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 Instructional Focus DocumentGrade 7 Mathematics
 TITLE : Unit 04: Graphs and Two-Variable Equations SUGGESTED DURATION : 14 days

Unit Overview

Introduction
This unit bundles student expectations that address the constant rate of change, the constant of proportionality, and various representations of linear relationships. 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.”

Prior to this Unit
In Unit 02, students modeled and solved one-variable, two-step equations and inequalities with concrete and pictorial models and algebraic representations. In Unit 03, students examined proportional reasoning with ratios and rates through the lens of constant rates of change. Students represented constant rates of change given pictorial, tabular, verbal, numeric, graphical, and algebraic representations. In Grade 6, students identified independent and dependent quantities from tables and graphs, wrote an equation that represents the relationship between independent and dependent quantities from a table, and represented a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b. Students also compared two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships.

During this Unit
Students use bivariate data, data with two variables, to reexamine constant rates of change given pictorial, tabular, verbal, numeric, graphical, and algebraic representations and extend their understandings of the constant of proportionality. Students are formally introduced to the slope intercept form of equations, y = mx + b, to represent linear relationships. Although students are not formally introduced to slope or y-intercept in linear proportional and non-proportional relationships until Grade 8, students are expected to relate the constant rate of change to m, and the y-coordinate, when the x-coordinate is zero, to b in equations that simplify to the form y = mx + b. This relationship is examined through the ratio of rise to run and the change in the y-values to the change in the x-values. Students represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.

After this Unit
In Grade 8, students will solve problems involving direct variation, distinguish between and represent proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0. Students will formally develop the concepts and applications of slope and y-intercept in proportional and non-proportional functions. Additionally, students will study systems of linear equations as they identify and verify values of x and y that simultaneously satisfy two linear equations in the form y = mx + b from the intersections of the graphed equations.

In Grade 7, representing constant rates of change is identified as STAAR Readiness Standard 7.4A, determining the constant of proportionality (k = y/x) is identified as STAAR Supporting Standard 7.4C. These standards are a foundational block of the Grade 7 Texas Response to Curriculum Focal Points (TxRCFP): Representing and applying proportional relationships. Representing linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b is identified as STAAR Readiness Standard 7.7A and is a part of the Grade 7 Focal Point: Using expressions and equations to describe relationships in a variety of contexts, including geometric problems (TxRCFP). All of these standards are subsumed within the Grade 7 STAAR Reporting Category 2: Computations and Algebraic Relationships. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, II. Algebraic Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Representation, and X. Connections.

Research
According to research, “Students in middle grades should develop an understanding of the multiple methods of expressing real-world functional relationships (words, graphs, equations, and tables). Working with these different representations of functions will allow students to develop a fuller understanding of functions.” (Van de Walle, Lovin, 2006, p. 284). As students begin to relate constant rates of change within multiple algebraic representations as a prerequisite for future coursework with slope and y-intercept, it should be noted that “Children need to learn that, in mathematics as in most subject areas, they should not do something a certain way because someone tells them to; rather they need to understand why doing it that way makes sense (or doesn’t make sense)” (Reyes, Lindquist, Lambdin & Smith, 2012, p. 318). Students need to develop the ability to move among algebraic representations flexibly, “Students are often not made aware of the power of mathematics. The realization of how the transfer of an equation to graphic form can reveal a whole set of possible solutions may be an eye-opening motivating factor for learning mathematics” (Solomon, 2007, p. 200).

Reyes, R. E., Lindquist, M., Lambdin, D. V., & Smith, N. L. (2012). Helping children learn mathematics. (10th ed.). Hoboken, NJ: Wiley.
Solomon, P. (2006). The math we need to know and do in grades 6 – 9. Thousand Oaks, CA: Corwin Press.
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., & Lovin, L. (2006). Teaching student-centered mathematics grades 5 – 8. 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) and can be reasoned up and down in situations involving invariant (constant) relationships builds flexible proportional reasoning in order to make predictions and critical judgements about the relationship.
• Constant rate of change and constant of proportionality represent equivalent values and are used to solve problems involving a proportional relationship.
• How can the …
• constant rate of change
• constant of proportionality
… be described and determined from a(n) …
• table?
• verbal description?
• graph?
• algebraic representation?
• What relationship exists between the constant of proportionality and constant rate of change?
• Equations can be modeled, written, and solved using various methods to gain insight into the context of the situation and make critical judgments about algebraic relationships and efficient strategies.
• What is the process of representing a linear relationship …
• verbally?
• with a table?
• with a graph?
• with an equation that simplifies to the form of y = mx + b?
• What is the purpose of using different representations, and how is the context of the problem highlighted in each representation?
• What are the characteristics of a linear relationship?
• 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?
• Proportionality
• Ratios and Rates
• Constant rate of change
• Relationships and Generalizations
• Equivalence
• Constant of proportionality
• Representations
• Expressions, Equations, and Relationships
• Algebraic Relationships
• Linear
• Independent and dependent quantities
• Numeric and Algebraic Representations
• Expressions
• Equations
• Equivalence
• Operations
• Properties of operations
• Order of operations
• 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:

• Some students may think the constant rate of change and the constant of proportionality are the same value rather than understanding the constant of proportionality is represented by k = and may equal the constant rate of change for the linear equation y = mx + b only if b = 0.
• Some students may not relate the constant rate of change to m in the equation y = mx + b.
• Some students may think that the ratio for rate of change in a linear relationship is m = , since the x-coordinate (horizontal) always comes before the y-coordinate (vertical) in an ordered pair, instead of the correct representation that rate of change in a linear relationship is m = .

Underdeveloped Concepts:

• Some students may think variables are letters representing an object as opposed to representing a number or quantity of objects.
• Some students may think that a variable is only a placeholder as it is in an equation.
• Some students may think the equal sign means, “solve this” or “the answer is” rather than understanding that equal sign represents a quantitative and balanced relationship.

Unit Vocabulary

• Coefficient – a number that is multiplied by a variable(s)
• Constant – a fixed value that does not appear with a variable(s)
• Constant of proportionality – a constant ratio between two proportional quantities denoted by the symbol k
• Constant rate of change – a ratio when the dependent, y-value, changes at a constant rate for each independent, x-value
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Unit rate – a ratio between two different units where one of the terms is 1

Related Vocabulary:

 Algebraic Dependent Equation Graphical Independent Linear Linear non-proportional Linear proportional Ordered pair Origin Positive Proportional Negative Rise Run Slope intercept form Tabular Undefined x-coordinate x-value y-coordinate y-value
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 Creator 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 7 Mathematics TEKS

Texas Instruments – Graphing Calculator Tutorials

TEKS# SE# Unit Level Taught Directly TEKS Unit Level Specificity

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.

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.
7.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
7.1A 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• X. Connections
7.1B 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• VIII. Problem Solving and Reasoning
7.1C

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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• VIII. Problem Solving and Reasoning
7.1D 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• IX. Communication and Representation
7.1E 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• IX. Communication and Representation
7.1F 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• X. Connections
7.1G 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:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• Representing and applying proportional relationships
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• Comparing sets of data
• TxCCRS:
• IX. Communication and Representation
7.4 Proportionality. The student applies mathematical process standards to represent and solve problems involving proportional relationships. The student is expected to:
7.4A

Represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d=rt.

Represent

CONSTANT RATES OF CHANGE IN MATHEMATICAL AND REAL-WORLD PROBLEMS GIVEN PICTORIAL, TABULAR, VERBAL, NUMERIC, GRAPHICAL, AND ALGEBRAIC REPRESENTATIONS

Including, but not limited to:

• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Various forms of positive and negative rational numbers
• Integers
• Decimals
• Fractions
• Constant rate of change – a ratio when the dependent, y-value, changes at a constant rate for each independent, x-value
• Proportional mathematical and real-world problems
• Various representations of constant rates of change in mathematical and real-world situations
• Pictorial
• Tabular (vertical/horizontal)
• Verbal
• Numeric
• Graphical
• Algebraic

Note(s):

• Grade 6 compared two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships.
• Grade 6 gave examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.
• Grade 6 represented mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing and applying proportional relationships
• TxCCRS:
• I. Numeric Reasoning
• II. Algebraic Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
7.4C Determine the constant of proportionality (k = y/x) within mathematical and real-world problems.
Supporting Standard

Determine

THE CONSTANT OF PROPORTIONALITY ( ) WITHIN MATHEMATICAL AND REAL-WORLD PROBLEMS

Including, but not limited to:

• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Various forms of positive and negative rational numbers
• Integers
• Decimals
• Fractions
• Percents converted to equivalent decimals or fractions for multiplying or dividing fluently
• Constant rate of change – a ratio when the dependent, y-value, changes at a constant rate for each independent, x-value
• Constant of proportionality – a constant ratio between two proportional quantities denoted by the symbol k
• Characteristics of the constant of proportionality
• A graphed proportional relationship where x represents the independent variable and y represents the dependent variable.
• Independent variables describe the input values in a relationship, normally represented by the x coordinate in the ordered pairs (xy)
• Dependent variables describe the output values in a relationship, normally represented by the y coordinate in the ordered pairs (xy).
• The constant of proportionality can never be zero.
• Unit rate – a ratio between two different units where one of the terms is 1
• Proportional mathematical and real-world problems
• Various representations of the constant of proportionality
• Tabular (vertical/horizontal)
• Verbal
• Numeric
• Graphical
• Algebraic

Note(s):

• Grade 6 compared two rules verbally, numerically, graphically, and symbolically in the form of y = ax or y = x + a in order to differentiate between additive and multiplicative relationships.
• Grade 8 will solve problems involving direct variation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing and applying proportional relationships
• TxCCRS:
• I. Numeric Reasoning
• II. Algebraic Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
7.7 Expressions, equations, and relationships. The student applies mathematical process standards to represent linear relationships using multiple representations. The student is expected to:
7.7A Represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b.

Represent

LINEAR RELATIONSHIPS USING VERBAL DESCRIPTIONS, TABLES, GRAPHS, AND EQUATIONS THAT SIMPLIFY TO THE FORM y = mx + b

Including, but not limited to:

• Rational numbers – the set of numbers that can be expressed as a fraction , where a and b are integers and b ≠ 0. The set of rational numbers is denoted by the symbol Q.
• Various forms of positive and negative rational numbers as constants and coefficients
• Coefficient – a number that is multiplied by a variable(s)
• Constant – a fixed value that does not appear with a variable(s)
• Integers
• Decimals
• Fractions
• Constant rate of change – a ratio when the dependent, y-value, changes at a constant rate for each independent, x-value
• 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 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 = m
• Linear non-proportional relationship
• Linear
• Does not pass through the origin (0, 0)
• Represented by y = mx + b, where b ≠ 0
• Constant rate of change represented as m = or m = • Rate of change is either positive, negative, zero, or undefined
• Various representations to describe algebraic relationships
• Verbal descriptions
• Tables
• Graphs
• Equations
• In the form y = mx + b(slope intercept form)

Note(s):

• Grade 6 identified independent and dependent quantities from tables and graphs.
• Grade 6 wrote an equation that represents the relationship between independent and dependent quantities from a table.
• Grade 6 represented a given situation using verbal descriptions, tables, graphs, and equations in the form y = kx or y = x + b.
• Grade 8 will represent linear non-proportional situations with tables, graphs, and equations in the form of y = mx + b, where b ≠ 0.
• Grade 8 will write an equation in the form y = mx + b to model a linear relationship between two quantities using verbal, numerical, tabular, and graphical representations.
• Grade 8 will distinguish between proportional and non-proportional situations using tables, graphs, and equations in the form y = kx or y = mx + b, where b ≠ 0.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Using expressions and equations to describe relationships in a variety of contexts, including geometric problems
• TxCCRS:
• I. Numeric Reasoning
• II. Algebraic Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation 