K.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


K.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 whole numbers
 Developing an understanding of addition and subtraction
 Identifying and using attributes of twodimensional shapes and threedimensional solids
 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.

K.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 whole numbers
 Developing an understanding of addition and subtraction
 Identifying and using attributes of twodimensional shapes and threedimensional solids
 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.

K.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 whole numbers
 Developing an understanding of addition and subtraction
 Identifying and using attributes of twodimensional shapes and threedimensional solids
 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.

K.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 whole numbers
 Developing an understanding of addition and subtraction
 Identifying and using attributes of twodimensional shapes and threedimensional solids
 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.

K.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 whole numbers
 Developing an understanding of addition and subtraction
 Identifying and using attributes of twodimensional shapes and threedimensional solids
 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.

K.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 whole numbers
 Developing an understanding of addition and subtraction
 Identifying and using attributes of twodimensional shapes and threedimensional solids
 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.

K.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 whole numbers
 Developing an understanding of addition and subtraction
 Identifying and using attributes of twodimensional shapes and threedimensional solids
 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.

K.3 
Number and operations. The student applies mathematical process standards to develop an understanding of addition and subtraction situations in order to solve problems. The student is expected to:


K.3A 
Model the action of joining to represent addition and the action of separating to represent subtraction.

Model
THE ACTION OF JOINING TO REPRESENT ADDITION
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
 Addend – a number being added or joined together with another number(s)
 Sum – the total when two or more addends are joined
 Addition of whole numbers up to sums of 10
 Connection between the action of joining situations and the concept of addition
 Joining situations in contexts that represent an action (e.g., Kristin had 2 pencils, and her teacher gave her 3 more pencils; etc.)
 Appropriate language for joining situations
 Addend, sum, start amount, change amount, result amount
 Connection between quantities and numbers in problem situations to objects and drawings used
 Concrete models to represent contextual joining situations (linking cubes, number path, counters, five frames, beaded number line, Rekenrek, etc.)
 Physical joining of concrete objects
 Pictorial models to represent contextual joining situations
 Simple sketches representing concrete models without unnecessary details
 Physical joining of pictorial representations by circling or connecting
 Acting out to represent contextual joining situations
Model
THE ACTION OF SEPARATING TO REPRESENT SUBTRACTION
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}
 Subtraction
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Subtraction of whole numbers up to minuends of 10
 Including 0 as the subtrahend
 Including 0 as the difference
 Connection between the action of separating and the concept of subtraction
 Separating situations in contexts that represent an action (e.g., Mark had 5 books, and then he gave 2 books away; etc.)
 Appropriate language for separating situations
 Start amount, change amount, result amount, difference, removed, separated from, taken away from, etc.
 Connection between quantities and numbers in problem situations to objects and drawings used
 Concrete models to represent contextual separating situations (linking cubes, number path, counters, five frames, beaded number line, Rekenrek, etc.)
 Physical separation of concrete objects
 Pictorial models to represent contextual separating situations
 Simple sketches representing concrete models without unnecessary details
 Physical separation of pictorial representations by crossing out or circling
 Acting out to represent contextual separating situations
Note(s):
 Grade Level(s):
 Grade 1 will use objects and pictorial models to solve word problems involving joining, separating, partpartwhole relationships, 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.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of addition and subtraction
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.
 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.2. Create and use representations to organize, record, and communicate mathematical ideas.

K.3B 
Solve word problems using objects and drawings to find sums up to 10 and differences within 10.

Solve
WORD PROBLEMS USING OBJECTS AND DRAWINGS TO FIND SUMS UP TO 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
 Addend – a number being added or joined together with another number(s)
 Sum – the total when two or more addends are joined
 Addition of whole numbers with sums up to 10
 Relationship between composing numbers and addition
 Mathematical and realworld problem situations
 Situational language
 Action words indicating joining of quantities
 Partpartwhole relationship of quantities, implied or mental joining
 Connection between quantities and numbers in problem situations to objects and drawings used
 Joining situations in contexts that represent an action (e.g., Kristin had 2 pencils, and her teacher gave her 3 more pencils; etc.)
 Start quantity (addend) given, change quantity (addend) given, result (sum) unknown
 Joining situations in contexts that represent no action (e.g., Kristin had 2 blue pencils and 3 red pencils; etc.)
 Both part quantities (addends) given, whole (sum) unknown
 Addition strategies based on counting
 Count all
 Onetoone correspondence
 Count out one quantity, count out the other quantity, and then count both quantities together.
 Count on strategies
 Onetoone correspondence
 Count on from the first number presented.
 Count on from the largest number.
 Connection to hierarchical inclusion
 Hierarchical inclusion – concept of nested numbers, meaning each prior number in the counting sequence is included in the set as the set increases (e.g., 18 is 17 increased by 1; 18 decreased by 1 is 17; etc.)
 Adding 1 does not require counting.
 Properties of addition
 Quantities may be joined in any order (commutative property).
 A number keeps its identity when 0 is added to it (additive identity property).
Solve
WORD PROBLEMS USING OBJECTS AND DRAWINGS TO FIND DIFFERENCES WITHIN 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}
 Subtraction
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Subtraction of whole numbers to find differences within 10
 Including 0 as the subtrahend
 Relationship between decomposing numbers and subtraction
 Mathematical and realworld problem situations
 Situational language
 Action words indicating separation of quantities
 Connection between quantities and numbers in problem situations to objects and drawings used
 Separating situations in contexts that represent an action (e.g., Mark had 5 books, and then he gave 2 books away; etc.)
 Start quantity (minuend) given, change quantity (subtrahend) given, result (difference) unknown
 Subtraction strategies based on counting
 Removing
 Onetoone correspondence
 Count out start quantity, count and remove change quantity, and then count remaining quantity.
 Count backward
 Onetoone correspondence
 Count the whole quantity and then count backward the amount of the change quantity, with the last number in sequence naming the difference.
Count on
 Onetoone correspondence
 Count on from the change quantity to the whole quantity and then recount the remaining quantity beginning with 1.
 Connection to hierarchical inclusion
 Hierarchical inclusion – concept of nested numbers, meaning each prior number in the counting sequence is included in the set as the set increases (e.g., 18 is 17 increased by 1; 18 decreased by 1 is 17; etc.)
 Subtracting 1 does not require counting.
 Properties of subtraction
 Commutative property does not apply to subtraction.
 A number keeps its identity when 0 is subtracted from it (additive identity property).
Note(s):
 Grade Level(s):
 Grade 1 will compose 10 with two or more addends with and without concrete objects.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of addition and subtraction
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.2. Perform computations with rational and irrational numbers.

K.3C 
Explain the strategies used to solve problems involving adding and subtracting within 10 using spoken words, concrete and pictorial models, and number sentences.

Explain
THE STRATEGIES USED TO SOLVE PROBLEMS INVOLVING ADDING AND SUBTRACTING WITHIN 10 USING SPOKEN WORDS, 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
 Addend – a number being added or joined together with another number(s)
 Sum – the total when two or more addends are joined
 Addition of whole numbers with sums up to 10
 Subtraction
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Subtraction of whole numbers to find differences within 10
 Including 0 as the subtrahend
 Including 0 as the difference
 Mathematical and realworld problem situations
 Detailed explanation of the solution process and strategy
 Addition strategies
 Count all
 Count on from the first number presented
 Count on from the largest number
 Subtraction strategies
 Removing
 Count backward
 Count on
 Connection between information in the problem and problem type
 Joining situations in contexts that represent an action (e.g., Kristin had 2 pencils, and her teacher gave her 3 more pencils; etc.)
 Joining situations in contexts that represent no action (e.g., Kristin had 2 blue pencils and 3 red pencils; etc.)
 Separating situations in contexts that represent an action (e.g., Mark had 5 books, and then he gave 2 books away; etc.)
 Relationship between quantities of objects used, pictures drawn and number sentences to the problem situation
 Explanation using spoken words
 Appropriate mathematical language for joining or separating situations
 Labels for quantities represented
 Explanation using objects
 Linking cubes, counters, etc.
 Explanation using pictorials
 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 indicates the same value being represented on both side(s)
Note(s):
 Grade Level(s):
 Kindergarten introduces number sentences.
 Grade 1 will explain strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentences.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of 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 mathematics
 VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
 VIII.C. Communication and Representation – Presentation and representation of mathematical work
 VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
