2.1 
Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:


2.1A 
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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:

2.1B 
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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 VIII. Problem Solving and Reasoning

2.1C 
Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems.

Select
TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND TECHNOLOGY AS APPROPRIATE, AND TECHNIQUES, INCLUDING MENTAL MATH AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS
Including, but not limited to:
 Appropriate selection of tool(s) and techniques to apply in order to solve problems
 Tools
 Real objects
 Manipulatives
 Paper and pencil
 Technology
 Techniques
Note(s):
 The mathematical process standards may be applied to all content standards as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 VIII. Problem Solving and Reasoning

2.1D 
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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 IX. Communication and Representation

2.1E 
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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 IX. Communication and Representation

2.1F 
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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:

2.1G 
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 proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 Measuring length
 Applying knowledge of twodimensional shapes and threedimensional solids, including exploration of early fraction concepts
 TxCCRS:
 IX. Communication and Representation

2.4 
Number and operations. The student applies mathematical process standards to develop and use strategies and methods for whole number computations in order to solve addition and subtraction problems with efficiency and accuracy. The student is expected to:


2.4A 
Recall basic facts to add and subtract within 20 with automaticity.

Recall With Automaticity
BASIC FACTS TO ADD WITHIN 20
Including, but not limited to:
 Whole numbers
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Automaticity – executing a basic fact with speed and accuracy with little or no conscious effort
 Addition
 Sum – the total when two or more addends are joined
 Addend – a number being added or joined together with another number(s)
 Addition of whole numbers within 20
 Solutions recorded with a number sentence
 Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
 Equal sign at beginning or end
 Decompose numbers – to break a number into parts or smaller values
 Compose numbers – to combine parts or smaller values to form a number
 Basic fact strategies for addition
 Counting all
 Count the amount of the first addend and count on the amount of the other addend.
 Counting on
 Begin with the first addend and count on the amount of the other addend.
 Begin with the largest addend and count on the amount of the other addend.
 Plus 1
 Adding 1 related to sequential counting
 Plus 2
 Adding 2 related to skip counting
 Plus 0 (additive identity)
 Adding zero to a number does not affect the total.
 Making 10
 Composing two addends to form a sum of 10
 Hidden tens
 Decomposing a number leading to a 10
 Plus 10
 Add 1 ten in the tens place and add 0 in the ones place.
 Doubles
 Adding two of the same addend
 Adding doubles always results in an even sum, regardless of whether the addends are even or odd.
 Double plus/minus 1
 Consecutive addends
 Double the smaller addend and add 1, or double the larger addend and subtract 1.
 Adding doubles plus/minus 1 always results in an odd sum.
 Hidden doubles
 Decompose an addend to form a doubles fact.
 Inbetweens
 Addends that have exactly one number between them consecutively.
 Double the number between the addends.
 Fact families – related number sentences using the same set of numbers
 Recognition of addition and subtraction as inverse operations
 Commutative property
 Sum does not change when the order of the addends are switched.
 Plus 9
 Adding 9 is equivalent to adding 10 and subtracting 1.
Recall With Automaticity
BASIC FACTS TO SUBTRACT WITHIN 20
Including, but not limited to:
 Whole numbers
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Automaticity – executing a basic fact with speed and accuracy with little or no conscious effort
 Subtraction
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Subtraction of whole numbers within 20
 Solutions recorded with a number sentence
 Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
 Equal sign at beginning or end
 Decompose numbers – to break a number into parts or smaller values
 Basic fact strategies for subtraction
 Counting back
 Begin with the minuend and count back the amount of the subtrahend.
 Counting up
 Begin with the subtrahend and count up to the minuend.
 Minus 1
 Subtracting 1 related to sequentially counting backward once
 Minus 2
 Subtracting 2 related to sequentially counting backward twice
 Minus 0 (additive identity)
 Subtracting 0 from a number does not affect the total.
 Fact families – related number sentences using the same set of numbers
 Recognition of addition and subtraction as inverse operations
 Inverse doubles
 The minuend will be even, and the subtrahend and difference will either both be even or both be odd.
 Inverse doubles plus/minus 1
 The minuend will be odd, and if the subtrahend is even, then the difference will be odd.
 The minuend will be odd, and if the subtrahend is odd, then the difference will be even.
 Decompose the subtrahend
 Decompose the subtrahend to form a known fact.
 Decompose the minuend
 Decompose the minuend to form a known fact.
 Minus 9
 Subtracting 9 is equivalent to subtracting 10 and adding 1.
Note(s):
 Grade Level(s):
 Grade 1 applied basic fact strategies to add and subtract within 20, including making 10 and decomposing a number leading to a 10.
 Grade 2 is accountable for recalling addition and subtraction facts within 20 with automaticity.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation

2.5 
Number and operations. The student applies mathematical process standards to determine the value of coins in order to solve monetary transactions. The student is expected to:


2.5A 
Determine the value of a collection of coins up to one dollar.

Determine
THE VALUE OF A COLLECTION OF COINS UP TO ONE DOLLAR
Including, but not limited to:
 Coins
 Penny: 1¢
 Nickel: 5¢
 Dime: 10¢
 Quarter: 25¢
 Halfdollar: 50¢
 Concrete and pictorial models
 Traditional and newly released designs
 Views of both sides of coins
 Counting to determine the value of a collection of coins up to one dollar
 Relationships represented using concrete or pictorial models
 Hundreds chart, number line, reallife objects, etc.
 Coins in like groups (e.g., halfdollars, quarters, dimes, nickels, pennies)
 Counting by ones or skip counting by twos to determine the value of a collection of pennies
 1¢, 2¢, 3¢, 4¢, …, 97¢, 98¢, 99¢, 100¢
 2¢, 4¢, 6¢, 8¢, …, 94¢, 96¢, 98¢, 100¢
 Skip counting by fives to determine the value of a collection of nickels
 5¢, 10¢, 15¢, 20¢, 25¢, 30¢, …, 95¢, 100¢
 Skip counting by tens to determine the value of a collection of dimes
 10¢, 20¢, 30¢, 40¢, 50¢, …, 80¢, 90¢, 100¢
 Skip counting by twentyfives to determine the value of a collection of quarters
 Skip counting by fifties to determine the value of a collection of halfdollars
 Compound counting to determine the value of a collection of mixed coins up to one dollar
 Separate coins into like groups prior to counting (e.g., halfdollars, quarters, dimes, nickels, pennies).
 Begin by counting the largest denomination of coins and then count on each denomination of coins in order from largest to smallest.
 Count halfdollars by fifties, count on quarters by twentyfives, count on dimes by tens, count on nickels by fives, count on pennies by twos or ones.
 Relationships by value
 Penny to nickel, dime, quarter, halfdollar
 5 pennies = 1 nickel; 10 pennies = 1 dime; 25 pennies = 1 quarter; 50 pennies = 1 halfdollar
 1 penny < 1 nickel; 1 penny < 1 dime; 1 penny < 1 quarter; 1 penny < 1 halfdollar
 Nickel to penny, dime, quarter, halfdollar
 1 nickel = 5 pennies; 2 nickels = 1 dime; 5 nickels = 1 quarter; 10 nickels = 1 halfdollar
 1 nickel > 1 penny; 1 nickel < 1 dime; 1 nickel < 1 quarter; 1 nickel < 1 halfdollar
 Dime to penny, nickel, quarter, halfdollar
 1 dime = 10 pennies; 1 dime = 2 nickels; 5 dimes = 2 quarters; 5 dimes = 1 halfdollar
 1 dime > 1 penny; 1 dime > 1 nickel; 1 dime < 1 quarter; 1 dime < 1 halfdollar
 Quarter to penny, nickel, dime, halfdollar
 1 quarter = 25 pennies; 1 quarter = 5 nickels; 2 quarters = 5 dimes; 2 quarters = 1 halfdollar
 1 quarter > 1 penny; 1 quarter > 1 nickel; 1 quarter > 1 dime; 1 quarter < 1 halfdollar
 Halfdollar to penny, nickel, dime, quarter
 1 halfdollar = 50 pennies; 1 halfdollar = 10 nickels; 1 halfdollar = 5 dimes; 1 halfdollar = 2 quarters
 1 halfdollar > 1 penny; 1 halfdollar > 1 nickel; 1 halfdollar > 1 dime; 1 halfdollar > 1 quarter
 Create a collection of coins for a given value.
 Comparison of the values of two collections of coins
 Number of coins may not be proportional to the value of the collection.
 Multiple combinations of the same value
Note(s):
 Grade Level(s):
 Grade 1 used relationships to count by twos, fives, and tens to determine the value of a collection of pennies, nickels, and/or dimes.
 Grade 3 will determine the value of a collection of coins and bills.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 TxCCRS:
 IX. Communication and Representation
 X. Connections

2.5B 
Use the cent symbol, dollar sign, and the decimal point to name the value of a collection of coins.

Use
THE CENT SYMBOL, DOLLAR SIGN, AND THE DECIMAL POINT TO NAME THE VALUE OF A COLLECTION OF COINS
Including, but not limited to:
 Value of a collection of coins named with numbers and symbols
 Cent symbol not used in conjunction with dollar symbol and decimal
 Cent symbol (¢)
 Cent symbol written to the right of the numerical value
 Cent label read and written after numerical value
 Values equal to or greater than 100 written with cent symbol not customary, but acceptable
 Dollar symbol ($) and decimal
 Dollar symbol written to the left of the dollar amount
 Decimal separates whole dollar amount from cent amount, or part of a dollar amount
 Dollar label read after dollar amount
 Decimal read as “and”
 Zero written for the dollar amount, but not read, if value is less than one dollar
 Multiple representations of the same value
Note(s):
 Grade Level(s):
 Grade 1 wrote a number with the cent symbol to describe the value of a coin.
 Grade 2 introduces the dollar sign and decimal point to name the value of a collection of coins.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 TxCCRS:
 IX. Communication and Representation
 X. Connections

2.7 
Algebraic reasoning. The student applies mathematical process standards to identify and apply number patterns within properties of numbers and operations in order to describe relationships. The student is expected to:


2.7A 
Determine whether a number up to 40 is even or odd using pairings of objects to represent the number.

Determine
WHETHER A NUMBER UP TO 40 IS EVEN OR ODD USING PAIRINGS OF OBJECTS TO REPRESENT THE NUMBER
Including, but not limited to:
 Whole numbers (0 – 40)
 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}
 Concrete objects organized in pairs to represent a number
 Even number – a number represented by objects that when paired have zero left over
 If the number of objects are paired with zero left over, the number represented by the objects is even.
 Numbers ending with the digit 0, 2, 4, 6, or 8 are even numbers.
 Odd number – a number represented by objects that when paired have one left over
 If the number of objects are paired with one left over, the number represented by the objects is odd.
 Numbers ending with the digit 1, 3, 5, 7, or 9 are odd numbers.
 Relationships in addition and subtraction
 Relationship between doubles facts and even numbers
 Adding doubles always results in an even sum, regardless of whether the addends are even or odd
 Inverse doubles
 The minuend will be even, and the subtrahend and difference will either both be even or both be odd.
 Relationship between doubles plus/minus 1 facts and odd numbers
 Adding doubles plus/minus 1 always results in an odd sum.
 Inverse doubles plus/minus 1
 The minuend will be odd, and if the subtrahend is even, then the difference will be odd.
 The minuend will be odd, and if the subtrahend is odd, then the difference will be even.
Note(s):
 Grade Level(s):
 Grade 1 skip counted by twos, fives, and tens to determine the total number of objects up to 120 in a set.
 Grade 2 introduces determining whether a number up to 40 is even or odd using pairings of objects to represent the number.
 Grade 3 will determine if a number is even or odd using divisibility rules.
 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:
 IX. Communication and Representation

2.7B 
Use an understanding of place value to determine the number that is 10 or 100 more or less than a given number up to 1,200.

Use
AN UNDERSTANDING OF PLACE VALUE TO DETERMINE THE NUMBER THAT IS 10 OR 100 MORE OR LESS THAN A GIVEN NUMBER UP TO 1,200
Including, but not limited to:
 Whole numbers (0 – 1,200)
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, etc.
 One thousands place
 Hundreds place
 Tens place
 Ones place
 Comparative language
 Greater than, more than
 Less than, fewer than
 Relationships based on place value
 10 more or 10 less
 Adding 10 to a number increases the digit in the tens place by 1.
 Subtracting 10 from a number decreases the digit in the tens place by 1.
 100 more or 100 less
 Adding 100 to a number increases the digit in the hundreds place by 1.
 Subtracting 100 from a number decreases the digit in the hundreds place by 1.
Note(s):
 Grade Level(s):
 Grade 1 used relationships to determine the number that is 10 more and 10 less than a given number up to 120.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Developing proficiency in the use of place value within the base10 numeration system
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation
 X. Connections

2.7C 
Represent and solve addition and subtraction word problems where unknowns may be any one of the terms in the problem.

Represent, Solve
ADDITION AND SUBTRACTION WORD PROBLEMS WHERE UNKNOWNS MAY BE ANY ONE OF THE TERMS IN THE PROBLEM
Including, but not limited to:
 Whole numbers
 Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
 Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
 Addition
 Sum – the total when two or more addends are joined
 Addend – a number being added or joined together with another number(s)
 Addition of whole numbers within 1,000
 With or without regrouping
 Subtraction
 Difference – the remaining amount after the subtrahend has been subtracted from the minuend
 Minuend – a number from which another number will be subtracted
 Subtrahend – a number to be subtracted from a minuend
 Subtraction of whole numbers within 1,000
 With or without regrouping
 Term – a number and/or an unknown in an expression separated by an operation symbol(s)
 Expression – a mathematical phrase, with no equal sign or comparison symbol, that may contain a number(s), an unknown(s), and/or an operator(s)
 Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
 Number sentences, or equations, with an equal sign at the beginning or end
 Represent mathematical and realworld problem situations
 Concrete models
 Objects represent the quantities described in the problem situation.
 Base10 blocks, place value disks, etc.
 Pictorial models
 Pictures drawn represent the quantities described in the problem situation.
 Base10 pictorials, number lines, strip diagrams, etc.
 Numbers
 Numbers represent the quantities described in the problem situation.
 Oral and written descriptions
 Explanation of relationship between objects, pictorials, and numbers and the information in the problem situation
 Solve mathematical and realworld problem situations with the result unknown.
 Onestep problems
 Connection between information in the problem and problem type
 Joining action result unknown
 Partpartwhole whole unknown
 Additive comparison compare quantity (larger quantity) unknown
 Separating action result unknown
 Partpartwhole part unknown
 Additive comparison difference unknown
 Additive comparison referent (smaller quantity) unknown
 Solve mathematical and realworld problem situations with the change unknown.
 Onestep problems
 Connection between information in the problem and problem type
 Connection between solution strategies for similar problem types
 Joining action change unknown
 a + __ = c
 Can be solved as c – a = __
 Partpartwhole part unknown
 a + __ = c
 Can be solved as c – a = __
 Additive comparison difference unknown
 a + __ = c
 Can be solved as c – a = __
 Separating action change unknown
 a – __ = c
 Can be solved as a – c = __
 Solve mathematical and realworld problem situations with the start unknown.
 Onestep problems
 Connection between information in the problem and problem type
 Connection between solution strategies for similar problem types
 Joining action start unknown
 __ + b = c
 Can be solved as c – b = __
 Separating action start unknown
 __ – b = c
 Can be solved as c + b = __
 Solve mathematical and realworld problem situations with multiple operations.
 Multistep problem situations
Note(s):
 Grade Level(s):
 Grade 1 determined the unknown whole number in an addition or subtraction equation when the unknown may be any one of the three or four terms in the equation.
 Grade 3 will represent one and twostep problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations.
 Various mathematical process standards will be applied to this student expectation as appropriate.
 TxRCFP:
 Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000
 TxCCRS:
 I. Numeric Reasoning
 II.D. Algebraic Reasoning – Representations
 VIII. Problem Solving and Reasoning
 IX. Communication and Representation
 X. Connections
