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Instructional Focus Document
Grade 8 Mathematics
TITLE : Unit 06: Statistics with Bivariate Data SUGGESTED DURATION : 10 days

Unit Overview

Introduction
This unit bundles student expectations that address representing bivariate sets of data with scatterplots and representations of linear situations. 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 Grade 5, students represented discrete paired data on a scatterplot. In Grade 8 Unit 04, students used similar right triangles to develop an understanding of slope. Students 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 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 (positive or negative trend), non-linear, or no association. 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 that represents a linear relationship, students use the trend line to make predictions and interpret the slope of the line that models the relationship as the unit rate of the scenario.

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 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.

Additional Notes
In Grade 8, graphing proportional relationships, interpreting the unit rate as the slope of the line that models the relationship and writing an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations are identified as STAAR Readiness Standards 8.4B and 8.5I. Representing linear proportional situations with tables, graphs, and equations in the form of y = kx and representing linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0 are identified as STAAR Supporting Standards 8.5A and 8.5B. All of these standards are subsumed under the Grade 8 Reporting Category: Computations and Algebraic Relationships. 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 is STAAR Supporting Standard 8.5C. Using a trend line that approximates the linear relationship between bivariate sets of data to make predictions is identified as STAAR Readiness Standard 8.5D. Both of these standards are part of the Grade 8 STAAR Reporting Category: Data Analysis and Personal Financial Literacy. All of these standards are subsumed under the Grade 8 Texas Response to Curriculum Focal Points (TxRCFP): Representing, applying and analyzing proportional relationships. Constructing a scatterplot and describe the observed data to address questions of association such as linear, non-linear, and no association between bivariate data is STAAR Supporting Standard 8.11A, part of the Grade 8 STAAR Reporting Category: Data Analysis and Personal Financial Literacy, and identified within the Grade 8 Focal Point: Making inferences from data (TxRCFP). This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, II. Algebraic Reasoning, III. Geometric Reasoning, VI. Statistical Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Representation, and X. Connections.

Research
According to Developing Essential Understanding of Statistics Grades 6 – 8 (2013) from the National Council of Teachers of Mathematics (NCTM), “The analysis of bivariate data focuses on identifying and describing patterns in the covariability in the data. For a collection of bivariate data on two quantitative variables, a scatterplot is a useful graphical display for illustrating the covariability and for identifying the general direction and form of a relationship” (p. 66). As students study scatterplots, NCTM (2010) suggests that “Teachers should point out that a trend line represents a set of biviariate data just as a measure of center or spread represent a set of univariate data…During their study of bivariate data, students should have the opportunity to analyze scatterplots and trend lines. Teachers should

 

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.
National Council of Teachers of Mathematics. (2013). Developing essential understanding of statistics grades 6 – 8. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/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


  • 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 between linear and non-linear proportional situations and identify attributes of linear relationships.
      • 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
    • Statistics
      • Predictions and inferences
      • Data
      • Statistical representations
    • Relationships and Generalizations
      • Linear
      • Non-linear
    • Representations
  • Measurement and Data
    • Coordinate Plane
    • 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.

  • 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?
      • 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?
    • 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 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?
      • 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?
  • Proportionality
    • Statistics
      • Predictions and inferences
      • Data
      • Statistical representations
    • Ratios and Rates
      • Unit rates
      • Slope
    • Relationships and Generalizations
      • Equivalence
      • Constant of proportionality
      • Independent and dependent quantities
      • Linear
      • Linear proportional
      • Linear non-proportional
    • 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.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

  • 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.

Underdeveloped Concepts:

  • 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 = .
  • Students may use (y, x) as the ordered pair instead of (x, y)
  • Some students may not associate the unit rate of a problem situation to the slope of the line that represents the problem situation.
  • 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.

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 – 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
  • 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:

  • Association
  • Constant
  • Constant rate of change
  • Correlation
  • Dependent
  • Independent
  • Linear association
  • Negative trend
  • No association
  • No trend
  • Non-linear association
  • Non-linear relationship
  • Non-proportional relationship
  • Ordered pair
  • Origin
  • Positive trend
  • Prediction
  • Proportional relationship
  • Rate of change
  • Scale factor
  • x-axis
  • y-axis
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 – 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:
    • X. Connections
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:
    • VIII. Problem Solving and Reasoning
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:
    • VIII. Problem Solving and Reasoning
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:
    • IX. Communication and Representation
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:
    • IX. Communication and Representation
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:
    • X. Connections
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:
    • IX. Communication and Representation
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.
Readiness Standard

 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):

  • Grade Level(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:
    • I. Numeric Reasoning
    • II. Algebraic Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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 Level(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:
    • I. Numeric Reasoning
    • II. Algebraic Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
    • X. Connections
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 Level(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:
    • I. Numeric Reasoning
    • II. Algebraic Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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 Level(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:
    • II. Algebraic Reasoning
    • VI. Statistical Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
8.5D Use a trend line that approximates the linear relationship between bivariate sets of data to make predictions.
Readiness Standard

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 Level(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:
    • II. Algebraic Reasoning
    • VI. Statistical Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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.
Readiness Standard

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 Level(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. Algebraic Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
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 Level(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:
    • II. Algebraic Reasoning
    • VI. Statistical Reasoning
    • VIII. Problem Solving and Reasoning
    • IX. Communication and Representation
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of each subject in the required curriculum.

School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.


School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.

http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4 


Choose appropriate ELPS to support instruction.

ELPS# Subsection C: Cross-curricular second language acquisition essential knowledge and skills.
Click here to collapse or expand this section.
ELPS.c.1 The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.1A use prior knowledge and experiences to understand meanings in English
ELPS.c.1B monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and grade-level vocabulary
ELPS.c.1D speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying ideas by defining or describing when exact English words are not known)
ELPS.c.1E internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept and language attainment
ELPS.c.1F use accessible language and learn new and essential language in the process
ELPS.c.1G demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one commensurate with grade-level learning expectations
ELPS.c.1H develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2 The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.2A distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters
ELPS.c.2C learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
ELPS.c.2E use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language attainment
ELPS.c.2G understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and contexts are familiar to unfamiliar
ELPS.c.2H understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3 The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce English words in a manner that is increasingly comprehensible
ELPS.c.3B expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using routine language needed for classroom communication
ELPS.c.3C speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as more English is acquired
ELPS.c.3D speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E share information in cooperative learning interactions
ELPS.c.3F ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended speaking assignments
ELPS.c.3G express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a variety of social and grade-appropriate academic topics
ELPS.c.3H narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and language attainment.
ELPS.c.4 The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used routinely in written classroom materials
ELPS.c.4D use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to enhance comprehension of written text
ELPS.c.4E read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly challenging language
ELPS.c.4G demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H read silently with increasing ease and comprehension for longer periods
ELPS.c.4I demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with content area needs
ELPS.c.4J demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content area needs
ELPS.c.4K demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5 The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet grade-level learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1, certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing system. The student is expected to:
ELPS.c.5A learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English is acquired
ELPS.c.5D edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses commensurate with grade-level expectations as more English is acquired
ELPS.c.5E employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in increasingly accurate ways as more English is acquired
ELPS.c.5G narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
Last Updated 08/01/2018
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