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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 11: Making Connections SUGGESTED DURATION : 10 days

Unit Overview

Introduction
This unit bundles the review of student expectations that address linear equations, inequalities, systems, and functions. The unit also includes student expectations that address the review of quadratic equations and functions and the review of exponential functions. 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 Units 01 – 05, students solved and applied linear equations, linear inequalities, and systems of linear equations and linear inequalities. Students also analyzed and applied linear functions in problem situations. In Unit 06, students investigated rules of exponents and operations and factoring of polynomial expressions. In Units 07 – 08, students solved and applied quadratic equations and analyzed and applied linear functions in problem situations.

During this Unit
Students review solving linear equations, inequalities, and systems of linear equations and inequalities. Students review writing linear equations and inequalities from given criteria and graphs. Students review application of linear functions to model real-world problem situations. Students review solving quadratic equations (taking square roots, factoring, quadratic formula, completing the square) and exponential equations (graphically). Students review writing and graphing quadratic and exponential functions from given criteria, including comparing the quadratic parent function with another function that undergoes parameter changes. Students review application of quadratic and exponential functions to model real-world problem situations.

After this Unit
Students will continue to apply their knowledge of linear, quadratic, and exponential functions and equations in subsequent courses in mathematics. Students will also extend their knowledge of functions and equations to additional function families.

Research

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. (2000). Principles and standards for school mathematics: Algebra standards for grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2002). Navigating through algebra in grades 9 – 12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2010). Developing essential understanding of functions for teaching mathematics grades 9-12. 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/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationships?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Linear functions are characterized by a constant rate of change and can be used to describe, model, and make predictions about situations.
• How can the rate of change of a linear function be determined?
• What kinds of mathematical and real-world situations can be modeled by linear functions?
• What graphs, key attributes, and characteristics are unique to linear functions?
• What pattern of covariation is associated with linear functions?
• How can the key attributes of linear functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of a linear function?
• What are the real-world meanings of the key attributes of a linear function model?
• How can the key attributes of a linear function be used to make predictions and critical judgments?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Equations and inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation or inequality.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can equations and inequalities be used to represent relationships between quantities?
• How do solutions to inequalities differ from solutions to equations?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write linear equations and linear inequalities?
• How does the given information and/or representation influence the selection of an efficient method for writing linear equations and linear inequalities?
• What methods can be used to solve linear equations and linear inequalities?
• How does the structure of the equation influence the selection of an efficient method for solving linear equations?
• How can the solutions to linear equations and linear inequalities be determined and represented?
• How are properties and operational understandings used to transform linear equations and linear inequalities?
• Systems of equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structures of the equations in the system.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can systems of equations be used to represent relationships between quantities?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write systems of equations?
• What methods can be used to solve systems of equations?
• How does the structure of the system influence the selection of an efficient method for solving the system of equations?
• How can the solutions to systems of equations be determined and represented?
• How are properties and operational understandings used to transform systems of equations?
• What kinds of algebraic and graphical relationships exist between equations in a system with …
• one solution?
• no solutions?
• infinitely many solutions?
• Systems of inequalities can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structures of the inequalities in the system.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can systems of inequalities be used to represent relationships between quantities?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write systems of inequalities?
• What methods can be used to solve systems of inequalities?
• How can the solution set of systems of inequalities be determined and represented?
• How do solution sets of systems of inequalities differ from solution sets of systems of equations?
• How are properties and operational understandings used to transform systems of inequalities?
• What kinds of algebraic and graphical relationships exist between inequalities in a system with …
• no solutions?
• infinitely many solutions?
• What relationships exist between the solution set of each inequality in a system and the solution set of the system of inequalities?
• How can the boundaries and the vertices of the solution set of a system of inequalities be used to make predictions and critical judgments in problem situations?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• Rate of change/Slope
• x- and y-intercept(s)
• Zeros
• Functions, Equations, and Inequalities
• Linear
• Patterns, Operations, and Properties
• Relations and Generalizations
• Systems of Linear Equations and Inequalities
• Linear
• Statistical Representations
• Regression methods
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy? Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life. What relationships exist within and between mathematical operations? How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies? Why is understanding the problem solving process an essential part of learning and working mathematically? Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgments in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• The ability to represent quantities in various forms develops the understanding of equivalence and allows for working flexibly with algebraic expressions in order to communicate and reason about quantities.
• How can expressions be used to represent situations?
• What mathematical conventions are used when representing expressions? Why?
• How can it be determined if two expressions are equivalent?
• How are properties and operational understandings used to generate equivalent expressions?
• Why can it be useful to simplify expressions?
• Why can it be useful to factor expressions?
• How does the structure of the expression influence the selection of an efficient method for factoring polynomial expressions?
• Equations can be written, transformed, and solved using various methods to make critical judgments, with different methods being more efficient or informative depending on the structure of the equation.
• How does knowing more than one solution strategy build mathematical flexibility?
• How can equations be used to represent relationships between quantities?
• Why must solutions be justified in terms of problem situations?
• What methods can be used to write …
• exponential equations?
• How does the given information and/or representation influence the selection of an efficient method for writing …
• exponential equations?
• What methods can be used to solve quadratic equations?
• How does the structure of the equation influence the selection of an efficient method for solving quadratic equations?
• How can the solutions to quadratic equations be determined and represented?
• How are properties and operational understandings used to transform quadratic equations?
• What kinds of algebraic and graphical relationships exist for quadratic equations with …
• two real solutions?
• one real solution?
• no real solutions?
• Functions can be combined and transformed in predictable ways to create new functions that can be used to describe, model, and make predictions about situations.
• How are functions …
• shifted?
• scaled?
• reflected?
• How do transformations affect the …
• representations
• key attributes
… of a function?
• Understanding how two quantities vary together (covariation) builds flexible functional reasoning in order to make predictions and critical judgments about the relationship.
• What are the strengths and limitations of a particular function model for a problem situation?
• How can functions be used to model problem situations efficiently?
• How can it be determined if a relationship between two variables can be represented by a function?
• How is function notation used to define and describe a function rule?
• How is function notation used to make predictions and critical judgements about the relationship?
• Different families of functions, each with their own unique characteristics, can be used to model problem situations to make predictions and critical judgments.
• Quadratic functions are characterized by a rate of change that changes at a constant rate and can be used to describe, model, and make predictions about problem situations.
• Exponential functions are characterized by a rate of change that is proportional to the value of the function and can be used to describe, model, and make predictions about problem situations.
• What kinds of mathematical and real-world situations can be modeled by …
• exponential functions?
• What graphs, key attributes, and characteristics are unique to …
• exponential functions?
• What patterns of covariation are associated with …
• exponential functions?
• How can the key attributes of quadratic and exponential functions be …
• determined?
• analyzed?
• described?
• How can key attributes be used to describe the behavior of quadratic and exponential functions?
• What are the real-world meanings of the key attributes of quadratic and exponential function models?
• How can the key attributes of quadratic and exponential functions be used to make predictions and critical judgments?
• Functions can be represented in various ways with different representations of the function highlighting different characteristics and being more useful than other representations depending on the context.
• How can functions be represented?
• What is the purpose of representing functions in various ways?
• How are function characteristics highlighted in different representations of the function?
• What are the limitations of different function representations?
• What connections can be made between multiple representations of a function?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• x- and y-intercept(s)
• Zeros
• Minimum or maximum value
• Vertex
• Axis of symmetry
• Asymptotes
• Functions and Equations
• Exponential
• Patterns, Operations, and Properties
• Relations and Generalizations
• Statistical Relationships
• Regression methods
• Transformations
• Parent functions
• Transformation effects
• Number and Algebraic Methods
• Expressions
• Polynomial
• Rational exponents
• Patterns, Operations, and Properties
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think that the negative in front of the parentheses is distributed only to the first term of the expression in parentheses rather than to all terms of the expression in parentheses.
• Some students may think that answers to both equations and inequalities are exact answers rather than correctly identifying the solutions to equations as exact answers and the solutions to inequalities as range of answers.
• Some students may think that whenever a negative is involved, the order of the inequality switches rather than only switching the order of inequality when multiplying or dividing by a negative.
• Some students may think that in the graph or table method of solving the equation, the y-value is the answer rather than the x-value.
• Some students may think that the ratio for rate of change (slope) in a linear function is , since the x variable (horizontal) always comes before the y variable (vertical), instead of the correct representation that rate of change (slope) in a linear function is .
• Some students may think that the intercept coordinate is the zero term instead of the non-zero term, since intercepts are associated with zeros. In other words, students may think (0, 4) would be the x-intercept because the 0 is in the x coordinate.
• Some students may think that the solution to a system of two linear equations in two variables is only the x-value rather than an ordered pair.
• Some students may think that the solution to a system of two linear inequalities in two variables is only one point rather than an infinite set of points in the identified region.
• Some students may think that the exponents combine when adding or subtracting two polynomials (e.g. (2x2 – 3x) + (3x2 + 9x) = (5x4 + 6x2)).
• Some students may add just half the coefficient of the middle term squared to the other side when completing the square rather than multiplying half the coefficient of the middle term squared times any value factored out before adding it to the other side.
• Some students may not connect that the root(s) or solution(s) of a quadratic equation set equal to zero is the same as the x-intercept(s) when the quadratic equation is graphed.
• Some students may think that when evaluating f(x) = –x2, the negative is included in the square rather than understanding it is a coefficient of –1 that reflects the graph of the quadratic parent function over the x-axis.
• Some students may think that the graph of the exponential parent function will at some point cross the x-axis rather than being asymptotic and only approaching the x-axis.

Unit Vocabulary

• Algebraic expression – a generalization that is a combination of variables, numbers (constants and coefficients), and operators
• Asymptote – a line that is approached and may or may not be crossed
• Axis of symmetry – line passing through the vertex of a parabola that divides the parabola into two congruent halves
• Binomial – two term expression
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Domain – set of input values for the independent variable over which the function is defined
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Linear equation in one variable – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear inequality in one variable – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Linear inequality in two variables – a relationship with a constant rate of change represented by a solution set denoted by the graph of a line, that may or may be not included in the solution, and the set of points above or below the line
• Maximum – graph opens downward, negative a value
• Minimum – graph opens upward, positive a value
• Parent functions – set of basic functions from which related functions are derived by transformations
• Perfect square trinomial - first term a perfect square, third term a perfect square, middle term double the product of the square root of the first and last terms
• Quadratic equation in one variable – a second-degree polynomial function that can be described in standard form by 0 = ax2 + bx + c, where a ≠ 0
• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic, or U-shaped
• Range – set of output values for the dependent variable over which the function is defined
• Slope of a line – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• Symmetric points – the image and pre-image points reflected across the axis of symmetry of the parabola
• Trinomial – three term expression
• Vertex – highest (maximum) or lowest (minimum) point on the graph of a parabola
• x-intercept(s)x coordinate of a point at which the relation crosses the x-axis, meaning the y coordinate equals zero, (x, 0)
• y-intercept(s)y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Zeros – the value(s) of x such that the y value of the relation equals zero

Related Vocabulary:

 Attributes of exponential functions Attributes of linear functions Attributes of quadratic functions Coinciding lines Completing the square Exponent Factoring Greatest common factor (GCF) Horizontal compression Horizontal shift Horizontal stretch Intersection Parabola Parallel Parameter change Perpendicular Point-slope form of a linear equation y – y1 = m(x – x1) Quadratic formula Scatterplot Slope-intercept form of a linear equation y = mx + b Solutions to systems of linear equations Solutions to systems of linear inequalities Standard form of a quadratic equation f(x) = ax2 + bx + c Standard form of a linear equation Ax + By = C Vertex form of a quadratic equation f(x) = a(x– h)2 + k Vertical compression Vertical shift Vertical stretch
Unit Assessment Items System Resources Other Resources

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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 Center if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway –Resources Aligned to Algebra I Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE 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.
• Student Expectations (TEKS) are labeled Process standards 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 strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific 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.
TEKS# SE# TEKS SPECIFICITY
A.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
A.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

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
A.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.
Process Standard

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
A.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.
Process Standard

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
A.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

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
A.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

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
A.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

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
A.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

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
A.2 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using properties of linear functions to write and represent in multiple ways, with and without technology, linear equations, inequalities, and systems of equations. The student is expected to:
A.2A Determine the domain and range of a linear function in mathematical problems; determine reasonable domain and range values for real-world situations, both continuous and discrete; and represent domain and range using inequalities.

Determine

THE DOMAIN AND RANGE OF A LINEAR FUNCTION IN MATHEMATICAL PROBLEMS AND REASONABLE DOMAIN AND RANGE VALUES FOR REAL-WORLD SITUATIONS, BOTH CONTINUOUS AND DISCRETE

Represent

THE DOMAIN AND RANGE OF A LINEAR FUNCTION USING INEQUALITIES

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Domain and range of linear functions in mathematical problem situations
• Domain – set of input values for the independent variable over which the function is defined
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Range – set of output values for the dependent variable over which the function is defined
• Inequality representations
• Verbal description
• Ex: x is all real numbers less than five.
• Ex: x is all real numbers.
• Ex: y is all real numbers greater than –3 and less than or equal to 6.
• Ex: y is all integers greater than or equal to zero.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Ex: x < 5, x ∈ ℜ
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6, y ∈ ℜ
• Ex: y ≥ 0, yΖ
• Domain and range of linear functions in real-world problem situations
• Reasonable domain and range for real-world problem situations
• Comparison of domain and range of function model to appropriate domain and range for a real-world problem situation

Note(s):

• The notation ℜ represents the set of real numbers, and the notation Ζ represents the set of integers.
• Grade 6 identified independent and dependent quantities.
• Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
• Algebra I introduces the concept of domain and range of a function.
• Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
• Algebra II will introduce representing domain and range using interval and set notation.
• Precalculus will introduce piecewise functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.2C Write linear equations in two variables given a table of values, a graph, and a verbal description.

Write

LINEAR EQUATIONS IN TWO VARIABLES GIVEN A TABLE OF VALUES, A GRAPH, AND A VERBAL DESCRIPTION

Including, but not limited to:

• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Various forms linear equations in two variables
• Slope-intercept form, y = mx + b
• m is the slope.
• b is the y-intercept.
• Point-slope form, y – y1 = m(xx1)
• m is the slope.
• (x1, y1) is a given point
• Standard form, Ax + By = C; A, B, CΖ, A ≥ 0
• x and y terms are on one side of the equation and the constant is on the other side.
• Given multiple representations
• Table of values
• Graph
• Verbal description

Note(s):

• Middle School introduced using multiple representations for linear relationships.
• Grade 8 represented linear proportional and non-proportional relationships in tables, graphs, and equations in the form y = mx + b.
• Algebra I introduces the use of standard form and point-slope form to represent linear relationships.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new 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
A.2E Write the equation of a line that contains a given point and is parallel to a given line.
Supporting Standard

Write

THE EQUATION OF A LINE THAT CONTAINS A GIVEN POINT AND IS PARALLEL TO A GIVEN LINE

Including, but not limited to:

• Multiple representations
• Graph
• Verbal description
• Characteristics of parallel lines
• In the same plane
• Do not intersect
• Same distance apart
• Slopes are equal, my1 = my2 , where my1 is the slope of line 1 and my2 is the slope of line 2.
• m = 0, y = #
• m = undefined, x = #
• Various forms of the equation of a line
• Slope-intercept form, y = mx + b
• Point-slope form, yy1 = m(xx1)
• Standard form, Ax + By = C

Note(s):

• Previous grade levels introduced slope and meaning of parallel separately.
• Algebra I introduces the concept of parallel lines in terms of slope.
• Geometry will write the equation of a line parallel to a given line passing through a given point to determine geometric relationships on a coordinate plane.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.2F Write the equation of a line that contains a given point and is perpendicular to a given line.
Supporting Standard

Write

THE EQUATION OF A LINE THAT CONTAINS A GIVEN POINT AND IS PERPENDICULAR TO A GIVEN LINE

Including, but not limited to:

• Multiple representations
• Graph
• Verbal description
• Characteristics of perpendicular lines
• In the same plane
• Intersecting lines
• Intersect to form four 90o angles
• Slopes are negated (opposite) reciprocals, my2 = • Given line y = #, where m = 0, the perpendicular line is x = # with an undefined slope.
• Given line x = #, where m = undefined, the perpendicular line is y = # with m = 0.
• Various forms of the equation of a line
• Slope-intercept form, y = mx + b
• Point-slope form, yy1 = m(xx1)
• Standard form, Ax + By = C

Note(s):

• Previous grade levels introduced slope and meaning of perpendicular separately.
• Algebra I introduces the concept of perpendicular lines in terms of slope.
• Geometry will write the equation of a line perpendicular to a given line passing through a given point to determine geometric relationships on a coordinate plane.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.2G Write an equation of a line that is parallel or perpendicular to the X or Y axis and determine whether the slope of the line is zero or undefined.
Supporting Standard

Write

AN EQUATION OF A LINE THAT IS PARALLEL OR PERPENDICULAR TO THE X OR Y AXIS

Determine

WHETHER THE SLOPE OF A LINE PARALLEL OR PERPENDICULAR TO THE X OR Y AXIS IS ZERO OR UNDEFINED

Including, but not limited to:

• Write equations for parallel or perpendicular lines
• Equations of lines parallel or perpendicular to the x-axis
• Parallel to the x-axis, y = #
• Perpendicular to the x-axis, x = #
• Equations of lines parallel or perpendicular to the y-axis
• Parallel to the y-axis, x = #
• Perpendicular to the y-axis, y = #
• Equations of lines parallel or perpendicular to lines that are parallel or perpendicular to the x-axis
• Parallel to a line parallel to the x-axis, y = #
• Parallel to a line perpendicular to the x-axis, x = #
• Perpendicular to a line parallel to the x-axis, x = #
• Perpendicular to a line perpendicular to the x-axis, y = #
• Equations of lines parallel or perpendicular to lines that are parallel or perpendicular to the y-axis
• Parallel to a line parallel to the y-axis, x = #
• Parallel to a line perpendicular to the y-axis, y = #
• Perpendicular to a line parallel to the y-axis, y = #
• Perpendicular to a line perpendicular to the y-axis, x = #
• Determine whether the slope of a parallel or perpendicular line is zero or undefined
• Slope of lines parallel to the x-axis, m = 0
• Slope of lines parallel to the y-axis, m is undefined
• Slope of lines perpendicular to the x-axis, m is undefined
• Slope of lines perpendicular to the y-axis, m = 0
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel or perpendicular to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel or perpendicular to the y-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is parallel to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is perpendicular to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is parallel to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is perpendicular to the x-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is parallel to the y-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is parallel to a line that is perpendicular to the y-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is parallel to the y-axis.
• Write the equation of a line and determine the slope of the line that passes through a point and is perpendicular to a line that is perpendicular to the y-axis.
• Generalizations
• A line parallel to the x-axis and perpendicular to the y-axis has a slope of zero.
• A line parallel to the y-axis and perpendicular to the x-axis has an undefined slope.

Note(s):

• Previous grade levels introduced slope and meaning of parallel and perpendicular separately.
• Algebra I introduces the concepts of parallel and perpendicular lines in terms of slope.
• Geometry will write the equation of a line parallel or perpendicular to a given line passing through a given point to determine geometric relationships on a coordinate plane.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.2H Write linear inequalities in two variables given a table of values, a graph, and a verbal description.
Supporting Standard

Write

LINEAR INEQUALITIES IN TWO VARIABLES GIVEN A TABLE OF VALUES, A GRAPH, AND A VERBAL DESCRIPTION

Including, but not limited to:

• Linear inequality in two variables – a relationship with a constant rate of change represented by a solution set denoted by the graph of a line, that may or may not be included in the solution, and the set of points above or below the line
• Inequality notation
• Less than, <, dashed line with shading below the graph of the line
• Greater than, >, dashed line with shading above the graph of the line
• Less than or equal to, ≤, solid line with shading below the graph of the line
• Greater than or equal to, ≥, solid line with shading above the graph of the line
• For vertical lines, greater than shades the right side of the graph and less than shades the left side of the graph.
• Given multiple representations
• Table of values
• Graph
• Verbal description

Note(s):

• Middle School used multiple representations for linear relationships.
• Grade 8 solved problems using one-variable inequalities.
• Algebra I introduces linear inequalities in two variables given various representations.
• Algebra II will continue systems of two linear inequalities in two variables and extend to systems of more than two linear inequalities in two variables.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B2 – Algebraically construct and analyze new functions.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.2I Write systems of two linear equations given a table of values, a graph, and a verbal description.

Write

SYSTEMS OF TWO LINEAR EQUATIONS GIVEN A TABLE OF VALUES, A GRAPH, AND A VERBAL DESCRIPTION

Including, but not limited to:

• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Characteristics of 2 × 2 systems of linear equations
• Two equations
• Two variables
• Given multiple representations
• Table of values
• Graph
• Verbal description

Note(s):

• Middle School used multiple representations for linear relationships.
• Algebra I formally introduces systems of two linear equations in two variables.
• Algebra II will introduce systems of three linear equations in three variables and systems of one linear equation and one quadratic equation.
• 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
A.3 Linear functions, equations, and inequalities. The student applies the mathematical process standards when using graphs of linear functions, key features, and related transformations to represent in multiple ways and solve, with and without technology, equations, inequalities, and systems of equations. The student is expected to:
A.3B Calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems.

Calculate

THE RATE OF CHANGE OF A LINEAR FUNCTION REPRESENTED TABULARLY, GRAPHICALLY, OR ALGEBRAICALLY IN CONTEXT OF MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Connections between slope and rate of change
• Rate of change by various methods
• Tabular method by analyzing rate of change in x and y values: m = = or m = • Graphical method by analyzing vertical and horizontal change: slope = • Algebraic method by analyzing m in y = mx + b form
• Solve equation for y
• Slope is represented by m
• Rate of change from multiple representations
• Tabular
• Graphical
• Algebraic
• Calculuation and comparison of the rate of change over specified intervals of a graph
• Meaning of rate of change in the context of real-world problem situations
• Emphasis on units of rate of change in relation to real-world problem situations

Note(s):

• Grade 8 introduced the concept of slope as a rate of change, including using the slope formula.
• Precalculus will introduce piecewise functions and their characteristics.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.3C Graph linear functions on the coordinate plane and identify key features, including x-intercept, y-intercept, zeros, and slope, in mathematical and real-world problems.

Graph

LINEAR FUNCTIONS ON THE COORDINATE PLANE

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Multiple representations
• Tabular
• Graphical
• Verbal
• Algebraic generalizations

Identify

KEY FEATURES OF LINEAR FUNCTIONS, INCLUDING x-INTERCEPT, y-INTERCEPT, ZEROS, AND SLOPE, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Linear functions in mathematical problem situations
• Linear functions in real-world problem situations
• Multiple representations
• Tabular
• Graphical
• Verbal
• Algebraic generalizations
• Characteristics of linear functions
• x-intercept – 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
• y-intercept – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• Slope of a line – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or = or • denoted as m in y = mx + b
• denoted as m in f(x) = mx + b
• Notation of linear functions
• Equation notation: y= mx + b
• Function notation: f(x) = mx + b

Note(s):

• Grades 7 and 8 introduced linear relationships using tables of data, graphs, and algebraic generalizations.
• Grade 8 introduced using tables of data and graphs to determine rate of change or slope and y-intercept.
• Algebra I introduces key attributes of linear, quadratic, and exponential functions.
• Algebra II will continue to analyze the key attributes of exponential functions and will introduce the key attributes of square root, cubic, cube root, absolute value, rational, and logarithmic functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.3D Graph the solution set of linear inequalities in two variables on the coordinate plane.

Graph

THE SOLUTION SET OF LINEAR INEQUALITIES IN TWO VARIABLES ON THE COORDINATE PLANE

Including, but not limited to:

• Inequality notation
• Less than, <, dashed line with shading below the graph of the line
• Greater than, >, dashed line with shading above the graph of the line
• Less than or equal to, ≤, solid line with shading below the graph of the line
• Greater than or equal to, ≥, solid line with shading above the graph of the line
• For vertical lines, greater than shades the right side of the graph and less than shades the left side of the graph.
• Given multiple representations
• Table of values
• Algebraic generalization
• Verbal description

Note(s):

• Algebra I introduces linear inequalities in two variables.
• Algebra II will continue systems of two linear inequalities in two variables and extend to systems of more than two linear inequalities in two variables.
• 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.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. 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
A.3H Graph the solution set of systems of two linear inequalities in two variables on the coordinate plane.
Supporting Standard

Graph

THE SOLUTION SET OF SYSTEMS OF TWO LINEAR INEQUALITIES IN TWO VARIABLES ON THE COORDINATE PLANE

Including, but not limited to:

• Linear inequality in two variables – a relationship with a constant rate of change represented by a solution set denoted by the graph of a line, that may or may not be included in the solution, and the set of points above or below the line
• Systems of inequalities
• Two unknowns
• Two inequalities
• Graphical analysis of the system of inequalities
• Graphing of each function
• Solid line
• Dashed line
• Shading of inequality region for each
• Representation of the solution as points in the region of intersection
• Justification of solution to systems of inequalities
• Substitution of various points in the solutions region into original functions

Note(s):

• Algebra I introduces linear inequalities in two variables given various representations.
• Algebra II will continue systems of two linear inequalities in two variables and extend to systems of more than two linear inequalities in two variables.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric reasoning
• C1 – Use estimation to check for errors and reasonableness of solutions.
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. 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
A.4 Linear functions, equations, and inequalities. The student applies the mathematical process standards to formulate statistical relationships and evaluate their reasonableness based on real-world data. The student is expected to:
A.4C Write, with and without technology, linear functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems.
Supporting Standard

Write

LINEAR FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, WITH AND WITHOUT TECHNOLOGY

Including, but not limited to:

• Linear function – a relationship with a constant rate of change represented by a graph that forms a straight line in which each element of the input (x) is paired with exactly one element of the output (y)
• Function models for sets of data
• Trend line by manipulating slope and y-intercept
• Regression equation, y = ax + b, using the graphing calculator

To Estimate, To Make

SOLUTIONS AND PREDICTIONS FOR REAL-WORLD PROBLEMS

Including, but not limited to:

• Function models for sets of data
• Trend line by manipulating slope and y-intercept
• Regression equation, y = ax + b, using the graphing calculator
• Correlation coefficient as an indicator of reliability of regression equations

Note(s):

• Grade 8 graphed scatterplots of bivariate data and used trend lines to analyze the correlation as linear, non-linear, or no association.
• Algebra I introduces calculation and interpretation of the correlation coefficient between two quantitative variables.
• Algebra I introduces the use of algebraic strategies and regression technology to determine the line of best fit.
• Algebra II will apply regression technology and will determine appropriate models between linear, quadratic, and exponential functions to make predictions and critical judgments.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI. Statistical Reasoning
• B1 – Determine types of data.
• B2 – Select and apply appropriate visual representations of data.
• B3 – Compute and describe summary statistics of data.
• B4 – Describe patterns and departure from patterns in a set of data.
• C1 – Make predictions and draw inferences using summary statistics.
• C2 – Analyze data sets using graphs and summary statistics.
• C3 – Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
• C4 – Recognize reliability of statistical results.
• VII. Functions
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.5 Linear functions, equations, and inequalities. The student applies the mathematical process standards to solve, with and without technology, linear equations and evaluate the reasonableness of their solutions. The student is expected to:
A.5A Solve linear equations in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.

Solve

LINEAR EQUATIONS IN ONE VARIABLE, INCLUDING THOSE FOR WHICH THE APPLICATION OF THE DISTRIBUTIVE PROPERTY IS NECESSARY AND FOR WHICH VARIABLES ARE INCLUDED ON BOTH SIDES

Including, but not limited to:

• Linear equation in one variable – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Linear equations in one variable including parentheses and variables on both sides of the equation
• Mathematical problem situations
• Real-world problem situations
• Multiple representations of mathematical and real-world problem situations
• Algebraic generalizations
• Missing coordinate of a solution point to a function
• Verbal
• Methods for solving equations
• Concrete and pictorial models (e.g., algebra tiles, etc.)
• Tables and graphs with and without technology
• Transformation of equations using properties of equality
• Distributive property
• Operational properties
• Possible solutions, including special cases
• No solution, empty set, ∅
• Infinite solutions, all real numbers, ℜ
• Relationships and connections between the methods of solution
• Justification of solutions to equations
• Justification of reasonableness of solutions in terms of mathematical and real-world problem situations

Note(s):

• Grade 5 used equations with variables to represent missing numbers.
• Grade 6 solved one-variable, one-step equations.
• Grade 7 solved one-variable, two-step equations.
• Grade 8 solved one-variable equations with variables on both sides.
• Algebra I introduces solving one-variable equations that include those for which the application of the distributive property is necessary and for which variables are included on both sides.
• Algebra II will introduce solving absolute value linear equations.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric reasoning
• C1 – Use estimation to check for errors and reasonableness of solutions.
• 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
A.5B Solve linear inequalities in one variable, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
Supporting Standard

Solve

LINEAR INEQUALITIES IN ONE VARIABLE, INCLUDING THOSE FOR WHICH THE APPLICATION OF THE DISTRIBUTIVE PROPERTY IS NECESSARY AND FOR WHICH VARIABLES ARE INCLUDED ON BOTH SIDES

Including, but not limited to:

• Linear inequality in one variable – a mathematical statement composed of algebraic and/or numeric expressions set apart by an inequality symbol
• Inequality symbols
• > (is greater than)
• < (is less than)
• ≥ (is greater than or equal to)
• ≤ (is less than or equal to)
• ≠ (is not equal to)
• Linear inequalities including parentheses and variables on both sides of the equation
• Mathematical problem situations
• Real-world problem situations
• Multiple representations of mathematical and real-world problem situations
• Algebraic generalizations
• Verbal
• Solutions to include numeric, graphic, and verbal representations
• Methods for solving inequalities
• Concrete and pictorial models (e.g., algebra tiles, etc.)
• Graphs and tables with and without technology
• Transformation of inequalities using properties of inequalities
• Distributive property
• Operational properties
• Special cases for empty set, Ø, and all real numbers, ℜ
• Relationships and connections between the methods of solution
• Justification of solutions to inequalities
• Differentiation between solutions of equations and inequalities
• Justification of reasonableness of solutions in terms of mathematical and real-world problem situations

Note(s):

• Grade 6 solved one-variable, one-step inequalities.
• Grade 7 solved one-variable, two-step inequalities.
• Grade 8 wrote one-variable inequalities with variables on both sides.
• Algebra I introduces solving one-variable inequalities, including those for which the application of the distributive property is necessary and for which variables are included on both sides.
• Algebra II will introduce solving absolute value linear inequalities.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric reasoning
• C1 – Use estimation to check for errors and reasonableness of solutions.
• II. Algebraic Reasoning
• C1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to solve equations, inequalities, and systems of linear equations.
• C2 – Explain the difference between the solution set of an equation and the solution set of an inequality.
• 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
A.5C Solve systems of two linear equations with two variables for mathematical and real-world problems.

Solve

SYSTEMS OF TWO LINEAR EQUATIONS WITH TWO VARIABLES FOR MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Linear equation in two variables – a relationship with a constant rate of change represented by a graph that forms a straight line
• Systems of 2 × 2 linear equations
• Two equations
• Two variables
• Solutions to systems of equations
• One common point of intersection, (x, y)
• Infinite set of points on a line
• Empty set, Ø
• Methods for solving systems of linear equations with and without technology
• Tables
• Graphs
• Concrete models
• Algebraic methods
• Substitution
• Linear combination (elimination)
• Special cases for empty set, Ø, and all real numbers, ℜ
• Relationships and connections between the methods of solution
• Justification of solutions to systems of equations with and without technology
• Systems of linear equations as models for real-world problem situations
• Interpretation of a solution point in terms of the real-world problem situation
• Justification of reasonableness of solution in terms of the real-world problem situation or data collection

Note(s):

• Algebra I formally introduces systems of two linear equations in two variables.
• Algebra II will introduce systems of three linear equations in three variables.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric reasoning
• C1 – Use estimation to check for errors and reasonableness of solutions.
• 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
A.6 Quadratic functions and equations. The student applies the mathematical process standards when using properties of quadratic functions to write and represent in multiple ways, with and without technology, quadratic equations. The student is expected to:
A.6A Determine the domain and range of quadratic functions and represent the domain and range using inequalities.

Determine, Represent

THE DOMAIN AND RANGE OF QUADRATIC FUNCTIONS USING INEQUALITIES

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• Domain and range of quadratic functions in mathematical problem situations
• Domain – set of input values for the independent variable over which the function is defined
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Discrete function – function whose values are distinct and separate and not connected; values are not continuous. Discrete functions are defined by their domain.
• Range – set of output values for the dependent variable over which the function is defined
• Domain and range of quadratic functions in real-world problem situations
• Reasonable domain and range for the real-world problem situation
• Comparison of domain and range of function model to appropriate domain and range for real-world problem situation
• Inequality representations
• Verbal description
• Ex: x is all real numbers less than five.
• Ex: x is all real numbers.
• Ex: y is all real numbers greater than –3 and less than or equal to 6.
• Ex: y is all integers greater than or equal to zero.
• Inequality notation – notation in which the solution is represented by an inequality statement
• Ex: x < 5
• Ex: x ∈ ℜ
• Ex: –3 < y ≤ 6
• Ex: y ≥ 0, y ∈ Ζ

Note(s):

• Grade 6 identified independent and dependent quantities.
• Grade 8 identified functions using sets of ordered pairs, tables, mappings, and graphs.
• Algebra I introduces quadratic functions.
• Algebra I introduces the concept of domain and range of a function.
• Algebra I represents domain and range using inequality verbal descriptions and inequality notation.
• Algebra II will extend the concept of domain and range.
• Algebra II will introduce representing domain and range using interval and set notation.
• Algebra II will continue to investigate quadratic functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.6B Write equations of quadratic functions given the vertex and another point on the graph, write the equation in vertex form (f(x) = a(x - h)2+ k), and rewrite the equation from vertex form to standard form (f(x) = ax2+ bx + c).
Supporting Standard

Write

EQUATIONS OF QUADRATIC FUNCTIONS GIVEN THE VERTEX AND ANOTHER POINT ON THE GRAPH IN VERTEX FORM (f(x) = a(xh)2 + k)

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 , and the graph of the function is always parabolic or U-shaped
• Vertex – highest (maximum) or lowest (minimum) point on the graph of a parabola
• Determination of an algebraic representation for a quadratic function in vertex form, y = a(xh)2 + k
• Given vertex (h, k)
• Given point (x, y) or (x, f(x))

Rewrite

THE EQUATION FROM VERTEX FORM TO STANDARD FORM (f(x) = ax2 + bx + c)

Including, but not limited to:

• Vertex form: f(x) = a(x – h)2 + k
• Standard form: f(x) = ax2 + bx + c
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra I introduces writing quadratic functions in vertex and standard form.
• Algebra II will transform quadratic functions from standard to vertex form.
• Algebra II will write equations of parabolas given the vertex and other attributes.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• 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
A.6C Write quadratic functions when given real solutions and graphs of their related equations.
Supporting Standard

Write

QUADRATIC FUNCTIONS WHEN GIVEN REAL SOLUTIONS AND GRAPHS OF THEIR RELATED EQUATIONS

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• Representations of quadratic functions
• Graphs
• Algebraic generalizations
• Comparisons of quadratic equations (0 = ax2 + bx + c) and quadratic functions (y = ax2 + bx + c)
• Comparisons of zeros/x-intercepts of quadratic functions and solutions/roots of quadratic equations
• Comparisons of solutions/roots and factors of the quadratic equation
• Solutions/roots: r1 and r2
• Factors: (xr1)(xr2)
• Quadratic equation: 0 = (xr1)(xr2)
• Zeros/x-intercepts and factors of the quadratic function
• Zeros/x-intercepts: z1 and z2 or (z1, 0) and (z2, 0)
• Factors: (xz1)(xz2)
• Quadratic function: f(x) = (xz1)(xz2)
• Multiple functions with the same solutions are possible depending on scalar multiples or the “a” value.

Note(s):

• Algebra I introduces writing quadratic functions from solutions/roots and zeros/x-intercepts.
• Algebra II will write quadratic functions given three points on the parabola.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• 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
A.7 Quadratic functions and equations. The student applies the mathematical process standards when using graphs of quadratic functions and their related transformations to represent in multiple ways and determine, with and without technology, the solutions to equations. The student is expected to:
A.7A Graph quadratic functions on the coordinate plane and use the graph to identify key attributes, if possible, including x-intercept, y-intercept, zeros, maximum value, minimum values, vertex, and the equation of the axis of symmetry.

Graph

QUADRATIC FUNCTIONS ON THE COORDINATE PLANE

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• Graphs of quadratic functions with and without technology
• Algebraic generalizations
• Real-world problem situations involving quadratic functions

Use

THE GRAPH OF A QUADRATIC FUNCTION TO IDENTIFY KEY ATTRIBUTES, IF POSSIBLE, INCLUDING x-INTERCEPT, y-INTERCEPT, ZEROS, MAXIMUM VALUE, MINIMUM VALUES, VERTEX, AND THE EQUATION OF THE AXIS OF SYMMETRY

Including, but not limited to:

• Representation of quadratic functions
• Standard form: f(x) = ax2 + bx + c
• Vertex form: f(x) = a(xh)2 + k
• Characteristics of quadratic functions
• Intercepts/Zeros
• x-intercept – 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
• y-intercept – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0,y)
• Denoted as c in f(x) = ax2 + bx + c
• Denoted as ah2 + k in f(x) = a(xh)2 + k
• Vertex  – highest (maximum) or lowest (minimum) point on the graph of a parabola
• Graphically, the maximum or minimum point of the parabola
• Algebraically x = and solving for y
• From standard form: x = and solving for y
• From vertex form: (h, k)
• Maximum – graph opens downward, negative a value
• Minimum – graph opens upward, positive a value
• Axis of symmetry – line passing through the vertex of a parabola that divides the parabola into two congruent halves
• Equation of the axis of symmetry
• From standard form: x = • Vertex form: x = h
• Symmetric points – the image and pre-image points reflected across the axis of symmetry of the parabola
• Real-world problem situations involving quadratic functions
• Analysis and conclusions for real-world problem situations using key attributes

Note(s):

• Algebra I introduces quadratic functions.
• Algebra I introduces the key attributes of a quadratic function.
• Algebra II will continue to investigate and apply quadratic functions and equations and write the equation of a parabola from given attributes, including direction of opening.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VII. Functions
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.7C Determine the effects on the graph of the parent function f(x) = x2 when f(x) is replaced by af(x), f(x) + d, f(x - c), f(bx) for specific values of a, b, c, and d.

Determine

THE EFFECTS ON THE GRAPH OF THE PARENT FUNCTION f(x) = x2 WHEN f(x) IS REPLACED BY af(x), f(x) + d, f(xc), f(bx) FOR SPECIFIC VALUES OF a, b, c, AND d

Including, but not limited to:

• Parent functions – set of basic functions from which related functions are derived by transformations
• General form of quadratic parent function (including equation and function notation)
• y = x2
• f(x) = x2
• Multiple representations
• Graphs
• Tables
• Verbal descriptions
• Algebraic generalizations (including equation and function notation)
• Changes in parameters a, b, c, and d on the graph of the parent function f(x) = x2
• Effects on the graph of the quadratic parent function f(x) = x2 when f(x) is replaced by af(x)
• a ≠ 0
• |a| > 1, stretches the graph vertically or makes the graph more narrow
• 0 < |a| < 1, compresses the graph vertically or makes the graph wider
• Opposite of a reflects the graph vertically over the horizontal axis (x-axis)
• Effects on the graph of the quadratic parent function f(x) = x2 when f(x) is replaced by f(bx)
• b ≠ 0
• |b| > 1, the graph compresses horizontally or makes the graph more narrow
• 0 < |b| < 1, the graph stretches horizontally or makes the graph wider
• b < 0, reflects horizontally over the y-axis
• Effects on the graph of the quadratic parent function f(x) = x2 when f(x) is replaced by f(xc)
• c = 0, no horizontal shift or translation
• Horizontal shift or translation left or right by |c| units
• Left shift or translation when c < 0
• For f(x + 2) → f(x – (–2)), c = –2, and the function moves to the left or shifts left or translates left two units.
• Right shift or translation when c > 0
• For f(x – 2), c = 2, and the function moves to the right or shifts right or translates right two units
• Effects on the graph of the quadratic parent function f(x) = x2 when f(x) is replaced by f(x) + d
• d = 0, no vertical shift or translation
• Vertical shift or translation up or down
• Shift or translation down when d < 0
• For f(x) – 2, d = –2, and the function moves down or shifts down or translates down two units.
• Shift or translation up when d > 0
• For f(x) + 2, d = 2, and the function moves up or shifts up or translates up two units.
• Generalizations of parameter changes to f(x) = x on the x-and y-intercepts
• For af(x) and f(bx), there are no changes to the x-and y-intercepts..
• For f(xc),
• If c < 0, then the x-intercept shifts or translates left by |c| units, (c, 0).
• If c > 0, then the x-intercept shifts or translates right by |c| units (c, 0).
• For f(x) + d,
• If d < 0, then the y-intercept shifts or translates up by |d| units (0, d).
• If d > 0, then the y-intercept shifts or translates down by |d| units (0, d).
• Graphical representation given the algebraic representation or parameter changes
• Algebraic representation given the graphical representation or parameter changes
• Descriptions of the effects on the domain and range by the parameter changes
• Combined parameter changes

Note(s):

• Algebra I introduces effects of parameter changes a, b, c, and d on the quadratic parent function.
• Algebra II will extend effects of parameter changes to other parent functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B1 – Identify and apply transformations to figures.
• C1 – Make connections between geometry and algebra.
• VII. Functions
• A1 – Recognize whether a relation is a function.
• A2 – Recognize and distinguish between different types of functions.
• B1 – Understand and analyze features of a function.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.8 Quadratic functions and equations. The student applies the mathematical process standards to solve, with and without technology, quadratic equations and evaluate the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:
A.8A Solve quadratic equations having real solutions by factoring, taking square roots, completing the square, and applying the quadratic formula.

Solve

QUADRATIC EQUATIONS HAVING REAL SOLUTIONS BY FACTORING, TAKING SQUARE ROOTS, COMPLETING THE SQUARE, AND APPLYING THE QUADRATIC FORMULA

Including, but not limited to:

• Quadratic equation in one variable – a second-degree polynomial function that can be described in standard form by 0 = ax2 + bx + c, where a ≠ 0
• Methods for solving quadratic equations with and without technology
• Concrete models
• Applicable only with quadratic equations that when set equal to zero the expression can be factored
• Algebraic methods
• Factoring
• Square roots
• Completing the square
• Quadratic formula, • Solution sets of quadratic equations
• Two solutions
• One solution (double root)
• No real solutions, Ø
• Real-world problem situations and/or data collection activity involving a quadratic function with and without technology
• Quadratic equation to represent the real-world problem situation
• Method of choice to solve

Note(s):

• Algebra I introduces solving quadratic equations.
• Algebra II will introduce solving equations involving absolute value (e.g., x2 = 25, = , |x| = 5; therefore, x = ±5) .
• Algebra II will continue to solve and apply quadratic equations, including imaginary solutions.
• Algebra II will solve quadratic inequalities.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric reasoning
• C1 – Use estimation to check for errors and reasonableness of solutions.
• 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
A.8B Write, using technology, quadratic functions that provide a reasonable fit to data to estimate solutions and make predictions for real-world problems.
Supporting Standard

Write

QUADRATIC FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, USING TECHNOLOGY

Including, but not limited to:

• Quadratic function – a second-degree polynomial function that can be described by f(x) = ax2 + bx + c, where a ≠ 0 and the graph of the function is always parabolic or U-shaped
• Data representations
• Real-world problem situations
• Data collections using technology
• Technology to determine a function model using quadratic regression

To Estimate, To Make

SOLUTIONS AND PREDICTIONS FOR REAL-WORLD PROBLEMS

Including, but not limited to:

• Data representations
• Real-world problem situations
• Data collections using technology
• Technology to determine a function model using quadratic regression
• Technology to determine key attributes of functions specific to the real-world problem situations
• Vertex (maximum, minimum)
• Intercepts (y-intercepts, x-intercepts, zeros)

Note(s):

• Algebra I introduces writing quadratic functions using technology to reasonably fit data.
• Algebra II will continue to investigate and apply quadratic equations.
• Algebra II will apply regression technology and will determine appropriate models between linear, quadratic, and exponential functions to make predictions and critical judgments.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• I. Numeric reasoning
• C1 – Use estimation to check for errors and reasonableness of solutions.
• III. Geometric Reasoning
• 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
A.9 Exponential functions and equations. The student applies the mathematical process standards when using properties of exponential functions and their related transformations to write, graph, and represent in multiple ways exponential equations and evaluate, with and without technology, the reasonableness of their solutions. The student formulates statistical relationships and evaluates their reasonableness based on real-world data. The student is expected to:
A.9C Write exponential functions in the form f(x) = abx (where b is a rational number) to describe problems arising from mathematical and real-world situations, including growth and decay.

Write

EXPONENTIAL FUNCTIONS IN THE FORM f(x) = abx (WHERE b IS A RATIONAL NUMBER) TO DESCRIBE PROBLEMS ARISING FROM MATHEMATICAL AND REAL-WORLD SITUATIONS, INCLUDING GROWTH AND DECAY

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• a value is the y-intercept, (0, a).
• b value is the successive ratio of range values.
• The b value is a rational number greater than 0.
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Exponential functions in real-world problem situations.
• Representative exponential function for the real-world problem situation
• Meaning of a and b in terms of the real-world problem situation
• Growth rate, r, is b – 1
• b = 1 + r, where r is in decimal form
• Rate of decay, r, is 1 – b
• b = 1 – r, where r is in decimal form

Note(s):

• Algebra I introduces exponential functions.
• Algebra II will continue to investigate exponential functions, including continuous growth and decay.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• 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.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.9D Graph exponential functions that model growth and decay and identify key features, including y-intercept and asymptote, in mathematical and real-world problems.

Graph

EXPONENTIAL FUNCTIONS THAT MODEL GROWTH AND DECAY

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• a value is the y-intercept, (0, a).
• b value is the successive ratio of range values.
• The b value is a rational number greater than 0.
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Mathematical problem situations
• Real-world problem situations

Identify

KEY FEATURES, INCLUDING y-INTERCEPT AND ASYMPTOTE, IN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
•  Key attributes
• y-intercept(s) – y coordinate of a point at which the relation crosses the y-axis, meaning the x coordinate equals zero, (0, y)
• y-intercept in an exponential function: (0, a) where a is the a value in f(x) = abx
• Asymptote – a line that is approached and may or may not be crossed
• Horizontal asymptote – horizontal line approached by the curve as the function approaches positive or negative infinity. Horizontal asymptotes may be crossed by the curve.
• Mathematical problem situations
• Real-world problem situations

Note(s):

• Algebra I introduces exponential functions.
• Algebra II will continue to investigate exponential functions, including continuous growth and decay.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• 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.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.9E Write, using technology, exponential functions that provide a reasonable fit to data and make predictions for real-world problems.
Supporting Standard

Write

EXPONENTIAL FUNCTIONS THAT PROVIDE A REASONABLE FIT TO DATA, USING TECHNOLOGY

Including, but not limited to:

• Exponential function – a function in which the independent variable, x, is in the exponent, denoted by f(x) = abx
• a value is the y-intercept, (0, a).
• b value is the successive ratio of range values.
• The b value is a rational number greater than 0.
• Exponential growth – an exponential function where b > 1 and as x increases, y increases exponentially
• Exponential decay – an exponential function where 0 < b < 1 and as x increases, y decreases exponentially
• Concrete models
• Data representations
• Real-world problem situations
• Data collections using technology
• Technology to determine a function model using exponential regression

Make

PREDICTIONS FOR REAL-WORLD PROBLEMS

Including, but not limited to:

• Real-world problem situations represented by exponential functions
• Exponential functions in the form f(x) = abx
• Predictions in terms of the real-world problem situation
• Reasonableness of predictions in terms of the real-world problem situation

Note(s):

• Algebra I introduces exponential functions.
• Algebra I predicts solutions for exponential functions.
• Algebra II will continue to investigate exponential functions, including continuous growth and decay.
• Aglebra II will formulate and solve exponential functions and equations.
• Algebra II will apply regression technology and will determine appropriate models between linear, quadratic, and exponential functions to make predictions and critical judgments.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• 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.
• C2 – Develop a function to model a situation.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.10 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite in equivalent forms and perform operations on polynomial expressions. The student is expected to:
A.10E Factor, if possible, trinomials with real factors in the form ax2 + bx + c, including perfect square trinomials of degree two.

Factor

TRINOMIALS WITH REAL FACTORS IN THE FORM ax2 + bx + c, INCLUDING PERFECT SQUARE TRINOMIALS OF DEGREE TWO, IF POSSIBLE

Including, but not limited to:

• Trinomial – three term expression
• Factorization of trinomials
• Form ax2 + bx + c
• a and b – coefficients of the variables
• c – the constant term
• Terms in descending order alphabetically and by degree
• First check for a greatest common factor (GCF).
• Leading coefficient, a, equal to 1
• Ex: x2 – 2x – 63; x2 + 5x + 25; p2 + 13pq + 40q2
• Leading coefficient, a, real number other than 1
• Ex: 3x2 – 24x + 36; 2x2 – 9x – 5; 15a2 + 11ab + 2b2
• Perfect square trinomial – first term a perfect square, third term a perfect square, middle term double the product of the square roots of the first and last terms
• Ex: 4x2 – 12x + 9; • Identification of factorable trinomials and non-factorable (prime) trinomials
• Factorization of factorable trinomials
• Leading coefficient of 1
• Factor tables
• Box method
• Leading coefficient other than 1
• Box method
• Grouping
• Multiplication/division method (Bottoms Up)

Note(s):

• Algebra I introduces factorization of polynomials of degree one and degree two.
• Algebra II will extend factorization to polynomials of degree three and degree four, including factoring by grouping.
• 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).
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.10F Decide if a binomial can be written as the difference of two squares and, if possible, use the structure of a difference of two squares to rewrite the binomial.
Supporting Standard

Decide

IF A BINOMIAL CAN BE WRITTEN AS THE DIFFERENCE OF TWO SQUARES

Including, but not limited to:

• Binomial – two term expression
• Binomial whose terms are the difference of two squares
• Both terms are perfect squares.
• The two terms have opposite signs.
• Difference of squares written as a2b2
• Ex: x2 – 9; 81x2 – 121y2; 25a2 – 9b2; Use

THE STRUCTURE OF A DIFFERENCE OF TWO SQUARES TO REWRITE THE BINOMIAL, IF POSSIBLE

Including, but not limited to:

• First check for a greatest common factor (GCF).
• Difference of squares written as a2b2
• Factorization of binomial that is the difference of two squares
• Factors into two binomials
• (square root of first term + square root of second term)(square root of first term – square root of second term)
• a2b2 = (a + b)(ab)
• Identification of factorable trinomials and non-factorable (prime) binomials
• Factorization of factorable trinomials

Note(s):

• Algebra I introduces factorization of polynomials of degree one and degree two.
• Algebra II will extend factorization to polynomials of degree three and degree four, including sum and difference of two cubes and factoring by grouping.
• 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).
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.11 Number and algebraic methods. The student applies the mathematical process standards and algebraic methods to rewrite algebraic expressions into equivalent forms. The student is expected to:
A.11B Simplify numeric and algebraic expressions using the laws of exponents, including integral and rational exponents.

Simplify

NUMERIC AND ALGEBRAIC EXPRESSIONS USING THE LAWS OF EXPONENTS, INCLUDING INTEGRAL AND RATIONAL EXPONENTS

Including, but not limited to:

• Algebraic expression – a generalization that is a combination of variables, numbers (constants and coefficients), and operators
• Laws (properties) of exponents
• Product of powers (multiplication when bases are the same): am • an = am+n
• Quotient of powers (division when bases are the same): = am–n
• Power to a power: (am)n = amn
• Negative exponent: a–n = • Zero exponent: a0 = 1
• Rational exponent: • Simplification of expressions using laws (properties) of exponents
• Numeric expressions, including scientific notation
• Algebraic expressions
• Variables can appear as either the base or the exponent, but in either case must be rational numbers.
• Applications of algebraic expressions involving exponents

Note(s):

• Prior grade levels simplified numeric expressions involving whole number exponents.
• Grade 8 introduced scientific notation.
• Algebra I introduces exponential functions.
• Algebra I applies laws (properties) of exponents to simplify numeric and algebraic expressions.
• Algebra II will introduce equations involving rational exponents.
• 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
• A1 – Explain and differentiate between expressions and equations using words such as “solve,” “evaluate,” and “simplify.”
• B1 – Recognize and use algebraic (field) properties, concepts, procedures, and algorithms to combine, transform, and evaluate expressions (e.g. polynomials, radicals, rational expressions).
• III. Geometric Reasoning
• C1 – Make connections between geometry and algebra.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections 