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 TITLE : Unit 03: Proportional Reasoning with Ratios and Rates SUGGESTED DURATION : 11 days

#### Unit Overview

Introduction
This unit bundles student expectations that address problem situations involving ratios, rates, conversions between measurement systems, and personal budgets. 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 Grade 6, students were formally introduced to proportional reasoning with ratios, rates, and proportions. Students examined and distinguished between ratios and rates as they gave examples of ratios as multiplicative comparisons of two quantities describing the same attribute and examples of rates as the comparison by division of two quantities having different attributes. They solved and represented problem situations involving ratios and rates, including percents, with scale factors, tables, graphs, and proportions. Students also represented real-world problems involving ratios and rates, including unit rates, while converting units within a customary measurement system and within a metric measurement system. In Grade 5, students balanced a simple budget.

During this Unit
Students examine proportional reasoning with ratios and rates through the lens of constant rates of change. Students are expected to represent constant rates of change given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt. The development of rates lends itself to students examining d = rt and reveals the amount of variation that can occur when various constant rates can be applied to a situation to reveal the same outcome (e.g., the distance traveled remains the same in a situation if the speed is doubled and the time is halved, etc.). Exploring the relationship between distance, rate, and time allows students to generalize the effects when rates within any problem situation are changed. Students solve problems involving ratios, rates, and percents. Computations with percents are now inclusive of solving problems involving percent increase, percent decrease, and financial literacy. Percents are also used as students identify the components of a personal budget and calculate what percentage each category comprises of the total budget. They also calculate unit rates from rates and determine the constant of proportionality in mathematical and real-world problems. Students use proportions and unit rates as they extend previous understandings of converting units within a measurement system to now include converting units between both customary and metric measurement systems.

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

After this Unit
In Unit 04, students will transfer from proportional thinking to solve problems to algebraic thinking as they examine two-variable equations. Students will represent linear relationships using verbal descriptions, tables, graphs, and equations that simplify to the form y = mx + b. In Grade 8, students will solve problems involving direct variation.

In Grade 7, constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical and algebraic representations, including d = rt and solving problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems are identified as STAAR Readiness Standard 7.4A and 7.4D. Calculating unit rates from rates and determining the constant of proportionality in mathematical and real-world problems are STAAR Supporting Standards 7.4B and 7.4C. These four standards are listed under the Grade 7 STAAR Reporting Category 2: Computations and Algebraic Relationships. Converting between measurement systems including the use of proportions and the use of unit rates is STAAR Supporting Standard 7.4E, and is part of the Grade 7 STAAR Reporting Category 3: Geometry and Measurement. All of these standards are a foundational block of the Grade 7 Texas Response to Curriculum Focal Points (TxRCFP): Representing and applying proportional relationships. Identifying the components of a personal budget and calculating what percentage each category comprises of the total budget is STAAR Supporting Standard 7.13B and is subsumed under the Grade 7 STAAR Reporting Category 4: Data Analysis and Personal Financial Literacy. This standard is a part of the Grade 7 Focal Point: Financial Literacy (TxRCFP). This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A2, B1, C2; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

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. When examining the relationships within the common context of distance-rate-time “we all encounter this relationship in some way every day of our lives, [yet] research has shown that it is not well understood by most people…Students need to encounter the distance-speed-time system of relationships to explicitly think about and discuss (a) ways to compare speeds of movement, (b) the characteristics of rate, (c) the meaning of constant speed (Lamon, 2006, p. 203 – 204). Proportional reasoning can be extending to current and future financial planning. Research from Gale and Levine (2011) that summarizes a study from the National Endowment of Financial Education that surveyed individuals about their financial literacy attained prior to entering college suggests that “respondents who were in a state with mandated financial education generally had higher financial literacy scores, as well as “better” financial behaviors including budgeting and use of credit” (p.10). Integrating financial literacy concepts within the curriculum as an extension of learning within a unit, allows for a variety of problem situations and contexts to deepen mathematical understandings.

Driscoll, M. (1999). Fostering algebraic thinking: A guide for teachers grades 6 – 10. Portsmouth, NH: Heinemann.
Gale, W. G., Levine, R. (2011). Financial literacy: what works? How could it be more effective? Boston, MA: Financial Security Project at Boston College.
Lamon, S. J. (2006). Teaching fractions and ratios for understanding: Essential content knowledge and instructional strategies for teachers. (2nd ed.). Mahwah, NJ: Lawrence Erlbaum Associates 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?
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.
• 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?
• How are …
• scale factors
• proportions
• unit rates
• conversion graphs
… used to convert from one measurement system to another?
• Proportionality
• Ratios and Rates
• Unit rates
• Constant rate of change
• Scale factors
• Relationships and Generalizations
• Equivalence
• Proportional
• Constant of proportionality
• Systems of Measurement
• Customary
• Metric
• Representations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life? Financial and economic knowledge leads to informed and rational decisions allowing for effective management of financial resources when planning for a lifetime of financial security.  Why is financial stability important in everyday life? What economic and financial knowledge is critical for planning for a lifetime of financial security? How can mapping one’s financial future lead to significant short and long-term benefits? How can current financial and economic factors in everyday life impact daily decisions and future opportunities?
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.
• Unit rate and constant rate of change represent equivalent values and are used to solve problems involving a proportional relationship.
• 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?
• Understanding components of a personal budget and how percentages of each category comprises the total budget helps one make informed financial management decisions, which promotes a more secured financial future.
• Why are budgets often displayed in a bar graph or circle graph?
• Why are categories in budgets represented with values and percents?
• How does income affect a budget?
• What are the components of a personal budget?
• What are fixed and variables expenses, and how are they represented in a budget?
• Why should retirement, college, taxes, and emergencies always be included in a budget?
• What is the process to determine the percentage of each category that comprise a total budget?
• How is a simple budget balanced, and why is it important to maintain a balanced budget?
• How is the total budget effected when one component …
• increases?
• decreases?
• What are the effects of the other categories in a personal budget when the amount in one category increases and the total budget remains the same?
• How does understanding a personal budget and the effects of change on the total budget help promote a more secured financial future?
• Proportionality
• Fractions and Decimals
• Percents
• Ratios and Rates
• Scale factors
• Relationships and Generalizations
• Equivalence
• Proportional
• Representations
• Personal Financial Literacy
• Budgets
• College
• Cost
• Savings plans
• Emergencies
• Expenses
• Fixed
• Variable
• Income
• Retirement
• Taxes
• 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 always 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 only use additive thinking rather than multiplicative thinking when solving proportions.
• Some students may think ratios and rates may not be represented on a graph rather than realizing all ratios and rates can be viewed as ordered pairs.

Underdeveloped Concepts:

• Some students may generate an “equivalent” ratio by exchanging the numbers in a ratio without their appropriate labels rather than interpreting the ratio as a comparison that must maintain the same relationship. (e.g., 2 girls:3 boys is not equivalent to 3 girls:2 boys)
• Some students may think that the order of the terms in a ratio or proportion is not important.
• Some students may think that generating an equivalent ratio is different from generating an equivalent fraction.
• Some students may think that all ratios are fractions rather than understanding that a ratio may represent a part-to-part or part-to-whole relationship.
• Some students may think that rates are not related to ratios.
• Some students may think that a unit rate must have a denominator of one rather than understanding that a unit rate is a ratio between two different units where one of the terms is one.
• Some students may not make the connection between the constant rate of change r, in d = rt, to the constant of proportionality, k, in y = kx.
• Some students may not connect the constant rate of change to m in the equation y = mx + b.

#### Unit Vocabulary

• Appreciation – the increase in value over time
• Budget – a monthly or yearly spending and savings plan for an individual, family, business, or organization
• Commission – pay based on a percentage of the sales or profit made by an employee or agent
• 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
• Expense – payment for goods and services
• Fixed expenses – expenses that are consistent from month to month
• Income – money earned or received
• Income tax – a percentage of money paid on the earned wages of an individual or business for the federal and/or state governments as required by law
• 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
• Payroll tax – a percentage of money that a company withholds from its employees for the federal government as required by law
• 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
• Property tax – a percentage of money collected on the value of a property for the local government as required by law
• 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.
• Sales tax – a percentage of money collected by a store (retailer), in addition to a good or service that was purchased, for the local government as required by law
• Savings for college – money saved for continuing education beyond high school
• Savings for emergencies – money saved for unexpected expenses (e.g., car repairs, emergency healthcare, etc.)
• Savings for retirement – money saved over the period of time an individual is employed to be spent once the individual retires from their occupation
• 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.
• Taxes – money paid to local, state, and federal governments to pay for things the government provides to its citizens
• 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
• Variable expenses – expenses that vary in cost from month to month

Related Vocabulary:

 Algebraic Conversion graph Customary Dimensional analysis Density Dependent Equivalent Graphical Independent Metric Number line Percent bar Percent graph Price Proportion Proportion method Proportional Scale factor Speed Strip diagram Tabular Unit conversion x-value 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.
7.4 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
• 12 to 3, 12 per 3, 12 parts to 3 parts, 12 for every 3, 12 out of every 3
• Units may or may not be included (e.g., 12 boys to 3 girls, 12 to 3, 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.
7.4E Convert between measurement systems, including the use of proportions and the use of unit rates.
Supporting Standard

Convert

BETWEEN MEASUREMENT SYSTEMS, INCLUDING THE USE OF PROPORTIONS AND THE USE OF UNIT RATES

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
• Convert units between measurement systems.
• Customary to metric
• Metric to customary
• Multiple solution strategies
• Dimensional analysis using unit rates
• Unit rates
• Scale factor between ratios
• Proportion method
• Conversion graph

Note(s):

• Grade 4 converted measurements within the same measurement system, customary or metric, from a smaller unit into a larger unit or a larger unit into a smaller unit when given other equivalent measures represented in a table.
• Grade 5 solved problems by calculating conversions within a measurement system, customary or metric.
• Grade 6 converted units within a measurement system, including the use of proportions and unit rates.
• 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.
• I.C. Numeric Reasoning – Systems of measurement
• I.C.2. Convert units within and between systems of measurement.
• 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.
7.13 Personal financial literacy. The student applies mathematical process standards to develop an economic way of thinking and problem solving useful in one's life as a knowledgeable consumer and investor. The student is expected to:
7.13B Identify the components of a personal budget, including income; planned savings for college, retirement, and emergencies; taxes; and fixed and variable expenses, and calculate what percentage each category comprises of the total budget.
Supporting Standard

Identify

THE COMPONENTS OF A PERSONAL BUDGET, INCLUDING INCOME; PLANNED SAVINGS FOR COLLEGE, RETIREMENT, AND EMERGENCIES; TAXES; AND FIXED AND VARIABLE EXPENSES

Including, but not limited to:

• Budget – a monthly or yearly spending and savings plan for an individual, family, business, or organization
• Budgets based on financial records help people plan and make choices about how to spend and save their money
• Components of a personal budget
• Income – money earned or received
• Savings for college – money saved for continuing education beyond high school
• Savings for retirement – money saved over the period of time an individual is employed to be spent once the individual retires from their occupation
• Savings for emergencies – money saved for unexpected expenses (e.g., car repairs, emergency healthcare, etc.)
• Taxes – money paid to local, state, and federal governments to pay for things the government provides to its citizens
• Various types of taxes
• Income tax – a percentage of money paid on the earned wages of an individual or business for the federal and/or state governments as required by law
• Payroll tax – a percentage of money that a company withholds from its employees for the federal government as required by law
• Sales tax – a percentage of money collected by a store (retailer), in addition to a good or service that was purchased, for the local government as required by law
• Property tax – a percentage of money collected on the value of a property for the local government as required by law
• Expense – payment for goods and services
• Fixed expenses – expenses that are consistent from month to month
• Variable expenses – expenses that vary in cost from month to month

Calculate

WHAT PERCENTAGE EACH CATEGORY OF A PERSONAL BUDGET COMPRISES OF THE TOTAL BUDGET

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
• Percents
• Proportional reasoning to determine percentages within a budget
• Proportional reasoning to determine amounts within a budget

Note(s): 