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 TITLE : Unit 08: Foundations of Numbers up to 120 SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address the understanding of whole numbers up to 120, comparing numbers using place value, and ordering these numbers using an open number line. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Unit 06, students explored base-10 place value system as they explored whole numbers up to 99. Students composed and decomposed numbers through 99 using concrete objects, pictorial models, and numerical representations. In addition, students used place value relationships and tools, such as a hundreds chart, as they generated numbers more or less than a given number. Students compared whole numbers up to 99 using comparison symbols and were introduced to using place value and open number lines to order whole numbers.

During this Unit
Students extend their understanding of the base-10 place value system to include the hundreds place as they continue exploring the foundations of whole numbers up to 120. Students compose and decompose numbers through 120 as a sum of 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, etc.), and numerical representations (e.g., expanded form and standard form). Students use place value relationships to generate numbers that are more or less than a given number using tools (e.g., a hundreds chart, calendar, base-10 blocks, etc.). Students use place value to compare whole numbers up to 120 and represent the comparison using comparative language and comparison symbols. Students also extend using place value and open number lines to order whole numbers up to 120.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 1

After this Unit
In Grade 2, students will extend their understanding of the base-10 place value system to include the thousands period and numbers up to 1,200. Students will compose and decompose numbers through 1,200 using concrete objects, pictorial models, and numerical representations. Students will use place value relationships in order to generate numbers that are more or less than a given number using tools such as base-10 blocks. Also, students will compare whole numbers up to 1,200 and represent the comparison using comparative language and symbols. Students will use number lines, including open number lines, to locate, name, and represent the order of these numbers.

In Grade 1, composing, decomposing, and representing numbers, and comparing and ordering numbers are foundational building blocks to the conceptual understanding of the Grade 1 Texas Response to Curriculum Focal Points (TxRCFP): Developing an understanding of place value. 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, B3, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to National Council of Teachers of Mathematics (2010), “An important extension of the concept of conservation is the idea that one or both of two quantities can be decomposed into parts or one of the quantities can be combined with another quantity to form a larger quantity without changing the equality relations between the two original quantities” (p.16). Griffin (2004) believes that “teachers must see mathematics as a set of conceptual relationships between numbers and number symbols rather than as numbers that are manipulated by rules.”

Griffin, Sharon. (2004). Teaching Number Sense. Educational Leadership 61, 39-42.
National Council of Teachers of Mathematics. (2010). Developing essential understanding of number and numeration pre-k – grade 2. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place. How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• 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 120).
• 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 120).
• 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 is the period (hundreds period) and the patterns within the period (hundreds place; tens place; ones place) 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 20).
• 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?
• 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 120).
• 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
• Composition and Decomposition of Numbers
• Number
• Counting (natural) numbers
• Whole numbers
• Number Recognition and Counting
• Sequence
• Conservation of set
• Hierarchical inclusion
• Magnitude
• Unitizing
• Number Representations
• Standard 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 1 in the number 120 represents the value 1 instead of the value 100 ones, 10 groups of 10, or 1 group of 100.
• Some students may think the decomposition of 115 is 1 + 1 + 5 instead of 100 + 10 + 5, not realizing the importance of the place value in the expanded representation.
• Some students may think a number can only be decomposed one way, when the number can actually be decomposed multiple ways (e.g., one hundred six could be represented as 10 groups of 10 and 6 ones, 106 ones, 8 groups of 10 and 26 ones, etc.).
• 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 89 and 112, the student may think that 89 is larger because the digits 8 and 9 are larger than any of the digits in the number 112).
• Some students may think numbers are always ordered from smallest to largest 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 think all number lines or open number lines must begin with zero rather than being able to visualize a number line or open number line that displays an isolated portion of a number line or open number line.
• Some students may think the less than and greater than comparison symbols are interchangeable rather than understanding the meaning of each symbol and how to appropriately read and write each symbol.

Underdeveloped Concepts:

• 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 10s.

#### 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., 119 as 100 + 10 + 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, etc.
• Standard form – the representation of a number using digits (e.g., 118)
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Base-10 place value system Comparative language Comparison symbols Decrease Equal to (=) Greater than (>) Hundreds place Increase Landmark (or anchor) numbers Less than (<) Magnitude (relative size) Ones place Tens place
System Resources Other Resources

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 1 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
1.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
1.1A 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 an understanding of place value
• Solving problems involving addition and subtraction
• Analyzing attributes of two-dimensional shapes and three-dimensional solids
• Developing the understanding of length
• 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.
1.1B 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 an understanding of place value
• Solving problems involving addition and subtraction
• Analyzing attributes of two-dimensional shapes and three-dimensional solids
• Developing the understanding of length
• 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.
1.1C

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 an understanding of place value
• Solving problems involving addition and subtraction
• Analyzing attributes of two-dimensional shapes and three-dimensional solids
• Developing the understanding of length
• 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.
1.1D

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 an understanding of place value
• Solving problems involving addition and subtraction
• Analyzing attributes of two-dimensional shapes and three-dimensional solids
• Developing the understanding of length
• 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.
1.1E 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 an understanding of place value
• Solving problems involving addition and subtraction
• Analyzing attributes of two-dimensional shapes and three-dimensional solids
• Developing the understanding of length
• 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.
1.1F 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 an understanding of place value
• Solving problems involving addition and subtraction
• Analyzing attributes of two-dimensional shapes and three-dimensional solids
• Developing the understanding of length
• 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.
1.1G 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 an understanding of place value
• Solving problems involving addition and subtraction
• Analyzing attributes of two-dimensional shapes and three-dimensional solids
• Developing the understanding of length
• 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.
1.2 Number and operations. The student applies mathematical process standards 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:
1.2B Use concrete and pictorial models to compose and decompose numbers up to 120 in more than one way as so many hundreds, so many tens, and so many ones.

Use

CONCRETE AND PICTORIAL MODELS OF NUMBERS UP TO 120

Including, but not limited to:

• Whole numbers (0 – 120)
• 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, etc.
• 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 hundred
• 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)
• Linking cubes (proportional representation of the magnitude of a number with 1-to-10 relationship)
• 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 120 IN MORE THAN ONE WAY AS SO MANY HUNDREDS, SO MANY TENS, AND SO MANY ONES

Including, but not limited to:

• Whole numbers (0 – 120)
• 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
• Compose a number in more than one way.
• As so many hundreds, so many tens, and so many ones
• Decompose a number in more than one way.
• As so many hundreds, so many tens, and so many ones

Note(s):

• Kindergarten composed and decomposed numbers up to 10 with objects and pictures.
• Grade 2 will 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.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of place value
• 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.
1.2C Use objects, pictures, and expanded and standard forms to represent numbers up to 120.

Use

OBJECTS, PICTURES, AND EXPANDED AND STANDARD FORMS TO REPRESENT NUMBERS UP TO 120

Including, but not limited to:

• Whole numbers (0 – 120)
• 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, etc.
• Objects
• 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)
• Linking cubes (proportional representation of the magnitude of a number with 1-to-10 relationship)
• 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)
• Beaded number line (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)
• Pictures
• 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
• Place value stacking cards
• Expanded form – the representation of a number as a sum of place values (e.g., 119 as 100 + 10 + 9)
• Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 107 as 100 + 0 + 7 or 100 + 7).
• 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.
• Standard form – the representation of a number using digits (e.g., 118)
• 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
• Units period is composed of the ones place, tens place, and hundreds place.
• 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., 107).
• 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).
• Multiple representations
• Number presented in concrete or pictorial form represented in expanded form
• Number presented in concrete or pictorial form represented in standard form
• Number presented in standard form represented in expanded form
• Number presented in expanded form represented in standard form

Note(s):

• Grade 1 introduces representing numbers in expanded and standard forms.
• Grade 2 will introduce representing numbers up to 1,200 in word form.
• Grade 2 will use standard, word, and expanded forms to represent numbers up to 1,200.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of place value
• TxCCRS:
• .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.
1.2D Generate a number that is greater than or less than a given whole number up to 120.

Generate

A NUMBER THAT IS GREATER THAN OR LESS THAN A GIVEN WHOLE NUMBER UP TO 120

Including, but not limited to:

• Whole numbers (0 – 120)
• 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
• Increasing the digit in the ones place by 1 will generate a number that is 1 more than the original number.
• Decreasing the digit in the ones place by 1 will generate a number that is 1 less than the original number.
• 10 more or 10 less
• Increasing the digit in the tens place by 10 will generate a number that is 10 more than the original number.
• Decreasing the digit in the tens place by 10 will generate a number that is 10 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.
• Number line
• Numbers increase from left to right.
• Numbers decrease from right to left.
• Calendar
• 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 7 more than the original number.
• Moving one row up will generate a number that is 7 less than the original number.

Note(s):

• Kindergarten generated a number that is one more than or one less than another number up to 20.
• Grade 2 will generate a number that is greater than or less than a given whole number up to 1,200.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of place value
• 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.
1.2E Use place value to compare whole numbers up to 120 using comparative language.

Use

PLACE VALUE OF WHOLE NUMBERS UP TO 120

Including, but not limited to:

• Whole numbers (0 – 120)
• 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, etc.
• Hundreds place
• Tens place
• Ones place

To Compare

WHOLE NUMBERS UP TO 120 USING COMPARATIVE LANGUAGE

Including, but not limited to:

• Whole numbers (0 – 120)
• 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
• Inequality language
• Greater than, more than
• Less than, fewer than
• Equality language
• Equal to, same as
• 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
• Concrete models
• Compare the amount modeled in the highest place value position first.
• 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
• Compare the amount represented in the highest place value position first.
• Base-10 block representations
• Place value disk representations
• Numerical
• 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 more than two numbers using place value to determine which number is the greatest.
• Compare digits in the greatest place value position to determine the number with the greatest value.
• If these digits are the same, continue to the next smallest place until the digits are different.
• Compare more than two numbers using place value to determine which number is the least.
• Compare digits in the greatest place value position to determine the number with the least value.
• If these digits are the same, continue to the next smallest place until the digits are different.

Note(s):

• Kindergarten used comparative language to describe two numbers up to 20 presented as written numerals.
• Grade 2 will use place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and the symbols >, <, or =.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of place value
• 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.
1.2F Order whole numbers up to 120 using place value and open number lines.

Order

WHOLE NUMBERS UP TO 120 USING PLACE VALUE AND OPEN NUMBER LINES

Including, but not limited to:

• Whole numbers (0 – 120)
• 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}
• 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.
• Order a set of numbers using place value.
• 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.)
• Order a set of numbers using open number lines.
• 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.
• Relative magnitude of a number describes the size of a number and its relationship to another number.
• Order a set of numbers on an open number line.

Note(s):

• Grade 1 introduces ordering whole numbers up to 120 using place value and open number lines.
• Grade 2 will locate the position of a given whole number on an open number line.
• Grade 2 will use place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and the symbols >, <, or =.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of place value
• 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.
1.2G Represent the comparison of two numbers to 100 using the symbols >, <, or =.

Represent

THE COMPARISON OF TWO NUMBERS TO 100 USING THE SYMBOLS >, <, OR =

Including, but not limited to:

• Whole numbers (0 – 100)
• 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}
• 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
• Comparative language and comparison symbols
• Inequality words and symbols
• Greater than (>)
• Less than (<)
• Equality words and symbol
• Equal to (=)

Note(s):

• Grade 1 introduces the comparison symbols >, <, and =.
• Grade 2 will use place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and the symbols >, <, or =.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of place value
• 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.