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, 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 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.2 
Number and operations. The student applies mathematical process standards to understand how to represent and compare whole numbers, the relative position and magnitude of whole numbers, and relationships within the numeration system. The student is expected to:


K.2A 
Count forward and backward to at least 20 with and without objects.

Count
FORWARD TO AT LEAST 20 WITH AND WITHOUT OBJECTS
Including, but not limited to:
 Counting numbers (1 – 20+)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Number word sequence has a correct order.
 Count forward orally by ones.
 With objects starting with one
 Onetoone correspondence – each object counted is matched accurately with a number word in correct sequence
 Tagging with synchrony, meaning when one object is touched it is matched with the correct word
 Arrangement and order of counting objects does not matter as long as the proper number sequence is used.
 Conservation of set – if the same number of objects are counted and then rearranged, the quantity of objects in the set does not change
 Cardinality – the last counting number identified represents the number of objects in the set regardless of which object was counted last
 Cardinal number – a number that names the quantity of objects in a set
 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.)
 Without objects starting with any counting number
 Proper number counting sequence
 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.)
Count
BACKWARD FROM AT LEAST 20 WITH AND WITHOUT OBJECTS
Including, but not limited to:
 Counting numbers (1 – 20+)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Number word sequence has a correct order.
 Count backward orally by ones.
 With objects starting from any given counting number
 Objects provided must match the number count (e.g., if counting backwards from 18, then provide 18 counters; etc.).
 Onetoone correspondence – each object counted is matched accurately with a number word in correct sequence
 Tagging with synchrony, meaning when one object is touched it is matched with the correct word
 Arrangement and order of counting objects does not matter as long as the proper number sequence is used.
 Conservation of set – if the same number of objects are counted and then rearranged, the quantity of objects in the set does not change
 Cardinality – the last counting number identified represents the number of objects in the set regardless of which object was counted last
 Cardinal number – a number that names the quantity of objects in a set
 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.)
 Without objects starting with any counting number
 Proper number counting sequence
 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.)
Note(s):
 Grade Level(s):
 Grade 1 will recite numbers forward and backward from any given number between 1 and 120.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers
 Developing an understanding of addition and subtraction
 TxCCRS:
 I.A. Numeric Reasoning – Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.

K.2B 
Read, write, and represent whole numbers from 0 to at least 20 with and without objects or pictures.

Read, Write, Represent
WHOLE NUMBERS FROM 0 TO AT LEAST 20 WITH AND WITHOUT OBJECTS OR PICTURES
Including, but not limited to:
 Whole numbers (0 – 20+)
 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}
 Numeric form
 Numerals represented using the digits 0 – 9
 With objects
 Number of objects in a set communicated orally
 Number of objects in a set written in numerals
 Number presented orally represented with a set of objects
 Number presented in writing represented with a set of objects
 Numbers presented out of sequence (e.g., represent 15; represent 9; represent 2; represent 17; etc.)
 Arrangement and order of counting objects does not matter as long as the proper number is used.
 Conservation of set – if the same number of objects are counted and then rearranged, the quantity of objects in the set does not change
 Relationship between number words and numerals to quantities
 Quantity in terms of “How many?”
 Concrete models begin to develop recognition of magnitude (relative size) of number.
 With pictures
 Number of objects in a picture communicated orally
 Number of objects in a picture written in numerals
 Number presented orally represented with a set of pictures
 Number presented in writing represented with a set of pictures
 Numbers presented out of sequence (e.g., represent 15; represent 9; represent 2; represent 17; etc.)
 Arrangement and order of pictures does not matter as long as the proper number is used.
 Conservation of set – if the same number of pictures are counted and then rearranged, the quantity of pictures in the set does not change
 Relationship between number words and numerals to quantities
 Quantity in terms of “How many?”
 Pictorial models begin to develop recognition of magnitude (relative size) of number.
 Without objects or pictures
 Number presented in written form communicated orally
 Number presented orally written in numerals
 Numbers presented out of sequence (e.g., write 15; write 9; write 2; write 17; etc.)
 Quantity in terms of “How many?”
Note(s):
 Grade Level(s):
 Kindergarten students read, write, and represent whole numbers numerically.
 Kindergarten students should be exposed to the word form of numbers along with the numeric form.
 Grade 1 students will begin reading numbers both in numeric and word form.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers

K.2C 
Count a set of objects up to at least 20 and demonstrate that the last number said tells the number of objects in the set regardless of their arrangement or order.

Count
A SET OF OBJECTS UP TO AT LEAST 20
Including, but not limited to:
 Set of objects (1 – 20+)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Number word sequence has a correct order.
 Arrangement and order of counting objects does not matter as long as the proper number is used.
 Onetoone correspondence – each object counted is matched accurately with a number word in correct sequence
 Tagging with synchrony, meaning when one object is touched it is matched with the correct word
Demonstrate
THE LAST NUMBER SAID TELLS THE NUMBER OF OBJECTS IN THE SET REGARDLESS OF THEIR ARRANGEMENT OR ORDER
Including, but not limited to:
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Cardinality – the last counting number identified represents the number of objects in the set regardless of which object was counted last
 Cardinal number – a number that names the quantity of objects in a set
 Conservation of set – if the same number of objects are counted and then rearranged, the quantity of objects in the set does not change
Note(s):
 Grade Level(s):
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers
 TxCCRS:
 I.A. Numeric Reasoning –Number representations and operations
 I.A.1. Compare relative magnitudes of rational and irrational numbers, and understand that numbers can be represented in different ways.
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.

K.2D 
Recognize instantly the quantity of a small group of objects in organized and random arrangements.

Recognize Instantly
THE QUANTITY OF A SMALL GROUP OF OBJECTS IN ORGANIZED AND RANDOM ARRANGEMENTS
Including, but not limited to:
 Group of objects (0 to 10)
 0 – 5 objects
 5 – 10 objects
 Subitizing– the ability to name the number of objects in a set without counting but rather by identifying the arrangement of objects
 Perceptual subitizing – the recognition of a quantity without using any other knowledge to determine the count
 Conceptual subitizing – recognition of a quantity based on a spatial arrangement, pattern, parts of the arrangement, etc.
 Organized arrangements
 Organization of objects aids in the instant recognition of the quantity based on the composition and decomposition of the parts.
 Various organized arrangements of objects (e.g., one or two five frame mats, a Rekenrek counting rack, fingers, number cubes, playing cards, dominoes, random number generators, etc.)
 Random arrangements
 Spatial arrangements of objects perceived in a variety of ways to aid in the instant recognition of a quantity based on the composition and decomposition of the parts
 Instant recognition of smaller quantities within the random arrangement aids in determining the total quantity of the random arrangement.
 Various random arrangements of objects
Note(s):
 Grade Level(s):
 Grade 1 recognizes instantly the quantity of structured arrangements.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers
 TxCCRS:
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.

K.2E 
Generate a set using concrete and pictorial models that represents a number that is more than, less than, and equal to a given number up to 20.

Generate
A SET USING CONCRETE AND PICTORIAL MODELS THAT REPRESENTS A NUMBER THAT IS MORE THAN, LESS THAN, AND EQUAL TO A GIVEN NUMBER UP TO 20
Including, but not limited to:
 Whole numbers (0 – 20)
 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}
 Quantity represented by concrete models, pictorial models, oral presentations, and symbolic representations
 Concrete and pictorial models begin to develop recognition of magnitude (relative size) of number.
 Concrete models
 Given number presented orally and symbolically
 Counting strategies used to create the set
 Relationship of the set to the given number
 Comparative language
 Describes the relationship between the concrete model and the given number
 Greater than, more than
 Less than, fewer than
 Equal to, same as
 Pictorial models
 Given number presented orally and symbolically
 Counting strategies used to create the set
 Relationship of the set to the given number
 Comparative language
 Describes the relationship between the pictorial model and the given number
 Greater than, more than
 Less than, fewer than
 Equal to, same as
Note(s):
 Grade Level(s):
 Grade 1 will generate a number that is greater than or less than a given whole number up to 120.
 Grade 1 will represent the comparison of two numbers to 100 using the symbols >, <, or =.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers
 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.

K.2F 
Generate a number that is one more than or one less than another number up to at least 20.

Generate
A NUMBER THAT IS ONE MORE THAN OR ONE LESS THAN ANOTHER NUMBER UP TO AT LEAST 20
Including, but not limited to:
 Whole numbers (0 – 20+)
 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}
 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.)
 Comparative language
 Describes the relationship between the number generated and the given number
 One more than a given number, including 1 more than 0 and 1 more than 20
 One less than a given number, including 1 less than 1 and 1 less than 21
 Quantity represented by concrete models, pictorial models, oral presentations, and symbolic representations
 Concrete and pictorial models begin to develop recognition of magnitude (relative size) of number.
 Counters, linking cubes, beans, calendar, hundreds chart, etc.
 Oral presentations and symbolic representations
 Verbal description, numerical recording using words and numbers
 Quantities presented out of correct sequence (e.g., 1 more than 10; 1 more than 4; 1 less than 18; 1 less than 6; etc.)
Note(s):
 Grade Level(s):
 Grade 1 will generate a number that is greater than or less than a given whole number to 120.
 Grade 2 will generate a number that is greater than or less than a given whole number to 1,200.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers
 Developing an understanding of addition and subtraction
 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.

K.2G 
Compare sets of objects up to at least 20 in each set using comparative language.

Compare
SETS OF OBJECTS UP TO AT LEAST 20 IN EACH SET USING COMPARATIVE LANGUAGE
Including, but not limited to:
 Whole numbers (0 – 20+)
 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}
 Quantity represented by concrete models, pictorial models, oral presentations, and symbolic representations
 Concrete and pictorial models begin to develop recognition of magnitude (relative size) of number.
 Counters, linking cubes, beans, calendar, hundreds chart, etc.
 Oral presentations and symbolic representations
 Verbal description, numerical recording using words and numbers
 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.)
 Compare sets – to consider the value of two sets to determine which set is greater or less in value or if the sets are equal in value
 Matching or counting strategies to compare sets
 Onetoone correspondence – each object counted is matched accurately with a number word in correct sequence
 Tagging with synchrony, meaning when one object is touched it is matched with the correct word
 Arrangement and order of counting objects does not matter as long as the proper number sequence is used.
 Conservation of set – if the same number of objects are counted and then rearranged, the quantity of objects in the set does not change
 Cardinality – the last counting number identified represents the number of objects in the set regardless of which object was counted last
 Cardinal number – a number that names the quantity of objects in a set
 Comparative language
 Describes the relationship between the quantities of each set
 Inequality language (greater than, more than, less than, fewer than, etc.)
 Equality language (equal to, same as, etc.)
 Compare two sets of objects up to at least 20.
 Recognition of the quantity represented by each set
 Comparative language describing the relationship between 2 sets
 Comparison of two organized sets
 Comparison of two unorganized sets
 Comparison of an organized set to an unorganized set
 Compare more than two sets of objects up to at least 20.
 Recognition of the quantity represented by each set
 Comparative language describing the relationship among more than 2 sets
 Comparison of organized sets and unorganized sets
Note(s):
 Grade Level(s):
 Kindergarten uses comparative language only.
 Grade 1 will use place value to compare whole numbers up to 120 using comparative language.
 Grade 1 introduces representing the comparison of two numbers to 100 using the symbols >, <, or =.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers
 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.

K.2H 
Use comparative language to describe two numbers up to 20 presented as written numerals.

Use
COMPARATIVE LANGUAGE
Including, but not limited to:
 Comparative language
 Describes the relationship between the value of each numeral
 Inequality language
 Greater than, more than
 Less than, fewer than
 Equality language
To Describe
TWO NUMBERS UP TO 20 PRESENTED AS WRITTEN NUMERALS
Including, but not limited to:
 Whole numbers (0 – 20)
 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}
 Numerals represent quantities
 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
 Compare two numbers
 Numerals presented out of sequence (e.g., compare 6 and 12; compare 19 and 5; etc.)
 Transition from comparing numbers by counting objects to comparing numbers without counting.
Note(s):
 Grade Level(s):
 Kindergarten uses comparative language only.
 Grade 1 will use place value to compare whole numbers up to 120 using comparative language.
 Grade 1 introduces representing the comparison of two numbers to 100 using the symbols >, <, or =.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers
 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.

K.2I 
Compose and decompose numbers up to 10 with objects and pictures.

Compose, Decompose
NUMBERS UP TO 10 WITH OBJECTS AND PICTURES
Including, but not limited to:
 Whole numbers (0 – 10)
 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
 Part to whole relationships
 Parts of a composed or decomposed number identified
 Correct number connected to appropriate parts
 Numeric relationship of one part to the other part
 Numeric relationship of each part to the whole
 Missing part determined
 Composition of a number in more than one way using objects and pictures
 Total of the parts conserved
 Composed parts may be listed in any order (commutative property).
 Relationship of composed parts to create a new set of composed parts
 Decomposition of a number in more than one way using objects and pictures
 Original decomposed number conserved
 Decomposed parts may be listed in any order (commutative property).
 Relationship of decomposed parts to create a new set of decomposed parts
Note(s):
 Grade Level(s):
 Grade 1 will 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.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers
 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.
 I.B. Numeric Reasoning – Number sense and number concepts
 I.B.2. Interpret the relationships between the different representations of numbers.

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.

K.5 
Algebraic reasoning. The student applies mathematical process standards to identify the pattern in the number word list. The student is expected to:


K.5A 
Recite numbers up to at least 100 by ones and tens beginning with any given number.

Recite
NUMBERS UP TO AT LEAST 100 BY ONES AND TENS BEGINNING WITH ANY GIVEN NUMBER
Including, but not limited to:
 Counting numbers (1 – 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}
 Number word sequence has a correct order
 Recite – to verbalize from memory
 Development of automaticity
 Relationship to counting
 Cardinal number – a number that names the quantity of objects in a set
 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.)
 Recite numbers forward up to at least 100
 Orally by ones beginning with 1
 Orally by ones beginning with any given number
 Orally by tens beginning with 10
 Orally by tens beginning with any given number between 10 and 100
 Beginning number is a multiple of 10.
Note(s):
 Grade Level(s):
 Kindergarten introduces reciting numbers by ten.
 Grade 1 will recite numbers forward and backward from any given number between 1 and 120.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing an understanding of whole numbers

K.8 
Data analysis. The student applies mathematical process standards to collect and organize data to make it useful for interpreting information. The student is expected to:


K.8C 
Draw conclusions from realobject and picture graphs.

Draw
CONCLUSIONS FROM REALOBJECT AND PICTURE 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
 Two to three categories
 Data values limited to whole numbers up to 20
 Data representations
 Realobject graph – a graphical representation to organize data that uses concrete or real objects evenly spaced or placed in individual cells, where each object represents one unit of data, to show the frequency (number of times) that each category occurs
 One unit of data represented by each object or picture
 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 object or picture
 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
 Comparisons of data represented
 Comparative language used without numbers (e.g., more than, less than, fewer than, the most, the least, the same as, equal to, etc.)
 Changes in orientation do not affect data values
Note(s):
 Grade Level(s):
 Grade 1 will draw conclusions and generate and answer questions using information from picture and bartype graphs.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Grade Level Connections (reinforces previous learning and/or provides development for future learning)
 TxCCRS:
 V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
 V.C.3. Make predictions using summary statistics.
 VIII.B. Communication and Representation – Interpretation of mathematical work
 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.
