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 Instructional Focus DocumentGrade 8 Mathematics
 TITLE : Unit 12: Essential Understandings of Algebra SUGGESTED DURATION : 20 days

#### Unit Overview

Introduction
This unit bundles student expectations that address problems involving proportional and non-proportional linear situations as well as representing bivariate sets of data. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, 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. The introduction to the grade level standards state, “While the use of all types of technology is important, the emphasis on algebra readiness skills necessitates the implementation of graphing technology.” Additionally, the availability of graphing technology is required during STAAR testing.

Prior to this Unit
In Unit 04, students used similar right triangles to develop an understanding of slope. Students determined that the ratio of the change in y-values and x-values is the same for any two points on the same line. They used data from a table or graph to determine the rate of change, or slope, and the y-intercept. In Unit 05, students distinguished between proportional and non-proportional linear situations and solved problems involving direct variation. Students also represented linear proportional and non-proportional situations with tables, graphs, and equations.

During this Unit
Students revisit and solidify essential understandings of algebra. Students extend their previous understandings of slope and y-intercept to represent proportional and non-proportional linear situations with tables, graphs, and equations. These representations are used as students distinguish between proportional and non-proportional linear situations. Students specifically examine the relationship between the unit rate and slope of a line that represents a proportional linear situation. Graphical representations of linear equations are examined closely as students begin to develop the understandings of systems of equations. Students are expected to identify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations. Students must also verify these values algebraically with the equations that represent the two graphed linear equations. Examining proportional and non-proportional linear relationships is extended to include identifying proportional and non-proportional linear functions in mathematical and real-world problems. A deep understanding of the characteristics of functions is essential to future mathematics coursework beyond Grade 8. Students continue to examine characteristics of linear relationships through the lens of trend lines that approximate the relationship between bivariate sets of data. Students contrast graphical representations of bivariate sets of data that suggest linear relationships with bivariate sets of data that do not suggest a linear relationship. Scatterplots are constructed from bivariate sets of data and used to describe the observed data. Observations include questions of association such as linear, non-linear, or no association. Students use trend lines that approximate the linear relationship between bivariate sets of data to make predictions. Students extend previous work with linear proportional and linear non-proportional situations to trend lines as they continue to represent situations with tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0, respectively. Within a scatterplot, students use the trend line of a linear proportional relationship to interpret the slope of the line that models the relationship as the unit rate of the scenario.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 8

After this Unit
In Algebra I, students will write linear equations in two variables in various forms and when given a table of values, a graph, and a verbal description. Students will write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1), given one point and the slope and given two points. In addition, students will also write and solve equations involving direct variation and calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in the context of mathematical and real-world problems. Students will calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association. Students will also graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist, solve systems of linear equations using concrete models, graphs, tables, and algebraic methods, and estimate graphically the solutions to systems of two linear equations with two variables in real-world problems.

Research
According to Van de Walle, Karp, and Bay-Williams (2010), “Algebraic thinking or algebraic reasoning involves forming generalizations from experiences with number and computation, formalizing these ideas with the use of a meaningful symbol system, and exploring the concepts of pattern and functions. Far from a topic with little real-world use, algebraic thinking pervades all of mathematics and is essential for making mathematics useful in daily life” (p. 254). According to research published by the National Center for Education Evaluation and Regional Assistance , Institute of Education Sciences, and U.S. Department of Education (2009), “A recent survey of algebra teachers associated with the report identified key deficiencies of students entering algebra, including aspects of whole number arithmetic, fractions, ratios, and proportions” (p. 4). The National Council of Teachers of Mathematics (2010) states that “A strong general mathematics course in grade 8, focused on building students’ skills in using symbols to represent their mathematical thinking, is essential for increasing these students’ readiness for algebra in high school” (p. 5).

Gersten, R., Beckmann, S., Clarke, B., Foegen, A., Marsh, L., Star, J. R., & Witzel, B. (2009). Assisting students struggling with mathematics: Response to Intervention (RtI) for elementary and middle schools (NCEE 2009-4060). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education. Retrieved from http://ies.ed.gov/ncee/wwc/publications/practiceguides/
National Council of Teachers of Mathematics. (2010). Focus in grade 8 teaching with curriculum focal points. 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
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements 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) in situations involving invariant (constant) relationships builds flexible proportional reasoning in order to make predictions and critical judgements about the relationship.
• The unit rate can be determined from the graph of a proportional relationship and used to describe the constant rate of change, the slope of the line.
• How can a graph of a proportional relationship be used to interpret the slope of a line and the unit rate?
• Proportional and non-proportional relationships can be presented using multiple representations, and those representations can be examined to distinguish between linear and non-linear proportional situations and identify attributes of linear relationships.
• What are the characteristics of a linear proportional situation in a(n) …
• table?
• graph?
• equation in the form of y = kx?
• What is the relationship between the slope of a line, the constant of proportionality, and the unit rate of a situation that represents a linear proportional relationship?
• How can the equation of a linear proportional situation be manipulated to prove that the constant of proportionality exists within the relationship?
• What are the key characteristics of a linear proportional and non-proportional situations?
• What are the similarities and differences between the …
• graphs
• tables
• equations
… of a linear proportional and linear non-proportional situations?
• What is the process for representing a linear relationship …
• verbally?
• with a table?
• with a graph?
• with an equation that simplifies to the form of y = mx + b?
• How are independent and dependent quantities related in a linear problem situation?
• What is the meaning of each of the variables in the equation y = mx + b?
• How are the table and graph of a linear problem situation related to an equation that simplifies to the form of y = mx + b?
• The ability to visualize the point of intersection of two simultaneously graphed linear equations aids in understanding how the point satisfies both equations, which introduces solutions to systems of equations.
• When two linear equations in the form of y = mx + b are graphed and the graphs intersect at one point, what does the point of intersection represent?
• What does the ordered pair at the point of intersection of two graphed linear equations in the form of y = mx + b represent in terms of the input and output values?
• How can the ordered pair at the point of intersection of two graphed linear equations in the form of y = mx + b be verified algebraically?
• What generalization can be made about the values of x and y that simultaneously satisfy two linear equations?
• Proportionality
• Statistics
• Predictions and inferences
• Data
• Statistical representations
• Relationships and Generalizations
• Independent and dependent quantities
• Linear proportional
• Linear non-proportional
• Representations
• Expressions, Equations, and Relationships
• Algebraic Relationships
• Linear
• Numeric and Algebraic Representations
• Equations
• Representations
• 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 judgements 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? Statistical displays often reveal patterns within data that can be analyzed to interpret information, inform understanding, make predictions, influence decisions, and solve problems in everyday life with degrees of confidence. How does society use or make sense of the enormous amount of data in our world available at our fingertips? How can data and data displays be purposeful and powerful? Why is it important to be aware of factors that may influence conclusions, predictions, and/or decisions derived from data?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) in situations involving invariant (constant) relationships builds flexible functional reasoning in order to make predictions and critical judgements about the relationship.
• Proportional and non-proportional relationships can be presented using multiple representations and those representations can be examined to distinguish linear and non-linear proportional relations and identify attributes of linear relations.
• What is bivariate data?
• What patterns are exhibited by the covariability in bivariate sets of data?
• What are the characteristics of bivariate data that shows a …
• linear
• non-linear
… relationship in a graphical representation?
• How can a trend line be used to …
• make predictions?
• describe a situation?
• What are the characteristics of a trend line that represents a …
• positive trend?
• negative trend?
• no trend?
• Data can be described in order to communicate and reason statically about the entire data set.
• What are the characteristics of a scatterplot?
• How does bivariate data that represents associations such as linear, non-linear, or no association appear in a graph?
• Proportionality
• Relationships and Generalizations
• Linear
• Non-linear
• Statistics
• Predictions and inferences
• Data
• Statistical representations
• Measurement and Data
• Graphical Representations
• Scatterplots
• 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

Underdeveloped Concepts:

• Some students may not relate the constant rate of change or unit rate to m in the equation y = mx + b.
• Some students may not relate the constant of proportionality or unit rate as k in the equation y = kx or m in the equation y = mx + b, when b = 0.
• Some students may think that a constant rate of change always means the situation is proportional.
• Some students may not associate a slope represented as whole number as a rational number that can be represented as .
• Some students may think that a function can have multiple outputs (y) for the same input (x).
• Some students may think that a function cannot have multiple inputs (x) that correspond to the same output (y).
• Students may think that the trend line has to begin at the origin rather than understanding that a trend line is not always proportional.
• Students may think that if both numbers in the data set are decreasing, then it represents a negative trend.
• Students may confuse a positive trend with a negative trend.
• Some students may attempt to connect the dots of a scatterplot rather than realizing the data is discrete and not continuous.
• Some students may think that the slope in a linear relationship is m = , since the x coordinate (horizontal) always comes before the y coordinate (vertical) in an ordered pair. Instead, the correct representation of slope in a linear relationship is m = .
• 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.

#### Unit Vocabulary

• Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
• Data – information that is collected about people, events, or objects
• Discrete paired data – data that involves only distinct values that are finite or countable
• Graph – a visual representation of the relationships between data collected
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• Scatterplot – a graphical representation used to display the relationship between discrete data pairs
• Slope – rate of change in y (vertical) compared to change in x (horizontal), or or , denoted as m in y = mx + b
• Trend line – the line that best fits the data points of a scatterplot
• Unit rate – a ratio between two different units where one of the terms is 1
• y-intercepty coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)

Related Vocabulary:

 Axis Constant of proportionality Dependent Independent Intersection Interval Linear No association Non-linear Non-proportional Origin Point Proportional
Unit Assessment Items System Resources Other Resources

Show this message:

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 – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

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 Grade 8 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.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
8.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
8.1A 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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
8.1B 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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
8.1C

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 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
• 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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
8.1D 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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
8.1E 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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
8.1F 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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
8.1G 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.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• Making inferences from data
• TxCCRS:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
8.4 Proportionality. The student applies mathematical process standards to explain proportional and non-proportional relationships involving slope. The student is expected to:
8.4B Graph proportional relationships, interpreting the unit rate as the slope of the line that models the relationship.

Graph

PROPORTIONAL RELATIONSHIPS, INTERPRETING THE UNIT RATE AS THE SLOPE OF THE LINE THAT MODELS THE RELATIONSHIP

Including, but not limited to:

• Unit rate – a ratio between two different units where one of the terms is 1
• Slope – 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
• 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
• Graphing unit rate from various representations
• Verbal
• Numeric
• Tabular(horizontal/vertical)
• Symbolic/algebraic
• Connections between unit rate in proportional relationships to the slope of a line

Note(s):

• Algebra I will calculate the rate of change of a linear function represented tabularly, graphically, or algebraically in context of mathematical and real-world problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
8.5 Proportionality. The student applies mathematical process standards to use proportional and non-proportional relationships to develop foundational concepts of functions. The student is expected to:
8.5A Represent linear proportional situations with tables, graphs, and equations in the form of y = kx.
Supporting Standard

Represent

LINEAR PROPORTIONAL SITUATIONS WITH TABLES, GRAPHS, AND EQUATIONS IN THE FORM OF y = kx

Including, but not limited to:

• Slope – 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
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Linear proportional problem situations
• 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.
• Multiple representations of linear proportional problem situations
• Verbal
• Table (horizontal/vertical)
• Graph
• Algebraic
• Both y = kx and kx = y forms
• Association of k as multiplication by a given constant factor (including unit rate)
• Rational number coefficients and constants
• Manipulation of equations

Note(s):

• Grade 7 represented constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.
• Grade 7 converted between measurement systems, including the use of proportions and the use of unit rates.
• Algebra I will write and solve equations involving direct variation.
• Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
• Algebra I will write linear equations with two variables given a table of values, a graph, and a verbal description.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
8.5B Represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b  0.
Supporting Standard

Represent

LINEAR NON-PROPORTIONAL SITUATIONS WITH TABLES, GRAPHS, AND EQUATIONS IN THE FORM OF y = mx + b, WHERE b ≠ 0

Including, but not limited to:

• Slope – 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
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Linear non-proportional problem situations
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m =
• Multiple representations of linear non-proportional problem situations
• Verbal
• Table (horizontal/vertical)
• Graph
• Algebraic
• Both y = mx + b and mx + b = y forms
• Rational number coefficients and constants
• Manipulation of equations

Note(s):

• Grade 7 represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
• Algebra I will write linear equations in two variables in various forms, including y = mx + b, Ax + By = C, and yy1 = m(xx1), given one point and the slope and given two points.
• Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
• Algebra I will write linear equations with two variables given a table of values, a graph, and a verbal description.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
8.5C Contrast bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation.
Supporting Standard

Contrast

BIVARIATE SETS OF DATA THAT SUGGEST A LINEAR RELATIONSHIP WITH BIVARIATE SETS OF DATA THAT DO NOT SUGGEST A LINEAR RELATIONSHIP FROM A GRAPHICAL REPRESENTATION

Including, but not limited to:

• Data – information that is collected about people, events, or objects
• Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
• Discrete paired data – data that involves only distinct values that are finite or countable
• Scatterplot – a graphical representation used to display the relationship between discrete data pairs
• Characteristics of a scatterplot
• Title clarifies the meaning of the data represented.
• Subtitles clarify the meaning of data represented on each axis.
• Numerical data represented with labels may be whole numbers, fractions, or decimals.
• Points are not connected by a line.
• Scale of the axes may be intervals of one or more, and scale intervals are proportionally displayed.
• The scales of the axes are number lines.
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Characteristics of bivariate data that suggests a linear relationship
• 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
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m =
• Characteristics of bivariate data that does not suggest a linear relationship
• Not linear
• Not represented by y = kx or y = mx + b
• No constant slope
• May or may not cross the origin (0, 0)

Note(s):

• Grade 8 introduces contrasting bivariate sets of data that suggest a linear relationship with bivariate sets of data that do not suggest a linear relationship from a graphical representation.
• Algebra I will calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• V.B. Statistical Reasoning – Describe data
• V.B.4. Describe patterns and departure from patterns in the study of data.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
8.5D Use a trend line that approximates the linear relationship between bivariate sets of data to make predictions.

Use

A TREND LINE THAT APPROXIMATES THE LINEAR RELATIONSHIP BETWEEN BIVARIATE SETS OF DATA TO MAKE PREDICTIONS

Including, but not limited to:

• Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
• Characteristics of bivariate data that suggests a linear relationship
• 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.
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m =
• Graph of data suggests a constant rate of change between the independent and dependent values
• Trend line – the line that best fits the data points of a scatterplot
• A tool for making predictions by approximating the linear relationship between bivariate sets of data
• A trend line contains most of the data points and/or is situated so that the data points are evenly distributed above and below the line.
• Given or collected data
• Analysis of parts of data representation
• Title
• Labels
• Scales
• Graphed data
• Predictions of independent value when given a dependent value using a trend line that approximates the linear relationship
• Predictions of dependent value when given an independent value using a trend line that approximates the linear relationship

Note(s):

• Grade 8 introduces using a trend line that approximates the linear relationship between bivariate sets of data to make predictions.
• Algebra I will calculate, using technology, the correlation coefficient between two quantitative variables and interpret this quantity as a measure of the strength of the linear association.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• V.B. Statistical Reasoning – Describe data
• V.B.4. Describe patterns and departure from patterns in the study of data.
• VI.C. Functions – Model real-world situations with functions
• VI.C.1. Apply known functions to model real-world situations.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
8.5I Write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.

Write

AN EQUATION IN THE FORM y = mx + b TO MODEL A LINEAR RELATIONSHIP BETWEEN TWO QUANTITIES USING VERBAL, NUMERICAL, TABULAR, AND GRAPHICAL REPRESENTATIONS

Including, but not limited to:

• Slope – 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
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• One quantity is dependent on the other
• Two quantities may be directly proportional to each other
• Can be classified as a positive or negative relationship
• Can be expressed as a pair of values that can be graphed as ordered pairs
• Graph of the ordered pairs matching the relationship will form a line
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant slope represented as m = or m = or m =
• Equations in the form y = mx + b to represent relationships between two quantities
• Rational number coefficients and constants
• Various representations
• Verbal
• Numerical
• Tabular (horizontal/vertical)
• Graphical

Note(s):

• Grade 7 represented linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.
• Algebra I will write linear equations in two variables given a table of values, a graph, and a verbal description.
• Algebra I will use, translate, and make connections among algebraic, tabular, graphical, or verbal descriptions of linear functions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing, applying, and analyzing proportional relationships
• TxCCRS:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
8.9 Expressions, equations, and relationships. The student applies mathematical process standards to use multiple representations to develop foundational concepts of simultaneous linear equations. The student is expected to:
8.9A Identify and verify the values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.
Supporting Standard

Identify, Verify

THE VALUES OF x AND y THAT SIMULTANEOUSLY SATISFY TWO LINEAR EQUATIONS IN THE FORM y = mx + b FROM THE INTERSECTIONS OF THE GRAPHED EQUATIONS

Including, but not limited to:

• Slope – 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
• y-intercept – y coordinate of a point at which the relationship crosses the y-axis meaning the x coordinate is equal to zero, denoted as b in y = mx + b and the ordered pair (0, b)
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• Linear proportional relationship
• Linear
• Represented by y = kx or y = mx + b, where b = 0
• For y = kx and y = mx + b, k = the slope, m
• Passes through the origin (0, 0), meaning the y-intercept, b, is 0
• Constant of proportionality represented as
• Constant slope represented as m = or m = or m =
• Linear non-proportional relationship
• Linear
• Represented by y = mx + b, where b ≠ 0
• Does not pass through the origin (0, 0), meaning the y-intercept, b, is not 0
• Constant slope represented as m = or m = or m =
• Values of x and y that simultaneously satisfy two linear equations from a graph
• Simultaneously satisfy both linear equations means the intersection point or solution will lie on both lines
• Algebraic verification of the intersection of graphed equations as ordered pairs
• Intersection point or solution, when substituted into each equation, will result in true equations. If both equations are true, then the point of intersection simultaneously satisfies both equations.
• Intersection point and any other points on the same line result in equivalent slopes. If two lines contain the same point, then the point simultaneously satisfies both equations.

Note(s):

• Grade 7 determined if the given value(s) make(s) one-variable, two-step equations and inequalities true.
• Algebra I will graph systems of two linear equations in two variables on the coordinate plane and determine the solutions if they exist.
• Algebra I will solve systems of linear equations using concrete models, graphs, tables, and algebraic methods.
• Algebra I will estimate graphically the solutions to systems of two linear equations with two variables in real-world problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships, including the Pythagorean Theorem
• TxCCRS:
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.1. Describe and interpret solution sets of equalities and inequalities.
• II.D. Algebraic Reasoning – Representing relationships
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
8.11 Measurement and data. The student applies mathematical process standards to use statistical procedures to describe data. The student is expected to:
8.11A Construct a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data.
Supporting Standard

Construct

A SCATTERPLOT

Including, but not limited to:

• Graph – a visual representation of the relationships between data collected
• Organization of data used to describe and summarize data
• Data – information that is collected about people, events, or objects
• Discrete paired data – data that involves only distinct values that are finite or countable
• Limitations
• Various forms of positive and negative rational numbers within related data pairs
• Integers
• Decimals
• Fractions
• Data representation
• Scatterplot – a graphical representation used to display the relationship between discrete data pairs
• Characteristics of a scatterplot
• Titles and subtitles
• Title represents the purpose of collected data
• Subtitles clarify the meaning of the data represented on each axis
• First quadrant of coordinate plane
• Number lines form x-axis and y-axis
• Proportional increments
• Intervals of one or more
• Break between 0 and the first marked interval indicated in one or both axes to accommodate large numbers if necessary
• Ordered pairs
• Pairs of data form each ordered pair
• Points not connected by a line
• Data pairs analyzed to find possible relationships between two sets of data
• Pairs of numbers collected to determine if a relationship exists between the two sets of data
• Relationship between each data pair is discrete although the data itself could be either continuous or discrete in nature
• Given or collected data
• Bivariate data – data relating two quantitative variables that can be represented by a scatterplot

Describe

THE OBSERVED DATA ON A SCATTERPLOT TO ADDRESS QUESTIONS OF ASSOCIATION SUCH AS LINEAR, NON-LINEAR, AND NO ASSOCIATION BETWEEN BIVARIATE DATA

Including, but not limited to:

• Discrete paired data – data that involves only distinct values that are finite or countable
• Limitations
• Various forms of positive and negative rational numbers within related data pairs
• Integers
• Decimals
• Fractions
• Data representation
• Scatterplot – a graphical representation used to display the relationship between discrete data pairs
• Data pairs analyzed to find possible relationships between two sets of data
• Pairs of numbers collected to determine if a relationship exists between the two sets of data
• Relationship between each data pair is discrete although the data itself could be either continuous or discrete in nature
• Given or collected data
• Bivariate data – data relating two quantitative variables that can be represented by a scatterplot
• Association within a scatterplot
• Linear trend
• Positive trend
• Negative trend
• Non-linear trend
• No trend or no association

Note(s):

• Grade 5 represented discrete paired data on a scatterplot.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Making inferences from data
• TxCCRS:
• III.C. Geometric and Spatial Reasoning – Connections between geometry and other mathematical content strands
• III.C.2. Make connections between geometry, statistics, and probability.
• V.B. Statistical Reasoning – Describe data
• V.B.2. Construct appropriate visual representations of data.
• V.B.4. Describe patterns and departure from patterns in the study of data.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.1. Analyze data sets using graphs and summary statistics.
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
• VI.A. Functions – Recognition and representation of functions
• VI.A.2. Recognize and distinguish between different types of functions.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.