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 Instructional Focus DocumentAlgebra I
 TITLE : Unit 04: Application of Linear Functions SUGGESTED DURATION : 15 days

Unit Overview

Introduction
This unit bundles student expectations that address the continued study of linear functions by exploring multiple processes for writing equations of lines and interpreting the meaning of key attributes in problem situations. In addition, students analyze data to determine a function model and the strength of the function model in making predictions. 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 Grade 6, students identified independent and dependent variables. In Grade 8, students graphed scatterplots of bivariate data and analyzed correlation as linear, non-linear, or no association. Students introduced slope and slope-intercept form to represent and compare proportional and non-proportional relationships. In Algebra I, Unit 03, students began an in-depth study of the representations and characteristics of linear functions, including domain and range, slope, intercepts, and transformations of the linear parent function.

During this Unit
Students write linear equations in two variables from given information, including a table of values, a graph, a verbal description, one point and the slope, two points, a point and parallel to a given line, a point and perpendicular to a given line, or a line parallel or perpendicular to the x- or y-axis, and represent the linear equations in various forms, including y = mx + b, Ax + By = C, and yy1 = m(x – x1). Students calculate the rate of change or slope as needed from a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1). Students write linear functions for real-world situations, and model the linear functions using various representations. Students identify domain (continuous or discrete), range, x-intercept, y-intercept, zeros, and slope and the meaning of the key attributes in terms of the situation. Students make predictions and critical judgments, and justify the solution in terms of the problem situation, including writing and solving problems involving direct variation. Students write, with and without technology, linear functions, analyze the strength of the linear function using scatterplots and linear correlations, compare association and causation between the variables, and estimate solutions and make predictions in terms of the problem situation.

After this Unit
In Units 08 and 09, students will apply similar concepts of representations and characteristics as they investigate quadratic and exponential functions. Students will continue to apply these concepts in subsequent courses in mathematics.

In Algebra I, determining the domain and range and representing it in inequality notation are identified as STAAR Readiness Standard A.2A. Writing linear equations from various representations is identified as STAAR Readiness Standard A.2C. These Readiness Standards are subsumed under STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. Graphing and identifying key features of linear functions, including determining rate of change, are identified as STAAR Readiness Standards A.3B and A.3C. These Readiness Standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Linear Functions, Equations, and Inequalities. Writing linear equations from various representations and writing equations of parallel and perpendicular lines are identified as STAAR Supporting Standards A.2B, A.2E, A.2F, and A.2G. Writing and solving equations of direct variation are identified as STAAR Supporting Standard A.2D. These Supporting Standards are subsumed under STAAR Reporting Category 3: Writing and Solving Linear Functions, Equations, and Inequalities. Determining slope from various representations is identified as STAAR Supporting Standards A.3A. Calculating the linear correlation coefficient and using it to write linear functions that best fit data to make predictions are identified as STAAR Supporting Standard A.4A and A.4C. Comparing and contrasting association and causation are identified as STAAR Supporting Standard A.4B. These Supporting Standards are subsumed under STAAR Reporting Category 2: Describing and Graphing Linear Functions, Equations, and Inequalities. This unit is supporting the development of Texas College and Career Readiness Standards (TxCCRS): II. Algebraic Reasoning; III. Geometric Reasoning B3, C1; VI. Statistical Reasoning B1, B2, B3, B4, C1, C2, C3, C4; VII. Functions B1, B2, C1, C2; VIII. Problem Solving and Reasoning; IX. Communication and Representation; X. Connections.

Research
According to the National Council of Teachers of Mathematics (NCTM), Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9 – 12 (2010), students in grades 9 – 12 need to understand functions well if they are to succeed in courses that build on quantitative thinking and relationships. According the National Council of Teachers of Mathematics (2000), students should have an opportunity to build on their earlier experiences, both deepening their understanding of relations and functions and expanding their repertoire of familiar functions. In middle school through the study of direct variation (proportional) and non-proportional linear situations, students explore the patterns that relate to linear functions, building a foundational understanding for development of slope and intercepts and writing equations of lines in high school. High school students’ algebra experience should enable them to create and use tabular, symbolic, graphical, and verbal representations and to analyze and understand patterns, relations, and functions with more sophistication than in the middle grades. High school algebra should provide students with insights into mathematical abstraction and structure for modeling real-world problem situations and making predictions and drawing conclusions.

National Council of Teachers of Mathematics. (2000). Developing essential understanding of functions for teaching mathematics grades 9-12. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. 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)
• 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 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 linear equations?
• How does the given information and/or representation influence the selection of an efficient method for writing linear equations?
• How can the solutions to linear equations be determined and represented?
• How are properties and operational understandings used to transform linear equations?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Rate of change/Slope
• x- and y-intercept(s)
• Functions and Equations
• Linear
• Patterns, Operations, and Properties
• Relations and Generalizations
• Associated Mathematical Processes
• 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.

 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?
• 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
• Linear
• Direct variation
• Patterns, Operations, and Properties
• Relations and Generalizations
• 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.

 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 can relationships between variables be analyzed for association and causation?
• 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?
• Functions, Equations, and Inequalities
• Attributes of Functions
• Domain and range
• Continuous or discrete
• Rate of change/Slope
• x- and y-intercept(s)
• Zeros
• Functions and Equations
• Linear
• Relations and Generalizations
• Statistical Relationships
• Regression methods
• Association and causation
• 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 if a set of bivariate data show a strong correlation and association, one variable must cause the other, rather than understanding that although the variables have an association, one may not be the cause of the other. e.g., When analyzing the number of shark bites as a function of the amount of ice cream consumed, a strong correlation and association is observed. However, eating ice cream is not what causes shark bites. The lurking variable is temperature. The hotter it is outside the more ice cream is consumed, and the more people go swimming at the beach.

Unit Vocabulary

• Association – a relationship or correlation between two measurable variables
• Quantitative bivariate data – data for two related (numeric) variables that can be represented by a scatterplot
• Causation – a relationship between two variables in which one variable directly causes change(s) in the other variable
• Continuous function – function whose values are continuous or unbroken over the specified domain
• Correlation – description of the linear relationship between the two variables in bivariate data
• Correlation coefficient (r-value) – numeric value that assesses the strength of the linear relationship between two quantitative variables in a set of bivariate data
• Direct variation – a linear relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = kx
• 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
• Inequality notation – notation in which the solution is represented by an inequality statement
• 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)
• Negative linear correlation – trend of points to descend from left to right
• No linear correlation – no trend observable in the data points
• Positive linear correlation – trend of points to ascend from left to right
• Range – set of output values for the dependent variable over which the function is defined
• Regression equation – line of best fit representing a set of bivariate data
• 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
• 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:

 Equation notation Equations of lines Function notation Parallel Parent function Perpendicular Point-slope form, y – y1 = m(x – x1) Proportionality Rate of change Scatterplot Slope-intercept form, y = mx + b Standard form, Ax + By = C Trend line
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.2B Write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and y - y1 = m(x - x1), given one point and the slope and given two points.
Supporting Standard

Write

LINEAR EQUATIONS IN TWO VARIABLES IN VARIOUS FORMS, INCLUDING y = mx + b, Ax + By = C, and y – y1 = m(x – x1), GIVEN ONE POINT AND THE SLOPE AND GIVEN TWO POINTS

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 of 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(x – x1)
• m is the slope.
• (x1, y1) is a given point
• Standard form, Ax + By = C
• Traditional format: 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 constraints
• Slope and a point
• Two points

Note(s):

• Grade 8 introduced slope and slope-intercept form to compare proportional and non-proportional relationships.
• 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.
• 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.2D Write and solve equations involving direct variation.
Supporting Standard

Write

EQUATIONS INVOLVING DIRECT VARIATION

Including, but not limited to:

• Direct variation – a linear relationship between two variables, x (independent) and y (dependent), that always has a constant, unchanged ratio, k, and can be represented by y = kx
• Multiple representations
• Table
• Graph
• Verbal description
• Characteristics of direct variation
• Linear proportional relationship
• Linear
• Passes through the origin (0, 0)
• Represented by y = kx
• Constant of proportionality represented as • When b = 0 in y = mx + b, then k = the slope, m
• Real-world problem situations

Solve

EQUATIONS INVOLVING DIRECT VARIATION

Including, but not limited to:

• Various solution methods
• Tabular
• Graphical
• Algebraic
• Real-world problem situations

Note(s):

• Grade 7 introduced representing and solving problems involving proportional relationships.
• Grade 8 solved problems involving direct variation.
• Algebra II will solve problems involving inverse variation.
• 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
• 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.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.3A Determine the slope of a line given a table of values, a graph, two points on the line, and an equation written in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1).
Supporting Standard

Determine

THE SLOPE OF A LINE GIVEN A TABLE OF VALUES, A GRAPH, TWO POINTS ON THE LINE, AND AN EQUATION WRITTEN IN VARIOUS FORMS, INCLUDING y = mx + b, Ax + By = C, and yy1 = m(xx1)

Including, but not limited to:

• Slope of a line – the steepness of a line; rate of change in y (vertical) compared to change in x (horizontal), or or or • Slope by various methods
• Equation method denoted as m in y = mx + b
• Table method by analyzing change in x and y values: m = or • Graph method by analyzing vertical and horizontal change: slope = • Formula method: For two points (x1, y1) and (x2, y2), m= • Slope from multiple representations
• Tables of values
• Graphs
• Two points on a line
• Linear equations in various forms
• Slope-intercept form, y = mx + b
• m is the slope.
• Point-slope form, yy1 = m(xx1)
• m is the slope.
• Standard form, Ax + By = C
• m = – Note(s):

• Grade 8 introduced the concept of slope through the use of proportionality using similar triangles, making connections between slope and proportional relationships, and determining slope from tables and graphs.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• III. Geometric Reasoning
• B3 – Use congruence transformations and dilations to investigate congruence, similarity, and symmetries of plane figures.
• 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.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.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.4A Calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.
Supporting Standard

Calculate

THE CORRELATION COEFFICIENT BETWEEN TWO QUANTITATIVE VARIABLES, USING TECHNOLOGY

Including, but not limited to:

• Quantitative bivariate data – data for two related numeric variables that can be represent by a scatterplot
• Correlation coefficient (r-value) – numeric value that assesses the strength of the linear relationship between two quantitative variables in a set of bivariate data
• Determined by analysis of bivariate data using the graphing calculator
• Range: –1 ≤ r ≤ 1

Interpret

THE CORRELATION COEFFICIENT AS A MEASURE OF THE STRENGTH OF THE LINEAR ASSOCIATION

Including, but not limited to:

• Quantitative bivariate data – data for two related numeric variables that can be represent by a scatterplot
• Correlation – description of the linear relationship between the two variables in bivariate data
• Positive linear correlation – trend of points to ascend from left to right
• Negative linear correlation – trend of points to descend from left to right
• No linear correlation – no trend observable in the data points
• Regression equation – line of best fit representing a set of bivariate data
• Correlation coefficient (r-value) – numeric value that assesses the strength of the linear relationship between two quantitative variables in a set of bivariate data
• The regression equation and correlation coefficient for a given set of bivariate data can be computed using a graphing calculator.
• Correlation for approximated r-values
• Weak, very weak, to no correlation as it approaches 0: 0 ≤ |r| < 0.33
• Moderate correlation: 0.33 ≤ |r| < 0.67
• Strong, very strong: 0.67 ≤ |r| < 1.00
• Data form a perfect line: ± 1

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 II will apply regression technology to determine appropriate models between linear, quadratic, and exponential functions.
• 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.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
A.4B Compare and contrast association and causation in real-world problems.
Supporting Standard

Compare, Contrast

ASSOCIATION AND CAUSATION IN REAL-WORLD PROBLEMS

Including, but not limited to:

• Association – a relationship or correlation between two measurable variables
• Causation – a relationship between two variables in which one variable directly causes change(s) in the other variable
• An association between two variables does not always imply causation

Note(s):

• Algebra I introduces comparison of association and causation in bivariate data.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxCCRS:
• VI. Statistical Reasoning
• C1 – Make predictions and draw inferences using summary statistics.
• C4 – Recognize reliability of statistical results.
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
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 