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 TITLE : Unit 01: Foundations of Number SUGGESTED DURATION : 15 days

#### Unit Overview

Introduction
This unit bundles student expectations that address the understanding of whole numbers up to 1,200 and comparing and ordering of these numbers using number lines, including open number lines. 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 Grade 1, students used place value to compose, decompose, and represent whole numbers up to 120. They also compared and ordered numbers using open number lines.

During this Unit
Students extend their understanding of the base-10 place value system to include the thousands period and numbers up to 1,200. Students compose and decompose numbers through 1,200 in more than one way as a sum of so many one thousands, so many hundreds, so many tens, and so many ones using concrete objects (e.g., proportional objects such as base-10 blocks, non-proportional objects such as place value disks, etc.), pictorial models (e.g., base-10 representations with place value charts, place value disk representations with place value charts, open number lines, etc.), and numerical representations (e.g., expanded form, word form, standard form, etc.). Students use place value relationships in order to generate numbers that are more or less than a given number using tools such as a hundreds chart or base-10 blocks. Students compare whole numbers up to 1,200 and represent the comparison using comparative language and symbols. Students use number lines, including open number lines, to locate, name, and represent the order of these numbers.

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

After this Unit
In Grade 3, students will extend their understanding of place value and the thousands period to include the ten thousand and hundred thousand places. Students will compose, decompose, represent, compare, and order whole numbers through 100,000. In addition to ordering numbers, students will use the number line, including the open number line, as a tool for rounding numbers and representing fractions.

In Grade 2, composing, decomposing, and representing numbers, and comparing and ordering numbers using number lines and open number lines are foundational building blocks to the conceptual understanding of the Grade 2 Texas Response to Curriculum Focal Points (TxRCFP): Developing proficiency in the use of place value within the base-10 numeration system. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, A2, 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
According to the National Research Council (2001),

Research indicates that students' experiences using physical models to represent hundreds, tens, and ones can be effective if the materials help them think about how to combine quantities and, eventually how these processes connect with written procedures. The models, however, are not automatically meaningful for students; the meaning must be constructed as they work with the materials. (p. 198)

In addition, conclusions in Chapter 4 of the Final Report of the National Mathematics Advisory Panel (2008) state,

Studies conducted in the United States have repeatedly demonstrated that many elementary-school children do not fully understand the base-10 structure of multidigit written numerals (e.g., understanding the place value meaning of the numeral) or number words (Fuson, 1990). As a result, many of these children are unable to effectively use this system when attempting to solve complex arithmetic problems. It appears that many children require instructional techniques that explicitly focus on the specifics of the repeating decade structure of the base-10 system and that focus on clarifying often confusing features of the associated notational system (Fuson & Briars, 1990; Varelas & Becker, 1997). (p. 4-31)

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Kilpatrick, J., Swafford, J., and Findell, B. (Eds.) Mathematics Learning Study Committee, Center for Education Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
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)
• 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 through 1,200).
• What relationships exist between numbers in the proper counting sequence?
• How are counting skills used to generate numbers that are greater or less than a given number?
• How are patterns in place value relationships used to determine a number that is greater or less than a given number?
• The base-10 place value system, based on 10 digits, allows for communicating very large and very small numbers efficiently (whole numbers through 1,200).
• In the base-10 place value system, a new place value unit is formed by grouping ten of the previous place value units, and this process can be repeated to create greater and greater place value units.
• How can any number be formed using only the digits 0 – 9?
• What patterns and relationships are found in the base-10 place value system?
• How are the periods (thousands period; hundreds period), the patterns within each period (hundreds place; tens place; ones place), and the comma(s) used to read and write whole numbers?
• What is the purpose of the digit zero in a number and when does the digit zero affect the value of a number?
• A digit’s position within the base-10 place value system determines the value of the number.
• How is the value of a digit within a number determined?
• How does the position of the digits determine the value of a number?
• 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 (whole numbers through 1,200).
• What are some ways a number can be represented?
• What is the relationship between the base-10 place value system language and the way numbers are represented in …
• standard form?
• word form?
• expanded form?
• Why can a number vary in representation but the value of the number stay the same?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a number using …
• expanded form
• concrete models
• pictorial models
… improve understanding and communicating about the value of a number and the equivalence of the representations?
• Number
• Base-10 Place Value System
• Composition and Decomposition of Numbers
• Number
• Counting (natural) numbers
• Whole numbers
• Number Recognition and Counting
• Sequence
• Hierarchical inclusion
• Magnitude
• Unitizing
• Number Representations
• Standard form
• Word form
• Expanded form
• 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.

 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 base-10 place value system, based on 10 digits, allows for communicating very large and very small numbers efficiently (whole numbers through 1,200).
• In the base-10 place value system, a new place value unit is formed by grouping ten of the previous place value units, and this process can be repeated to create greater and greater place value units.
• How can any number be formed using only the digits 0 – 9?
• What patterns and relationships are found in the base-10 place value system?
• How are the periods (thousands period; hundreds period), the patterns within each period (hundreds place; tens place; ones place), and the comma(s) used to read and write whole numbers?
• What is the purpose of the digit zero in a number and when does the digit zero affect the value of a number?
• A digit’s position within the base-10 place value system determines the value of the number.
• How is the value of a digit within a number determined?
• How does the position of the digits determine the value of a number?
• 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 (whole numbers through 1,200).
• What are some ways a number can be represented?
• What is the relationship between the base-10 place value system language and the way numbers are represented in …
• standard form?
• word form?
• expanded form?
• Why can a number vary in representation but the value of the number stay the same?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a number using …
• expanded form
• concrete models
• pictorial models
… improve understanding and communicating about the value of a number and the equivalence of the representations?
• Quantities are compared and ordered to determine magnitude of number and equality or inequality relations (whole numbers through 1,200).
• Why is it important to identify the unit or attribute being described by numbers before comparing or ordering the numbers?
• How can …
• place value
• numeric representations
• concrete representations
• pictorial representations
… aid in comparing and/or ordering numbers?
• How can the comparison of two numbers be described and represented?
• How are quantifying descriptors used to determine the order of a set of numbers?
• Number
• Base-10 Place Value System
• Compare and Order
• Comparative language
• Comparison symbols
• Number
• Counting (natural) numbers
• Whole numbers
• Number Recognition
• Sequence
• Hierarchical inclusion
• Magnitude
• Unitizing
• Number Representations
• Standard form
• Word form
• Expanded form
• 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 the digit 3 in the number 326 represents the value 3 instead of the value 300.
• Some students may think if two numbers are composed of the same digits, they have the same value even if the digits’ place value locations within the two numbers are different.
• Some students may think the total value of a number changes when the number is represented using different decompositions, not realizing that the sum of the addends in each decomposition remains the same.
• Some students may think, when comparing numbers, a number value is only dependent on the largest digit, regardless of the place value location within the number (e.g., when comparing 169 and 243, the student may think that 169 is larger because the digit 6 and/or 9 are/is larger than any of the digits in the number 243).
• When ordering numbers, some students may think if the first digit is larger, it is the largest number despite its place value location (e.g., when ordering 91, 135, and 456, the student may think that 91 is the largest because the digit 9 is larger than both the digits 1 and 4).
• Some students may be confused about directionality when ordering by place value rather than understanding that quantifying descriptors determine the order of numbers as they are read from left to right (e.g., largest to smallest, smallest to largest, etc.).
• Some students may not realize that a zero in a place value position does not have to be written in expanded or word form rather than understanding that it is only in standard form that the zero must always be represented.

Underdeveloped Concepts:

• Although some students may be able to decompose a number one way, they may not recognize the relationship between the place values that will allow for multiple decompositions.
• Some students may still be in the one-to-one correspondence counting stage making it difficult to use the base-10 blocks or other manipulatives used for representing 10’s and 100’s.
• Some students may have an incorrect number concept of multi-digit numbers and see them as numbers without place value such as decomposing two hundred five as 2 + 5 instead of 200 + 5.
• Directionality may pose a challenge for some students as numbers are read left to right, but place values, and number periods with commas are identified from right to left.
• Some students may have difficulty visualizing a number line or open number line that does not begin with zero.
• Some students may have difficulty reading, writing, and/or understanding the meaning of the appropriate comparison symbol.

#### Unit Vocabulary

• Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
• Compose numbers – to combine parts or smaller values to form a number
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Decompose numbers – to break a number into parts or smaller values
• Digit – any numeral from 0 – 9
• Expanded form – the representation of a number as a sum of place values (e.g., 1,189 as –1,000 + 100 + 80 + 9)
• Numeral – a symbol used to name a number
• Open number line – an empty number line where tick marks are added to represent landmarks of numbers, often indicated with arcs above the number line (referred to as jumps) demonstrating approximate proportional distances
• Order numbers – to arrange a set of numbers based on their numerical value
• Period – a three-digit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, etc.
• Standard form – the representation of a number using digits (e.g., 1,200)
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Word form – the representation of a number using written words (e.g., 1,152 as one thousand, one hundred fifty-two)

Related Vocabulary:

 Base-10 place value system Comparative language Comparison symbols Equal to (=) Greater than (>) Hundreds place Increment Landmark (or anchor) numbers Less than (<) Magnitude (relative size) Number sequence One thousands place Ones place Tens place Thousands period Tick mark Units period
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 MENTAL MATH 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
• 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, 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:
• 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.
2.2 Number and operations. The student applies mathematical process standards to understand how to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system related to place value. The student is expected to:
2.2A Use concrete and pictorial models to compose and decompose numbers up to 1,200 in more than one way as a sum of so many thousands, hundreds, tens, and ones.

Use

CONCRETE AND PICTORIAL MODELS OF NUMBERS UP TO 1,200

Including, but not limited to:

• Whole numbers (0 – 1,200)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Numeral – a symbol used to name a number
• Digit – any numeral from 0 – 9
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, etc.
• One thousands place
• Hundreds place
• Tens place
• Ones place
• Base-10 place value system
• A number system using ten digits 0 – 9
• Relationships between places are based on multiples of 10.
• The magnitude (relative size) of one thousand
• Concrete models
• Proportional models – a visual representation that demonstrates the relative size of each place value using models with proportional dimensions, meaning the model of each place value is exactly 10 times larger than the place value model to the right (e.g., the base-10 long is exactly 10 times as big as the unit showing that one 10 is equal to ten ones)
• Bundled sticks (proportional representation of the magnitude of a number with 1-to-10 relationship)
• Base-10 blocks (proportional representation of the magnitude of a number with 1-to-10 relationship)
• Non-proportional models – a visual representation that does not maintain the proportional relationship of size, meaning the size of each place value model is not 10 times larger than the place value model to the right (e.g., the value of each place value disk is indicated by the numerical label and color but does not change in size)
• Place value disks (non-proportional representation with a 1-to-10 relationship)
• Pictorial models
• Base-10 block representations
• Place value disk representations
• Open number line – an empty number line where tick marks are added to represent landmarks of numbers, often indicated with arcs above the number line (referred to as jumps) demonstrating approximate proportional distances

To Compose, To Decompose

NUMBERS UP TO 1,200 IN MORE THAN ONE WAY AS A SUM OF SO MANY THOUSANDS, HUNDREDS, TENS, AND ONES

Including, but not limited to:

• Whole numbers (0 – 1,200)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Compose numbers – to combine parts or smaller values to form a number
• Decompose numbers – to break a number into parts or smaller values
• Decomposition of whole numbers in regards to place value does not involve carrying digits to the next placeholder, meaning each addend of the decomposition should only have one non-zero digit (e.g., 789 may be decomposed as 700 + 80 + 9 or 500 + 200 + 50 + 30 + 9 but not decomposed as 600 + 90 + 90 + 9 or 600 + 180 + 9, etc.).
• Compose a number in more than one way using concrete and pictorial models.
• As a sum of so many thousands, hundreds, tens, and ones
• Decompose a number in more than one way using concrete and pictorial models.
• As a sum of so many thousands, hundreds, tens, and ones

Note(s):

• Grade 1 used concrete and pictorial models to compose and decompose numbers up to 120 and used objects, pictures and expanded and standard forms to represent numbers.
• Grade 3 will compose and decompose numbers up to 100,000 using objects, pictorial models, and numbers, including expanded notation as appropriate.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• 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.
2.2B Use standard, word, and expanded forms to represent numbers up to 1,200.

Use

STANDARD, WORD, AND EXPANDED FORMS TO REPRESENT NUMBERS UP TO 1,200

Including, but not limited to:

• Whole numbers (0 – 1,200)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, etc.
• Standard form – the representation of a number using digits (e.g., 1,200)
• Period – a three-digit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
• Thousands period is composed of the one thousands place, ten thousands place, and hundred thousands place.
• Units period is composed of the ones place, tens place, and hundreds place.
• The word “thousand” after the numerical value of the thousands period is stated when read.
• A comma between the thousands period and the units period is recorded when written but not stated when read.
• The word “unit” after the numerical value of the units period is not stated when read.
• The word “hundred” in each period is stated when read.
• The words “ten” and “one” in each period are not stated when read.
• The tens place digit and ones place digit in each period are stated as a two-digit number when read.
• Zeros are used as place holders between digits as needed to maintain the value of each digit (e.g., 1,075).
• Leading zeros in a whole number are not commonly used in standard form, but are not incorrect and do not change the value of the number (e.g., 037 equals 37).
• Word form – the representation of a number using written words (e.g., 1,152 as one thousand, one hundred fifty-two)
• The word “thousand” after the numerical value of the thousands period is stated when read and recorded when written.
• A comma between the thousands period and the units period is not stated when read but is recorded when written.
• The word “unit” after the numerical value of the units period is not stated when read and not recorded when written.
• The word “hundred” in each period is stated when read and recorded when written.
• The words “ten” and “one” in each period are not stated when read and not recorded when written.
• The tens place digit and ones place digit in each period are stated as a two-digit number when read and recorded using a hyphen, where appropriate, when written (e.g., twenty-three, thirteen, etc.).
• The zeros in a whole number are not stated when read and are not recorded when written (e.g., 1,005 in standard form is read and written as one thousand, five in word form).
• Expanded form – the representation of a number as a sum of place values (e.g., 1,189 as 1,000 + 100 + 80 + 9)
• Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 1,075 as 1,000 + 0 + 70 + 5 or 1,000 + 70 + 5).
• Expanded form is written following the order of place value.
• The sum of place values written in random order is an expression but not expanded form.
• Multiple representations

Note(s):

• Grade 1 used objects, pictures, and expanded and standard forms to represent numbers up to 120.
• Grade 3 will compose and decompose numbers up to 100,000 using objects, pictorial models, and numbers, including expanded notation as appropriate.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• 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.
2.2C Generate a number that is greater than or less than a given whole number up to 1,200.

Generate

A NUMBER THAT IS GREATER THAN OR LESS THAN A GIVEN WHOLE NUMBER UP TO 1,200

Including, but not limited to:

• Whole numbers (0 – 1,200)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Comparative language
• Greater than, more than
• Less than, fewer than
• Place value relationships
• 1 more or 1 less
• Adding 1 to the digit in the ones place will generate a number that is 1 more than the original number.
• Subtracting 1 from the digit in the ones place will generate a number that is 1 less than the original number.
• 10 more or 10 less
• Adding 1 to the digit in the tens place will generate a number that is 10 more than the original number.
• Subtracting 1 from the digit in the tens place will generate a number that is 10 less than the original number.
• 100 more or 100 less
• Adding 1 to the digit from the digit in the hundreds place will generate a number that is 100 more than the original number.
• Subtracting 1 from the digit in the hundreds place will generate a number that is 100 less than the original number.
• Numerical relationships
• Counting order
• Skip counting
• Doubles
• Concrete and pictorial models
• Hundreds chart
• Moving one place to the right will generate a number that is 1 more than the original number.
• Moving one place to the left will generate a number that is 1 less than the original number.
• Moving one row down will generate a number that is 10 more than the original number.
• Moving one row up will generate a number that is 10 less than the original number.
• Base-10 blocks
• Adding unit cubes will increase a number by increments of 1.
• Removing unit cubes will decrease a number by increments of 1.
• Adding longs will increase a number by increments of 10.
• Removing longs will decrease a number by increments of 10.
• Adding flats will increase a number by increments of 100.
• Removing flats will decrease a number by increments of 100.
• Number line
• Numbers increase from left to right.
• Numbers decrease from right to left.

Note(s):

• Grade 1 generated a number that is greater than or less than a given whole number up to 120.
• Grade 1 used relationships to determine the number that is 10 more and 10 less than a given number up to 120.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• 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.
2.2D Use place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and symbols (>, <, or =).

Use

PLACE VALUE FOR WHOLE NUMBERS UP TO 1,200

Including, but not limited to:

• Whole numbers (0 – 1,200)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, etc.
• One thousands place
• Hundreds place
• Tens place
• Ones place
• Period – a three-digit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
• Thousands period is composed of the one thousands place, ten thousands place, and hundred thousands place.
• Units period is composed of the ones place, tens place, and hundreds place.

To Compare, To Order

WHOLE NUMBERS UP TO 1,200 USING COMPARATIVE LANGUAGE, NUMBERS, AND SYMBOLS (>, <, OR =)

Including, but not limited to:

• Whole numbers (0 – 1,200)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, etc.
• Comparative language and comparison symbols
• Inequality words and symbols
• Greater than (>)
• Less than (<)
• Equality words and symbol
• Equal to (=)
• Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
• Relative magnitude of a number describes the size of a number and its relationship to another number.
• Compare two numbers using place value charts.
• Compare digits in the same place value position beginning with the greatest place value.
• If these digits are the same, continue to the next smallest place until the digits are different.
• Numbers that have common digits but are not equal in value (different place values)
• Numbers that have a different number of digits
• Compare two numbers using a number line.
• Number lines (horizontal/vertical)
• Proportional number lines (pre-determined intervals with at least two labeled numbers)
• Open number lines (no marked intervals)
• Order numbers – to arrange a set of numbers based on their numerical value
• A set of numbers can be compared in pairs in the process of ordering numbers.
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Order a set of numbers on a number line.
• Order a set of numbers on an open number line.
• Quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)

Note(s):

• Grade 1 used place value to compare numbers up to 120 and represented the comparison of two numbers to 100 using the symbols >, <, or =.
• Grade 3 will use compare and order whole numbers up to 100,000 and represent the comparisons using the symbols >, <, or =.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• 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.
2.2E Locate the position of a given whole number on an open number line.

Locate

THE POSITION OF A GIVEN WHOLE NUMBER ON AN OPEN NUMBER LINE

Including, but not limited to:

• Whole numbers (0 – 1,200)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Characteristics of an open number line
• An open number line begins as a line with no intervals (or tick marks) and no positions/numbers labeled.
• Numbers/positions are placed on the empty number line only as they are needed.
• When reasoning on an open number line, the position of zero is often not placed.
• When working with larger numbers, an open number line without the constraint of distance from zero allows the ability to “zoom-in” on the relevant section of the number line.
• The placement of the first two numbers on an open number line determines the scale of the number line.
• Once the scale of the number line has been established by the placement of the first two numbers, intervals between additional numbers placed are approximately proportional.
• The differences between numbers are approximated by the distance between the positions on the number line.
• Open number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Landmark (or anchor) numbers may be placed on the open number line to help locate other numbers.
• Purpose of open number line
• Open number lines allow for the consideration of the magnitude of numbers and the place-value relationships among numbers when locating a given whole number
• Open number line given

Note(s):

• Grade 1 ordered whole numbers up to 120 using place value and open number lines.
• Grade 3 will represent a number on a number line as being between two consecutive multiples of 10; 100; 1,000; or 10,000 and use words to describe relative size of numbers in order to round whole numbers.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• 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.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers.
2.2F Name the whole number that corresponds to a specific point on a number line.

Name

THE WHOLE NUMBER THAT CORRESPONDS TO A SPECIFIC POINT ON A NUMBER LINE

Including, but not limited to:

• Whole numbers (0 – 1,200)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Characteristics of a number line
• A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
• A minimum of two positions/numbers should be labeled.
• Numbers on a number line represent the distance from zero.
• The distance between the tick marks is counted rather than the tick marks themselves.
• The placement of the labeled positions/numbers on a number line determines the scale of the number line.
• Intervals between position/numbers are proportional.
• When reasoning on a number line, the position of zero may or may not be placed.
• When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoom-in” on the relevant section of the number line.
• Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
• Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
• Points to the left of a specified point on a horizontal number line are less than points to the right.
• Points to the right of a specified point on a horizontal number line are greater than points to the left.
• Points below a specified point on a vertical number line are less than points above.
• Points above a specified point on a vertical number line are greater than points below.
• Number lines can be horizontal, vertical, or circular.
• Intervals and partial labels given
• Partial intervals and labels given

Note(s):

• Grade 2 introduces naming the whole number that corresponds to a specific point on a number line.
• Grade 3 will determine the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing proficiency in the use of place value within the base-10 numeration system
• 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.
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.2. Interpret the relationships between the different representations of numbers. 