INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
UNIT OVERVIEW
This unit bundles student expectations that address writing equations of parabolas given various characteristics; writing quadratic functions given three points; formulating,
solving, and determining the reasonableness of solutions to a system of equations consisting of a linear equation and a quadratic equation; and solving quadratic equations
and inequalities. Concepts are incorporated into both mathematical and real-world problem situations. According to the Texas Education Agency, mathematical process
standards including application, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content
knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.
Prior to this unit, in Algebra I Units 07 and 08, students investigated quadratic functions and equations. Students also formulated quadratic models to represent problem
situations and applied various methods to solve quadratic equations. In Algebra I Unit 05 and Algebra II Unit 03, students investigated systems of linear equations.
During this unit, students use a system of three equations in three variables to write quadratic functions given three specified points in a plane and justify the quadratic function
using the graphing calculator. Students transform quadratic functions from standard form, f(x) = ax2 + bx + c, to vertex form, f(x) = a(x – h)2 + k , and identify attributes of f(x),
including vertex, symmetries, maximum and minimum. Students write equations of parabolas from attributes including vertex, focus, directrix, axis of symmetry, and direction
of opening. Students define the complex number system and its subsets as well as perform operations (addition, subtraction, multiplication) with complex numbers. Students
solve quadratic equations using various methods, including graphing, factoring, completing the square, and the quadratic formula, and verify solutions by graphing and
multiplying factors created by roots. Students solve quadratic inequalities graphically and algebraically. Students formulate quadratic equations from tables of data and realworld problem situations, solve the quadratic equations by a method of choice, and justify the solution in terms of the problem situation. Students formulate systems of
equations consisting of two equations, the first linear and the second quadratic, solve the system algebraically, and determine the reasonableness of the solution in terms of
the problem situation.
After this unit, in Algebra 2 Units 06, 07, 08 and 11, students will continue to apply the concepts of quadratic functions, equations, and inequalities. In subsequent
mathematics courses, students will also continue to apply these concepts when quadratic functions, equations, and inequalities arise in problem situations.
In Algebra II, analysis of quadratic relations, including the equations and attributes of parabolas, is identified as STAAR Readiness Standard 2A.4B and is subsumed under
STAAR Reporting Category 4: Quadratic and Square Root Functions, Equations, and Inequalities. Formulating system of equations and solving quadratic and square root
equations are identified as STAAR Readiness Standards 2A.3A and 2A.4F and are subsumed under STAAR Reporting Category 3: Writing and Solving Systems of Equations
and Inequalities and STAAR Reporting Category 4: Quadratic and Square Root Functions, Equations, and Inequalities. Performing operations with complex numbers is
identified as STAAR Supporting Standard 2A.7A and is subsumed under STAAR Reporting Category 1: Number and Algebraic Methods. Writing quadratic functions,
formulating and solving quadratic equations, and solving quadratic inequalities are identified as STAAR Supporting Standards 2A.4A, 2A.4D, 2A.4E, and 2A.4H. These STAAR
Supporting Standards are subsumed under STAAR Reporting Category 4: Quadratic and Square Root Functions, Equations, and Inequalities. Solving and determining the
reasonableness to a system of a linear equation and a quadratic equation is identified as STAAR Supporting Standards 2A.3C and 2A.3D and are subsumed under STAAR
Reporting Category 3: Writing and Solving Systems of Equations and Inequalities. This unit is supporting the development of Texas College and Career Readiness Standards
(TxCCRS): I. Numeric Reasoning A2, B1; II. Algebraic Reasoning A1, B1, C1, D1, D2; III. Geometric Reasoning B2, C1; VII. Functions A1, A2, B1, B2, C2; VIII. Problem
Last Updated 08/17/2015
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Page 1 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
Solving and Reasoning; IX. Communication and Representation; X. Connections.
According to the National Council of Teachers of Mathematics (NCTM), Developing Essential Understanding of Functions, Grades 9-12, understanding of the function concept
is essential to describing and analyzing quantities which vary with respect to one another. According to research from the National Council of Teachers of Mathematics (2000),
high school algebra should provide students with insights into mathematical abstraction and structure. High school students’ algebra experience should enable them to create
and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with a higher degree of sophistication.
Students should develop an understanding of the algebraic properties that govern manipulation of symbols in expressions, equations, and inequalities.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2011). Developing essential understanding of expressions, equations, and functions, grades 6-8. Reston, VA: National Council of
Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from
http://www.thecb.state.tx.us/collegereadiness/crs.pdf
OVERARCHING UNDERSTANDINGS AND QUESTIONS
The complex number system is a way to encompass all number relationships.
Why is the complex number system used to represent number relationships?
How are different sets and subsets of numbers related in the complex number system?
How are sets of numbers within the complex number system used in problem situations?
Equations and inequalities can model problem situations and be solved using various methods.
Why are equations and inequalities used to model problem situations?
How are equations and inequalities used to model problem situations?
What methods can be used to solve equations and inequalities?
Why is it essential to solve equations and inequalities using various methods?
How can solutions to equations and inequalities be represented?
How do the representations of solutions to equations and solutions to inequalities compare?
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Page 2 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
Relations are algebraic models that describe how two quantities relate to one another. Functions are a subset of relations.
What are types of relations?
How can relations be represented?
Why do some relations not define a function?
Why do some relations define a function?
Why can function models describe how two variable quantities change in relation to one another?
Systems of equations can model problem situations and be solved using various methods.
Why are systems of equations used to model problem situations?
How are systems of equations used to model problem situations?
What methods can be used to solve systems of equations?
Why is it essential to solve systems of equations using various methods?
How can solutions to systems of equations be represented?
Functions can be classified into different families with each function family having its own unique graphs, attributes, and relationships.
Why are functions classified into families of functions?
How are functions classified as a family of functions?
What graphs, key attributes, and characteristics are unique to each family of functions?
What patterns of covariation are associated with the different families of functions?
How are the parent functions and their families used to model real-world situations?
Function models for problem situations can be determined by collecting and analyzing data using a variety of representations and applied to make predictions and critical
judgments in terms of the problem situation.
Why is it important to determine and apply function models for problem situations?
What representations can be used to analyze collected data and how are the representations interrelated?
Why is it important to analyze various representations of data when determining appropriate function models for problem situations?
How can function models be used to evaluate one or more elements in their domains?
How do the key attributes and characteristics of the function differ from the key attributes and characteristics of the function model for the problem situation?
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Page 3 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
How does technology aid in the analysis and application of modeling and solving problem situations?
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
Algebraic Reasoning
Algebra II Unit 05 PA 01
An engineer has been commissioned to create a
parabolic arch to be built as a focal point of a new city
park. He sketched the blueprint of the design on a
coordinate grid with the origin representing ground level
at the center of the park. The arch of the bridge passes
through the points (3, 2), (5, 6), and (6.5, 3.75) on the
coordinate grid.
Create a report to present to the park committee that
contains the blueprint diagram and the representative
equations, in both standard form and vertex form, of the
parabolic arch.
1. Using the three given points, formulate a system of
three linear equations in three variables, to
determine the coefficients of the equation that
models the parabolic arch on the coordinate grid.
Equations
Expressions
Multiple Representations
Solve
Systems of Equations
UNIT UNDERSTANDINGS
A quadratic function can be determined using a three by three system
of equations when given three points in a plane through which the
function passes.
How are the three points used to formulate a three by three
system of equations?
What methods can be used to solve the three by three system
of equations?
Functions
Attributes of Functions
Non-Linear Functions
Associated Mathematical Processes
Application
Tools and Techniques
Problem Solving Model
Communication
Representations
Relationships Justification
A quadratic function in standard form, f(x) = ax2 + bx + c, can be
transformed to vertex form, f(x) = a(x – h)2 + k .
How are quadratic equations transformed from standard form to
vertex form?
How are quadratic equations transformed from vertex form to
standard form?
What attributes can be determined from the standard form, f(x)
= ax2 + bx + c?
What attributes can be determined from the vertex form, f(x) =
a(x – h)2 + k ?
2. Solve the system for a, b, and c, and write the
Last Updated 08/17/2015
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Page 4 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
SUGGESTED DURATION : 14 days
UNIT UNDERSTANDINGS
quadratic function in the form of y = ax2 + bx + c
that models the parabolic arch on the coordinate
grid.
3. Transform the quadratic function to the vertex form,
f(x) = a(x – h)2 + k .
4. What attributes of the representative model of the
parabola can be determined from the standard
form?
5. What attributes of the representative model of the
parabola can be determined from the vertex form?
Standard(s): 2A.1A , 2A.1B , 2A.1C , 2A.1D , 2A.1E , 2A.1F , 2A.1G , 2A.3A , 2A.4 , 2A.4D
ELPS.c.1E , ELPS.c.1G , ELPS.c.2D , ELPS.c.3D
, ELPS.c.4H , ELPS.c.4K , ELPS.c.5B
Last Updated 08/17/2015
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Page 5 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
Algebraic Reasoning
Algebra II Unit 05 PA 02
Using the given attributes, create an organizer table
that includes a diagram of the parabola with labeled
attributes (vertex, directrix, focus, axis of symmetry)
and the equation of the parabola in both standard and
vertex form.
1. Vertex: (4, 0); Focus: (0, 0)
2. Directrix: y = 5 ; Focus: Standard(s): 2A.1B , 2A.1C , 2A.1D , 2A.1E , 2A.1F , 2A.1G , 2A.4B , 2A.4D ELPS.c.1C , ELPS.c.2D , ELPS.c.4H , ELPS.c.4K , ELPS.c.5B
Equations
Multiple Representations
Relations
Functions
Attributes of Functions
Non-Linear Functions
Associated Mathematical Processes
Tools and Techniques
Problem Solving Model
Communication
Representations
Relationships Justification
Numeric Reasoning
Algebra II Unit 05 PA 03
1. Create a graphic organizer on complex numbers
that includes:
Addition
Complex numbers
Imaginary Numbers
Multiplication
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SUGGESTED DURATION : 14 days
UNIT UNDERSTANDINGS
The representative equation of a parabola can be determined by
analyzing its attributes.
How is a parabola defined as a locus of points? Explain.
What are the attributes of a parabola?
In what directions can a parabola open?
How can the equation of a parabola be used to determine the
way the parabola opens?
Why does a parabola always represent a relation but not always
represent a function?
What is the connection between p in the formula (x – h)2 = 4p(y
– k) and a in the formula y = a(x – h)2 + k ?
The complex number system encompasses real numbers, imaginary
numbers, and their subsets.
How are the number systems interrelated?
What makes up a complex number?
What is an imaginary number?
Page 6 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
a. Venn diagram representing the complex number
system and its subsets
b. Explanation of an imaginary number
c. Definition of a complex number
d. Description and example of how to add complex
numbers
e. Description and example of how to subtract
complex numbers
f. Description and example of how to multiply
complex numbers
OVERARCHING CONCEPTS
UNIT CONCEPTS
Subtraction
Algebraic Reasoning
Equations
Equivalence
Expressions
Inequalities
Simplify
Solve
Associated Mathematical Processes
Tools and Techniques
Problem Solving Model
Communication
Representations
Relationships Justification
SUGGESTED DURATION : 14 days
UNIT UNDERSTANDINGS
What operations can be performed with complex numbers?
How are complex numbers applicable in quadratic functions and
equations?
Equations and inequalities can be used to model and solve
mathematical problem situations.
What methods can be used to solve quadratic equations?
What methods can be used to solve quadratic inequalities?
What are the advantages and disadvantages of various methods
used to solve quadratic equations and inequalities?
What methods can be used to justify the reasonableness of
solutions to quadratic equations and inequalities?
How can roots and their factors be used to determine models for
quadratic equations?
2. Solve the quadratic equations algebraically and
check the solutions by writing the factors from the
solutions and multiplying the factors.
a. 15x2 + 8x = 12
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Page 7 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
SUGGESTED DURATION : 14 days
UNIT UNDERSTANDINGS
b. 37 = 5 – 2x2
c. 12x2 = 5x + 9
3. Solve the quadratic inequalities algebraically and
check the solutions by graphing.
a. 2x2 ≤ x + 3
b. 14x + 15 < 8x2
Standard(s): 2A.1B , 2A.1C , 2A.1D , 2A.1E , 2A.1F , 2A.1G , 2A.4F , 2A.4H , 2A.7A
ELPS.c.1C , ELPS.c.1E , ELPS.c.2D , ELPS.c.4H
, ELPS.c.4K , ELPS.c.5B , ELPS.c.5G
Algebraic Reasoning
Algebra II Unit 05 PA 04
For each problem create a graphic organizer that
includes a table, graph, and representative quadratic
equation or inequality. Also include appropriate
calculations and solutions to answer questions and
make predictions.
Equations
Expressions
Inequalities
Solve
Functions
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Quadratic functions can be used to model real-world problem situations
by analyzing collected data, key attributes, and various representations
in order to interpret and make predictions and critical judgments.
What representations can be used to display quadratic function
models?
What key attributes identify a quadratic parent function model?
What are the connections between the key attributes of a
Page 8 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
1. Greg was practicing his diving on the 3 meter
spring board into a 5 meter pool. When he left the
board, he was 3 meters above the water. The table
below shows his height above the water as a
function of the elapsed time.
OVERARCHING CONCEPTS
UNIT CONCEPTS
Attributes of Functions
Non-Linear Functions
Associated Mathematical Processes
Application
Tools and Techniques
Problem Solving Model
Communication
Representations
Relationships Justification
a. How far from the water will Greg be after 1.25
seconds?
SUGGESTED DURATION : 14 days
UNIT UNDERSTANDINGS
quadratic function model and the real-world problem situation?
How can quadratic function representations be used to interpret
and make predictions and critical judgments in terms of the
problem situation?
Equations and inequalities can be used to model and solve real-world
problem situations.
How are real-world problem situations identified as ones that
can be modeled by quadratic equations and inequalities?
How are quadratic equations used to model problem situations?
How are quadratic inequalities used to model problem
situations?
What methods can be used to solve quadratic equations?
What methods can be used to solve quadratic inequalities?
What are the advantages and disadvantages of various methods
used to solve quadratic equations and inequalities?
What methods can be used to justify the reasonableness of
solutions to quadratic equations and inequalities?
b. When will Greg be 4 meters above the water?
c. When will Greg hit the water?
2. A reflecting pool in the new park is to be 30 feet
wide and 90 feet long. A uniform walkway is to be
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Page 9 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
SUGGESTED DURATION : 14 days
UNIT UNDERSTANDINGS
built around the entire pool. If the recreation
committee does not want the combined area to
exceed 4000 square feet, what are the possible
widths of the walkway?
Standard(s): 2A.1A , 2A.1B , 2A.1C , 2A.1D , 2A.1E , 2A.1F , 2A.1G , 2A.4E , 2A.4F , 2A.4H
ELPS.c.1C , ELPS.c.2D , ELPS.c.3D , ELPS.c.4H
, ELPS.c.4K , ELPS.c.5B
Algebraic Reasoning
Algebra II Unit 05 PA 05
For each problem create a graphic organizer that
includes various representations of the system of
equations (a linear equation and a quadratic equation),
including a table, graph, and representative equations.
Also include appropriate calculations and solutions to
answer questions and make predictions and justify the
reasonableness of the solutions in terms of the
problem situation.
Equations
Solve
Systems of Equations
Systems of equations can be used to model and solve real-world
problem situations.
How are systems of equations used to model problem
situations?
What methods can be used to solve systems of equations?
Functions
Attributes of Functions
Linear Functions
Non-Linear Functions
Associated Mathematical Processes
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Systems of equations in two variables consisting of a linear equation
and a quadratic equation can be used to model real-world problem
situations by analyzing the problem situation and various
representations in order to interpret and make predictions and critical
judgments.
Page 10 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
1. When two twin brothers graduated from high school
in 1990, their grandfather gave each grandson
$2,500 to invest. Being competitive, the siblings
decided to invest their money individually, and
come back years later to see whose investment
was the most successful. The oldest twin brother
put the $2,500 into the bank, adding $550 each
year thereafter. The youngest twin brother put his
money into technology stocks whose value climbed
according to a parabolic curve until it peaked at
$45,600 in 2002. By 2014, the stocks had fallen
back to the original value of $2,500.
OVERARCHING CONCEPTS
UNIT CONCEPTS
Application
Tools and Techniques
Problem Solving Model
Communication
Representations
Relationships Justification
SUGGESTED DURATION : 14 days
UNIT UNDERSTANDINGS
What representations can be used to display the system of
equations?
How can the representations of the system of equations be
used to interpret and make predictions and critical judgments in
terms of the problem situation?
How can solutions be justified for reasonableness in terms of
the problem situation?
a. Besides the initial investment at time zero,
approximately when were the twin brothers’
investment values the same, and approximately
what was the investment value?
b. When was the investment value of the younger
twin brother higher than the investment value of
the older twin brother?
c. When was the investment value of the older twin
brother higher than the investment value of the
younger twin brother?
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Page 11 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
SUGGESTED DURATION : 14 days
UNIT UNDERSTANDINGS
2. Tammy is going skydiving and plans to have her
best friend Sukie shoot a pillow to her using a
pillow cannon from the roof of a 1,000 foot high-rise
building so that Tammy could use the pillow to
cushion her landing. When the plane reaches an
altitude of 6,000 feet, Tammy jumps and deploys
her parachute, falling at a constant rate of
approximately 50 feet per second. Simultaneously,
Sukie fires the pillow cannon. The pillow leaves the
cannon at an initial velocity of 550 feet per second.
The height of the pillow over time could be found
using the formula, h = –16x2 + 550x + 1000.
Tammy missed the pillow as it went up, but caught
it on its way down. At what elapsed time and height
did Tammy catch the pillow?
Standard(s): 2A.1A , 2A.1B , 2A.1C , 2A.1D , 2A.1E , 2A.1F , 2A.1G , 2A.3A , 2A.3C , 2A.3D
ELPS.c.1C , ELPS.c.2D , ELPS.c.3D , ELPS.c.4H
, ELPS.c.4K , ELPS.c.5B
MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS
Misconceptions:
Some students may think that the terms zeros, x-intercepts, roots, and solutions are all the same and can be used interchangeably rather than understanding that when
an equation is set equal to zero, these will be equivalent, but not at other times. Roots and solutions pertain to equations, while x-intercepts and zeros pertain to
functions.
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Page 12 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
Some students may think that in order to be a complex number, the number must contain an imaginary part rather than that all numbers can be written in complex form,
e.g., 25 can be written as 25 + 0i, and its conjugate is 25 – 0i.
UNIT VOCABULARY
Complex conjugates – complex numbers having the same real part but an opposite imaginary part
Complex number – sum of a real number and an imaginary number, usually written in the form a + bi
Directrix – horizontal or vertical line not passing through the focus whose distance from the vertex is |p| and is perpendicular to the axis of symmetry
Focus – point not on the directrix whose distance from the vertex is |p| and lies on the axis of symmetry
Imaginary number – a number in the form of bi where b is a real number and i = |p| – distance from vertex to directrix or distance from vertex to focus
Parabola – the locus of points, P, such that the distance from P to a point F (the focus) is equal to the distance from P to a line q (the directrix)
x-intercept(s) – x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
Zeros – the value(s) of x such that the y value of the relation equals zero; the x-intercepts
Related Vocabulary:
Axis of symmetry
Completing the square
Discriminant
Factoring
Fundamental Theorem of Algebra
Gaussian method
Horizontal shift
Inverse matrix
Locus of points
Maximum
Minimum
Operations of complex numbers
Quadratic equation
Quadratic formula
Quadratic function
Quadratic inequality
Quadratic regression
rref
Real numbers
Roots
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Solutions
Standard form, f(x)= ax2 + bx + c
Substitution method
Symmetric point
Transformation
Vertex
Vertex form, f(x)= a(x - h)2 + k
Vertical compression
Vertical shift
Vertical stretch
Page 13 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
UNIT ASSESSMENT ITEMS
Unit Assessment Items that have been published by
your district may be accessed through Search All
Components in the District Resources tab.
Assessment items may also be found using the
Assessment Creator if your district has granted access
to that tool.
SUGGESTED DURATION : 14 days
SYSTEM RESOURCES
OTHER RESOURCES
Mathematics Algebra II TEKS Supporting
Information
Mathematics Concepts Tree
STAAR Algebra II Mathematics Enhanced Blue
Print
Texas Education Agency – Revised Mathematics
TEKS: Side-by-Side TEKS Comparison
Texas Education Agency – Texas College and
Career Readiness Standards
Texas Education Agency – Algebra I Reference
Materials
Texas Education Agency - Revised Mathematics
TEKS: Vertical Alignment Charts
Texas Instruments - Graphing Calculator Tutorials
Texas Education Agency – Mathematics Curriculum
Texas Education Agency – Assessment
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Bold black text in italics: Knowledge and
Skills Statement (TEKS)
Bold black text: Student Expectation (TEKS)
Bold red text in italics: Student Expectation
identified by TEA as a Readiness Standard for
STAAR
Bold green text in italics: Student Expectation
identified by TEA as a Supporting Standard for
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Blue text: Supporting information / Clarifications from TCMPC (Specificity)
Blue text in italics: Unit-specific clarification
Black text: Texas Education Agency (TEA); Texas College and Career Readiness Standards
(TxCCRS)
Page 14 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
STAAR
Strike-through: Indicates portions of the Student
Expectation that are not included in this unit but
are taught in previous or future unit(s)
2A.1
Mathematical process standards. The student uses
mathematical processes to acquire and
demonstrate mathematical understanding. The
student is expected to:
2A.1A
Apply mathematics to problems arising in
everyday life, society, and the workplace.
Apply
MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Including, but not limited to:
Mathematical problem situations within and between disciplines
Everyday life
Society
Workplace
Note(s): The mathematical process standards may be applied to all content standards as appropriate.
TxCCRS:
X. Connections
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Page 15 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
2A.1B
Use a problem-solving model that incorporates
analyzing given information, formulating a plan
or strategy, determining a solution, justifying the
solution, and evaluating the problem-solving
process and the reasonableness of the solution.
Use
A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION,
FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION,
AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE
SOLUTION
Including, but not limited to:
Problem-solving model
Analyze given information
Formulate a plan or strategy
Determine a solution
Justify the solution
Evaluate the problem-solving process and the reasonableness of the solution
Note(s): The mathematical process standards may be applied to all content standards as appropriate.
TxCCRS:
VIII. Problem Solving and Reasoning
2A.1C
Select tools, including real objects,
manipulatives, paper and pencil, and technology
as appropriate, and techniques, including mental
math, estimation, and number sense as
appropriate, to solve problems.
Select
TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY
AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER
SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Including, but not limited to:
Appropriate selection of tool(s) and techniques to apply in order to solve problems
Tools
Real objects
Manipulatives
Paper and pencil
Technology
Techniques
Mental math
Estimation
Number sense
Note(s): The mathematical process standards may be applied to all content standards as appropriate.
TxCCRS:
VIII. Problem Solving and Reasoning
2A.1D
Communicate mathematical ideas, reasoning,
and their implications using multiple
representations, including symbols, diagrams,
graphs, and language as appropriate.
Communicate
MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE
REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS
APPROPRIATE
Including, but not limited to:
Mathematical ideas, reasoning, and their implications
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Multiple representations, as appropriate
Symbols
Diagrams
Graphs
Language
Note(s): The mathematical process standards may be applied to all content standards as appropriate.
TxCCRS:
IX. Communication and Representation
2A.1E
Create and use representations to organize,
record, and communicate mathematical ideas.
Create, Use
REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
Representations of mathematical ideas
Organize
Record
Communicate
Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being
communicated
Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
The mathematical process standards may be applied to all content standards as appropriate.
TxCCRS:
IX. Communication and Representation
2A.1F
Analyze mathematical relationships to connect
and communicate mathematical ideas.
Analyze
MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS
Including, but not limited to:
Mathematical relationships
Connect and communicate mathematical ideas
Conjectures and generalizations from sets of examples and non-examples,
patterns, etc.
Current knowledge to new learning
Note(s): The mathematical process standards may be applied to all content standards as appropriate.
TxCCRS:
X. Connections
2A.1G
Display, explain, or justify mathematical ideas
and arguments using precise mathematical
language in written or oral communication.
Display, Explain, Justify
MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN
WRITTEN OR ORAL COMMUNICATION
Including, but not limited to:
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Mathematical relationships
Connect and communicate mathematical ideas
Conjectures and generalizations from sets of examples and non-examples,
patterns, etc.
Current knowledge to new learning
Note(s): The mathematical process standards may be applied to all content standards as appropriate.
TxCCRS:
IX. Communication and Representation
2A.3
Systems of equations and inequalities. The student
applies mathematical processes to formulate
systems of equations and inequalities, use a variety
of methods to solve, and analyze reasonableness of
solutions. The student is expected to:
2A.3A
Formulate systems of equations, including
systems consisting of three linear equations in
three variables and systems consisting of two
equations, the first linear and the second
quadratic. Readiness Standard
Formulate
SYSTEMS OF EQUATIONS, INCLUDING SYSTEMS CONSISTING OF THREE LINEAR EQUATIONS
IN THREE VARIABLES AND SYSTEMS CONSISTING OF TWO EQUATIONS, THE FIRST LINEAR
AND THE SECOND QUADRATIC
Including, but not limited to:
Systems of linear equations
Two equations in two variables
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Three equations in three variables
Systems of one linear equation and one quadratic equation in two variables
Real-world problem situations
Note(s):
Grade Level(s):
Algebra I solved systems of two linear equations in two variables using graphs, tables,
and algebraic methods.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
II. Algebraic Reasoning
D1 – Interpret multiple representations of equations and relationships.
D2 – Translate among multiple representations of equations and relationships.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
2A.3C
Solve, algebraically, systems of two equations in
two variables consisting of a linear equation and
a quadratic equation. Supporting Standard
Solve
SYSTEMS OF TWO EQUATIONS IN TWO VARIABLES CONSISTING OF A LINEAR EQUATION AND
A QUADRATIC EQUATION, ALGEBRAICALLY
Including, but not limited to:
Two equations in two variables
One linear equation
One quadratic equation
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Methods for solving systems of equations consisting of one linear equation and one quadratic
equation
Tables
Common points on tables
Graphs
Identification of possible solutions in terms of points of intersection
Algebraic methods
Substitution of linear equation into quadratic
Solve by factoring
Solve by quadratic formula
Solve by completing the square
Note(s):
Grade Level(s):
Algebra I solved systems of two linear equations in two variables using graphs, tables,
and algebraic methods.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
II. Algebraic Reasoning
A1 – Explain and differentiate between expressions and equations using words
such as “solve,” “evaluate,” and “simplify.”
C1 – Recognize and use algebraic (field) properties, concepts, procedures, and
algorithms to solve equations, inequalities, and systems of linear equations.
D1 – Interpret multiple representations of equations and relationships.
D2 – Translate among multiple representations of equations and relationships.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
2A.3D
Determine the reasonableness of solutions to
systems of a linear equation and a quadratic
equation in two variables. Supporting Standard
Determine
THE REASONABLENESS OF SOLUTIONS TO SYSTEMS OF A LINEAR EQUATION AND A
QUADRATIC EQUATION IN TWO VARIABLES
Including, but not limited to:
Types of equations in system
Two equations in two variables
One linear equation
One quadratic equation
Justification of reasonableness of solutions to systems of equations
Tables
Graphs
Substitution of solutions into original functions
Restriction of solutions in terms of real-world problem situations
Verbal description in terms of real-world problem situations
Note(s):
Grade Level(s):
Algebra I solved systems of two linear equations in two variables using graphs, tables,
and algebraic methods.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
II. Algebraic Reasoning
C1 – Recognize and use algebraic (field) properties, concepts, procedures, and
algorithms to solve equations, inequalities, and systems of linear equations.
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
D1 – Interpret multiple representations of equations and relationships.
D2 – Translate among multiple representations of equations and relationships.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections 2A.4
Quadratic and square root functions, equations,
and inequalities. The student applies mathematical
processes to understand that quadratic and square
root functions, equations, and quadratic
inequalities can be used to model situations, solve
problems, and make predictions. The student is
expected to:
2A.4A
Write the quadratic function given three specified
points in the plane. Supporting Standard
Write
THE QUADRATIC FUNCTION GIVEN THREE SPECIFIED POINTS IN THE PLANE
Including, but not limited to:
3 x 3 system of three linear equations in three variables
Determination of a linear system of three equations in three variables using the three
points and the standard form of the quadratic function, ax2 + bx + c = y
Methods for solving the linear system of three equations in three variables
Substitution
Gaussian elimination
Graphing calculator technology
Inverse matrix
rref
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Quadratic regression using the graphing calculator
Three points required
Correlation of determination, or r2 value, closer to ±1, the better the fit of the regression
equation
Note(s):
Grade Level(s):
Algebra II solves systems of three linear equations in three variables using various
methods.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
VII. Functions
A2 – Recognize and distinguish between different types of functions.
C1 – Apply known function models.
C2 – Develop a function to model a situation.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
2A.4B
Write the equation of a parabola using given
attributes, including vertex, focus, directrix, axis
of symmetry, and direction of opening. Readiness Standard
Write
THE EQUATION OF A PARABOLA USING GIVEN ATTRIBUTES, INCLUDING VERTEX, FOCUS,
DIRECTRIX, AXIS OF SYMMETRY, AND DIRECTION OF OPENING
Including, but not limited to:
Parabola – the locus of points, P, such that the distance from P to a point F (the focus) is equal
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
to the distance from P to a line q (the directrix)
Quadratic equation representations
Standard form
Vertical axis of symmetry: y = ax2 + bx + c
Horizontal axis of symmetry: x = ay2 + by + c
Vertex form
Vertical axis of symmetry: y = a(x – h)2 + k
Horizontal axis of symmetry: x = a(y – k )2 + h
Parabola (conic form)
Vertical axis of symmetry: (x – h)2 = 4p(y – k )
Horizontal axis of symmetry: (y – k )2 = 4p(x – h)
Connection between a and p in the vertex form and parabola (conic form)
a = Attributes of a parabola
Vertex: (h, k )
Axis of symmetry
Vertical axis of symmetry for a parabola that opens up or down: x = h
Horizontal axis of symmetry for a parabola that opens to the right or to the
left: y = k
Positive value of a or p, the parabola opens up or to the right
Negative value of a or p, the parabola opens down or to the left
|p| = distance from vertex to directrix or distance from vertex to focus
Directrix – horizontal or vertical line not passing through the focus whose distance from
the vertex is |p| and is perpendicular to the axis of symmetry
Focus – point not on the directrix whose distance from the vertex is |p| and lies on the
axis of symmetry
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Note(s):
Grade Level(s):
Algebra I wrote quadratic equations in vertex form (f(x) = a(x – h)2 + k), and rewrote from
vertex form to standard form (f(x) = ax2 + bx + c).
Precalculus will address parabolas as conic sections.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
III. Geometric Reasoning
B2 – Identify the symmetries of a plane figure.
C1 – Make connections between geometry and algebra.
VII. Functions
A2 – Recognize and distinguish between different types of functions.
B1 – Understand and analyze features of a function.
B2 – Algebraically construct and analyze new functions.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
2A.4D
Transform a quadratic function f(x) = ax2 + bx + c
to the form f(x) = a(x - h)2 + k to identify the
different attributes of f(x). Supporting Standard
Transform
A QUADRATIC FUNCTION f(x)= ax2 + bx + c TO THE FORM f(x)= a(x – h)2 + k
Including, but not limited to:
Forms of quadratic functions
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Standard form: f(x) = ax2 + bx + c
Vertex form: f(x) = a(x – h)2 + k
Completing the square to transform from the standard form f(x) = ax2 + bx + c to vertex form f(x)
= a(x – h)2 + k
To Identify
THE DIFFERENT ATTRIBUTES OF f(x)
Including, but not limited to:
Attributes from the vertex form, f(x) = a(x – h)2 + k
Vertex of the function, (h, k )
Minimum point of function if a > 0
Maximum point of function if a < 0
Axis of symmetry, x = h
Attributes from the standard form, f(x) = ax2 + bx + c
Vertex of the function, Minimum point of function if a > 0
Maximum point of function if a < 0
y-intercept, c
Axis of symmetry, x = Note(s):
Grade Level(s):
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Page 28 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Algebra I wrote quadratic equations in vertex form (f(x) = a(x – h)2 + k), and rewrote from
vertex form to standard form (f(x) = ax2 + bx + c).
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
III. Geometric Reasoning
B2 – Identify the symmetries of a plane figure.
C1 – Make connections between geometry and algebra.
VII. Functions
B1 – Understand and analyze features of a function.
B2 – Algebraically construct and analyze new functions.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
2A.4E
Formulate quadratic and square root equations
using technology given a table of data.
Supporting Standard
Formulate
QUADRATIC EQUATIONS USING TECHNOLOGY GIVEN A TABLE OF DATA
Including, but not limited to:
Data collection activities with and without technology
Data modeled by quadratic functions
Real-world problem situations
Real-world problem situations modeled by quadratic functions
Data tables with at least three data points
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Technology methods
Transformations of f(x) = x2
Solving three by three matrix to determine a, b, and c for f(x) = ax2 + bx + c
Quadratic regression
Note(s):
Grade Level(s):
Algebra I solved quadratic equations having real solutions using tables, graphs, factoring,
completing the square, quadratic formula and technology.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
VII. Functions
B1 – Understand and analyze features of a function.
B2 – Algebraically construct and analyze new functions.
C2 – Develop a function to model a situation.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
2A.4F
Solve quadratic and square root equations.
Solve
Readiness Standard
QUADRATIC EQUATIONS
Including, but not limited to:
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Methods for solving quadratic equations with and without technology
Tables
Zeros – the values of x such that f(x) = 0; the x-intercepts
Domain values with equal range values
Graphs
x-intercept – x-coordinate of a point at which the relationship crosses the x-axis,
meaning the y-coordinate equals zero, (x, 0)
Zeros – the value(s) of x such that the y value of the relation equals zero
Algebraic methods
Factoring
Completing the square
Quadratic formula, x = The discriminant, b 2 – 4ac, can be used to analyze types of solutions for
quadratic equations.
b 2 – 4ac = 0, one rational double root
b 2 – 4ac > 0 and perfect square, two rational roots
b 2 – 4ac > 0 and not perfect square, two irrational roots
(conjugates)
b 2 – 4ac < 0, two imaginary roots (conjugates)
Connections between solutions and roots of quadratic equations to the zeros and x-intercepts of
the related function
Complex number system
Complex number – sum of a real number and an imaginary number, usually written in the
form a + bi
Imaginary number – a number in the fomr of bi where b is a real number and i = i 2 = –1
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
i = Complex conjugates – complex numbers having the same real part but an
opposite imaginary part
a + bi and a – bi
Operations with complex numbers, with and without technology
Complex solutions for quadratic equations
One real solution
One rational double root
Two real solutions
Two rational roots
Two irrational root conjugates
Two complex solutions
Two complex root conjugates
Reasonableness of solutions
Note(s):
Grade Level(s):
Algebra I solved quadratic equations having real solutions using tables, graphs, factoring,
completing the square, and the quadratic formula.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
I. Numeric Reasoning
A2 – Define and give examples of complex numbers.
B1 – Perform computations with real and complex numbers.
II. Algebraic Reasoning
A1 – Explain and differentiate between expressions and equations using words
such as “solve,” “evaluate,” and “simplify.”
C1 – Recognize and use algebraic (field) properties, concepts, procedures, and
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
algorithms to solve equations, inequalities, and systems of linear equations.
D1 – Interpret multiple representations of equations and relationships.
D2 – Translate among multiple representations of equations and relationships.
III. Geometric Reasoning
C1 – Make connections between geometry and algebra.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
2A.4H
Solve quadratic inequalities. Supporting Standard
Solve
QUADRATIC INEQUALITIES
Including, but not limited to:
Methods for solving quadratic inequalities with and without technology
Graphs
Tables
Algebraic methods
Factoring
Completing the square
Quadratic formula
Testing and identifying acceptable regions on a number line
Graphical analysis of solution sets for quadratic inequalities
One-dimensional on a number line
Two-dimensional on a coordinate plane
Comparison of solution sets of equations and inequalities
Comparison of one-dimensional solutions and two-dimensional solutions, e.g. intervals versus
points
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Reasonableness of solutions
Note(s):
Grade Level(s):
Algebra I solved quadratic equations.
Algebra II introduces quadratic inequalities.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
II. Algebraic Reasoning
A1 – Explain and differentiate between expressions and equations using words
such as “solve,” “evaluate,” and “simplify.”
C1 – Recognize and use algebraic (field) properties, concepts, procedures, and
algorithms to solve equations, inequalities, and systems of linear equations.
D1 – Interpret multiple representations of equations and relationships.
D2 – Translate among multiple representations of equations and relationships.
III. Geometric Reasoning
C1 – Make connections between geometry and algebra.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
2A.7
Number and algebraic methods. The student
applies mathematical processes to simplify and
perform operations on expressions and to solve
equations. The student is expected to:
2A.7A
Add, subtract, and multiply complex numbers. Supporting Standard
Add, Subtract, Multiply
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
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TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
COMPLEX NUMBERS
Including, but not limited to:
Complex number system
The complex number system, C, is composed of real and imaginary numbers.
Real numbers, , are composed of rational numbers, Q, and irrational numbers, – Q.
Rational numbers, Q, are composed of integers, , whole numbers, N 0,
and natural numbers, N.
Complex number – sum of a real number and an imaginary number, usually written in the
form a + bi
Real part of a complex number, a
Imaginary part of a complex number, b
Imaginary number – a number in the fomr of bi where b is a real number
and i = Imaginary number unit, i, is a number whose square equals –1; therefore,
the = i.
If x is a non-negative, real number = i
.
Complex conjugates – complex numbers having the same real part but an opposite
imaginary part
When complex conjugates are added or multiplied the imaginary part equals 0.
Operations with complex numbers
Addition/subtraction of complex numbers
Real parts combine with real parts and imaginary parts combine with imaginary
parts.
Multiplication of complex numbers
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INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
SUGGESTED DURATION : 14 days
TEKS#
TEKS
UNIT LEVEL SPECIFICITY
SE#
Distribute and collect like terms.
The imaginary unit, i, can only have a power of 1.
Any i² units must be converted to –1.
Note(s):
Grade Level(s):
Algebra II introduces the system of complex numbers and operations with complex
numbers.
Various mathematical process standards will be applied to this student expectation as
appropriate.
TxCCRS:
I. Numeric Reasoning
B1 – Perform computations with real and complex numbers.
II. Algebraic Reasoning
B1 – Recognize and use algebraic (field) properties, concepts, procedures, and
algorithms to combine, transform, and evaluate expressions (e.g. polynomials,
radicals, rational expressions).
D1 – Interpret multiple representations of equations and relationships.
VIII. Problem Solving and Reasoning
IX. Communication and Representation
X. Connections
ELPS#
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language
proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of
each subject in the required curriculum.
School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated
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Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
ELPS#
SUGGESTED DURATION : 14 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.
School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the
ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.
http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 Choose appropriate ELPS to support instruction.
ELPS.c.1
The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL
to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be
linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency.
The student is expected to:
ELPS.c.1A
use prior knowledge and experiences to understand meanings in English
ELPS.c.1B
monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C
use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and
grade-level vocabulary
ELPS.c.1D
speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying
ideas by defining or describing when exact English words are not known)
ELPS.c.1E
internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept
and language attainment
ELPS.c.1F
use accessible language and learn new and essential language in the process
ELPS.c.1G
demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one
commensurate with grade-level learning expectations
ELPS.c.1H
develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and
analyzing sayings and expressions commensurate with grade-level learning expectations.
Last Updated 08/17/2015
Print Date 08/21/2016 Printed By Kendall Price, CORSICANA H IGH SCHOOL
Page 37 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
ELPS#
SUGGESTED DURATION : 14 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
ELPS.c.2
The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly
acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language
acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all
instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the
student's level of English language proficiency. The student is expected to:
ELPS.c.2A
distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B
recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant
clusters
ELPS.c.2C
learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D
monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E
use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F
listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and
language attainment
ELPS.c.2G
understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language,
and contexts are familiar to unfamiliar
ELPS.c.2H
understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I
demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken
messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level
needs.
ELPS.c.3
The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary
with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced
high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and
enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded)
commensurate with the student's level of English language proficiency. The student is expected to:
Last Updated 08/17/2015
Print Date 08/21/2016 Printed By Kendall Price, CORSICANA H IGH SCHOOL
Page 38 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
ELPS#
SUGGESTED DURATION : 14 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
ELPS.c.3A
practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce
English words in a manner that is increasingly comprehensible
ELPS.c.3B
expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing
people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using
routine language needed for classroom communication
ELPS.c.3C
speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as
more English is acquired
ELPS.c.3D
speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E
share information in cooperative learning interactions
ELPS.c.3F
ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and
expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during
extended speaking assignments
ELPS.c.3G
express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on
a variety of social and grade-appropriate academic topics
ELPS.c.3H
narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I
adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J
respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and
language attainment.
ELPS.c.4
The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the
beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level
learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated
(communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1,
certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
Last Updated 08/17/2015
Print Date 08/21/2016 Printed By Kendall Price, CORSICANA H IGH SCHOOL
Page 39 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
ELPS#
SUGGESTED DURATION : 14 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
ELPS.c.4A
learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as
recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B
recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C
develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used
routinely in written classroom materials
ELPS.c.4D
use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to
enhance comprehension of written text
ELPS.c.4E
read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F
use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm
understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly
challenging language
ELPS.c.4G
demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding
to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H
read silently with increasing ease and comprehension for longer periods
ELPS.c.4I
demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of
supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with
content area needs
ELPS.c.4J
demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between
ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content
area needs
ELPS.c.4K
demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and
performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5
The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may
be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-
Last Updated 08/17/2015
Print Date 08/21/2016 Printed By Kendall Price, CORSICANA H IGH SCHOOL
Page 40 of 41
INSTRUCTIONAL FOCUS DOCUMENT
Algebra II
TITLE : Unit 05: Quadratic Relations, Equations, and Inequalities
ELPS#
SUGGESTED DURATION : 14 days
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated
(communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1,
certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard
writing system. The student is expected to:
ELPS.c.5A
learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B
write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C
spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English
is acquired
ELPS.c.5D
edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses
commensurate with grade-level expectations as more English is acquired
ELPS.c.5E
employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F
write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in
increasingly accurate ways as more English is acquired
ELPS.c.5G
narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/17/2015
Last Updated 08/17/2015
Print Date 08/21/2016 Printed By Kendall Price, CORSICANA H IGH SCHOOL
Page 41 of 41