Download Algebra 2 Mathematics Curriculum Guide

Algebra 2 Mathematics Curriculum Guide
Unit 2: Polynomials – Quadratic Equations and Expressions
2015 – 2016
Time Frame: 15 Days
Primary Focus
There are 3 primary areas that will be focused on in this unit:
A. Students will learn to represent quadratics functions algebraically, in tables, and graphically. Students will learn to rewrite quadratics in different forms
through completing the square and factoring and will be able to use those forms to identify key features of the function and its graph.
B. Students can analyze quadratic functions and show intercepts, maxima, and minima. Students can use the process of factoring or completing the square
to show zeros, extreme values, and symmetry. Students can rearrange quadratic equations to reveal new information about the function.
C. Students will learn how to solve quadratic equations, including those with complex solutions, and to relate the solution of a quadratic equation to the
zeros of a quadratic function.
Common Core State Standards for Mathematical Practice
Standards for Mathematical Practice
How It Applies to this Topic…
MP1 - Make sense of problems and persevere in solving
them.
Analyze given information to develop possible strategies for solving the problem.
MP3 - Construct viable arguments and critique the
reasoning of others.
MP4 - Model with mathematics.
MP5 - Use appropriate tools strategically.
Unit 1
Justify (orally and in written form) how a particular function fits the context from which
the problem arose.
Use a variety of methods to model, represent, and solve real-world problems.
Select and use appropriate tools to best model/solve problems.
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Stage 1 Desired Results
Transfer Goals
Students will be able to independently use their learning to…
Use the skills and understandings developed here to model with quadratics and solve quadratic equations.
The use of complex numbers is spread throughout mathematics and its applications to science, such as electrical engineering, physics, statistics and aeronautical
engineering.
Meaning Goals
UNDERSTANDINGS
ESSENTIAL QUESTIONS
Students will understand that…
• What do the factors of a quadratic reveal about the function?
• The factors of a quadratic can be used to reveal the zeros of the quadratic.
• What does completing the square reveal about a quadratic function?
• The process of completing the square can be used to reveal the vertex of the • What is the graph of a quadratic function? What are its properties?
graph of a quadratic (and consequently the minimum or maximum of the
• What do the key features of a quadratic graph represent in a modeling
function).
situation?
• The graph of a quadratic function is a curve called a parabola which will have • What new information will be revealed if this equation is written in a
an interval of increase, an interval of decrease, a minimum or maximum, a ydifferent but equivalent form?
intercept, and which may or may not have x-intercepts.
• How can a quadratic equation be solved?
• Quadratic expressions have equivalent forms that can reveal new
• What are complex numbers, and why do they exist?
information to aid in solving problems
• How is the quadratic formula derived?
• A quadratic function has a domain that provides information to the function, • How do the factors of a quadratic determine the x-intercepts of the graph
graph and situation that it describes.
and vice versa?
• Applied problems using quadratics can be answered by either solving a
quadratic equation or re-writing the quadratic in a more useful form
(factoring to find the zeros, or completing the square to find the maximum
or minimum, for instance).
• There are several ways to solve a quadratic equation (square roots,
completing the square, quadratic formula, and factoring), and that the most
efficient route to solving can often be determined by the initial form of the
equation.
• The quadratic formula is derived from the process of completing the square.
• Complex numbers exist and can arise in the solutions of quadratic equations.
• A quadratic function that does not intersect the x-axis has complex zeros.
• The relationship between the factors of a quadratic and the x-intercepts of
the graph of the quadratic.
Acquisition Goals
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Students will know and be skilled at…
Identifying zeros and understanding their meaning in the related graph.
Solving quadratic equations by graphing.
Identifying the key graphical features of quadratics including vertex (max/min) and intercepts.
Appreciating the symmetry of the parabola and use symmetry to graph efficiently.
Factoring trinomials and special cases.
Identifying and producing equivalent forms of quadratic expressions.
Rewriting a quadratic function in standard, vertex, and factored forms.
Solving a quadratic equation by using a square root.
Solving a quadratic equation by completing the square.
Solving a quadratic equation using the quadratic formula.
Solving a quadratic equation by factoring (zero product property).
Connecting solutions to graphical representations and interpret within a context.
Solving quadratic equations with complex solutions.
Recognizing graphs of quadratic functions with complex solutions.
Adding, subtracting, and multiplying complex numbers.
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Algebra 2 Mathematics Curriculum Guide
Cluster: Standard(s)
2015 – 2016
Stage 1 Established Goals: Common Core State Standards for Mathematics
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
Explanations, Examples, and Comments
Every complex number can be written in the form a + bi where a and b are real numbers. The square root of a negative number is a complex number. Complex
numbers can be added, subtracted, and multiplied like binomials.
The commutative, associative, and distributive properties hold true when adding, subtracting, and multiplying complex numbers.
Before introducing complex numbers, revisit simpler examples demonstrating how number systems can be seen as “expanding” from other number systems in
order to solve more equations. For example, the equation x + 5 = 3 has no solution as a whole numbers, but it has a solution x = -2 as an integers. Similarly, although
7x = 5 has no solution in the integers, it has a solution x = 5/7 in the rational numbers. The linear equation ax + b = c, where a, b, and c are rational numbers, always
(𝑐−𝑏)
.
has a solution x in the rational numbers: 𝑥 =
𝑎
When moving to quadratic equations, once again some equations do not have solutions, creating a need for larger number systems. For example,𝑥 2 − 2 = 0 has no
solution in the rational numbers. But it has solutions ±√2 in the real numbers. (The real number line augments the rational numbers, completing the line with the
irrational numbers.)
Cluster: Standard(s)
Use complex numbers in polynomial identities and equations.
N.CN.7 Solve quadratic equations with real coefficients that have complex solutions.
Explanations, Examples, and Comments
Limit to polynomials with real coefficients.
This standard has a direct connection to the standard A.REI.4 in the Algebra conceptual category. A solid understanding of number systems, including complex
numbers, is foundational for advancing in solving various types of equations, investigating functions and sketching their graphs.
Solve quadratic equations with real coefficients that have solutions of the form a + bi and a – bi.
Determine when a quadratic equation in standard form, 𝑎𝑥 2 + 𝑏𝑏 + 𝑐 = 0, has complex roots by looking at a graph of 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑏 + 𝑐 or by calculating
the discriminant.
Solve equations and inequalities in one variable.
A.REI.4 Solve quadratic equations in one variable.
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Algebra 2 Mathematics Curriculum Guide
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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.
Derive the quadratic formula from this form.
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.
In Algebra 1, student should be able to recognize when the solution to a quadratic equation yields a complex solution; however writing the solution in the complex
form a ± bi for real numbers a and b will be addressed in Algebra 2.
Transform a quadratic equation written in standard form to an equation in vertex form (x - p)2 = q by completing the square.
Derive the quadratic formula by completing the square on the standard form of a quadratic equation.
Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square.
Understand why taking the square root of both sides of an equation yields two solutions.
Students should solve by factoring, completing the square, and using the quadratic formula. The zero product property is used to explain why the factors are set
equal to zero. Students should relate the value of the discriminant to the type of root to expect. A natural extension would be to relate the type of solutions to ax2 +
bx + c = 0 to the behavior of the graph of y = ax2 + bx + c .
Completing the square is usually introduced for several reasons to find the vertex of a parabola whose equation has been expanded; to look at the parabola through
the lenses of translations of a “parent” parabola 𝑦 = 𝑥 2 ; and to derive a quadratic formula. Completing the square is a very useful tool that will be used repeatedly
by students in many areas of mathematics. Teachers should carefully balance traditional paper-pencil skills of manipulating quadratic expressions and solving
quadratic equations along with an analysis of the relationship between parameters of quadratic equations and properties of their graphs.
Value of Discriminant
Nature of Roots
Nature of Graph
b2 – 4ac = 0
1 real roots
intersects x-axis once
2
b – 4ac > 0
2 real roots
intersects x-axis twice
b2 – 4ac < 0
2 complex roots
does not intersect x-axis
2
Are the roots of 2x + 5 = 2x real or complex? How many roots does it have? Find all solutions of the equation.
What is the nature of the roots of x2 + 6x + 10 = 0?
Solve the equation using the quadratic formula and completing the square. How are the two methods related?
Interpret the structure of expressions.
★
A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
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a. Interpret parts of an expression, such as terms, factors, and coefficients.
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not
depending on P.
A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can
be factored as (x2 – y2)(x2 + y2).
Linear, quadratic, and exponential expressions are the focus in Algebra I, and integer exponents are extended to rational exponents (only those with square or
cubed roots). In Algebra 2, the expectation is to extend to polynomial and rational expressions.
Identify the different parts of the expression and explain their meaning within the context of a problem.
Decompose expressions and make sense of the multiple factors and terms by explaining the meaning of the individual parts.
Students should understand the vocabulary for the parts that make up the whole expression and be able to identify those parts and interpret there meaning in
terms of a context. Students should understand the vocabulary for the parts that make up the whole expression and be able to identify those parts and interpret
there meaning in terms of a context.
Rewrite algebraic expressions in different equivalent forms such as factoring or combining like terms.
Use factoring techniques such as common factors, grouping, the difference of two squares, the sum or difference of two cubes, or a combination of methods to
factor completely. Students should extract the greatest common factor (whether a constant, a variable, or a combination of each). If the remaining expression is
quadratic, students should factor the expression further.
Write expressions in equivalent forms to solve problems.
A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★
a) Factor a quadratic expression to reveal the zeros of the function it defines.
b) Complete the square in a quadratic expression to reveal the maximum value of the function it defines.
It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and
completing the square goes hand-in-hand with understanding what different forms of a quadratic expression reveal.
(a) Write expressions in equivalent forms by factoring to find the zeros of a quadratic function and explain the meaning of the zeros.
(b) Write expressions in equivalent forms by completing the square to convey the vertex form, to find the maximum or minimum value of a quadratic function, and
to explain the meaning of the vertex. Students will use the properties of operations to create equivalent expressions.
Analyze functions using different representations.
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Algebra 2 Mathematics Curriculum Guide
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F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
a) Graph linear and quadratic functions and show intercepts, maxima, and minima.
This is a standard that should be used to scaffold material in this unit. Common misconceptions to be aware of are:
• Students may believe that each family of functions (e.g., quadratic, square root, etc.) is independent of the others, so they may not recognize commonalities
among all functions and their graphs.
• Students may also believe that skills such as factoring a trinomial or completing the square are isolated within a unit on polynomials, and that they will come to
understand the usefulness of these skills in the context of examining characteristics of functions.
• Additionally, student may believe that the process of rewriting equations into various forms is simply an algebra symbol manipulation exercise, rather than
serving a purpose of allowing different features of the function to be exhibited.
Performance Tasks and Formative Assessment Lessons:
MATERIALS BY STANDARD(S):
Teacher should use assessment data to determine which of the materials below best
meet student instructional needs. All materials listed may not be needed
Holt Algebra 2 Chapter 5 Lesson 1 Using Transformations to Graph Quadratic Functions
Holt Algebra 2 Chapter 5 Lesson 2 Properties of Quadratic Functions in Standard Form
Holt Algebra 2 Chapter 5 Lesson 3 Solving Quadratic Equations by Graphing and
Factoring
Holt Algebra 2 Chapter 5 Lesson 4 Completing the Square
Holt Algebra 2 Chapter 5 Lesson 5 Complex Numbers and Roots
Holt Algebra 2 Chapter 5 Lesson 6 The Quadratic Formula
Holt Algebra 2 Chapter 5 Lesson 9 Operations with Complex Numbers
Used to reinforce and develop prerequisite skills required for this unit. Use a preassessment to determine which lessons would be needed.
Holt Algebra 1 Lesson 1 Factors and Greatest Common Factors
Holt Algebra 1 Lesson 2 Factoring by GCF
Holt Algebra 1 Lesson 3 Factoring 𝑥 2 + 𝑏𝑏 + 𝑐
Holt Algebra 1 Lesson 4 Factoring 𝑎𝑥 2 + 𝑏𝑏 + 𝑐
Holt Algebra 1 Lesson 5 Factoring Special Products
Holt Algebra 1 Lesson 6 Choosing a Factoring Method
Unit 1
MAP: Forming Quadratics
MAP: Equations and Identities
MAP: Solving Quadratic Equation: Cutting Corners
MAP: Reasoning with Equations and Inequalities
MAP: Seeing Structure in Expressions
Georgia CCGPS: Parabola Investigation
Georgia CCGPS: Henley’s Chocolates
Georgia CCGPS: Protein Bar Toss
Georgia CCGPS: Protein Bar Toss, Part 2
Georgia CCGPS: Just the Right Border
Georgia CCGPS: Concept Map-The Quadratic Formula
Georgia CCGPS: Paula’s Peaches
Georgia CCGPS: Paula’s Peaches The Sequel
MVP: Building the Perfect Square
MVP: Factor Fixin’
MVP: Lining Up Quadratics
MVP: I’ve got a Fill-in
MVP: Throwing an Interception
MVP: Curbside Rivarly
MVP: Perfecting my Quads
MVP: To be determined…
MVP: My Irrational and Imaginary Friends
MVP: iNumbers
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Algebra 2 Mathematics Curriculum Guide
Evaluative Criteria/Assessment Level
Descriptors (ALDs):
Claim 1 Clusters:
• Interpret the structure of expressions
• Write expressions in equivalent forms to
solve problems
• Solve equations and inequalities in one
variable
• Analyze functions using different
representations.
2015 – 2016
Stage 2 - Evidence
Examples of Assessment Evidence
Concepts and Procedures
Level 3 students should be able to…
• recognize equivalent forms of expressions and use the structure of an expression to identify ways to rewrite it.
They should be able to interpret complicated expressions by viewing one or more of their parts as a single entity.
• solve multi-step linear equations and inequalities and quadratic equations in one variable with real roots.
• write a quadratic expression with rational coefficients in an equivalent form by factoring and by completing the
square. They should be able to identify and use the zeroes to solve or explain familiar problems, and they should
be able to use properties of exponents to write equivalent forms of exponential functions with one or more
variables, integer coefficients, and nonnegative rational exponents involving operations of addition, subtraction,
and multiplication, including distributing an exponent across terms within parentheses.
• analyze and compare properties of two functions of different types represented in different ways and
understand equivalent forms of functions.
Level 4 students should be able to…
• look for and use structure and repeated reasoning to make generalizations about the possible equivalent forms
expressions can have, e.g., a quadratic expression can always be represented as the product of two factors
containing its roots.
• find the maximum or minimum values of a quadratic function. They should be able to choose an appropriate
equivalent form of an expression in order to reveal a property of interest when solving problems.
• graph a quadratic function, square root by hand and by using technology.
• solve quadratic equations in one variable with complex roots.
Claim 2 Clusters:
• Interpret the structure of expressions
• Write expressions in equivalent forms to
solve problems
• Solve equations and inequalities in one
variable
• Analyze functions using different
representations.
Unit 1
Problem Solving
Level 3 students should be able to map, display, and identify relationships, use appropriate tools strategically, and
apply mathematics accurately in everyday life, society, and the workplace. They should be able to interpret
information and results in the context of an unfamiliar situation.
Level 4 students should be able to analyze and interpret the context of an unfamiliar situation for problems of
increasing complexity and solve problems with optimal solutions.
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Algebra 2 Mathematics Curriculum Guide
Claim 3 Clusters:
• Interpret the structure of expressions
• Analyze functions using different
representations.
Go here for Sample SBAC items
2015 – 2016
Communicating Reasoning
Level 3 students should be able to use stated assumptions, definitions, and previously established results and
examples to test and support their reasoning or to identify, explain, and repair the flaw in an argument. Students
should be able to break an argument into cases to determine when the argument does or does not hold.
Level 4 students should be able to use stated assumptions, definitions, and previously established results to support
their reasoning or repair and explain the flaw in an argument. They should be able to construct a chain of logic to
justify or refute a proposition or conjecture and to determine the conditions under which an argument does or does
not apply.
Go here for more information about the Achievement Level Descriptors for Mathematics:
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Stage 3 – Learning Plan: Sample
Summary of Key Learning Events and Instruction that serves as a guide to a detailed lesson planning
Graphing
Quadratics
Factoring
Quadratic
Expression
Charatacteristics
of Quadratics
Solving
Quadratics:
Factoring and
One Step
LEARNING ACTIVITIES:
This is a suggested lesson flow:
NOTES:
Begin with a graphical approach (Holt Algebra 2 chapter 5-1&2)
Day 1: Parabolas and Transformations (1 Day)
Day 2: Properties of quadratics (1 Day)
Methods of solving (Holt Algebra 2: 5-3)
Day 3: Solving Graphically (1 Day)
This is where we will embed a sub-unit on factoring. Students need to have a fluency on factoring quadratics
with an a=1, using GCF, and Special Products (such as Perfect Square Trinomials and Difference of Squares)
Days 4-6: Factoring Unit (Holt Algebra 1- Chapter 8) (3 Days)
Days 7 & 8: Solving using Factoring (Holt Algebra 1- Chapter 8) (2 days)
Understanding Completing the Square (Holt Algebra 2: 5-4)
Day 9 & 10) Emphasis should be on completing the square as a process to get a quadratic into vertex form. Also
It may be best to split up the unit into two halves.
Unit 1
Clover Park School District 2015-2016
Many of the topics are reviewed from Algebra 1 in
the first half. See the Algebra 1 book for
scaffolding lessons.
Operations with complex numbers should be
monomial or binomial additions, subtraction and
multiplication. Also, students should understand
the pattern of 𝑖, 𝑖 2 , 𝑖 3 , 𝑖 4 …
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Algebra 2 Mathematics Curriculum Guide
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Stage 3 – Learning Plan: Sample
students should be able to recognize the process if they see it. (2 Days)
Quadratic Formula and Complex Numbers (Holt Algebra 2: 5-5/5-6)
Day 11: Quadratic Formula (1 Day)
Day 12 & 13: Complex Numbers: including introducing complex and solving quadratics with complex (2 Days)
Operations with Complex Numbers (Holt Algebra 2:5-9)
Day 14: Operations with Complex Numbers (Focus only on add subtract, multiply and powers of i) (1 Day)
Daily Lesson Components
Learning Target
Warm-up
Activities
• Whole Group:
• Small Group/Guided/Collaborative/Independent:
• Whole Group:
Checking for Understanding (before, during and after):
Assessments
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