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

2.2 
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 related to place value. The student is expected to:


2.2A 
Use concrete and pictorial models to compose and decompose numbers up to 1,200 in more than one way as a sum of so many thousands, hundreds, tens, and ones.

Use
CONCRETE AND PICTORIAL MODELS OF NUMBERS 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}
 Numeral – a symbol used to name a number
 Digit – any numeral from 0 – 9
 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
 Base10 place value system
 A number system using ten digits 0 – 9
 Relationships between places are based on multiples of 10.
 The magnitude (relative size) of one thousand
 Concrete models
 Proportional models – a visual representation that demonstrates the relative size of each place value using models with proportional dimensions, meaning the model of each place value is exactly 10 times larger than the place value model to the right (e.g., the base10 long is exactly 10 times as big as the unit showing that one 10 is equal to ten ones)
 Bundled sticks (proportional representation of the magnitude of a number with 1to10 relationship)
 Base10 blocks (proportional representation of the magnitude of a number with 1to10 relationship)
 Nonproportional models – a visual representation that does not maintain the proportional relationship of size, meaning the size of each place value model is not 10 times larger than the place value model to the right (e.g., the value of each place value disk is indicated by the numerical label and color but does not change in size)
 Place value disks (nonproportional representation with a 1to10 relationship)
 Pictorial models
 Base10 block representations
 Place value disk representations
 Open number line – an empty number line where tick marks are added to represent landmarks of numbers, often indicated with arcs above the number line (referred to as jumps) demonstrating approximate proportional distances
To Compose, To Decompose
NUMBERS UP TO 1,200 IN MORE THAN ONE WAY AS A SUM OF SO MANY THOUSANDS, HUNDREDS, TENS, AND ONES
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}
 Compose numbers – to combine parts or smaller values to form a number
 Decompose numbers – to break a number into parts or smaller values
 Decomposition of whole numbers in regards to place value does not involve carrying digits to the next placeholder, meaning each addend of the decomposition should only have one nonzero digit (e.g., 789 may be decomposed as 700 + 80 + 9 or 500 + 200 + 50 + 30 + 9 but not decomposed as 600 + 90 + 90 + 9 or 600 + 180 + 9, etc.).
 Compose a number in more than one way using concrete and pictorial models.
 As a sum of so many thousands, hundreds, tens, and ones
 Decompose a number in more than one way using concrete and pictorial models.
 As a sum of so many thousands, hundreds, tens, and ones
Note(s):
 Grade Level(s):
 Grade 1 used concrete and pictorial models to compose and decompose numbers up to 120 and used objects, pictures and expanded and standard forms to represent numbers.
 Grade 3 will compose and decompose numbers up to 100,000 using objects, pictorial models, and numbers, including expanded notation as appropriate.
 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:
 I. Numeric Reasoning
 IX. Communication and Representation

2.2B 
Use standard, word, and expanded forms to represent numbers up to 1,200.

Use
STANDARD, WORD, AND EXPANDED FORMS TO REPRESENT NUMBERS 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.
 Standard form – the representation of a number using digits (e.g., 1,200)
 Period – a threedigit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
 Thousands period is composed of the one thousands place, ten thousands place, and hundred thousands place.
 Units period is composed of the ones place, tens place, and hundreds place.
 The word “thousand” after the numerical value of the thousands period is stated when read.
 A comma between the thousands period and the units period is recorded when written but not stated when read.
 The word “unit” after the numerical value of the units period is not stated when read.
 The word “hundred” in each period is stated when read.
 The words “ten” and “one” in each period are not stated when read.
 The tens place digit and ones place digit in each period are stated as a twodigit number when read.
 Zeros are used as place holders between digits as needed to maintain the value of each digit (e.g., 1,075).
 Leading zeros in a whole number are not commonly used in standard form, but are not incorrect and do not change the value of the number (e.g., 037 equals 37).
 Word form – the representation of a number using written words (e.g., 1,152 as one thousand, one hundred fiftytwo)
 The word “thousand” after the numerical value of the thousands period is stated when read and recorded when written.
 A comma between the thousands period and the units period is not stated when read but is recorded when written.
 The word “unit” after the numerical value of the units period is not stated when read and not recorded when written.
 The word “hundred” in each period is stated when read and recorded when written.
 The words “ten” and “one” in each period are not stated when read and not recorded when written.
 The tens place digit and ones place digit in each period are stated as a twodigit number when read and recorded using a hyphen, where appropriate, when written (e.g., twentythree, thirteen, etc.).
 The zeros in a whole number are not stated when read and are not recorded when written (e.g., 1,005 in standard form is read and written as one thousand, five in word form).
 Expanded form – the representation of a number as a sum of place values (e.g., 1,189 as 1,000 + 100 + 80 + 9)
 Zero may or may not be written as an addend to represent the digit 0 in a number (e.g., 1,075 as 1,000 + 0 + 70 + 5 or 1,000 + 70 + 5).
 Expanded form is written following the order of place value.
 The sum of place values written in random order is an expression but not expanded form.
 Multiple representations
Note(s):
 Grade Level(s):
 Grade 1 used objects, pictures, and expanded and standard forms to represent numbers up to 120.
 Grade 3 will compose and decompose numbers up to 100,000 using objects, pictorial models, and numbers, including expanded notation as appropriate.
 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:
 I. Numeric Reasoning
 IX. Communication and Representation

2.2C 
Generate a number that is greater than or less than a given whole number up to 1,200.

Generate
A NUMBER THAT IS GREATER THAN OR LESS THAN A GIVEN WHOLE 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}
 Comparative language
 Greater than, more than
 Less than, fewer than
 Place value relationships
 1 more or 1 less
 Adding 1 to the digit in the ones place will generate a number that is 1 more than the original number.
 Subtracting 1 from the digit in the ones place will generate a number that is 1 less than the original number.
 10 more or 10 less
 Adding 1 to the digit in the tens place will generate a number that is 10 more than the original number.
 Subtracting 1 from the digit in the tens place will generate a number that is 10 less than the original number.
 100 more or 100 less
 Adding 1 to the digit from the digit in the hundreds place will generate a number that is 100 more than the original number.
 Subtracting 1 from the digit in the hundreds place will generate a number that is 100 less than the original number.
 Numerical relationships
 Counting order
 Skip counting
 Doubles
 Concrete and pictorial models
 Hundreds chart
 Moving one place to the right will generate a number that is 1 more than the original number.
 Moving one place to the left will generate a number that is 1 less than the original number.
 Moving one row down will generate a number that is 10 more than the original number.
 Moving one row up will generate a number that is 10 less than the original number.
 Base10 blocks
 Adding unit cubes will increase a number by increments of 1.
 Removing unit cubes will decrease a number by increments of 1.
 Adding longs will increase a number by increments of 10.
 Removing longs will decrease a number by increments of 10.
 Adding flats will increase a number by increments of 100.
 Removing flats will decrease a number by increments of 100.
 Number line
 Numbers increase from left to right.
 Numbers decrease from right to left.
Note(s):
 Grade Level(s):
 Grade 1 generated a number that is greater than or less than a given whole number up to 120.
 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
 TxCCRS:
 I. Numeric Reasoning
 IX. Communication and Representation

2.2D 
Use place value to compare and order whole numbers up to 1,200 using comparative language, numbers, and symbols (>, <, or =).

Use
PLACE VALUE FOR WHOLE NUMBERS 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
 Period – a threedigit grouping of whole numbers where each grouping is composed of a ones place, a tens place, and a hundreds place, and each grouping is separated by a comma
 Thousands period is composed of the one thousands place, ten thousands place, and hundred thousands place.
 Units period is composed of the ones place, tens place, and hundreds place.
To Compare, To Order
WHOLE NUMBERS UP TO 1,200 USING COMPARATIVE LANGUAGE, NUMBERS, AND SYMBOLS (>, <, OR =)
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.
 Comparative language and comparison symbols
 Inequality words and symbols
 Greater than (>)
 Less than (<)
 Equality words and symbol
 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
 Relative magnitude of a number describes the size of a number and its relationship to another number.
 Compare two numbers using place value charts.
 Compare digits in the same place value position beginning with the greatest place value.
 If these digits are the same, continue to the next smallest place until the digits are different.
 Numbers that have common digits but are not equal in value (different place values)
 Numbers that have a different number of digits
 Compare two numbers using a number line.
 Number lines (horizontal/vertical)
 Proportional number lines (predetermined intervals with at least two labeled numbers)
 Open number lines (no marked intervals)
 Order numbers – to arrange a set of numbers based on their numerical value
 A set of numbers can be compared in pairs in the process of ordering numbers.
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Order a set of numbers on a number line.
 Order a set of numbers on an open number line.
 Quantifying descriptors (e.g., between two given numbers, greatest/least, ascending/descending, tallest/shortest, warmest/coldest, fastest/slowest, longest/shortest, heaviest/lightest, closest/farthest, oldest/youngest, etc.)
Note(s):
 Grade Level(s):
 Grade 1 used place value to compare numbers up to 120 and represented the comparison of two numbers to 100 using the symbols >, <, or =.
 Grade 3 will use compare and order whole numbers up to 100,000 and represent the comparisons using the symbols >, <, or =.
 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:
 I. Numeric Reasoning
 IX. Communication and Representation

2.2E 
Locate the position of a given whole number on an open number line.

Locate
THE POSITION OF A GIVEN WHOLE NUMBER ON AN OPEN NUMBER LINE
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}
 Characteristics of an open number line
 An open number line begins as a line with no intervals (or tick marks) and no positions/numbers labeled.
 Numbers/positions are placed on the empty number line only as they are needed.
 When reasoning on an open number line, the position of zero is often not placed.
 When working with larger numbers, an open number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 The placement of the first two numbers on an open number line determines the scale of the number line.
 Once the scale of the number line has been established by the placement of the first two numbers, intervals between additional numbers placed are approximately proportional.
 The differences between numbers are approximated by the distance between the positions on the number line.
 Open number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Landmark (or anchor) numbers may be placed on the open number line to help locate other numbers.
 Purpose of open number line
 Open number lines allow for the consideration of the magnitude of numbers and the placevalue relationships among numbers when locating a given whole number
 Open number line given
Note(s):
 Grade Level(s):
 Grade 1 ordered whole numbers up to 120 using place value and open number lines.
 Grade 3 will represent a number on a number line as being between two consecutive multiples of 10; 100; 1,000; or 10,000 and use words to describe relative size of numbers in order to round whole numbers.
 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:
 I. Numeric Reasoning
 IX. Communication and Representation

2.2F 
Name the whole number that corresponds to a specific point on a number line.

Name
THE WHOLE NUMBER THAT CORRESPONDS TO A SPECIFIC POINT ON A NUMBER LINE
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}
 Characteristics of a number line
 A number line begins as a line with predetermined intervals (or tick marks) with positions/numbers labeled.
 A minimum of two positions/numbers should be labeled.
 Numbers on a number line represent the distance from zero.
 The distance between the tick marks is counted rather than the tick marks themselves.
 The placement of the labeled positions/numbers on a number line determines the scale of the number line.
 Intervals between position/numbers are proportional.
 When reasoning on a number line, the position of zero may or may not be placed.
 When working with larger numbers, a number line without the constraint of distance from zero allows the ability to “zoomin” on the relevant section of the number line.
 Number lines extend infinitely in both directions (arrows indicate the number line continues infinitely).
 Numbers increase from left to right on a horizontal number line and from bottom to top on a vertical number line.
 Points to the left of a specified point on a horizontal number line are less than points to the right.
 Points to the right of a specified point on a horizontal number line are greater than points to the left.
 Points below a specified point on a vertical number line are less than points above.
 Points above a specified point on a vertical number line are greater than points below.
 Number lines can be horizontal, vertical, or circular.
 Intervals and partial labels given
 Partial intervals and labels given
Note(s):
 Grade Level(s):
 Grade 2 introduces naming the whole number that corresponds to a specific point on a number line.
 Grade 3 will determine the corresponding fraction greater than zero and less than or equal to one with denominators of 2, 3, 4, 6, and 8 given a specified point on a number line.
 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:
 I. Numeric Reasoning
 IX. Communication and Representation
