expressions and functions
were considered critical areas in Algebra I. Students worked intensively to master quadratic functions,
both from an algebraic and formal perspective as well as in the context of modeling. The work that
students did with quadratic functions was connected with and reinforced their work in quadratic
equations, polynomial arithmetic and seeing structure in expressions. Students learned how to reexpress quadratic functions in equivalent forms to reveal different properties. They compared the
properties of quadratic functions from multiple representations. Students wrote a quadratic function
that described a quantitative relationship by determining an explicit expression, a recursive process, or
steps for calculation. In Geometry, students used quadratic equations to solve problems involving area
and special right triangles.
Current Learning
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Algebra II, Quarter 1, Unit 1.3
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The study of functions is a critical area for Algebra II students. In this unit, students study quadratic
functions and also compare quadratic and linear. Students learn to understand the graph of a quadratic
function by interpreting various forms of quadratic expressions. In particular, they identify the real
solutions of a quadratic equation as the zeros of a related quadratic function. Algebra II students solve
quadratic equations in one variable. In this unit, students also develop conceptual understanding of the
meaning of i and perform arithmetic operations with complex numbers. Students become fluent with
solving quadratic functions by completing the square and factoring quadratics with complex numbers.
They recognize when the quadratic formula gives complex solutions. They explain why the x-coordinates
of the intersection points of two graphs are the solutions to the equation. Students use technology to
graph two functions, make tables of values, and find successive approximations. They also interpret
solutions as viable or nonviable options in a modeling context. According to the PARCC Model Content
Frameworks for Algebra 2, A-RE1.1 is classified as major content for Algebra 2. N-Q.2, A-REI.4b, F-IF.9
and A-CED.1 are defined as supporting standards. N-CN.1, N-CN.2, N-CN.7, and A-REI.7 are classified as
additional standards for Algebra 2.
Future Learning
In Precalculus, students will perform arithmetic operations with complex numbers. They will also
represent complex numbers and their operations on the complex plane. Students will also graph
rational functions, identify zeros and asymptotes when suitable factorizations are available, and show
end behavior. Mastery of these concepts will be required in Precalculus and AP Calculus.
Additional Findings
According to Principles and Standards for School Mathematics, high school students should use real
numbers and learn enough about complex numbers to interpret them as solutions to quadratic
equations. They should fully understand the concept of a number system, how different numbers are
related, and whether the properties in one system hold to another (pp. 291–294).
Electronic computation technologies provide opportunities for students to work on realistic problems
and perform difficult computations.
According to Principles and Standards for School Mathematics, students should be able to solve
problems involving complex numbers using a pencil in some cases and technology in all cases (p. 395).
Assessment
When constructing an end-of-unit assessment, be aware that the assessment should measure your
students’ understanding of the big ideas indicated within the standards. The CCSS for Mathematical
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Content and the CCSS for Mathematical Practice should be considered when designing assessments.
Standards-based mathematics assessment items should vary in difficulty, content, and type. The
assessment should comprise a mix of items, which could include multiple choice items, short and
extended response items, and performance-based tasks. When creating your assessment, you should be
mindful when an item could be differentiated to address the needs of students in your class.
The mathematical concepts below are not a prioritized list of assessment items, and your assessment is
not limited to these concepts. However, care should be given to assess the skills the students have
developed within this unit. The assessment should provide you with credible evidence as to your
students’ attainment of the mathematics within the unit.

Demonstrate knowledge of the meaning of complex numbers and their mathematical importance.

Simplify expressions that involve square roots of negative numbers.

Apply properties of complex numbers to addition, subtraction and multiplication.

Connect understanding of the complex number system with the real number system.

Use quadratic equations and inequalities in one variable to solve problems.

Solve quadratic equations by taking square roots, completing the square, using the quadratic
formula and factoring with real coefficients that have solutions of the forms a + bi and a – bi.

Compare properties of two quadratic functions each represented in different ways i.e.
algebraically, graphically, and numerically in tables or by verbal description.

Solve simple systems of equations consisting of a linear equation and a quadratic equation in 2
variables.
Instruction
Learning Objectives
Students will be able to:

Write the square root of a negative number as a pure imaginary number.

Know and apply the value of the first four powers of i.

Add, subtract, and multiply complex numbers.

Extend operations and properties to work with complex numbers.
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Algebra II, Quarter 1, Unit 1.3
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
Solve simple systems of two equations, consisting of a linear equation and a quadratic equation in 2
variables both algebraically and graphically.

Interpret the intersection of two graphs in a real world context.

Recognize when the quadratic formula gives complex solutions.

Solve quadratic equations by using the Quadratic Formula, factoring, taking square roots and
completing the square with complex roots.

Use graphing technology to find the solution(s) of quadratics.

Compare properties of two functions each represented in different ways i.e. algebraically,
graphically, numerically in tables or by verbal description.

Create quadratic equations and inequalities in one variable and use them to solve problems.

Demonstrate understanding of the concepts taught in this unit.
Resources
Algebra 2, Glencoe McGraw-Hill, 2010, Student/Teacher Editions

Sections 5-1 through 5-3 (pp. 249-275)

Section 5-4 (pp. 276-282)

Sections 5.5 through 5-6 (pp. 284-299)

Section 5-8 (pp. 312-319)

Section 10-7 (pp. 662-665, systems of linear-quadratic)

Chapter 5 Resources Masters

Chapter 10 Resources Masters

http://connected.mcgraw-hill.com/connected/login.do: Glencoe McGraw-Hill Online

Interactive Classroom CD (PowerPoint Presentations)

Teacher Works CD-ROM
Exam View Assessment Suite
Forming Quadratics: http://map.mathshell.org/materials/lessons.php?taskid=224&subpage=concept
TI-Nspire Teacher Software
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Quadratic Function: Complex Numbers (12-14 days)
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Education.ti.com: Complex Numbers (ID: 10887). See the Supplementary Unit Materials section of this
binder for the student and teacher notes for this activity
Graphing Technology Lab: Systems of Linear and Quadratic Equations. See the Supplementary Unit
Materials section of the binder for this resource. From Algebra 1, (Glencoe McGraw Hill) 2010
CCSS Lesson 5-3: Solving Quadratic Equations by Factoring. See the Supplementary Unit Materials
section of the binder for this resource. Glencoe McGraw-Hill Online: CCSS Supplement
Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the
Planning for Effective Instructional Design and Delivery section below for specific recommendations.
Materials
Rulers, graph paper, TI-Nspire graphing calculator
Instructional Considerations
Key Vocabulary
factored form
factoring
completing the square
imaginary number
complex number
definition of i
Planning for Effective Instructional Design and Delivery
Reinforced vocabulary taught in previous grades or units: quadratic expression, quadratic equation,
standard form, quadratic function.
Students must have ample opportunity to explore imaginary numbers by performing arithmetic
operations with the powers and other complex forms of i. The variable-like property of i when adding
and subtracting complex numbers supports its “unknown” quality, but, unlike most variables, i is clearly
defined. This concept is fairly abstract and difficult for students to grasp.
Have students make a graphic organizer such as a foldable to organize their learning of the powers of i.
Students can use an identifying similarities and differences strategy to represent the cyclical nature of
imaginary numbers. They can make a comparison matrix to show how multiplying 1 by itself x
number of times gives a pattern:
i0
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i1
i
i2
–1
i3
–i
i4
1
i5
i
The Complex Numbers (ID: 10887) activity on education.ti.com provides the opportunity for students to
incorporate technology into their study of operations with imaginary numbers. However, be sure to limit
students to adding, subtracting, and multiplying complex numbers.
Use scaffolding techniques, additional examples, and differentiated instructional guidelines as suggested
by the Glencoe resource in section 5-1 and 5-2. These sections are to be used to activate prior
knowledge. Living word walls will assist all students in developing content language. Word walls should
be visible to all students, focus on the current unit’s vocabulary, both new and reinforced, and have
pictures, examples, and/or diagrams to accompany the definitions. In section 10-7 focus on the sections
involving linear-quadratic systems.
The 5-minute check transparencies can be used as a cue, questions, and advance organizers strategy as
students will be activating prior knowledge. Some 5-minute checks may take longer than the allotted
time, so consider choosing only problems that activate prior knowledge and use the rest for
differentiation, to formatively assess student learning, as an exit ticket, or assigning for homework.
For planning considerations read through the teacher edition for suggestions about scaffolding
techniques, using additional examples, and differentiated instructional guidelines as suggested by the
Glencoe resource. In the classroom, have a blog where students can write entries, explaining how they
decide when to use each of the different methods for solving systems of equations. One thing they can
do is write a pros and cons list for each method. See Study Notebook Chapter 5 graphic organizer (pp. 73
and 77) in the Teacher Edition.
Have students work in pairs for graphing systems of equations, taking turns between graphing the
equation and checking the solution set. Discuss with students when a system of equations has one
solution, no solution, or infinitely many solutions. Write examples of systems on the board and have
students answer: one, none, or infinite solutions.
The Graphing Technology Lab: Systems of Linear and Quadratic Equations may be used to deepen
content knowledge of linear-quadratic systems using technology. The lesson resources for this lab have
been provided in the supplementary materials section of this curriculum frameworks binder. Use section
10-7 in the Glencoe resource for additional problems related to solving linear-quadratic systems.
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Algebra II, Quarter 1, Unit 1.3
Quadratic Function: Complex Numbers (12-14 days)
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District resources for Solving Quadratic Equations by Factoring are also available in the Supplemental
Unit Materials Section of this binder. They are also accessible online on the Glencoe McGraw-Hill Math –
Algebra 2 Student and Teacher textbooks. First, select the CCSS icon on the homepage of the online
textbook and then select the CCSS Supplement to access supplementary Glencoe lessons identified in
the resource section.
Graphing technology will assist students with modeling real-world problems involving systems of
equations and inequalities as well as identifying solution(s) to system of equations.
The Mathematics Assessment Project, also known as Math Shell, provides a variety of projects for
formative and summative assessment that may help make knowledge and reasoning visible to students.
These CCSSM aligned resources include a variety of lessons focused on concept development and
problem solving. It also includes numerous tasks designed for expert, apprentice and novice learners.
The Forming Quadratics lesson is intended to help teachers assess how well students are able to
understand what the different algebraic forms of a quadratic function reveal about the properties of its
graphical representation. In particular, the lesson will help you identify and help students who have the
following difficulties:

Understanding how the factored form of the function can identify a graph’s roots.

Understanding how the completed square form of the function can identify a graph’s maximum
or minimum point.

Understanding how the standard form of the function can identify a graph’s intercept.
Use the link below to access detailed information for the Forming Quadratics lesson:
http://map.mathshell.org/materials/lessons.php?taskid=224&subpage=concept. The lesson resources
are also provided in the supplementary materials section of this curriculum frameworks binder.
Incorporate the Essential Questions as part of the daily lesson. Options include using them as a “do
now” to activate prior knowledge of the previous day’s lesson, using them as an exit ticket by having
students respond to it and post it, or hand it in as they exit the classroom, or using them as other
formative assessments. Essential questions should be included in the unit assessment.
Notes
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Algebra II, Quarter 1, Unit 1.3
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