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 TITLE : Unit 11: Fractions – Equivalency and Comparisons SUGGESTED DURATION : 12 days

#### Unit Overview

Introduction
This unit bundles student expectations that address representing and explaining equivalent fractions and comparing fractions. 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.

Prior to this Unit
In Unit 06, students represented fractions using objects, pictorial models, and number lines. Students also composed and decomposed fractions as a sum of unit fractions and solved problems involving partitioning an object or set of objects using pictorial representations of fractions.

During this Unit
Students represent equivalent fractions with denominators of 2, 3, 4, 6, and 8 using objects, pictorial models (including strip diagrams and area models), and number lines. Students explain that two fractions are equivalent if and only if they are both represented by the same point on the number line or represent the same portion of a same size whole for an area model. Students learn the role of the numerator and the role of the denominator. Understandings of the numerator and denominator assist students when comparing fractions with denominators of 2, 3, 4, 6, and 8 but not limited to these values. Strategies that students begin to develop when comparing fractions include comparing the size of the numerators when the denominators are the same, comparing the size of the denominators when the numerators are the same, and comparing the size of parts and the number of equal sized parts considered when the numerators and/or denominators are not the same. With extensive exploration, students develop fractional reasoning skills about the size of a fraction. For instance, students realize the smaller the number in the denominator, the larger the size of equal pieces; whereas, the larger the number in the numerator, the more equal size pieces being considered. A common misunderstanding when comparing fractions is to compare the numerators of the fractions only with no consideration of the denominators of the fractions or vice versa. Students develop an understanding that although a fraction is composed of a number in the numerator and a number in the denominator, together they represent a single value. Students also justify the comparison of fractions using symbols, words, objects, and pictorial models.

After this Unit
In Grade 4, students will relate their understanding of fractions that name tenths and hundredths to decimal numbers and represent both fractions and decimals as distances from zero on a number line. Grade 4 students will also represent fractions (including those that represent values greater than one) as sums of unit fractions and decompose fractions into sums of fractions with the same denominator using concrete and pictorial models.

In Grade 3, representing equivalent fractions and comparing fractions are identified as STAAR Readiness Standards 3.3F and 3.3H, while explaining equivalent fractions is identified as STAAR Supporting Standard 3.3G. These standards are incorporated within the Grade 3 Texas Response to Curriculum Focal Points (TxRCFP): Understanding fractions as numbers and representing equivalent fractions and are included in Grade 3 STAAR Reporting Category 1: Numerical Representations and Relationships. These standards support the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning and IX. Communication and Reasoning.

Research
According to Empson and Levi (2011), teachers play an active role in helping students understand fraction equivalence by asking probing questions and choosing number combinations for problems with specific purposes in mind (p. 115). The National Council of Teachers of Mathematics (2009) states that area, bar, and number line models provide purposeful visual experiences that aid students in their understanding of equivalent fractions (p. 42-43).

Empson, S. and Levi, L. (2011). Extending children’s mathematics fractions and decimals. Portsmouth, NH: Heinemann.
National Council of Teachers of Mathematics. (2009). Focus in grade 3: Teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, 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

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place. How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• The ability to recognize and represent numbers in various forms develops the understanding of equivalence and allows for working flexibly with numbers in order to communicate and reason about the value of the number.
• Fractions are an extension of the number system used to communicate and reason about the infinite values that exist between whole values.
• When might fractions be used in real life?
• Why is it important to be able to identify or work with fractional parts of a whole?
• What relationships exist between …
• whole numbers and fractions?
• proper fractions, improper fractions, and mixed numbers?
• Why is it important to determine the unit or whole when working with fractions?
• How can a set of objects be described as a whole?
• How does changing the size of the whole affect the size or amount of a fractional partition?
• How are names of fractional parts determined?
• How is the …
• denominator
• numerator
… of a fraction determined?
• Why can the denominator not be represented as zero?
• How can the numerator and the denominator of a fraction be described as a multiplicative relationship?
• What are some ways a fraction can be represented?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a fraction …
• using words
• using concrete models
• using pictorial models
• using numerals and symbols
• in simplified form
… improve understanding and communicating about the value of a fraction and the equivalence of the representations?
• What strategies can be used to …
• generate equivalent fractions?
• determine if two fractions are equivalent?
• compare fractions having the same numerator or denominator?
• What relationship exists between the number of partitions in a whole and the size of the partition?
• Why can equal partitions of identical wholes look different?
• Why can a fraction vary in representation but the value of the fraction stay the same?
• Number
• Compare and Order
• Comparative language
• Comparison symbols
• Number
• Counting (natural) numbers
• Whole numbers
• Fractions
• Number Representations
• Relationships
• Numerical
• Equivalence
• 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 of equivalency and comparison of fractions as strictly a numerical consideration rather than realizing equivalency and comparison of fractions is only valid when referring to the same size whole.

Underdeveloped Concepts:

• Some students may struggle recording the denominator as the number of parts in the whole regardless of the number of parts being considered in the numerator.
• Some students may continue to struggle with the inverse relationship between the number of fractional pieces in a whole (the denominator) and the size of each piece (e.g., the larger the denominator the smaller the fractional piece; the smaller the denominator the larger the fractional piece).

#### Unit Vocabulary

• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Equivalent fractions – fractions that have the same value
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object or part of a set of objects.
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered

Related Vocabulary:

 Compare Equal (=) Equal sized parts Fractional part Fraction bar Greater than (>) Less than (<) Simpler form Whole
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Creator if your district has granted access to that tool.

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Grade 3 Mathematics TEKS

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

Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
3.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
3.1A 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:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• X. Connections
3.1B 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:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• VIII. Problem Solving and Reasoning
3.1C 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:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• VIII. Problem Solving and Reasoning
3.1D 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:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• IX. Communication and Representation
3.1E 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:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• IX. Communication and Representation
3.1F 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:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• X. Connections
3.1G 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:
• Understanding and applying place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Solving problems with multiplication and division within 100
• Understanding fractions as numbers and representing equivalent fractions
• Describing characteristics of two-dimensional and three-dimensional geometric figures, including measurable attributes
• TxCCRS:
• IX. Communication and Representation
3.3 Number and operations. The student applies mathematical process standards to represent and explain fractional units. The student is expected to:
3.3F Represent equivalent fractions with denominators of 2, 3, 4, 6, and 8 using a variety of objects and pictorial models, including number lines.

Represent

EQUIVALENT FRACTIONS WITH DENOMINATORS OF 2, 3, 4, 6, AND 8 USING A VARIETY OF OBJECTS AND PICTORIAL MODELS, INCLUDING NUMBER LINES

Including, but not limited to:

• Fractions greater than zero and less than or equal to one
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object or part of a set of objects.
• Equivalent fractions – fractions that have the same value
• Whole number denominators of 2, 3, 4, 6, and 8
• The number 1 as a fraction
• Comparisons of fractions are only valid when referring to the same size whole.
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Concrete models of whole objects
• Linear models
• Cuisenaire rods, fraction bars, customary rulers, linking cube trains, folded paper strips, etc.
• Area models
• Fraction circles or squares, pattern blocks, geoboards, etc.
• Concrete models of a set of objects
• Pattern blocks, color tiles, counters, real-world objects, etc.
• Pictorial models
• Fraction strips, fraction bar models, number lines, etc.
• Real-world situations involving equivalent fractions

Note(s):

• Grade 2 used concrete models, pictorials, and words to represent and name fractional parts (e.g., halves, one-half, fourths, one-fourth, etc.).
• Grade 3 introduces the fraction symbol.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding fractions as numbers and representing equivalent fractions
• TxCCRS:
• I. Numeric Reasoning
• IX. Communication and Representation
3.3G Explain that two fractions are equivalent if and only if they are both represented by the same point on the number line or represent the same portion of a same size whole for an area model.
Supporting Standard

Explain

THAT TWO FRACTIONS ARE EQUIVALENT IF AND ONLY IF THEY ARE BOTH REPRESENTED BY THE SAME POINT ON THE NUMBER LINE OR REPRESENT THE SAME PORTION OF A SAME SIZE WHOLE FOR AN AREA MODEL

Including, but not limited to:

• Fractions greater than zero and less than or equal to one
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object or part of a set of objects.
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Whole number denominators of 2, 3, 4, 6, and 8  but not limited to these values
• Equivalent fractions – fractions that have the same value
• Comparisons of fractions are only valid when referring to the same size whole.
• Equivalency using a number line
• Equivalency using an area model
• Real-world situations involving equivalent fraction values of two different area models

Note(s):

• Grade 2 used concrete models, pictorials, and words to represent and name fractional parts (e.g., halves, one-half, fourths, one-fourth, etc.).
• Grade 3 introduces the fraction symbol.
• Grade 4 will determine if two given fractions are equivalent using a variety of methods.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding fractions as numbers and representing equivalent fractions
• TxCCRS:
• I. Numeric Reasoning
• IX. Communication and Representation
3.3H Compare two fractions having the same numerator or denominator in problems by reasoning about their sizes and justifying the conclusion using symbols, words, objects, and pictorial models.

Compare

TWO FRACTIONS HAVING THE SAME NUMERATOR OR DENOMINATOR IN PROBLEMS BY REASONING ABOUT THEIR SIZES AND JUSTIFYING THE CONCLUSION USING SYMBOLS, WORDS, OBJECTS, AND PICTORIAL MODELS

Including, but not limited to:

• Fractions greater than zero and less than or equal to one
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object or part of a set of objects.
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Whole number denominators of 2, 3, 4, 6, and 8 but not limited to these values
• Comparisons of fractions are only valid when referring to the same size whole.
• Comparison of two fractions with the same numerator
• Common numerators standardize the number of pieces; therefore, compare the size of each piece (denominator).
• Reason about the sizes of fractions with the same numerator.
• Larger denominator → smaller fractional piece → smaller fraction
• Smaller denominator → larger fractional piece → larger fraction
• Comparison of two fractions with the same denominator
• Common denominators standardize the size of the pieces; therefore, compare the number of pieces (numerator).
• Reason about the sizes of fractions with the same denominator.
• Larger numerator → more equal-size fractional pieces → larger fraction
• Smaller numerator → fewer equal-size fractional pieces → smaller fraction
• Justification of comparison
• Comparative language and comparison symbols
• Inequality words and symbols
• Greater than (>)
• Less than (<)
• Equality words and symbol
• Equal to (=)
• Words
• Reasoning related to the size of the parts and/or the number of parts
• Objects
• Cuisenaire rods, fraction bars, customary rulers, linking cube trains, folded paper strips, fraction circles or squares, geoboards, pattern blocks, color tiles, counters, etc.
• Pictorial models
• Fraction strips, fraction bar models, number lines, etc.
• Comparison of two fractions in mathematical and real-world problem situations

Note(s):

• Grade 2 used concrete models, pictorials, and words to represent and name fractional parts (e.g., halves, one-half, fourths, one-fourth, etc.).
• Grade 3 introduces the fraction symbol.
• Grade 4 will compare two fractions with different numerators and different denominators and represent the comparison using the symbols >, =, or <.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Understanding fractions as numbers and representing equivalent fractions
• TxCCRS:
• I. Numeric Reasoning
• IX. Communication and Representation