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 TITLE : Unit 05: Proportional Reasoning with Ratios and Rates SUGGESTED DURATION : 15 days

#### Unit Overview

Introduction
This unit bundles student expectations that address representing and solving problems with ratios and rates, including those involving percents and converting units within a measurement system using proportions and unit rates. 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 4 and Grade 5, students represented unknown quantities in equations using a letter standing for the unknown. In Grade 6 Unit 01, students extended their mathematical foundations of equivalency within rational numbers, including percents. Students applied their understanding of percents to solve real-world problems. Methods for solving real-world problem situations involving percents, such as the use of proportions or scale factors between ratios, were not included in Unit 01. In Unit 03, students performed operations with positive fractions and decimals.

During this Unit
Students are formally introduced to proportional reasoning with the building blocks of ratios, rates, and proportions. Students examine and distinguish between ratios and rates as they give 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. Students extend previous work with representing percents using concrete models and fractions. Additionally, students are introduced to generating equivalent forms of fractions, decimals, and percents using ratios, including problems that involve money. Students solve and represent problem situations involving ratios and rates with scale factors, tables, graphs, and proportions. Students also represent real-world problems involving ratios and rates, including unit rates, while converting units within a measurement system. These representations allow students to develop a sense of covariation when using proportional reasoning to solve problems, which means they are able to determine and analyze how related quantities change together. Students use both qualitative and quantitative reasoning to make both predictions and comparisons in problem situations involving ratios and rates. Students revisit solving 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. Methods for solving real-world problem situations involving percents, such as the use of proportions (using a letter standing for the unknown quantity) or scale factors between ratios, are included within this unit. Extensive and deliberate development of proportional reasoning skills is foundational for all future mathematics coursework, much of which concentrates on the concept of proportionality.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 6

After this Unit
In Units 06 – 07, students will apply their understanding of proportional reasoning to solve one-variable, one-step equations and inequalities. In Unit 08, students will transfer from proportional thinking to solve problems to algebraic thinking as they examine two-variable additive and multiplicative equations. In Grade 7, students will solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, as well as financial literacy problems. Students will also convert between measurement systems, using proportions and unit rates, as well as represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.

In Grade 6, representing mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions is identified as Supporting Standard 6.5A. Applying qualitative and quantitative reasoning to solve prediction and comparison of real-world problems involving ratios and rates as well as real-world problems involving percents are described as Readiness Standard 6.4B and 6.5B. All of these standards are subsumed under the Grade 6 STAAR Reporting Category 2: Computations and Algebraic Relationships. Giving examples of ratios as multiplicative comparisons of two quantities describing the same attribute, giving examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients, and representing ratios and percents with concrete models, fractions, and decimals are identified as Supporting Standards 6.4C, 6.4D, and 6.4E. Generating equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money is classified as Readiness Standard 6.4G. All of these standards are subsumed under the Grade 6 STAAR Reporting Category 1: Numerical Representations and Relationships. Readiness Standard 6.4H, convert units within a measurement system, including the use of proportions and unit rates, is part of the Grade 6 STAAR Reporting Category 3: Geometry and Measurement. All of the standards in this unit are subsumed under Grade 6 Texas Response to Curriculum Focal Points (TxRCFP): Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS):  I. Numeric Reasoning A2, B1, C2; II. Algebraic Reasoning C3, D1, D2; III. Geometric and Spatial Reasoning D3; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, C2, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to Van de Walle, Karp, and Bay-Williams (2010), proportional reasoning “begins with the ability to understand multiplicative relationships, distinguishing them from the relationships that are additive. The development of proportional reasoning is one of the most important goals of the 5 – 8 curriculum. Proportional reasoning goes well beyond the notion of setting up a proportion to solve a problem—it is a way of reasoning about multiplicative situations. In fact, proportional reasoning, like equivalence, is considered a unifying theme in mathematics” (p. 348). When introducing ratios and rates, Lamon (2006) suggests that “when children prefer the ratio interpretation and classroom instruction that builds on their intuitive knowledge of comparisons, they develop a richer understanding of rational numbers and employ proportional reasoning sooner than those children whose curriculum used the part-whole comparison as the primary interpretation of rational numbers” (p.184). The use of proportional reasoning as a means to solve problems involving ratios and rates is notable as “Proportions provide a way to find answers to problems  where the numbers are relational…Proportional reasoning is complex, both in terms of mathematics and the developmental experiences it requires. Yet it is an important skill for students to gain because it facilitates algebraic thinking” (Reyes, Lindquist, Lambdin & Smith, 2012, p. 297).

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.
Reyes, R. E., Lindquist, M., Lambdin, D. V., & Smith, N. L. (2012). Helping children learn mathematics. (10th ed.). Hoboken, NJ: Wiley.
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., Karp, K., & Bay-Williams, J. (2010). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) 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.
• Ratios and rates are modeled and described to develop an understanding of proportional relationships and these relationships are applied to represent equivalence.
• How can ratios and rates be described …
• qualitatively?
• quantitatively?
• How can …
• qualitative
• quantitative
… reasoning be used to make predictions and comparisons in problems involving ratios and rates?
• What is the difference between qualitative and quantitative reasoning?
• What relationships exist between ratios describing quantities of the same attribute?
• What relationships exist between rates describing quantities of different attributes?
• How are ratios and rates similar and different?
• Proportionality
• Ratios and Rates
• Scale factors
• Relationships and Generalizations
• Equivalence
• Qualitative and quantitative reasoning
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Understanding how two quantities vary together (covariation) 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.
• Ratios and rates are modeled and described to develop an understanding of proportional relationships and these relationships are applied to represent equivalence and solve problem situations involving ratios and rates.
• How can representing ratios and percents using …
• concrete models
• fractions
• decimals
… aid in problem solving?
• How can an equivalent …
• fraction be generated when given a decimal or percent?
• decimal be generated when given a fraction or percent?
• percent be generated when given a decimal or fraction?
• What types of models and strategies can be used to generate equivalent forms of fractions, decimals, and percents?
• Why is the ability to model numbers in a variety of ways essential to solving problems in everyday life?
• How can benchmark fractions and percents be used to solve problems involving money?
• How are …
• scale factors
• proportions
• unit rates
… used to convert within a measurement system?
• How can …
• scale factors
• tables
• graphs
• proportions
… be used to solve problems involving …
• ratios?
• rates?
• How can …
• concrete and pictorial models
• proportions or scale factors between ratios
… be used to determine the …
• whole when given a part and the percent?
• part when given the whole and the percent?
• percent when given the part and the whole?
• Proportionality
• Fractions and Decimals
• Percents
• Ratios and Rates
• Unit rates
• Scale factors
• Relationships and Generalizations
• Equivalence
• Proportions
• Representations
• Solution Strategies
• Systems of Measurement
• Customary
• Metric
• 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 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 ratios and rates may not be represented on a graph rather than realizing all ratios and rates can be viewed as ordered pairs.
• 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 only use additive thinking rather than multiplicative thinking when solving proportions.

Underdeveloped Concepts:

• Some students may forget to multiply or divide both of the terms in a ratio by the same number to find an equivalent ratio.
• Some students may think the value of 43% of 35 is the same value of 43% of 45 because the percents are the same rather than considering that the wholes of 35 and 45 are different, so 43% of each value will be different.
• Some students may not realize which operation is easier to use when converting between number forms.
• Some students may have difficulty recognizing the part and the whole in problem situations.
• Some students may believe every fraction relates to a different rational number instead of realizing equivalent fractions relate to the same relative amount.
• Some students may try to convert a fraction to a decimal by placing the denominator in the dividend rather than the numerator.
• Some students may think that a fraction can be converted to a decimal by simply writing the numerator and denominator as digits after a decimal (e.g., is equivalent to 0.78).

#### Unit Vocabulary

• Comparison by division of two quantities – a proportional comparison in which one quantity can be described as a ratio of the other
• Multiplicative comparison of two quantities – a proportional comparison in which one quantity can be described as a multiple of the other
• Percent – a part of a whole expressed in hundredths
• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Qualitative – a broad subjective description (e.g., The speed of car A is slower than the speed of car B.)
• Quantitative – a narrowed objective description associated with a quantity (e.g., The ratio of blue cars to red cars is 6:3; therefore, there are twice as many blue cars as red cars.)
• Rate – a multiplicative comparison of two different quantities where the measuring unit is different for each quantity
• Ratio– a multiplicative comparison of two quantities
• Scale factor – the common multiplicative ratio between pairs of related data which may be represented as a unit rate
• Strip diagram – a linear model used to illustrate number relationships
• Unit rate – a ratio between two different units where one of the terms is 1

Related Vocabulary:

 10 by 10 grid Compare Decimal Decimal notation Denominator Equivalent Fraction Fraction circle Fraction notation Improper fraction Mixed number Number line Numerator Proper fraction Proportion Proportion method Ratio table Unit fraction
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 6 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
6.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
6.1A 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:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1B 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:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1C 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 REAL OBJECTS, MANIPULATIVES, 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
• Real objects
• Manipulatives
• 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:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1D 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:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1E 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:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1F 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:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.1G 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:
• Using operations with integers and positive rational numbers to solve problems
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• Using expressions and equations to represent relationships in a variety of contexts
• Understanding data representation
• 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.
6.4 Proportionality. The student applies mathematical process standards to develop an understanding of proportional relationships in problem situations. The student is expected to:
6.4B Apply qualitative and quantitative reasoning to solve prediction and comparison of real-world problems involving ratios and rates.

Apply

QUALITATIVE AND QUANTITATIVE REASONING TO SOLVE PREDICTION AND COMPARISON OF REAL-WORLD PROBLEMS INVOLVING RATIOS AND 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
• Percents converted to equivalent decimals or fractions for multiplying or dividing fluently
• Qualitative – a broad subjective description (e.g., The speed of car A is slower than the speed of car B.)
• Qualitative reasoning to compare and predict
• Ratio – a multiplicative comparison of two quantities
• Comparing ratios (e.g., color is brighter, taste is sweeter, pace is slower, etc.)
• Predictions from ratios
• Qualitative reasoning to compare and predict in real-world problem situations involving rates
• Rate – a multiplicative comparison of two different quantities where the measuring unit is different for each quantity
• Comparing rates (e.g., decreases faster, more per pound, etc.)
• Predictions from rates
• Quantitative – a narrowed objective description associated with a quantity (e.g., The ratio of blue cars to red cars is 6:3; therefore, there are twice as many blue cars as red cars.)
• Quantitative reasoning to compare and predict in real-world problem situations involving ratios
• Comparing ratios (e.g., twice as much, half as sweet)
• Predictions from ratios
• Comparing rates (e.g., decreases half as fast, three times more per pound, etc.)
• Predictions from rates

Note(s):

• Grade 6 introduces applying qualitative and quantitative reasoning to solve prediction and comparison of real-world problems involving ratios and rates.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• 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.2. Understand attributes and relationships with inductive and deductive reasoning.
• 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.
6.4C Give examples of ratios as multiplicative comparisons of two quantities describing the same attribute.
Supporting Standard

Give

EXAMPLES OF RATIOS AS MULTIPLICATIVE COMPARISONS OF TWO QUANTITIES DESCRIBING THE SAME ATTRIBUTE

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 fluently
• Multiplicative comparison of two quantities – a proportional comparison in which one quantity can be described as a multiple of the other
• 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.).
• Quantities describing the same attribute
• Proportional representations of multiplicative comparisons
• Strip diagram – a linear model used to illustrate number relationships
• Ratio table
• Double number lines

Note(s):

• Grade 6 introduces giving examples of ratios as multiplicative comparisons of two quantities describing the same attribute.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• 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.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.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.
6.4D Give examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.
Supporting Standard

Give

EXAMPLES OF RATES AS THE COMPARISON BY DIVISION OF TWO QUANTITIES HAVING DIFFERENT ATTRIBUTES, INCLUDING RATES AS QUOTIENTS

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 fluently
• Comparison by division of two quantities – a proportional comparison in which one quantity can be described as a ratio of the other
• 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
• Quantities describing different attributes
• Proportional representations of comparisons by division
• Ratio table
• Double number lines
• Rates as quotients

Note(s):

• Grade 6 introduces giving examples of rates as the comparison by division of two quantities having different attributes, including rates as quotients.
• Grade 7 will represent constant rates of change in mathematical and real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.
• Grade 7 will calculate unit rates from rates in mathematical and real-world problems.
• Grade 7 will determine the constant of proportionality ( ) within mathematical and real-world problems.
• Grade 7 will solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.
• Grade 7 will convert between measurement systems, including the use of proportions and the use of unit rates.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• 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.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.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.
6.4E Represent ratios and percents with concrete models, fractions, and decimals.
Supporting Standard

Represent

RATIOS AND PERCENTS WITH CONCRETE MODELS, FRACTIONS, AND DECIMALS

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
• Percent – a part of a whole expressed in hundredths
• 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.)
• Concrete and pictorial models of ratios
• Objects
• Fraction circle
• Strip diagram – a linear model used to illustrate number relationships
• 10 by 10 grid
• Number line
• Numeric representation of ratios
• Fraction notation
• Decimal notation
• Percent
• Numeric forms
• Concrete and pictorial models of percents
• Objects
• Fraction circle
• Strip diagram – a linear model used to illustrate number relationships
• 10 by 10 grid
• Number line
• Numeric representation of percents
• Fraction notation
• Decimal notation

Note(s):

• Grade 6 introduces representing ratios and percents with concrete models, fractions, and decimals.
• Grade 7 will solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
6.4G Generate equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money.

Generate

EQUIVALENT FORMS OF FRACTIONS, DECIMALS, AND PERCENTS USING REAL-WORLD PROBLEMS, INCLUDING PROBLEMS THAT INVOLVE MONEY

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
• Percent – a part of a whole expressed in hundredths
• Equivalent forms of positive rational numbers in real-world problem situations, including money
• Given a fraction, generate a decimal and percent
• Given a decimal, generate a fraction and percent
• Given a percent, generate a fraction and decimal

Note(s):

• Grade 6 introduces generating equivalent forms of fractions, decimals, and percents using real-world problems, including problems that involve money.
• Grade 6 introduces ordering a set of rational numbers arising from mathematical and real-world contexts.
• Grade 7 will solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• 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.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• 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.
6.4H Convert units within a measurement system, including the use of proportions and unit rates.

Convert

UNITS WITHIN A MEASUREMENT SYSTEM, INCLUDING THE USE OF PROPORTIONS AND 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
• Unit conversions within systems
• Customary
• Metric
• Unit rate – a ratio between two different units where one of the terms is 1
• Multiple solution strategies
• Scale factor between ratios
• Proportion method

Note(s):

• Grade 4 converted measurements within the same measurement system, customary or metric, from a smaller unit into a large unit or a large unit into a smaller unit when given other equivalent measures represented in a table.
• Grade 6 introduces using proportions and unit rates to convert units within a measurement system.
• Grade 7 will convert between measurement systems, including the use of proportions and the use of unit rates.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• 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.
6.5 Proportionality. The student applies mathematical process standards to solve problems involving proportional relationships. The student is expected to:
6.5A Represent mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions.
Supporting Standard

Represent

MATHEMATICAL AND REAL-WORLD PROBLEMS INVOLVING RATIOS AND RATES USING SCALE FACTORS, TABLES, GRAPHS, AND PROPORTIONS

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
• 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.)
• Scale factor – the common multiplicative ratio between pairs of related data which may be represented as a unit rate
• Various representations of scale factor involving ratios in mathematical and real-world problem situations
• Tables
• Graphs
• Proportions
• Rate – a multiplicative comparison of two different quantities where the measuring unit is different for each quantity
• Various representations of scale factor involving rates in mathematical and real-world problem situations
• Tables
• Graphs
• Proportions

Note(s):

• Grade 6 introduces representing mathematical and real-world problems involving ratios and rates using scale factors, tables, graphs, and proportions.
• Grade 7 will represent constant rates of change in mathematical an real-world problems given pictorial, tabular, verbal, numeric, graphical, and algebraic representations, including d = rt.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent proportional relationships
• TxCCRS:
• II.C. Algebraic Reasoning – Solving equations, inequalities, and systems of equations and inequalities
• II.C.3. Recognize and use algebraic properties, concepts, and algorithms to solve equations, inequalities, and systems of linear equations and inequalities.
• 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.
• 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.
6.5B Solve 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.

Solve

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

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
• Percent – a part of a whole expressed in hundredths
• Relationship between part, whole, and percent
• • Multiple methods for solving real-world problem situations involving percent
• Concrete and pictorial models (e.g., objects, area model, strip diagram, 10 by 10 grid, number line, etc.)
• Proportion method
• Scale factor between ratios
• Various types of real-world problem situations involving percent
• Limited to situations in which the parts and percents are less than the whole
• Finding the whole given a part and a percent
• Finding the part given the whole and a percent
• Finding the percent given the part and the whole

Note(s):

• Grade 6 introduces solving 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 7 will solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding and applying ratios and rates and using equivalent ratios to represent 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.
• 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. 