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 Instructional Focus DocumentGrade 2 Mathematics
 TITLE : Unit 02: Number Relationships SUGGESTED DURATION : 10 days

Unit Overview

Introduction
This unit bundles student expectations that address patterns and relationships in numbers, including basic addition and subtraction facts, fact families, odd and even numbers, place value, and the value of coins in a collection. 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 Grade 1, students used concrete models to explore basic addition and subtraction fact strategies and fact families. Students explored patterns within numbers as they generated numbers greater than or less than a given number and experienced skip counting by 2, 5, and 10. Students used these skip counting relationships to determine the value of a collection of coins, and recorded the value of the collection using the cent symbol notation. Students also explored determining an unknown in any position of an equation.

During this Unit
Students focus on developing mathematical strategies based on patterns and number sense to strengthen their understanding of number relationships and fluency with computations. Students explore number relationships in strategies based on place value and properties of operations in order to develop automaticity in the recall of basic addition and subtraction facts, meaning executing the fact with speed and accuracy with little or no conscious effort. Students use fact family relationships to solve problems with an unknown in any position, such as start unknown, change unknown, and result unknown problems. Students also mentally calculate sums and differences for numbers using place value as they explore “10/100 more/less” relationships. Students discover patterns in odd and even numbers through the pairing of objects and determining if the number can be paired without leftovers. Students use skip counting patterns and relationships between the values of coins to determine the value of a collection of like or mixed coins up to one dollar. Students extend their representation of the value of coins to include either the cent symbol notation or the dollar sign and decimal point notation.

After this Unit
In Unit 03, students will extend and apply the patterns and relationships in numbers and operations to solve a variety of addition and subtraction problem situations, including situations involving the value of a collection of coins and bills.

In Grade 2, recalling basic addition and subtraction facts, using fact families to solve problems, and place value relationships within numbers are foundational building blocks to the conceptual understanding of the Grade 2 Texas Response to Curriculum Focal Points (TxRCFP): Using place value and properties of operations to solve problems involving addition and subtraction of whole numbers within 1,000. Place value relationships within numbers is also included within the Grade 2 Texas Response to Curriculum Focal Points (TxRCFP): Developing proficiency in the use of place value within the base-10 numeration system along with the standards that address determining and representing the value of a collection of coins. Determining if a number is odd or even is identified within the Grade 2 Texas Response to Curriculum Focal Points (TxRCFP): Grade level connections. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning A1, A2, B1; 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 many researchers, patterns and relationships are found in numerous mathematical and numerical concepts including the value of numbers, operations, and money. In discussing the concept of patterns, Heddens and Speer (2006) state,

The study of patterns is central to all mathematical learning. If students have developed an appreciation for patterns and can recognize them in different contexts, then transfer of learning will proceed more smoothly. Students who understand the importance of patterns begin to look for patterns in places that others might not look. (p. 109)

Understanding the patterns in skip counting and the relationships within and between values of 5, 10, and 25 are essential to students’ success in determining the value of a collection of coins. Van de Walle (2006) states,

For [coin] values to make sense, students must have an understanding of 5, 10, and 25. More than that, they need to be able to think of these quantities without seeing countable objects… A child whose number concepts remain tied to counts of objects [one object is one count] is not going to be able to understand the value of coins. (p. 150)

To support students in developing essential understandings and skills needed for future success in mathematics, the foundational skills related to basic facts and patterns are critical. The National Mathematics Advisory Panel (2008) advises,

Research has demonstrated that declarative knowledge (e.g., memory for addition facts), procedural knowledge (or skills), and conceptual knowledge are mutually reinforcing, as opposed to being pedagogical alternatives… to obtain the maximal benefits of automaticity in support of complex problem solving, arithmetic facts and fundamental algorithms should be thoroughly mastered, and indeed, over-learned, rather than merely learned to a moderate degree of proficiency. (p. 34)

Heddens, J. W. & Speer, W. R. (2006). Today’s mathematics, concepts, classroom methods, and instructional activities. Hoboken, NJ: Wiley Jossey-Bass Education
National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the national mathematics advisory panel. Washington, DC: U.S. Department of Education.
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
Van de Walle, J., & Lovin, L. (2006). Teaching student-centered mathematics grades k – 3. Boston, MA: Pearson Education, Inc.

 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)
• Recognizing the distinct attributes, values, and relationships between the values of U.S. coins leads to efficient and accurate determination of the value of a collection of money (collection of coins up to \$1.00).
• Why is it important to be able to identify US coins?
• What are the distinct attributes of the …
• penny?
• nickel?
• dime?
• quarter?
• half-dollar?
• What is the value of the …
• penny?
• nickel?
• dime?
• quarter?
• half-dollar?
• How can the value of a coin or a collection of coins be represented using words, numbers, and/or symbols?
• What does the decimal point represent when used with a dollar sign?
• What relationships exist between the values of coins?
• What numerical patterns and counting strategies could be used to efficiently determine the value of a collection of coins?
• How is skip counting related to determining the value of a collection of coins?
• How can different collections of coins equal the same amount?
• Recognizing and understanding numerical patterns and operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (odd and even whole numbers up to 40; place value patterns in whole numbers up to 1,200; addition and subtraction of whole numbers within 1,000).
• How does the context of a problem situation affect the representation, operation(s), and/or solution strategy that could be used to solve the problem?
• How can representing a problem situation using …
• concrete models
• pictorial models
• a number sentence(s)
... aid in problem solving?
• What patterns and relationships can be found within and between the words, concrete objects, pictorial models, and number sentences used to represent a problem situation?
• How does understanding …
• relationships within and between operations
• place value
• properties of operations
… aid in determining an efficient strategy or representation to investigate problem situations?
• What strategies can be used to determine …
• the sum
• the difference
• any unknown
… in an addition or subtraction situation?
• What relationships exist between addition and subtraction?
• What patterns and place value relationships exist between numbers that are …
• 10 more
• 100 more
• 10 less
• 100 less
… than a given number?
• What patterns exist within and between odd and even numbers?
• What strategies can be used to determine if a number is even or odd?
• When using addition to solve a problem situation, why can the order of the addends be changed?
• When using subtraction to solve a problem situation, why can the order of the minuend and subtrahend not be changed?
• Number
• Base-10 Place Value System
• Money
• Coin identification
• Value of a coin
• Value of a collection
• Symbolic notation
• Number
• Counting (natural) numbers
• Whole numbers
• Algebraic Reasoning
• Equivalence
• Patterns and Relationships
• Multiples
• Even and odd numbers
• Representations
• Concrete models
• Pictorial models
• Expressions
• Equations
• 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.

 Understanding and generalizing operational relationships leads to more sophisticated representations and solution strategies in order to investigate or solve problem situations in everyday life. What relationships exist within and between mathematical operations? How does generalizing operational relationships lead to developing more flexible, efficient representations and/or solution strategies? Why is understanding the problem solving process an essential part of learning and working mathematically?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Recognizing and understanding operational relationships in basic facts are essential for future work with more complex numbers and sophisticated solution strategies.
• What strategies and patterns can be used to solve basic …
• subtraction facts?
• Why is it important to be able to compose and decompose a number in more than one way?
• What relationships exist between …
• counting strategies and addition?
• counting strategies and subtraction?
• addition and subtraction?
• How does understanding …
• relationships within and between operations
• properties of operations
… aid in determining an efficient strategy or representation to investigate basic fact problems?
• Why is it important to able to recall addition and subtraction facts automatically?
• Number and Operations
• Number
• Counting (natural) numbers
• Whole numbers
• Operations
• Subtraction
• Solution Strategies
• 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.

MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• Some students may think you can use the dollar symbol, decimal, and cent symbol in the same representation because the labels “dollars” and “cents” are both stated when describing the value of coins and bills rather than either using the dollar symbol with a decimal or using the cent symbol.
• Some students may think they must add or subtract in the order that the numbers are presented in the problem rather than performing the operation based on the meaning and action(s) of the problem situation.
• Some students may think subtraction is commutative rather than recognizing the minuend as the total amount and the subtrahend as the amount being subtracted (e.g., 5 – 3 is not the same as 3 – 5, etc.).
• Some students may think the cent symbol is only used to record values less than one dollar and the dollar symbol is only used to record values greater than or equal to one dollar rather than realizing that equivalent coin values can be recorded using different symbolic representations (e.g., 75¢ or \$0.75).
• Some students may think the concept of even and odd numbers is related only to skip counting rather than the relationships in pairs and doubles.

Underdeveloped Concepts:

• Some students may correctly recall the sum of two basic fact addends but have difficulty applying the commutative property of addition to recall the sum when the addends are reversed (e.g., 4 + 5 and 5 + 4 seen as different facts).
• Some students may correctly determine related addition number sentences but have difficulty determining the subtraction number sentences within a fact family.
• Some students may recognize the traditional views of coins but not recognize new or commemorative views (e.g., state quarters, buffalo nickels, etc.).
• Some students may not realize why collections of coins that look different may have the same value.
• Students may not have grasped the inverse relationships of addition and subtraction, which may cause confusion when solving problems involving the start unknown or change unknown.
• Without fluency in grouping strategies, some students may not be able to take advantage of mental math strategies for successful fact retrieval, composing/decomposing numbers, number line representations, and compound counting with coins.

Unit Vocabulary

• Addend – a number being added or joined together with another number(s)
• 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
• Difference – the remaining amount after the subtrahend has been subtracted from the minuend
• Even number – a number represented by objects that when paired have zero left over
• 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)
• Fact families – related number sentences using the same set of numbers
• Minuend – a number from which another number will be subtracted
• Number sentence – a mathematical statement composed of numbers, and/or an unknown(s), and/or an operator(s), and an equality or inequality symbol
• Odd number – a number represented by objects that when paired have one left over
• Place value – the value of a digit as determined by its location in a number such as ones, tens, hundreds, one thousands, etc.
• Subtrahend – a number to be subtracted from a minuend
• Sum – the total when two or more addends are joined
• Term – a number and/or an unknown in an expression separated by an operation symbol(s)
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Addition Cent symbol (¢) Change unknown Decimal point Dime Dollar Dollar sign (\$) Greater than, more than Half-dollar Hundreds place Less than, fewer than Nickel Number sequence Ones place One thousands place Pairing Part-part-whole Penny Quarter Result unknown Skip counting Start unknown Subtraction Tens place Value
Unit Assessment Items System Resources Other Resources

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Unit Assessment Items that have been published by your district may be accessed through Search All Components in the District Resources tab. Assessment items may also be found using the Assessment Center if your district has granted access to that tool.

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 Grade 2 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
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 base-10 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 two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• X. Connections
2.1B 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 proficiency in the use of place value within the base-10 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 two-dimensional shapes and three-dimensional 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
• Mental math
• Number sense

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 base-10 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 two-dimensional shapes and three-dimensional 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 base-10 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 two-dimensional shapes and three-dimensional 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 base-10 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 two-dimensional shapes and three-dimensional 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 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 proficiency in the use of place value within the base-10 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 two-dimensional shapes and three-dimensional solids, including exploration of early fraction concepts
• TxCCRS:
• X. Connections
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 base-10 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 two-dimensional shapes and three-dimensional 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
• 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
• 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.
• In-betweens
• 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 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¢
• Half-dollar: 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, real-life objects, etc.
• Coins in like groups (e.g., half-dollars, 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 twenty-fives to determine the value of a collection of quarters
• 25¢, 50¢, 75¢, \$1.00
• Skip counting by fifties to determine the value of a collection of half-dollars
• 50¢, \$1.00
• 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., half-dollars, 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 half-dollars by fifties, count on quarters by twenty-fives, 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, half-dollar
• 5 pennies = 1 nickel; 10 pennies = 1 dime; 25 pennies = 1 quarter; 50 pennies = 1 half-dollar
• 1 penny < 1 nickel; 1 penny < 1 dime; 1 penny < 1 quarter; 1 penny < 1 half-dollar
• Nickel to penny, dime, quarter, half-dollar
• 1 nickel = 5 pennies; 2 nickels = 1 dime; 5 nickels = 1 quarter; 10 nickels = 1 half-dollar
• 1 nickel > 1 penny; 1 nickel < 1 dime; 1 nickel < 1 quarter; 1 nickel < 1 half-dollar
• Dime to penny, nickel, quarter, half-dollar
• 1 dime = 10 pennies; 1 dime = 2 nickels; 5 dimes = 2 quarters; 5 dimes = 1 half-dollar
• 1 dime > 1 penny; 1 dime > 1 nickel; 1 dime < 1 quarter; 1 dime < 1 half-dollar
• Quarter to penny, nickel, dime, half-dollar
• 1 quarter = 25 pennies; 1 quarter = 5 nickels; 2 quarters = 5 dimes; 2 quarters = 1 half-dollar
• 1 quarter > 1 penny; 1 quarter > 1 nickel; 1 quarter > 1 dime; 1 quarter < 1 half-dollar
• Half-dollar to penny, nickel, dime, quarter
• 1 half-dollar = 50 pennies; 1 half-dollar = 10 nickels; 1 half-dollar = 5 dimes; 1 half-dollar = 2 quarters
• 1 half-dollar > 1 penny; 1 half-dollar > 1 nickel; 1 half-dollar > 1 dime; 1 half-dollar > 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 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 base-10 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 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 base-10 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 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 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 base-10 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}
• 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 real-world problem situations
• Concrete models
• Objects represent the quantities described in the problem situation.
• Base-10 blocks, place value disks, etc.
• Pictorial models
• Pictures drawn represent the quantities described in the problem situation.
• Base-10 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 real-world problem situations with the result unknown.
• One-step problems
• Connection between information in the problem and problem type
• Joining action result unknown
• a + b = __
• Part-part-whole whole unknown
• a + b = __
• Additive comparison compare quantity (larger quantity) unknown
• a + b = __
• Separating action result unknown
• ab = __
• Part-part-whole part unknown
• ab = __
• Additive comparison difference unknown
• ab = __
• Additive comparison referent (smaller quantity) unknown
• ab = __
• Solve mathematical and real-world problem situations with the change unknown.
• One-step 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 = __
• Part-part-whole part unknown
• a + __ = c
• Can be solved as c a = __
• Additive comparison difference unknown
• a + __ = c
• Can be solved as ca = __
• Separating action change unknown
• a – __ = c
• Can be solved as ac = __
• Solve mathematical and real-world problem situations with the start unknown.
• One-step 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 cb = __
• Separating action start unknown
• __ – b = c
• Can be solved as c + b = __
• Solve mathematical and real-world problem situations with multiple operations.
• Multi-step problem situations

Note(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 two-step 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 