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 TITLE : Unit 11: Essential Fractional Understandings SUGGESTED DURATION : 5 days

#### Unit Overview

Introduction
This unit bundles student expectations that address constructing, describing, and naming fractional parts of a whole using models. 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 05, students decomposed two-dimensional figures and partitioned objects into two, four, or eight equal parts. Students discovered and explained the relationship between the number of fractional parts used to make a whole and the size of the parts. Using concrete models, students counted fractional parts beyond one whole.

During this Unit
Students revisit and solidify essential understandings of fractions. Students partition objects or sets of objects into equal parts and name the parts, including halves, fourths, and eighths, using words rather than symbols (e.g., one half or 1 out of 2 equal parts rather than ). Students use direct comparison to understand that the equality of fractional parts of an object is determined based on the size in area of the parts, meaning fractional parts of an object are the same size in area and may or may not be the same shape. Students may also experience fractional parts of a set of objects when the set is defined as the whole. Through repeated practice modeling and naming fractions, students recognize the inverse relationship between the number of parts and the size of each part and explain this relationship using appropriate mathematical language. Students determine how many parts it takes to equal one whole and use this understanding to count fractional parts. Students further their understanding of hierarchical inclusion (each prior number in the counting sequence is included in the set as the set increases) as they apply their understanding of counting whole numbers to counting fractional parts beyond one whole using words.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 2

After this Unit
In Grade 3, students will extend their understanding of fractions to include representing fractions using symbolic notation. Students will compose and decompose fractions and represent equivalent fractions with denominators 2, 3, 4, 6, and 8 using a variety objects and pictorial models, including strip diagrams and number lines. Students will solve problems involving partitioning an object or a set of objects among two or more recipients. Grade 3 students will also decompose two congruent two-dimensional figures into parts with equal areas and recognize that equal shares of identical wholes need not have the same shape.

In Grade 2, constructing, describing, naming, and counting fractions are all included in Grade 2 Texas Response to Curriculum Focal Points (TxRCFP): Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, B1, B2; 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
Understanding fraction concepts and relationships has posed a considerable challenge to students. In order to establish a mathematically sound foundation, Van de Walle’s (2006) research suggests, the first goal in the development of fractions should be to help children construct the “idea of fractional parts of the whole – the parts that result when the whole or unit has been partitioned into equal-sized portions or fair shares. Children seem to understand the idea of separating a quantity into two or more parts to be shared fairly among friends. They eventually make connections between the idea of fair shares and fractional parts” (p. 131-132). “In the discussion of sharing, all of the tasks involved sharing something that could be cut into smaller parts. The fractions are based on parts of an area or region. This is a good place to begin and is almost essential when doing sharing tasks” (p.134). In Grade 2, continuing exploration of fractional parts relies on the use of concrete models to count parts and create wholes. The concept of “one,” the original whole unit, is critical to understanding the fractional part. Without this solid understanding, students will not be able to correctly write fractional notations or understand the meaning of the parts. The National Council of Teachers of Mathematics (2011) states, “The concept of unit is fundamental to the interpretation of rational numbers. Why is the unit so important? To describe the size of some quantity with a rational number, the first step is to determine what serves as the unit, the whole. Consequently, the rational number that results must be interpreted with respect to that unit” (p. 19). Students need an organized andconsistent approach to working with fractional parts as numbers.Students need to first be successful with naming fractional parts prior to using the numerical form to name fractions. Bergeson (2000) supports this gradual building to fraction notation. He states, “Students need to work first with the verbal form of fractions (e.g., two-fourths) before they work with the numerical form (e.g., 2/4), as students’ informal language skills can enhance their understanding of fractions” (p. 8). These foundational fractional relationships: partitioning into equal parts, recognizing the unit, and naming fractions are the building blocks for fractional relationships and successful mathematical understanding.

Bergeson, T. (2000). Teaching and Learning Mathematics Using Research to Shift From the ‘Yesterday’ Mind to the ‘Tomorrow’ Mind. Retrieved from: https://www.k12.wa.us/research/pubdocs/pdf/MathBook.pdf
National Council of Teachers of Mathematics. (2011). Developing essential understanding of rational numbers grades 3 – 5. 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
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades 3 – 5. Boston, MA: Pearson Education, Inc.

 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)
• A thorough understanding of counting involves integrating different skills or characteristics of numbers and is foundational and essential for continued work with numbers (whole numbers and fractions).
• Why is it important to determine the unit or whole before counting fractional parts?
• What strategies can be used to determine the number of fractional parts in the whole?
• How is counting fractions …
• similar
• different
… to counting whole numbers?
• What patterns exist when counting fractions?
• What relationships exist between …
• fractions
• whole numbers and fractions
… in the proper counting sequence?
• 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 (halves, fourths, eighths).
• Fractions are an extension of the number system used to represent quantities that exist when a whole is partitioned into equal parts.
• 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?
• Why is it important to determine the unit or whole when working with fractions?
• How does changing the size of the whole affect the size or amount of a fractional partition?
• What strategies can be used to …
• partition a whole into equal parts?
• determine if partitions of a whole are equal or unequal?
• Why can equal partitions of identical wholes look different?
• How are names of fractional parts determined?
• How can a fraction be named in more than one way?
• How is the fractional name of the partition related to the number of equal partitions in the whole?
• How could representing a fraction using …
• words
• concrete models
• pictorial models
… improve understanding and communicating about the value of a fraction?
• What relationship exists between the number of partitions in a whole and the size of the partition?
• Number and Operations
• Number
• Counting (natural) numbers
• Whole numbers
• Fractions
• Number Recognition and Counting
• Sequence
• Cardinality
• Conservation of set
• Hierarchical inclusion
• Magnitude
• Number Representations
• Word form
• 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 one eighth must be greater than one fourth of the same whole because eight is greater than four rather than understanding that the greater the number of parts of a whole, the smaller the parts.
• Some students may think all fractional parts named “one half” must be equal rather than understanding that the size of the half depends on the size of the whole.
• Some students may think partitioning a shape into any 2 parts means that these parts are halves rather than understanding that parts of a shape must be two equal parts in order to be halves.
• Some students may think the first part of a fraction names the number of parts being considered, and the second part of the fraction names the number of parts remaining rather than the second part of the fraction naming the total number of parts (e.g., 2 parts out of 8 total incorrectly named two sixths because there are 6 parts remaining, etc.).
• Some students may think the term “one quarter” refers to 1 out of 25 parts due to the connection between a quarter and 25¢ rather than 1 out of 4 parts.
• Some students may think fractions can only be represented using commercial manipulatives rather than applying the concept of fractions to other models (e.g., a students may be familiar with a green triangle representing one sixth of a hexagon using pattern blocks but may struggle identifying one sixth of a rectangle).
• Some students may think fractional parts can only be counted up to one whole rather than recognizing the counting sequence of fractional parts beyond one whole.

#### Unit Vocabulary

• Equal parts – fractional parts that are the same size in area and may or may not be the same shape
• Fraction – a number that can be used to name part of an object or part of a set of objects
• Partition – separation or division of an object into parts

Related Vocabulary:

 Counting sequence Eighths Fourths Halves Quarters Unequal parts 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 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 2 Mathematics TEKS

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.
• 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
2.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
2.1A Apply mathematics to problems arising in everyday life, society, and the workplace.

Apply

MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE

Including, but not limited to:

• Mathematical problem situations within and between disciplines
• Everyday life
• Society
• Workplace

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• 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.
2.1B 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.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• 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.
2.1C

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.

Select

TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING 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
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• 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.
2.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• 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.
2.1E Create and use representations to organize, record, and communicate mathematical ideas.

Create, Use

REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Representations of mathematical ideas
• Organize
• Record
• Communicate
• Evaluation of the effectiveness of representations to ensure clarity of mathematical ideas being communicated
• Appropriate mathematical vocabulary and phrasing when communicating mathematical ideas

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• 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.
2.1F Analyze mathematical relationships to connect and communicate mathematical ideas.

Analyze

MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE MATHEMATICAL IDEAS

Including, but not limited to:

• Mathematical relationships
• Connect and communicate mathematical ideas
• Conjectures and generalizations from sets of examples and non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• 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.
2.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.

Display, Explain, Justify

MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN WRITTEN OR ORAL COMMUNICATION

Including, but not limited to:

• Mathematical ideas and arguments
• Validation of conclusions
• Displays to make work visible to others
• Diagrams, visual aids, written work, etc.
• Explanations and justifications
• Precise mathematical language in written or oral communication

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
• Measuring length
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• 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.
2.3 Number and operations. The student applies mathematical process standards to recognize and represent fractional units and communicates how they are used to name parts of a whole. The student is expected to:
2.3A Partition objects into equal parts and name the parts, including halves, fourths, and eighths, using words.

Partition

OBJECTS INTO EQUAL PARTS, INCLUDING HALVES, FOURTHS, AND EIGHTHS

Name

THE EQUAL PARTS OF PARTITIONED OBJECTS, INCLUDING HALVES, FOURTHS, AND EIGHTHS, USING WORDS

Including, but not limited to:

• Fraction – a number that can be used to name part of an object or part of a set of objects
• Equal parts – fractional parts that are the same size in area and may or may not be the same shape
• Partition – separation or division of an object into parts
• Determination of the whole
• One object or shape defined as the whole
• Multiple connected shapes or objects defined as the whole
• A set of separate objects defined as the whole
• Whole divided into two, four, or eight equal parts
• Appropriate oral and written mathematical language to name equal parts, including halves, fourths, and eighths
• Fractions written in word form do not include hyphens.
• Two equal parts
• One half, two halves or one whole
• Four equal parts
• One fourth, two fourths, three fourths, four fourths or one whole
• One quarter, two quarters, three quarters, four quarters or one whole
• Eight equal parts
• One eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, eight eighths or one whole
• Hyphen is used only when the fraction written in word form modifies another word
• Each fractional part of an object is the same size in area and the same shape.
• Orientation of the partitioned parts does not affect the size in area or shape of the parts.
• A whole may be partitioned different ways.
• Equal parts of a whole that are the same size in area may not be the same shape.
• Direct comparison of parts to justify equal size in area
• Equal parts of non-identical wholes may have the same fractional name but may not have equal size in area or the same shape.
• Concrete models of whole objects
• Linear models
• Cuisenaire rods, fraction bars, 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.
• Relationship and distinction between ordinal numbers and the number of parts named in a fraction

Note(s):

• Grade 1 partitioned two-dimensional figures into two and four fair shares or equal parts and described the parts using words.
• Grade 2 is not expected to identify the relationship between equivalent fractions (e.g., two-fourths is the same as one-half, etc.).
• Grade 3 will introduce fraction symbols to represent pictorial models of fractions and fractional parts of a set of objects.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
2.3B Explain that the more fractional parts used to make a whole, the smaller the part; and the fewer the fractional parts, the larger the part.

Explain

THE MORE FRACTIONAL PARTS USED TO MAKE A WHOLE, THE SMALLER THE PART; AND THE FEWER THE FRACTIONAL PARTS, THE LARGER THE PART

Including, but not limited to:

• Fraction – a number that can be used to name part of an object or part of a set of objects
• Inverse relationship between the size of the fractional part and the number of equal parts in the whole when given the same size whole
• The greater the number of parts, the smaller the size of the parts
• The smaller the number of parts, the greater the size of the parts
• Whole divided into two, four, or eight equal parts
• Concrete models of whole objects
• Linear models
• Cuisenaire rods, fraction bars, linking cube trains, folded paper strips, etc.
• Area models
• Fraction circles or squares, pattern blocks, geoboards, etc.
• Real-world problem situations

Note(s):

• Grade 1 partitioned two-dimensional figures into two and four fair shares or equal parts and described the parts using words.
• Grade 3 will introduce fraction symbols to represent pictorial models of fractions and fractional parts of a set of objects.
• Grade 3 will explain that the unit fraction  represents the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• 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.
2.3C Use concrete models to count fractional parts beyond one whole using words and recognize how many parts it takes to equal one whole.

Use

CONCRETE MODELS TO COUNT FRACTIONAL PARTS BEYOND ONE WHOLE USING WORDS

Recognize

HOW MANY PARTS IT TAKES TO EQUAL ONE WHOLE

Including, but not limited to:

• Fraction – a number that can be used to name part of an object or part of a set of objects
• Relationship between counting whole numbers and counting fractional parts of a whole
• Hierarchical inclusion – concept of nested numbers, meaning each prior number in the counting sequence is included in the set as the set increases (e.g., 18 is 17 increased by 1; 18 decreased by 1 is 17; etc.)
• Appropriate oral and written mathematical language
• Determination of the whole
• One object or shape defined as the whole
• Multiple connected shapes or objects defined as the whole
• A set of separate objects defined as the whole
• Wholes divided into two, four, or eight equal parts
• Recognition of the number of parts that equal one whole
• Two halves equal one whole; four fourths equal one whole; eight eighths equal one whole
• Recognition of the number of parts being considered
• Number of parts being considered within one whole
• Number of parts being considered beyond one whole
• Concrete models of the whole
• Linear models
• Cuisenaire rods, fraction bars, linking cube trains, folded paper strips, etc.
• Area models
• Fraction circles or squares, pattern blocks, geoboards, etc.
• Set models
• Pattern blocks, color tiles, counters, real-world objects, etc.
• Count fractional parts up to one whole using concrete models
• Correct sequence of fractional names
• Two equal parts
• One half, two halves or one whole
• Four equal parts
• One fourth, two fourths, three fourths, four fourths or one whole
• One quarter, two quarters, three quarters, four quarters or one whole
• Eight equal parts
• One eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, eight eighths or one whole
• Count fractional parts beyond one whole using concrete models
• Correct sequence of fractional names
• Two equal parts
• One half, two halves, three halves, four halves, five halves, etc.
• One half, one whole, one and one half, two wholes, two and one half, etc.
• Four equal parts
• One fourth, two fourths, three fourths, four fourths, five fourths, six fourths, seven fourths, eight fourths, nine fourths, etc.
• One fourth, two fourths, three fourths, one whole, one and one fourth, one and two fourths, one and three fourths, two wholes, two and one fourth, etc.
• One quarter, two quarters, three quarters, four quarters, five quarters, six quarters, seven quarters, eight quarters, nine quarters, etc.
• One quarter, two- quarters, three quarters, one whole, one and one quarter, one and two quarters, one and three quarters, two wholes, two and one quarter, etc.
• Eight equal parts
• One eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, eight eighths, nine eighths, ten eighths, eleven eighths, twelve eighths, thirteen eighths, fourteen eighths, fifteen eighths, sixteen eighths, seventeen eighths, etc.
• One eighth, two eighths, three eighths, four eighths, five eighths, six eighths, seven eighths, one whole, one and one eighth, one and two eighths, one and three eighths, one and four eighths, one and five eighths, one and six eighths, one and seven eighths, two wholes, two and one eighth, etc.

Note(s):

• Grade 3 will represent fractions greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 using concrete objects and pictorial models, including strip diagrams and number lines.
• Grade 3 will introduce fraction symbols to represent pictorial models of fractions and fractional parts of a set of objects.
• Grade 3 will solve problems involving partitioning an object or a set of objects among two or more recipients using pictorial representations of fractions with denominators of 2, 3, 4, 6, and 8.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Applying knowledge of two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.