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 TITLE : Unit 01: Equivalent Forms of Fractions, Decimals, and Percents SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address representing and generating equivalent forms of fractions, decimals, and percents as well as solving real-world problems involving fractions, decimals, and percents. 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, students determined if two given fractions were equivalent using a variety of methods. Additionally, in Grade 5 students added and subtracted positive rational numbers fluently, including problems involving mixed forms of rational numbers.

During this Unit
Students extend their mathematical foundations of equivalency within rational numbers, including percents. Concrete and pictorial models, including 10 by 10 grids, strip diagrams, and number lines are used to represent multiples of benchmark fractions and percents. Additionally, percents are represented with concrete and pictorial models, fractions, and decimals. Students continue their understanding of equivalency by generating and using equivalent forms of fractions, decimals, and percents to solve real-world problems, including those involving money. Percents less than or greater than 100%, including percents with fractional or decimal values such as 8.25% or  are encompassed within this unit. Students apply their understandings of percents to solve real-world problems that involve finding the whole given a part and the percent, the part given the whole and a percent, and the percent given the part and the whole. Methods for solving real-world problem situations involving percents, such as the use of proportions or scale factors between ratios, are not included in this unit. Computations within this unit are restricted to operational capabilities from Grade 5 which include sums and differences with any positive rational numbers, products with factors limited to a whole number by a whole number, a decimal by a decimal, or a whole number by a fraction, and quotients limited to whole number dividends and divisors, a decimal dividend by a whole number divisor, or whole number and unit fraction dividends and divisors.

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

After this Unit
In Unit 02, students will further their understanding of equivalency by ordering whole numbers, positive and negative rational numbers, and integers. In Unit 03, students will add, subtract, multiply, and divide whole numbers, positive rational numbers, and integers. In Unit 05, students will examine percents again through the lens of proportional reasoning with ratios and rates. At that time, the proportion method and utilizing scale factors between ratios will be appropriate methods to solve real-world problem situations involving percent. In Grade 7, students will solve problems involving ratios, rates, and percents, including multi-step problems involving percent increase and percent decrease, and financial literacy problems.

In Grade 6, generating equivalent forms of fractions, decimals, and percents is identified as STAAR Readiness Standard 6.4G. Representing percents with concrete models, fractions and decimals is STAAR Supporting Standard 6.4E. Representing benchmark fractions and percents with concrete and pictorial models and numbers, as well as using equivalent fractions, decimals, and percents to show equal parts of the same whole are identified as STAAR Supporting Standards 6.4F and 6.5C. All of these standards are subsumed under the Grade 6 STAAR Reporting Category 1:  Numerical Representations and Relationships. Solving real-world problems involving percents is identified as STAAR Readiness Standard 6.5B and part of the Grade 6 STAAR Reporting Category 2: Computations and Algebraic Relationships. All of these standards are subsumed under the 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 A1, A2, B1; 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 Van de Walle, Bay-Williams, Lovin, and Karp (2013), “Physical models provide the main link between fractions, decimal, and percents. Students should develop an understanding of the percent equivalence of familiar fractions (halves, thirds, fourths, fifths, and eighths)” (p. 162). The essential understanding of percent is built on the concept of 100. “Students who understand that percent means parts out of one hundred and have a good pictorial representation of percent are more successful in solving percent problems than those who do not” (Reyes, Lindquist, Lambdin & Smith, 2012, p. 292). Additionally, as students initially begin to solve problems involving percents instruction should include “activities that center on direct translation of experiences involving 100” (p. 293). When determining equivalent forms of fractions, decimals, and percents, Van de Walle et al. (2013) suggests that rather than, “rush[ing] to develop rules or procedures for different types of problems – encourage students to notice patterns…Require student to use manipulatives, drawings, and contexts to explain their solutions” (p. 166).

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., Bay-Williams, J., Lovin, L., & Karp, K., (2013). Teaching student-centered mathematics: Developmentally appropriate instruction for grades 6 - 8. Boston, MA: Pearson.

 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 numeric reasoning in order to make predictions and critical judgements about the relationship (fractions; decimals; percents).
• Fractions, decimals, and percents are modeled and described to develop an understanding of proportional relationships and these relationships are applied to represent equivalence and solve problem situations.
• How can representing percents using …
• concrete models
• fractions
• decimals
… aid in problem solving?
• How can representing benchmark fractions and percents using …
• 10 by 10 grid
• strip diagrams
• number lines
• numbers
… aid in problem solving?
• How are the models for fractions and percents …
• alike?
• different?
• What is the relationship between benchmark fractions, their multiples, and unit fractions?
• 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 can concrete and pictorial models aid in understanding and determining the whole, part, or percent when two of the three are given?
• Proportionality
• Fractions and Decimals
• Percents
• Relationships and Generalizations
• Equivalence
• Benchmark fractions and percents
• Representations
• Solution Strategies
• 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 that a percent may not exceed 100%.
• Some students may think that a percent may not be less than 1%.
• Some students may divide a decimal by 100 by moving the decimal two places to the left when trying to convert it to a percent rather than multiplying by 100 and moving the decimal two places to the right.
• 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 quantity will be different.

Underdeveloped Concepts:

• Some students may not realize which operation is easier to use when converting between number forms.
• Some students may confuse decimal place values when converting decimals to fractions.
• 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

• 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
• Strip diagram – a linear model used to illustrate number relationships

Related Vocabulary:

 10 by 10 grid Area model Place value Benchmark fraction Benchmark percent Decimal Decimal notation Denominator Equivalent Fraction Fraction circle Fraction notation Improper fraction Mixed number Multiple Number line Numerator Part Proper fraction Whole Whole number
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, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• 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:
6.4E

Represent ratios and percents with concrete models, fractions, and decimals.

Supporting Standard

Represent

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
• 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.
6.4F Represent benchmark fractions and percents such as 1%, 10%, 25%, 33 1/3%, and multiples of these values using 10 by 10 grids, strip diagrams, number lines, and numbers.
Supporting Standard

Represent

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

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
• Benchmark fractions ()
• Multiples of benchmark fractions
• Benchmark percents (1%, 10%, 25%, 33%)
• Multiples of benchmark percents
• Various representations of benchmark fractions and percents and their multiples
• 10 by 10 grid
• Strip diagram – a linear model used to illustrate number relationships
• Number line
• Numerically

Note(s):

• Grade 6 introduces representing 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 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.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
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.
6.5 Proportionality. The student applies mathematical process standards to solve problems involving proportional relationships. The student is expected to:
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
• 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.)
• 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.
6.5C Use equivalent fractions, decimals, and percents to show equal parts of the same whole.
Supporting Standard

Use

EQUIVALENT FRACTIONS, DECIMALS, AND PERCENTS TO SHOW EQUAL PARTS OF THE SAME WHOLE

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
• Various representations to show equal parts of the same whole
• 10 by 10 grid
• Strip diagram – a linear model used to illustrate number relationships
• Number line

Note(s):