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 Instructional Focus DocumentKindergarten Mathematics
 TITLE : Unit 06: Introducing and Developing Numbers 11 – 15 and Reciting Numbers to 90 SUGGESTED DURATION : 15 days

#### Unit Overview

Introduction
This unit bundles student expectations that address the foundational skills for developing an understanding of numbers 0 – 15, counting forward and backward 1 – 15, cardinality, subitizing, conservation of set, comparing numbers and sets of objects using comparative language, and generating numbers or set of objects less than or greater than a given amount. This unit also includes the student expectation that addresses reciting numbers up to 90 by ones beginning with any number. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Unit 03, students continued to investigate the foundational skills for understanding and using numbers from 0 to 10 and recited numbers up to 60 by ones beginning with any given number.

During this Unit
Students are introduced to the number 11 – 15. They use sets of objects up to 15 to develop an understanding of the concepts of cardinality, meaning that the last number said when counting a set of objects names the number of objects; hierarchical inclusion, meaning each prior number in the counting sequence is included in the set as the set increases; and conservation of set, meaning if the same number of objects are counted and then rearranged, the quantity of objects in the set does not change. Students apply cardinality, hierarchical inclusion, and conservation of set as they continue to explore the true meaning of numbers. Students count forward and backward to 15 with and without objects, as well as read, write, and represent the numbers. Students also compose and decompose numbers up to 10 using objects and pictures, which parallels the development of subitizing, meaning instantly recognizing the number being represented by a small quantity of objects in random and organized arrangements. Students apply all of these skills as they consider magnitude, or relative size, to compare sets of objects up to 15 and generate a set of objects and pictures that is more than, less than, or equal to a given number. Students use comparative language to describe the comparison of numbers represented using objects, pictures, or numerals. When given a number up to 15, students are expected to generate a number that is one more than or one less than a given number. Along with the investigation of number and quantity, students are expected to recite numbers up to 90 by tens beginning with 10 and by ones beginning with any number. Practice with rote reciting of numbers and learning the correct sequence of numbers aids in developing the foundation for meaningful counting strategies.

After this Unit
In Unit 08, students will continue to develop the foundations of number as they extend their number set to include 15 to 20. Students will also extend reciting numbers up to 100 by tens beginning with any multiple of ten and by ones beginning with any number.

In Kindergarten, reciting numbers up to 90, reading, writing, and representing numbers, cardinality, subitizing, and comparing and describing sets of objects are foundational concepts that are subsumed within the Kindergarten Texas Response to Curriculum Focal Points (TxRCFP): Developing an understanding of whole numbers. Counting forward and backward with and without objects, composing and decomposing numbers, and generating numbers and sets of objects that are more than, less than, or equal to an original quantity are also subsumed within the Kindergarten Texas Response to Curriculum Focal Points (TxRCFP): Developing an understanding of whole numbers as well as the Kindergarten Texas Response to Curriculum Focal Points (TxRCFP): Developing an understanding of addition and subtraction. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, A2, B1, B2; II. Algebraic Reasoning D1, D2; V. Statistical Reasoning A1, C2; VII. Problem Solving and Reasoning A1, A2, A3, A4, A5, B1, C1, D1, D2; VIII. Communication and Representation A1, A2, A3, B1, B2, C1, C2, C3; IX. Connections A1, A2, B1, B2, B3.

Research
According to Copley (2010), “The teen numbers are often the most difficult for children, at least in English” (p. 57). The National Council of Teachers of Mathematics (2010) also acknowledges the difficulties students encounter with teen numbers due to the irregularities in the language pattern when paired with the written numerals. NCTM identifies several of the irregularities that students encounter. One of these irregularities is that the written symbol does not match the verbalization of the word. NCTM also states, “This ones-before-tens structure of the teen words is opposite to the tens-before-ones structure in the written teen number symbols. We say “four” first in “fourteen” but write 4 second in 14 (1 ten 4 ones)” (p. 18). This often leads to reversals when writing numerals. Copley further states, “To progress in counting, children must recognize the patterns involved in counting numbers greater than 9 — for example, after a number ending in 9, a new decade (10, 20, 30…) begins; or after a new decade number (20), subsequent numbers require the addition of the numbers 1 through 9 (21, 22, 23…)” (p. 57).

Copley, J. (2010). The young child and mathematics. Washington, DC: National Association for the Education of Young Children
National Council of Teachers of Mathematics. (2010). Focus in kindergarten teaching with curriculum focal points. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• A thorough understanding of counting involves integrating different skills or characteristics of numbers and is foundational and essential for continued work with numbers (counting numbers through 15).
• What relationships exist between numbers in the proper counting sequence?
• What strategies can be used to keep track of the count when counting a set of objects?
• Why are tracking strategies important in counting a set of objects?
• How does starting the count with a different object affect the count?
• How does rearranging the set of objects affect the count?
• The ability to recognize and represent numbers in various forms develops the understanding of equivalence and allows for working flexibly with numbers in order to communicate and reason about the value of the number (whole numbers through 15; decomposition within 10).
• What are some ways a number can be represented?
• Why can a number vary in representation but the value of the number stay the same?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a number using …
• concrete models
• pictorial models
… improve understanding and communicating about the value of a number and the equivalence of the representations?
• Number
• Composition and Decomposition of Numbers
• Number
• Counting (natural) numbers
• Whole numbers
• Number Recognition and Counting
• Sequence
• Cardinality
• Conservation of set
• Hierarchical inclusion
• Magnitude
• Number Representations
• Standard form
• Relationships
• Numerical
• Equivalence
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• A thorough understanding of counting involves integrating different skills or characteristics of numbers and is foundational and essential for continued work with numbers (whole numbers through 15).
• What relationships exist between numbers in the proper counting sequence?
• What strategies can be used to keep track of the count when counting a set of objects?
• Why are tracking strategies important in counting a set of objects?
• How does starting the count with a different object affect the count?
• How are counting skills used to generate numbers that are greater or less than a given number?
• The ability to recognize and represent numbers in various forms develops the understanding of equivalence and allows for working flexibly with numbers in order to communicate and reason about the value of the number (whole numbers through 15).
• What are some ways a number can be represented?
• Why can a number vary in representation but the value of the number stay the same?
• Why is it important to be able to recognize and create a variety of representations for a quantity?
• How could representing a number using …
• concrete models
• pictorial models
… improve understanding and communicating about the value of a number and the equivalence of the representations?
• Quantities are compared to determine magnitude of number and equality or inequality relations (whole numbers through 15).
• Why is it important to identify the unit or attribute being described by numbers before comparing the numbers?
• How can …
• numeric representations
• concrete representations
• pictorial representations
… aid in comparing numbers?
• How can the comparison of two numbers be described and represented?
• Number
• Compare
• Comparative language
• Number
• Counting (natural) numbers
• Whole numbers
• Number Recognition and Counting
• Sequence
• Cardinality
• Conservation of set
• Hierarchical inclusion
• Magnitude
• Number Representations
• Standard form
• Relationships
• Numerical
• Equivalence
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• A thorough understanding of counting involves integrating different skills or characteristics of numbers and is foundational and essential for continued work with numbers (whole numbers through 15).
• What relationships exist between numbers in the proper counting sequence?
• How are counting skills used to generate numbers that are greater or less than a given number?
• What relationships exist between numerals and the quantities?
• Quantities are compared to determine magnitude of number and equality or inequality relations (whole numbers through 15).
• Why is it important to identify the unit or attribute being described by numbers before comparing the numbers?
• How can …
• numeric representations
• concrete representations
• pictorial representations
… aid in comparing numbers?
• How can the comparison of two numbers be described and represented?
• Number
• Compare
• Comparative language
• Number
• Counting (natural) numbers
• Whole numbers
• Number Recognition and Counting
• Hierarchical inclusion
• Magnitude
• Number Representations
• Standard form
• Relationships
• Numerical
• Equivalence
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• A thorough understanding of counting involves integrating different skills or characteristics of numbers and is foundational and essential for continued work with numbers (whole numbers through 10).
• Why are visualizing and instantly recognizing small quantities beneficial when …
• working with larger quantities of objects?
• composing or decomposing numbers?
• Number
• Composition and Decomposition of Numbers
• Number
• Whole numbers
• Number Recognition and Counting
• Subitizing
• Associated Mathematical Processes
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

 Numeracy requires the ability to work flexibly with quantities in order to recognize, reason, and solve situations of varying contexts in everyday life, society, and the work place.  How is numeracy like literacy? What are some examples of numeracy in everyday life, society, and the work place? How does context influence understanding of a quantity? Why is the ability to work flexibly with quantities essential to developing the foundations of numeracy? Quantitative relationships model problem situations efficiently and can be used to make generalizations, predictions, and critical judgements in everyday life. What patterns exist within different types of quantitative relationships and where are they found in everyday life? Why is the ability to model quantitative relationships in a variety of ways essential to solving problems in everyday life?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• A thorough understanding of counting involves integrating different skills or characteristics of numbers and is foundational and essential for continued work with numbers (counting numbers forward and backward through 15).
• What relationships exist between numbers in the counting sequence when …
• counting forward from one number to the next number?
• counting backward from one number to the previous number?
• Recognition of patterns in the number word sequence, which are repeated with every grouping of ten, leads to efficient and accurate reciting of numbers (reciting numbers to 90).
• What patterns can be found between each grouping of ten when reciting numbers in sequence by ones?
• Number
• Number
• Counting (natural) numbers
• Number Recognition and Counting
• Sequence
• Cardinality
• Hierarchical inclusion
• Magnitude
• Algebraic Reasoning
• Patterns and Relationships
• Reciting numbers
• Associated Mathematical Processes
• Problem Solving Model
• Tools and Techniques
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think the last number said when counting a set of objects represents the last object counted rather than the quantity of all objects in the set.
• Some students may think a change in the arrangement of objects changes the number of objects in the set rather than recognizing that the quantity does not change if the objects are rearranged or counted in a different order.
• Some students may think that a number can be composed or decomposed in only one way rather than understanding that a number can be composed or decomposed in many ways as long as the quantity of the whole remains the same.
• Some students may think of naming or reciting counting numbers in sequence as a memorization task rather than associating each number with a single object in the set and understanding the tagging of objects to demonstrate one-to-one correspondence.
• Some students may think of naming or reciting counting numbers in sequence as a memorization task rather than understanding that each number represents a quantity and that each number in the counting sequence represents a quantity of one more than the previous number.
• Some students may think there is no pattern or connection between the sequence of number words and the decade words in sequence rather than seeing the pattern or relationship as numbers in sequence move to the next decade (e.g., 49 to 50; 59 to 60; 69 to 70; etc.).
• Some students may think the comparison of two numbers has no relationship to other comparisons rather than realizing that if a given number is greater than another number, then the given number is also greater than all numbers before that number in numerical sequence (e.g., if 14 is greater than 12, it is also greater than 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, and 0).
• Some students may think the comparison of two numbers has no relationship to other comparisons rather than realizing that if a given number is greater than another number, then the given number is also greater than all numbers that could compose that number (e.g., 12 is greater than 11 and greater than 1, 12 is greater than 10 and greater than 2, 12 is greater than 9 and greater than 3, 12 is greater than 8 and greater than 2, 12 is greater than 7 and greater than 5, 12 is greater than 6, and 12 is greater than 0).
• Some students may think that the comparison of two sets of objects has no relationship to other comparisons rather than realizing that the same comparison of sets of objects applies to the numerals representing the sets of objects.
• Some students may think that numbers with the same digits represent the same numbers rather than recognizing that digits in different positions represent different numbers (e.g., thinking that 51 is 15 because both numbers have the same digits).
• Some students may auditorily confuse teen words with decade words (e.g., fifteen and fifty) when reciting numbers.
• Some students may auditorily confuse number words with similar sounds (e.g., seven and eleven) when reciting numbers.
• Some students may pronounce teen words incorrectly (e.g., saying eleventeen for eleven) when reciting numbers.

Underdeveloped Concepts:

• Some students may not associate the idea of “none” with the number zero.

#### Unit Vocabulary

• Compare numbers – to consider the value of two numbers to determine which number is greater or less or if the numbers are equal in value
• 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
• Compose numbers – to combine parts or smaller values to form a number
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Decompose numbers – to break a number into parts or smaller values
• One-to-one correspondence – each object counted is matched accurately with a number word in correct sequence
• Recite – to verbalize from memory
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Backward Comparative language Count Counting by ones Counting order Decrease Digit Eleven Equal to, same as Fifteen Fourteen Forward Greater than, more than Increase Less than, fewer than Model Number Numeral Part Quantity Sequence Set Thirteen Twelve Whole

System Resources Other Resources

System Resources may be accessed through Search All Components in the District Resources Tab.

Texas Higher Education Coordinating Board – Texas College and Career Readiness Standards

Texas Education Agency – Texas Response to Curriculum Focal Points for K-8 Mathematics Revised 2013

Texas Education Agency – Mathematics Curriculum

Texas Education Agency – STAAR Mathematics Resources

Texas Education Agency Texas Gateway – Revised Mathematics TEKS: Vertical Alignment Charts

Texas Education Agency Texas Gateway – Mathematics TEKS: Supporting Information

Texas Education Agency Texas Gateway – Interactive Mathematics Glossary

Texas Education Agency Texas Gateway – Resources Aligned to Kindergarten Mathematics TEKS

TAUGHT DIRECTLY TEKS

TEKS intended to be explicitly taught in this unit.

TEKS/SE Legend:

• Knowledge and Skills Statements (TEKS) identified by TEA are in italicized, bolded, black text.
• Student Expectations (TEKS) identified by TEA are in bolded, black text.
• Portions of the Student Expectations (TEKS) that are not included in this unit but are taught in previous or future units are indicated by a strike-through.

Specificity Legend:

• Supporting information / clarifications (specificity) written by TEKS Resource System are in blue text.
• Unit-specific clarifications are in italicized, blue text.
• Information from Texas Education Agency (TEA), Texas College and Career Readiness Standards (TxCCRS), Texas Response to Curriculum Focal Points (TxRCFP) is labeled.
• A Partial Specificity label indicates that a portion of the specificity not aligned to this unit has been removed.
TEKS# SE# TEKS SPECIFICITY
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 two-dimensional shapes and three-dimensional solids
• TxCCRS:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
K.1B Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution.

Use

A PROBLEM-SOLVING MODEL THAT INCORPORATES ANALYZING GIVEN INFORMATION, FORMULATING A PLAN OR STRATEGY, DETERMINING A SOLUTION, JUSTIFYING THE SOLUTION, AND EVALUATING THE PROBLEM-SOLVING PROCESS AND THE REASONABLENESS OF THE SOLUTION

Including, but not limited to:

• Problem-solving model
• Analyze given information
• Formulate a plan or strategy
• Determine a solution
• Justify the solution
• Evaluate the problem-solving process and the reasonableness of the solution

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of whole numbers
• Developing an understanding of addition and subtraction
• Identifying and using attributes of two-dimensional shapes and three-dimensional 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 problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving 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 two-dimensional shapes and three-dimensional 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 two-dimensional shapes and three-dimensional 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, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world 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 two-dimensional shapes and three-dimensional 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 non-examples, patterns, etc.
• Current knowledge to new learning

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing an understanding of whole numbers
• Developing an understanding of addition and subtraction
• Identifying and using attributes of two-dimensional shapes and three-dimensional 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. ConnectionsConnections 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 two-dimensional shapes and three-dimensional solids
• TxCCRS:
• VII.A. Problem Solving and ReasoningMathematical 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 15 WITH AND WITHOUT OBJECTS

Including, but not limited to:

• Counting numbers (1 – 15)
• 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
• One-to-one 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., 15 is 14 increased by 1; 15 decreased by 1 is 14; 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., 15 is 14 increased by 1; 15 decreased by 1 is 14; etc.)

Count

BACKWARD FROM AT LEAST 15 WITH AND WITHOUT OBJECTS

Including, but not limited to:

• Counting numbers (1 – 15)
• 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 15, then provide 15 counters; etc.).
• One-to-one 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., 15 is 14 increased by 1; 15 decreased by 1 is 14; 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., 15 is 14 increased by 1; 15 decreased by 1 is 14; etc.)

Note(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.

WHOLE NUMBERS FROM 0 TO AT LEAST 15 WITH AND WITHOUT OBJECTS OR PICTURES

Including, but not limited to:

• Whole numbers (0 – 15)
• 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 7; 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 7; 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 7; etc.)
• Quantity in terms of “How many?”

Note(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 15

Including, but not limited to:

• Set of objects (1 – 15)
• 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.
• One-to-one 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):

• 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
• Quantities of 5 or fewer
• 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 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 15

Including, but not limited to:

• Whole numbers (0 – 15)
• 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 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 15

Including, but not limited to:

• Whole numbers (0 – 15)
• 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., 15 is 14 increased by 1; 15 decreased by 1 is 14; 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 14
• One less than a given number, including 1 less than 1 and 1 less than 15
• 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 14; 1 less than 6; etc.)

Note(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 15 IN EACH SET USING COMPARATIVE LANGUAGE

Including, but not limited to:

• Whole numbers (0 – 15)
• 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., 15 is 14 increased by 1; 15 decreased by 1 is 14; 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
• One-to-one 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 15.
• 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 15.
• 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):

• 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
• Equal to, same as

To Describe

TWO NUMBERS UP TO 15 PRESENTED AS WRITTEN NUMERALS

Including, but not limited to:

• Whole numbers (0 – 15)
• 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 15 and 5; etc.)
• Transition from comparing numbers by counting objects to comparing numbers without counting.

Note(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 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.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 90 BY ONES AND TENS BEGINNING WITH ANY GIVEN NUMBER

Including, but not limited to:

• Counting numbers (1 – 90)
• 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., 15 is 14 increased by 1; 15 decreased by 1 is 14; etc.)
• Recite numbers forward up to at least 90
• Orally by ones beginning with 1
• Orally by ones beginning with any given number
• By tens beginning with 10

Note(s):