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 twodimensional shapes and threedimensional solids
 Developing the understanding of length
 TxCCRS:
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.1. Interpret results of the mathematical problem in terms of the original realworld 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, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld 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 problemsolving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problemsolving process and the reasonableness of the solution.

Use
A PROBLEMSOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEMSOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION
Including, but not limited to:
 Problemsolving model
 Analyze given information
 Formulate a plan or strategy
 Determine a solution
 Justify the solution
 Evaluate the problemsolving 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 twodimensional shapes and threedimensional 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 problemsolving process.
 VII.D. Problem Solving and Reasoning – Realworld problem solving
 VII.D.2. Evaluate the problemsolving 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, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:
 Appropriate selection of tool(s) and techniques to apply in order to solve problems
 Tools
 Real objects
 Manipulatives
 Paper and pencil
 Technology
 Techniques
 Mental math
 Estimation
 Number sense
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional 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, GRAPHS, AND LANGUAGE AS APPROPRIATE
Including, but not limited to:
 Mathematical ideas, reasoning, and their implications
 Multiple representations, as appropriate
 Symbols
 Diagrams
 Graphs
 Language
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing an understanding of place value
 Solving problems involving addition and subtraction
 Analyzing attributes of twodimensional shapes and threedimensional 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, realworld situations, and everyday life
 IX.B.1. Use multiple representations to demonstrate links between mathematical and realworld 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 twodimensional shapes and threedimensional 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 nonexamples, 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 twodimensional shapes and threedimensional 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 twodimensional shapes and threedimensional 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.

1.3 
Number and operations. The student applies mathematical process standards to develop and use strategies for whole number addition and subtraction computations in order to solve problems. The student is expected to:


1.3B 
Use objects and pictorial models to solve word problems involving joining, separating, and comparing sets within 20 and unknowns as any one of the terms in the problem such as 2 + 4 = [ ]; 3 + [ ] = 7; and 5 = [ ] – 3.

Use
OBJECTS AND PICTORIAL MODELS TO SOLVE WORD PROBLEMS INVOLVING JOINING, SEPARATING, AND COMPARING SETS WITHIN 20 AND UNKNOWNS AS ANY ONE OF THE TERMS IN THE PROBLEM
Including, but not limited to:
 Whole numbers
 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}
 Addition
 Sum – the total when two or more addends are joined
 Addend – a number being added or joined together with another number(s)
 Addition of whole numbers within 20
 Subtraction
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Subtraction of whole numbers within 20
 Solutions recorded with a number sentence
 Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
 Number sentences, or equations, with equal sign at beginning or end
 Unknown in any position
 Concrete models
 Sets of objects within 20
 Base10 blocks, linking cubes, counters, etc.
 Pictorial models
 Base10 pictorials, number lines, strip diagrams, etc.
 Strip diagram – a linear model used to illustrate number relationships
 Mathematical and realworld problem situations
 Problems involving action
 Joining action result unknown
 Joining action change unknown
 Joining action start unknown
 Separating action result unknown
 Separating action change unknown
 Separating action start unknown
 Problems with no action
 Partpartwhole whole unknown
 Partpartwhole part unknown
 Additive comparison difference unknown
 Additive comparison compare quantity (larger quantity) unknown
 Additive comparison referent (smaller quantity) unknown
 Recognition of addition and subtraction as inverse operations
 Addition can be reversed by subtraction.
 Subtraction can be reversed by addition.
 Fact families – related number sentences using the same set of numbers
Note(s):
 Grade Level(s):
 Kindergarten modeled the action of joining to represent addition and the action of separating to represent subtraction in problems with the result unknown.
 Grade 1 introduces comparison problems.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Solving problems involving addition and subtraction
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.

1.3D 
Apply basic fact strategies to add and subtract within 20, including making 10 and decomposing a number leading to a 10.

Apply
BASIC FACT STRATEGIES TO ADD AND SUBTRACT WITHIN 20, INCLUDING MAKING 10 AND DECOMPOSING A NUMBER LEADING TO A 10
Including, but not limited to:
 Whole numbers
 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}
 Addition
 Sum – the total when two or more addends are joined
 Addend – a number being added or joined together with another number(s)
 Addition of whole numbers within 20
 Subtraction
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Subtraction of whole numbers within 20
 Solutions recorded with a number sentence
 Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
 Number sentences, or equations, with equal sign at beginning or end
 Decompose numbers – to break a number into parts or smaller values
 Compose numbers – to combine parts or smaller values to form a number
 Basic fact strategies for addition
 Counting all
 Count the amount of the first addend and count on the amount of the other addend.
 Counting on
 Begin with the first addend and count on the amount of the other addend.
 Begin with the largest addend and count on the amount of the other addend.
 Plus 1
 Adding 1 related to sequential counting
 Plus 2
 Adding 2 related to skip counting
 Plus 0 (additive identity)
 Adding zero to a number does not affect the total.
 Making 10
 Composing two addends to form a sum of 10
 Hidden tens
 Decomposing a number leading to a 10
 Plus 10
 Add 1 ten in the tens place and add 0 in the ones place.
 Doubles
 Adding two of the same addend
 Double plus/minus 1
 Consecutive addends
 Double the smaller addend and add 1, or double the larger addend and subtract 1.
 Hidden doubles
 Decompose an addend to form a doubles fact.
 Inbetweens
 Addends that have exactly one number between them consecutively.
 Double the number between the addends.
 Fact families – related number sentences using the same set of numbers
 Recognition of addition and subtraction as inverse operations
 Commutative property
 Sum does not change when the order of the addends are switched.
 Plus 9
 Adding 9 is equivalent to adding 10 and subtracting 1.
 Basic fact strategies for subtraction
 Counting back
 Begin with the minuend and count back the amount of the subtrahend.
 Counting up
 Begin with the subtrahend and count up to the minuend.
 Minus 1
 Subtracting 1 related to sequentially counting backward once
 Minus 2
 Subtracting 2 related to sequentially counting backward twice
 Minus 0 (additive identity)
 Subtracting 0 from a number does not affect the total.
 Fact families – related number sentences using the same set of numbers
 Recognition of addition and subtraction as inverse operations
 Decompose the subtrahend
 Decompose the subtrahend to form a known fact.
 Decompose the minuend
 Decompose the minuend to form a known fact.
 Minus 9
 Subtracting 9 is equivalent to subtracting 10 and adding 1.
Note(s):
 Grade Level(s):
 Grade 1 introduces applying basic fact strategies to add and subtract within 20.
 Grade 2 will recall basic facts to add and subtract within 20 with automaticity.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Solving problems involving addition and subtraction
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.

1.3E 
Explain strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentences.

Explain
STRATEGIES USED TO SOLVE ADDITION AND SUBTRACTION PROBLEMS UP TO 20 USING SPOKEN WORDS, OBJECTS, PICTORIAL MODELS, AND NUMBER SENTENCES
Including, but not limited to:
 Whole numbers
 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}
 Mathematical and realworld problem situations
 Addition
 Sum – the total when two or more addends are joined
 Addend – a number being added or joined together with another number(s)
 Addition of whole numbers within 20
 Subtraction
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Subtraction of whole numbers within 20
 Detailed explanation of solution process and strategy
 Addition strategies
 Counting all
 Counting on
 Plus 1
 Plus 2
 Plus 0 (additive identity)
 Making 10
 Hidden tens
 Plus 10
 Doubles
 Doubles plus/minus 1
 Hidden doubles
 Inbetweens
 Fact families
 Commutative property
 Plus 9
 Subtraction strategies
 Counting back
 Counting up
 Minus 1
 Minus 2
 Minus 0 (additive identity)
 Fact families
 Decompose the subtrahend
 Decompose the minuend
 Minus 9
 Connection between information in the problem and problem type
 Joining action situations
 Result unknown
 Change unknown
 Start unknown
 Separating action situations
 Result unknown
 Change unknown
 Start unknown
 Partpartwhole situations
 Whole unknown
 Part unknown
 Additive comparison situations
 Difference unknown
 Compare quantity (larger quantity) unknown
 Referent (smaller quantity) unknown
 Relationship between quantities of objects used, pictures drawn, and number sentences to the problem situation
 Explanation using spoken words
 Appropriate mathematical language for addition and subtraction situations
 Labels for quantities represented
 Explanation using objects
 Base10 blocks, linking cubes, counters, etc.
 Explanation using pictorials
 Base10 pictorials, number lines, strip diagrams, etc.
 Strip diagram – a linear model used to illustrate number relationships
 Explanation using number sentences
 Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
 Addition symbol represents joining
 Addend + addend = sum
 Sum = addend + addend
 Subtraction symbol represents separating
 Minuend – subtrahend = difference
 Difference = minuend – subtrahend
 Equal symbol represents a relationship where expressions on each side of the equal sign represent the same value
Note(s):
 Grade Level(s):
 Kindergarten explained the strategies used to solve problems involving adding and subtracting within 10 using spoken words, concrete and pictorial models, and number sentences.
 Grade 2 will add up to four twodigit numbers and subtract twodigit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Solving problems involving addition and subtraction
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 VIII.A. Communication and Representation – Language, terms, and symbols of mathematic
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 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.

1.3F 
Generate and solve problem situations when given a number sentence involving addition or subtraction of numbers within 20.

Generate, Solve
PROBLEM SITUATIONS WHEN GIVEN A NUMBER SENTENCE INVOLVING ADDITION OR SUBTRACTION OF NUMBERS WITHIN 20
Including, but not limited to:
 Whole numbers
 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}
 Addition
 Sum – the total when two or more addends are joined
 Addend – a number being added or joined together with another number(s)
 Addition of whole numbers within 20
 Subtraction
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Subtraction of whole numbers within 20
 Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
 Number sentences, or equations, with an equal sign at the beginning or end
 Unknown in any position
 Generate and solve mathematical and realworld problem situations when given an addition number sentence.
 Appropriate mathematical language
 Connection between information in the problem and problem type
 Addition situations
 Joining action result unknown
 Joining action change unknown
 Joining action start unknown
 Partpartwhole whole unknown
 Partpartwhole part unknown
 Additive comparison difference unknown
 Additive comparison compare quantity (larger quantity) unknown
 Generate and solve mathematical and realworld problem situations when given a subtraction number sentence.
 Appropriate mathematical language
 Connection between information in the problem and problem type
 Subtraction situations
 Separating action result unknown
 Separating action change unknown
 Separating action start unknown
 Partpartwhole part unknown
 Additive comparison difference unknown
 Additive comparison referent (smaller quantity) unknown
Note(s):
 Grade Level(s):
 Grade 1 introduces generating and solving problem situations when given a number sentence involving addition and subtraction of whole numbers within 20.
 Grade 2 will generate and solve problem situations for a given mathematical number sentence involving addition and subtraction of whole numbers within 1,000.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Solving problems involving addition and subtraction
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.

1.5 
Algebraic reasoning. The student applies mathematical process standards to identify and apply number patterns within properties of numbers and operations in order to describe relationships. The student is expected to:


1.5D 
Represent word problems involving addition and subtraction of whole numbers up to 20 using concrete and pictorial models and number sentences.

Represent
WORD PROBLEMS INVOLVING ADDITION AND SUBTRACTION OF WHOLE NUMBERS UP TO 20 USING CONCRETE AND PICTORIAL MODELS AND NUMBER SENTENCES
Including, but not limited to:
 Whole numbers
 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}
 Addition
 Sum – the total when two or more addends are joined
 Addend – a number being added or joined together with another number(s)
 Addition of whole numbers within 20
 Subtraction
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Subtraction of whole numbers within 20
 Represent mathematical and real world problem situations
 Concrete models
 Objects represent the quantities described in the problem situation.
 Base10 blocks, linking cubes, counters, etc.
 Pictorial models
 Pictures drawn represent the quantities described in the problem situation.
 Base10 pictorials, number lines, strip diagrams, etc.
 Solutions recorded with a number sentence
 Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
 Numbers represent the quantities described in the problem situation.
 Number sentences, or equations, with an equal sign at the beginning or end
 Unknown in any position
 Oral and written descriptions
 Explanation of relationship between objects, pictorials, and numbers and the information in the problem situation
 Addition problem types
 Joining action result unknown
 Joining action change unknown
 Joining action start unknown
 Partpartwhole whole unknown
 Partpartwhole part unknown
 Additive comparison difference unknown
 Additive comparison compare quantity (larger quantity) unknown
 Subtraction problem types
 Separating action result unknown
 Separating action start unknown
 Separating action change unknown
 Partpartwhole part unknown
 Additive comparison difference unknown
 Additive comparison referent (smaller quantity) unknown
Note(s):
 Grade Level(s):
 Grade 1 introduces representing word problems involving addition and subtraction of whole numbers up to 20 using concrete and pictorial models and number sentences.
 Grade 2 will represent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Solving problems involving addition and subtraction
 TxCCRS:
 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.

1.5E 
Understand that the equal sign represents a relationship where expressions on each side of the equal sign represent the same value(s).

Understand
THE EQUAL SIGN REPRESENTS A RELATIONSHIP WHERE EXPRESSIONS ON EACH SIDE OF THE EQUAL SIGN REPRESENT THE SAME VALUE(S)
Including, but not limited to:
 Term – a number and/or an unknown in an expression separated by an operation symbol(s)
 Expression – a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s)
 Equal sign – a mathematical symbol representing equivalence
 Equation – a mathematical statement composed of equivalent expressions separated by an equal sign
 Multistep solutions represented with one number sentence, or equation, per step
 All expressions separated by equal signs must be equivalent.
 Equal sign does not necessarily mean “find the answer”
 Equations with operations on both sides of the equal sign
 Equations with the unknown in any position
Note(s):
 Grade Level(s):
 Grade 1 introduces an understanding that the equal sign represents a relationship where expressions on each side of the equal sign represent the same value(s).
 Grade 2 will represent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Solving problems involving addition and subtraction
 TxCCRS:
 II.A. Algebraic Reasoning – Identifying expressions and equations
 II.A.1. Explain the difference between expressions and equations.

1.5G 
Apply properties of operations to add and subtract two or three numbers.

Apply
PROPERTIES OF OPERATIONS TO ADD AND SUBTRACT TWO OR THREE NUMBERS
Including, but not limited to:
 Whole numbers
 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}
 Addition
 Sum – the total when two or more addends are joined
 Addend – a number being added or joined together with another number(s)
 Addition of whole numbers within 20
 Subtraction
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Subtraction of whole numbers within 20
 Recognition of addition and subtraction as inverse operations
 Fact families – related number sentences using the same set of numbers
 Additive identity – the sum/difference is not affected when zero is added/subtracted to a number
 Commutative property of addition – if the order of the addends are changed, the sum will remain the same
 Two addends
 Three addends
 Subtraction is not commutative even though addition is commutative.
 Associative property of addition – if three or more addends are added, they can be grouped in any order, and the sum will remain the same
 Two addends
 Hidden tens
 Decompose an addend to form a tens fact.
 Hidden doubles
 Decompose an addend to form a doubles fact.
 Three addends
 Modeling properties of operations to add and subtract two or three numbers
Note(s):
 Grade Level(s):
 Grade 1 introduces applying properties of operations to add and subtract two or three numbers.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Solving problems involving addition and subtraction
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.

1.8 
Data analysis. The student applies mathematical process standards to organize data to make it useful for interpreting information and solving problems. The student is expected to:


1.8B 
Use data to create picture and bartype graphs.

Use
DATA
To Create
PICTURE AND BARTYPE GRAPHS
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 May include numbers or ranges of numbers
 Data collected in the form of responses to a question with up to three categories
 Survey – to ask a group of people a question in order to collect information about their opinions or answers
 Common characteristics in a collection of objects sorted into up to three categories
 Limitations
 Up to three categories
 Data values limited to whole numbers up to 120
 Data representations
 Picture graph – a graphical representation to organize data that uses pictures or symbols evenly spaced or placed in individual cells, where each picture or symbol represents one unit of data, to show the frequency (number of times) that each category occurs
 Characteristics of picture graphs
 Titles, subtitles, and labels
 Title represents the purpose of collected data
 Subtitle clarifies the meaning of categories
 Labels identify each category below the line
 Representation of categorical data
 Pictures or symbols
 Placed in a horizontal or vertical linear arrangement
 Vertical graph beginning at the bottom and progressing up above the line
 Horizontal graph beginning at the left and progressing to the right of the line
 Spaced approximately equal distances apart or placed in individual cells within each category
 Different picture or symbol used to represent each category
 Every piece of data represented using a onetoone correspondence
 One unit of data represented by each picture or symbol
 Value of the data represented by the pictures
 Determined by the total number of pictures or symbols in that category
 Represents the frequency of each category
 Bartype graph – a graphical representation to organize data that uses bars divided into individual cells, where each cell represents one unit of data, to show the frequency (number of times) that each category occurs
 Characteristics of bartype graphs
 Titles, subtitles, and labels
 Title represents the purpose of collected data
 Subtitle clarifies the meaning of categories
 Labels identify each category below the line
 Representation of categorical data
 Bars composed of individual cells
 Placed in a horizontal or vertical linear arrangement
 Vertical graph beginning at the bottom and progressing up above the line
 Horizontal graph beginning at the left and progressing to the right of the line
 Bars divided into equalsized cells with no gaps between cells
 Bars equal in width
 Independent bars that do not touch
 May use a different color to represent each category
 Does not include a scaled axis
 Every piece of data represented using a onetoone correspondence
 One unit of data represented by each shaded cell
 Value of the data represented by the bar
 Determined by the total number of cells in the bar for each category
 Represents the frequency for that category
 Connection between graphs representing the same data
 Picture graph to bartype graph
 Bartype graph to picture graph
 Same data represented using a picture graph and a bartype graph
Note(s):
 Grade Level(s):
 Kindergarten used data to create realobject and picture graphs.
 Grade 2 will explain that the length of a bar in a bar graph or the number of pictures in a pictograph represents the number of data points for a given category.
 Grade 2 will organize a collection of data with up to four categories using pictographs and bar graphs with intervals of one or more.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of place value
 TxCCRS:
 V.B. Statistical Reasoning – Design a study
 V.B.2. Construct appropriate visual representations of data.
 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.

1.8C 
Draw conclusions and generate and answer questions using information from picture and bartype graphs.

Draw
CONCLUSIONS USING INFORMATION FROM PICTURE AND BARTYPE GRAPHS
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 Limitations
 Up to three categories
 Data values limited to whole numbers up to 120
 Data values limited to addition or subtraction within 20
 Data representations
 Picture graph – a graphical representation to organize data that uses pictures or symbols evenly spaced or placed in individual cells, where each picture or symbol represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each picture or symbol
 Bartype graph – a graphical representation to organize data that uses bars divided into individual cells, where each cell represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each shaded cell
 Description of data represented
 Identification of title and category labels
 Explanation of what the graph represents
 Conclusions related to the question that led to the data collection
 Numerical conclusions in the data
 Quantities represented by the data
 Number in each category
 Number in a category(s) may be zero
 Combined total(s)
 Comparisons of data represented
 Comparative language used without numbers
 Comparative language used with numbers
 Changes in orientation do not affect data values
Generate, Answer
QUESTIONS USING INFORMATION FROM PICTURE AND BARTYPE GRAPHS
Including, but not limited to:
 Graph – a visual representation of the relationships between data collected
 Organization of data used to interpret data, draw conclusions, and make comparisons
 Data – information that is collected about people, events, or objects
 Categorical data – data that represents the attributes of a group of people, events, or objects
 Limitations
 Up to three categories
 Data values limited to whole numbers up to 120
 Operations limited to onestep addition or subtraction within 20
 Data representations
 Picture graph – a graphical representation to organize data that uses pictures or symbols evenly spaced or placed in individual cells, where each picture or symbol represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each picture or symbol
 Bartype graph – a graphical representation to organize data that uses bars divided into individual cells, where each cell represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each shaded cell
 Generate and answer questions using data in in picture graphs and bartype graphs
 Description of data
 Comparison using data values
 Operations using data values
Note(s):
 Grade Level(s):
 Kindergarten drew conclusions from realobject and picture graphs.
 Grade 2 will draw conclusions and make predictions from information in a graph.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of place value
 TxCCRS:
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.3. Make predictions using summary statistics.
 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.
