
Legend:  Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
 Student Expectations (TEKS) identified by TEA are in bolded, black text.
 Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
 Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
 Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a
strikethrough.

Legend:  Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
 Unitspecific clarifications are in italicized, blue text.
 Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
 A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.

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:

2A.1B 
Use a problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving 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
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
 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
Note(s):
 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 nonexamples, patterns, etc.
 Current knowledge to new learning
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxCCRS:

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:
 Mathematical ideas and arguments
 Validation of conclusions
 Displays to make work visible to others
 Diagrams, visual aids, written work, etc.
 Explanations and justifications
 Precise mathematical language in written or oral communication
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
 Three equations in three variables
 Systems of one linear equation and one quadratic equation in two variables
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
 Methods for solving systems of equations consisting of one linear equation and one quadratic equation
 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

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 realworld problem situations
 Verbal description in terms of realworld 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.
 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 × 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, ax^{2} + bx + c = y
 Methods for solving the linear system of three equations in three variables
 Substitution
 Gaussian elimination
 Graphing calculator technology
 Quadratic regression using the graphing calculator
 Three points required
 Correlation of determination, or r^{2} value, closer to ±1, the better the fit of the regression equation
Note(s):
 Grade Level(s):
 Algebra I wrote quadractic functions given real solutions and graphs of their related equations.
 Algebra I formulated a quadratic function using technology.
 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 to the distance from P to a line q(the directrix)
 Quadratic equation representations
 Standard form
 Vertical axis of symmetry: y = ax^{2} + bx + c
 Horizontal axis of symmetry: x = ay^{2} + 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
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) = ax^{2} + 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) = ax^{2} + 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)= ax^{2} + bx + c TO THE FORM f(x)= a(x – h)^{2} + k
Including, but not limited to:
 Forms of quadratic functions
 Standard form: f(x) = ax^{2} + bx + c
 Vertex form: f(x) = a(x – h)^{2} + k
 Completing the square to transform from the standard form f(x) = ax^{2} + 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
 yintercept, ah^{2} + k
 Axis of symmetry, x = h
 Attributes from the standard form, f(x) = ax^{2} + bx + c
 Vertex of the function,
 Minimum point of function if a > 0
 Maximum point of function if a < 0
 yintercept, c
 Axis of symmetry, x =
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) = ax^{2} + bx + c).
 Algebra I solved quadratic equations having real solutions by completeing the square.
 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
 Realworld problem situations
 Realworld problem situations modeled by quadratic functions
 Data tables with at least three data points
 Technology methods
 Transformations of f(x) = x^{2}
 Solving three by three matrix to determine a, b, and c for f(x) = ax^{2} + 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.
 Algebra I wrote, using technology, quadratic functions that provide a reasonable fit to date to estimate solutions and make predictions for realworld problems.
 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.
Readiness Standard

Solve
QUADRATIC EQUATIONS
Including, but not limited to:
 Methods for solving quadratic equations with and without technology
 Tables
 Zeros – the value(s) of x such that the y value of the relation equals zero
 Domain values with equal range values
 Graphs
 xintercept(s) – x coordinate of a point at which the relation crosses the xaxis, 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
 Solving equations by taking square roots
 Solving quadratic equations using absolute value
 x^{2} = 25, therefore, x = ±5
 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 xintercepts 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 form of bi where b is a real number and i =
 i ^{2} = –1
 i =
 Complex conjugates – complex numbers having the same real part but an opposite imaginary part
 Operations with complex numbers, with and without technology
 Complex solutions for quadratic equations
 One real solution
 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 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
 Solving inequalities by taking square roots
 Solving quadratic inequalities using absolute value
 x^{2} ≤ 25, x ≤ 5; therefore, –5 ≤ x ≤ 5
 Completing the square
 Quadratic formula
 Testing and identifying acceptable regions on a number line
 Graphical analysis of solution sets for quadratic inequalities
 Onedimensional on a number line
 Twodimensional on a coordinate plane
 Comparison of solution sets of equations and inequalities
 Comparison of onedimensional solutions and twodimensional solutions, e.g. intervals versus points
 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
COMPLEX NUMBERS
Including, but not limited to:
 Complex number system
 The complex number system, C, is composed of real and imaginary numbers.
 Real numbers, ℜ, are composed of rational numbers, Q, and irrational numbers, ℜ – Q.
 Rational numbers, Q, are composed of integers, Ζ, 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 form 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 nonnegative, real number .
 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
 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
