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 TITLE : Unit 10: Making Connections SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address operations with rational numbers regarding sales tax, ratios and rates, constant rate of change, constant of proportionality, linear relationships using various representations, problems involving percents, and similar shapes. 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 Units 01 – 09, students solved problems using addition, subtraction, multiplication, and division of rational numbers. In Unit 03, students represented and solved problems involving proportional relationships. This included representing constant rates of change, calculating unit rates, determining the constant of proportionality, and solving problems involving ratios, rates, and percents. In Unit 04, students represented linear relationships using multiple representations including equations that simplify to the form y = mx + b. In Unit 05, students continued to apply concepts of proportionality to solve problems involving similar shapes and scale drawings.

During this Unit
Students continue to apply operations with rational numbers to calculate unit rates from rates and determine the constant of proportionality within mathematical and real-world problems. Students are expected to represent constant rates of change given pictorial, tabular, verbal, numeric, graphical, and algebraic representations. Students use verbal descriptions, tables, graphs, and the slope-intercept form of equations, y = mx + b, to represent linear relationships. Students revisit and apply concepts of proportionality to two-dimensional figures as they solve mathematical and real-world problems involving similar shapes.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 7

After this Unit
In Grade 8, students will solve problems involving direct variation and 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. Students will solve problems involving direct variation. Also, students will generalize the ratio of corresponding sides of similar shapes are proportional, compare and contrast the attributes of a shape and its dilations on a coordinate plane, and use algebraic representations to explain the effect of a given scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of the dilation.

Research
According to Driscoll (1999), “It is not so much exposure to different representations that is important for students entering and learning algebra, as it is the linking of representations and translation among them” (p. 144). Relating the pictorial, tabular, verbal, numeric, and graphical representations of constant rate of change is essential to future coursework involving proportional and non-proportional relationships.” Additional research states, “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).

Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6 – 10. Portsmouth, NH: Heinemann.
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. (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.

 Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life. What relationships exist within and between mathematical operations? How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies? Why is understanding the problem solving process an essential part of learning and working mathematically? Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical 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)
• Estimation strategies can be used to mentally approximate solutions and determine reasonableness of solutions (rational numbers).
• What strategies can be used to estimate solutions to problems?
• When might an estimated answer be preferable to an exact answer?
• How can an estimation aid in determining the reasonableness of an actual solution?
• When might one estimation strategy be more beneficial than another?
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (rational numbers).
• How does the context of a problem situation affect the representation, operation(s), and/or solution strategy that could be used to solve the problem?
• How does understanding …
• relationships within and between operations
• properties of operations
… aid in determining an efficient strategy or representation to investigate and solve problem situations?
• Why is it important to understand when and how to use standard algorithms?
• Why is important to be able to perform operations with rational numbers fluently?
• Operational understandings lead to generalizations that aid in determining the reasonableness of solutions (rational numbers).
• When adding two non-zero positive rational numbers, why is the sum always greater than each of the addends?
• When two non-zero rational numbers have the same sign, why does the sum always have the same sign as both addends?
• When two non-zero rational numbers have different signs, why does the sum always have the sign of the addend with the greatest absolute value?
• When subtracting two non-zero positive rational numbers with the minuend greater than the subtrahend, why is the difference always less than the minuend?
• When subtracting two non-zero positive rational numbers with the subtrahend greater than the minuend, why is the difference always less than the minuend and negative?
• When subtracting two rational numbers with opposite signs, why can the subtraction problem be rewritten as an addition problem by adding the opposite of the rational number following the subtraction symbol and applying the rules for adding rational numbers?
• When multiplying two non-zero positive rational numbers with one of the factors greater than one, why is the product always greater than the smallest factor?
• When multiplying two non-zero positive rational numbers with both factors less than one, why is the product less than the smallest factor?
• When multiplying or dividing two rational numbers with the same sign, why is the product or quotient positive?
• When multiplying or dividing two rational numbers with different signs, why is the product or quotient negative?
• When dividing two non-zero positive rational numbers with the dividend less than the divisor, why is the quotient always greater than zero and less than one?
• When dividing two non-zero positive rational numbers with the dividend greater than the divisor, why is the quotient always greater than one?
• Why is dividing by a non-zero rational number equivalent to multiplying by its reciprocal?
• When multiplying or dividing two or more rational numbers with no negative signs or an even number of negative signs, why is the product or quotient positive?
• When multiplying or dividing two or more rational numbers with one negative sign or an odd number of negative signs, why is the product or quotient negative?
• 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.
• Unit rate, 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
• unit rate
• constant of proportionality
… be described and determined from a(n) …
• table?
• verbal description?
• graph?
• algebraic representation, including d = rt?
• Why would you describe a rate where either of the quantities has a value of 1 as a unit rate?
• What relationship exists between the constant of proportionality and constant rate of change?
• Various strategies may be used to solve problems involving ratios, rates, and percents.
• How can …
• scale factors
• tables
• proportions
… model equivalence and be used to solve problems involving …
• ratios?
• rates?
• percents?
• What is the process to solve a problem involving …
• percent increase?
• percent decrease?
• Analyzing geometric relationships in models aids in representing the attributes and quantifiable measures to generalize proportional geometric relationships and solve problems.
• How does the scale factor affect the size of two similar shapes?
• What is the relationship between the scale factor and linear measures of similar shapes?
• What is the process for solving problems involving similar shapes and scale drawings?
• 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?
• Number and Operations
• Number
• Rational numbers
• Operations
• Subtraction
• Multiplication
• Division
• Proportionality
• Fractions and Decimals
• Percents
• Ratios and Rates
• Unit rates
• Constant rate of change
• Scale factors
• Relationships and Generalizations
• Equivalence
• Constant of proportionality
• Geometric similarity
• Scale drawings
• Representations
• Solution Strategies and Algorithms
• Expressions, Equations, and Relationships
• Algebraic Relationships
• Linear
• 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

Underdeveloped Concepts:

• Some students may think the constant rate of change and the constant of proportionality are always the same value rather than understanding the constant of proportionality is represented by and may equal the constant rate of change for the linear equation y = mx + b only if b = 0.
• 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 = .
• Some students may not understand that a scale factor must be applied to all dimensions of a shape to maintain similarity.
• Some students may attempt to perform computations with percents without converting them to equivalent decimals or fractions for multiplying or dividing.
• Some students may not connect the constant rate of change to m in the equation y = mx + b.
• Some students may think that the order of the terms in a ratio or proportion is not important.
• Some students may think variables are letters representing an object as opposed to representing a number or quantity of objects.
• Some students may think proportionality is an additive relationship instead of a multiplicative relationship.

#### Unit Vocabulary

• Appreciation – the increase in value over time
• Coefficient – a number that is multiplied by a variable(s)
• Commission – pay based on a percentage of the sales or profit made by an employee or agent
• 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
• Depreciation – the decrease in value over time
• Linear relationship – a relationship with a constant rate of change represented by a graph that forms a straight line
• Markdown – the difference between the original price of an item and its current price
• Markup – the difference between the purchase price of an item and its sales price
• Percent – a part of a whole expressed in hundredths
• Percent decrease – a change in percentage where the value decreases
• Percent increase – a change in percentage where the value increases
• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Principal – the original amount invested or borrowed
• Rate – a multiplicative comparison of two different quantities where the measuring unit is different for each quantity
• Ratio – a multiplicative comparison of two quantities
• 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.
• Similar shapes – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• Simple interest – interest paid or earned on the original principal amount, disregarding any previously paid or earned interest
• Tax – a financial charge, usually a percentage applied to goods, property, sales, etc.
• Tip – an amount of money rendered for a service, gratuity
• Unit rate – a ratio between two different units where one of the terms is 1

Related Vocabulary:

 Customary Independent Integer Dependent Density Dimensional analysis Linear Metric Negative Rational Number Number line Non-proportional Ordered pair Percent graph Positive Proportion Proportional Rate Ratio Rational Number Scale drawing Scale factor Slope-intercept form Speed Strip diagram 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 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 7 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
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:
• 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.
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:
• 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.
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:
• 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.
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:
• 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.
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:
• 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.
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:
• 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.
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:
• 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.
7.3 Number and operations. The student applies mathematical process standards to add, subtract, multiply, and divide while solving problems and justifying solutions. The student is expected to:
7.3B Apply and extend previous understandings of operations to solve problems using addition, subtraction, multiplication, and division of rational numbers.

Apply, Extend

PREVIOUS UNDERSTANDINGS OF OPERATIONS TO SOLVE PROBLEMS USING ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION OF RATIONAL NUMBERS

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
• Equivalent representations of a negative number
• Generalizations of integer operations
• If a pair of addends has the same sign, then the sum will have the sign of both addends.
• If a pair of addends has opposite signs, then the sum will have the sign of the addend with the greatest absolute value.
• A subtraction problem may be rewritten as an addition problem by adding the opposite of the integer following the subtraction symbol, and then applying the rules for addition.
• Multiplication and division
• If two rational numbers have the same sign, then the product or quotient is positive.
• If two rational numbers have opposite signs, then the product or quotient is negative.
• When multiplying or dividing two or more rational numbers, the product or quotient is positive if there are no negative signs or an even number of negative signs.
• When multiplying or dividing two or more rational numbers, the product or quotient is negative if there is one negative sign or an odd number of negative signs.
• Connections between generalizations for integer operations to rational number operations for addition and subtraction
• Recognize that dividing by a rational number and multiplying by its reciprocal result in equivalent values.
• Connections between generalizations for integer operations to rational number operations for multiplication and division
• Mathematical and real-world problem situations
• Multi-step problems
• Multiple operations

Note(s):

• Grade 6 multiplied and divided positive rational numbers fluently.
• Grade 6 determined, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with rational numbers and operations to solve problems in a variety of contexts
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
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 d = rt

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
• Unit conversions within and between systems
• Customary
• Metric
• d = rt
• In d = rt, the d represents distance, the r represents rate, and the t represents time.
• Connections between constant rate of change r, in d = rt, to the constant of proportionality, k, in y = kx
• 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:
• 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.
• 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.
7.4B Calculate unit rates from rates in mathematical and real-world problems.
Supporting Standard

Calculate

UNIT RATES FROM RATES IN 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
• Unit rate – a ratio between two different units where one of the terms is 1
• Rate – a multiplicative comparison of two different quantities where the measuring unit is different for each quantity
• Various representations of rates
• Verbal (e.g., for every, per, for each, to, etc.)
• Symbolic (e.g., , 2 to 7, etc.)
• Multiplication/division to determine unit rate from mathematical and real-world problems
• Speed
• Density ()
• Price
• Measurement in recipes
• Student–teacher ratios
• Unit conversions within and between systems
• Customary
• Metric

Note(s):

• Grade 7 introduces calculating unit rates from rates in mathematical and real-world problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing and applying proportional relationships
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret 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.
• 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.
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 (x, y)
• Dependent variables describe the output values in a relationship, normally represented by the y coordinate in the ordered pairs (x, y).
• 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
• Unit conversions within and between same system
• Customary
• Metric
• d = rt
• In d = rt, the d represents distance, the r represents rate, and the t represents time
• Connections between constant rate of change r, in d = rt, to the constant of proportionality, k, in y = kx
• 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.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret 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.
• 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.
7.4D Solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.

Solve

PROBLEMS INVOLVING RATIOS, RATES, AND PERCENTS INCLUDING MULTI-STEP PROBLEMS INVOLVING PERCENT INCREASE AND PERCENT DECREASE, AND FINANCIAL LITERACY PROBLEMS

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents converted to equivalent decimals or fractions for multiplying or dividing
• Ratio – a multiplicative comparison of two quantities
• Symbolic representations of ratios
• a to b, a:b, or
• Verbal representations of ratios
• 3 to 12, 3 per 12, 3 parts to 12 parts, 3 for every 12, 3 out of every 12
• Units may or may not be included (e.g., 3 boys to 12 girls, 3 to 12, etc.)
• Rate – a multiplicative comparison of two different quantities where the measuring unit is different for each quantity
• Relationship between ratios and rates
• All ratios have associated rates.
• Percent – a part of a whole expressed in hundredths
• Numeric forms
• Algebraic notation as a decimal
• Multi-step problems
• Multiple methods for solving problems involving ratios, rates, and percents
• Models (e.g., percent bars, hundredths grid, strip diagram, number line, etc.)
• Decimal method (algebraic)
• Dimensional analysis
• Proportion method
• Scale factors between ratios
• Equivalent representations of ratios, rates and percents
• Various representations of ratios, rates, percents
• Pictorial
• Tabular (vertical/horizontal)
• Verbal
• Numeric
• Graphical
• Algebraic
• Situations involving ratios, rates, or percents
• Ratios
• Rates
• Percent increase – a change in percentage where the value increases
• Percent decrease – a change in percentage where the value decreases
• Financial literacy problems
• Principal – the original amount invested or borrowed
• Simple interest – interest paid or earned on the original principal amount, disregarding any previously paid or earned interest
• Formula for simple interest from STAAR Grade 7 Mathematics Reference Materials
• I = Prt, where I represents the interest, P represents the principal amount, r represents the interest rate in decimal form, and t represents the number of years the amount is deposited or borrowed
• Tax – a financial charge, usually a percentage applied to goods, property, sales, etc.
• Tip – an amount of money rendered for a service, gratuity
• Commission – pay based on a percentage of the sales or profit made by an employee or agent
• Markup – the difference between the purchase price of an item and its sales price
• Markdown – the difference between the original price of an item and its current price
• Appreciation – the increase in value over time
• Depreciation – the decrease in value over time

Note(s):

• Grade 6 represented ratios and percents with concrete models, fractions, and decimals.
• Grade 6 represented benchmark fractions and percents such as 1%, 10%, 25%, 33% and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers.
• Grade 6 generated equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money.
• Grade 6 solved real-world problems to find the whole given a part and the percent, to find the part given the whole and the percent, and to find the percent given the part and the whole, including the use of concrete and pictorial models.
• Grade 6 used equivalent fractions, decimals, and percents to show equal parts of the same whole.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing and applying proportional relationships
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• 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.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.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.
7.5 Proportionality. The student applies mathematical process standards to use geometry to describe or solve problems involving proportional relationships. The student is expected to:
7.5C Solve mathematical and real-world problems involving similar shape and scale drawings.

Solve

MATHEMATICAL AND REAL-WORLD PROBLEMS INVOLVING SIMILAR SHAPE AND SCALE DRAWINGS

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Various forms of positive rational numbers
• Counting (natural) numbers
• Decimals
• Fractions
• Percents converted to equivalent decimals or fractions for multiplying or dividing
• Mathematical and real-world problems
• Similar shapes – shapes whose angles are congruent and side lengths are proportional (equal scale factor)
• Similar shapes are proportional when a scale factor is applied to the linear measures, creating a dilated (enlarged or reduced) shape.
• Scale drawings
• Scale drawings are proportional when a scale factor is applied to the linear measures, creating a dilated (enlarged or reduced) scale drawing.

Note(s):

• Grade 6 represented mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions.
• Grade 7 introduces solving mathematical and real-world problems involving similar shape and scale drawings.
• Grade 8 will generalize that the ratio of corresponding sides of similar shapes are proportional, including a shape and its dilation.
• Grade 8 will compare and contrast the attributes of a shape and its dilation(s) on a coordinate plane.
• Grade 8 will use an algebraic representation to explain the effect of a given positive rational scale factor applied to two-dimensional figures on a coordinate plane with the origin as the center of dilation.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Representing and applying proportional relationships
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• III.D. Geometric and Spatial Reasoning – Measurements involving geometry and algebra
• III.D.3. Determine indirect measurements of geometric figures using a variety of methods.
• 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.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.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.
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:
• 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.
• VI.B. Functions – Analysis of functions
• VI.B.1. Understand and analyze features of functions.
• VI.C. Functions – Model real-world situations with functions
• VI.C.2. Develop a function to model a situation.
• 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.