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 TITLE : Unit 06: Addition and Subtraction with Algorithms SUGGESTED DURATION : 15 days

#### Unit Overview

Introduction
This unit bundles student expectations that address generating, representing and solving addition and subtraction problem situations using algorithms, including the standard algorithm, and recalling basic facts. Problems include situations related to calculating savings over time. According to the Texas Education Agency, mathematical process standards including application, a problem-solving model, tools and techniques, communication, representations, relationships, and justifications should be integrated (when applicable) with content knowledge and skills so that students are prepared to use mathematics in everyday life, society, and the workplace.

Prior to this Unit
In Unit 03, students recalled basic facts and applied strategies and properties of operations based on place value to add up to four two-digit numbers or subtract two-digit numbers without algorithms. Students explored flexible methods and models to solve and represent addition and subtraction situations within 1,000, including mental math, concrete models, pictorial representations, number sentences, and open number lines.

During this Unit
Students apply strategies based on place value and properties of operations to add up to four two-digit numbers or subtract two-digit numbers within 1,000 in real-world problem situations. Problem situations include distinguishing between deposits and withdrawals and calculating how money saved accumulates over time with deposits. Addition and subtraction situations, where the unknown may be any one of the terms in the problem, should include numbers that require regrouping to solve the problem. Students make connections between representing and solving addition and subtractions problems using flexible methods, concrete and pictorial models, and number sentences to mental strategies and algorithms based on knowledge of place value and properties of operations. Students generate and solve problem situations for a given number sentence involving addition and subtraction of whole numbers within 1,000. Students continue to apply basic addition and subtraction fact strategies to solve problems while working toward automatic recall and fact fluency, thus increasing efficiency with mental and algorithmic problem solving.

Other considerations: Reference the Mathematics COVID-19 Gap Implementation Tool Grade 2

After this Unit
In Grade 3, students will solve one- and two-step addition and subtraction problems fluently using a variety of strategies based on place value and properties of operations. Students will also use the estimation strategies of rounding and compatible numbers to justify the reasonableness of their solutions.

In Grade 2, generating, representing, and solving addition and subtraction situations, and recalling basic facts are subsumed within 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. Calculating savings over time and distinguishing between a deposit and a withdrawal are identified within the Grade 2 Texas Response to Curriculum Focal Points (TxRCFP):  Financial Literacy. This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning 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
In the development of competent math thinkers, research supports the conclusion that fluency with algorithms creates an efficient method for accurately solving math problems. In the book Adding it Up: Helping Children Learn Mathematics (2001), the authors state,

More simply, an algorithm is a recipe for computation. The more transparent an algorithm, the easier it is to understand, and a child who understands an algorithm can reconstruct it after months or even years of not using it…Algorithms are important in school mathematics because they can help students understand better the fundamental operations of arithmetic and important concepts such as place value, and they pave the way for learning more advanced topics. (p. 103)

However, caution is advised when teaching algorithms. NCTM (2010) warns us that “algorithms tend to deemphasize the place value of each digit. When children look at separate digits in a number, rather than looking at the number as a whole, they are less likely to be able to determine the reasonableness of their answer. The teacher can help students write down the process of adding the two numbers in ways that follow their thinking” (page 58). Algorithms are precise steps designed to lead to an answer, but we are advised when teaching the process to not overlook the reasoning behind the rules or students will not be able to explain why the process works.

National Council of Teachers of Mathematics. (2010). Developing essential understanding of number and numeration pre-k – grade 2. Reston, VA: National Council of Teachers of Mathematics, Inc.
National Research Council. (2001). Adding it up: Helping children learn mathematics. Kilpatrick, J., Swafford, J., and Findell, B. (Eds.) Mathematics Learning Study Committee, Center for Education Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.
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

 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? Financial and economic knowledge leads to informed and rational decisions allowing for effective management of financial resources when planning for a lifetime of financial security. Why is financial stability important in everyday life? What economic and financial knowledge is critical for planning for a lifetime of financial security? How can mapping one’s financial future lead to significant short and long-term benefits? How can current financial and economic factors in everyday life impact daily decisions and future opportunities?
Unit Understandings
and Questions
Overarching Concepts
and Unit Concepts
Performance Assessment(s)
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (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 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?
• Why is it important to understand when and how to use standard algorithms?
• What relationships exist between …
• basic facts and addition or subtraction of multi-digit numbers?
• place value and addition or subtraction of multi-digit numbers?
• 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?
• Understanding saving, deposits, and withdrawals aids in making informed financial management decisions, which promotes a more secured financial future.
• What are some examples of saving money?
• What is the benefit of saving money over time?
• How does money saved grow to a larger amount over time?
• What is the difference between a deposit and a withdrawal?
• Number and Operations
• Base-10 Place Value System
• Number
• Counting (natural) numbers
• Whole numbers
• Operations
• Subtraction
• Problem Types
• Properties of Operations
• Relationships and Generalizations
• Operational
• Equivalence
• Solution Strategies and Algorithms
• Personal Financial Literacy
• Deposits and Withdrawals
• Saving
• 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? 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 and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (addition and subtraction of whole numbers within 1,000).
• How does the operation(s) in a number sentence determine the context of a problem situation that can be represented by the number sentence?
• 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?
• What strategies can be used to determine …
• the sum
• the difference
• any unknown
… in an addition or subtraction situation?
• What relationships exist between …
• basic facts and addition or subtraction of multi-digit numbers?
• place value and addition or subtraction of multi-digit numbers?
• 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 and Operations
• Number
• Counting (natural) numbers
• Whole numbers
• Operations
• Subtraction
• Problem Types
• Relationships and Generalizations
• Operational
• Equivalence
• Solution Strategies and Algorithms
• Algebraic Reasoning
• Equivalence
• Representations
• 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.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Misconceptions:

• 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, when using the standard addition algorithm, they can record the sum of the digits in each column regardless of the amount rather than recognizing that amounts of 10 or more require regrouping since only a single digit is recorded in each column.
• Some students may think, when using the standard subtraction algorithm, they can subtract the smaller digit from the larger digit regardless of the location rather than recognizing that if the digit on the top (minuend) is larger than the digit on the bottom (subtrahend) regrouping will be necessary prior to subtracting.

Underdeveloped Concepts:

• Some students may not recognize the difference between an addition situation and a subtraction situation based on the context of the problem.
• Some students may recognize the inverse relationship of addition and subtraction in basic fact families but do not apply the same understanding to multi-digit numbers (e.g., if 7 + 3 = 10 is the inverse of 10 – 3 = 7, then 17 + 13 = 30 is the inverse of 30 – 13 = 17).
• Some students may interpret the equal sign to mean that an operation must be performed on the numbers on one side and the result of this operation is recorded on the other side of the equal sign rather than understanding the equal sign as representing equivalent values (e.g., 10 + 8 = 13 + 5).

#### Unit Vocabulary

• Addend – a number being added or joined together with another number(s)
• Automaticity – executing a basic fact with speed and accuracy with little or no conscious effort
• Balance – the amount of money that is in a bank account after a deposit or withdrawal
• 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
• Deposit – money put into an account
• Difference – the remaining amount after the subtrahend has been subtracted from the minuend
• Digit – any numeral from 0 – 9
• 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
• Saving – setting aside money earned or received for future use
• 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}
• Withdrawal – money taken out of an account

Related Vocabulary:

 Addition Algorithm Bank account (checking/savings account) Calculate Change unknown Comparison problem Decrease Increase Inverse operations Mental strategies Operator/operation Partial sums/differences Part-part-whole Properties of operations Regrouping Result unknown Solution Start unknown Strategy Subtraction Transaction
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:
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.1. Interpret results of the mathematical problem in terms of the original real-world situation.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
• IX.B.2. Understand and use appropriate mathematical models in the natural, physical, and social sciences.
• IX.B.3. Know and understand the use of mathematics in a variety of careers and professions.
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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.A. Statistical Reasoning – Design a study
• V.A.1. Formulate a statistical question, plan an investigation, and collect data.
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VII.A.2. Formulate a plan or strategy.
• VII.A.3. Determine a solution.
• VII.A.4. Justify the solution.
• VII.A.5. Evaluate the problem-solving process.
• VII.D. Problem Solving and Reasoning – Real-world problem solving
• VII.D.2. Evaluate the problem-solving process.
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, ESTIMATION, AND NUMBER SENSE AS APPROPRIATE, TO SOLVE PROBLEMS

Including, but not limited to:

• Appropriate selection of tool(s) and techniques to apply in order to solve problems
• Tools
• Real objects
• Manipulatives
• Paper and pencil
• Technology
• Techniques
• Mental math
• Estimation
• Number sense

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing 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:
• I.B. Numeric Reasoning – Number sense and number concepts
• I.B.1. Use estimation to check for errors and reasonableness of solutions.
• V.C. Statistical Reasoning – Analyze, interpret, and draw conclusions from data
• V.C.2. Analyze relationships between paired data using spreadsheets, graphing calculators, or statistical software.
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:
• II.D. Algebraic Reasoning – Representing relationships
• II.D.1. Interpret multiple representations of equations, inequalities, and relationships.
• II.D.2. Convert among multiple representations of equations, inequalities, and relationships.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
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:
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
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:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.1. Analyze given information.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.A.2. Use mathematical language to represent and communicate the mathematical concepts in a problem.
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
•  VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
• VIII.C.2. Create and use representations to organize, record, and communicate mathematical ideas.
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
• IX.A. Connections – Connections among the strands of mathematics
• IX.A.1. Connect and use multiple key concepts of mathematics in situations and problems.
• IX.A.2. Connect mathematics to the study of other disciplines.
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:
• VII.A. Problem Solving and Reasoning – Mathematical problem solving
• VII.A.4. Justify the solution.
• VII.B. Problem Solving and Reasoning – Proportional reasoning
• VII.B.1. Use proportional reasoning to solve problems that require fractions, ratios, percentages, decimals, and proportions in a variety of contexts using multiple representations.
• VII.C. Problem Solving and Reasoning – Logical reasoning
• VII.C.1. Develop and evaluate convincing arguments.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.3. Use mathematical language for reasoning, problem solving, making connections, and generalizing.
• VIII.B. Communication and Representation – Interpretation of mathematical work
• VIII.B.1. Model and interpret mathematical ideas and concepts using multiple representations.
• VIII.B.2. Summarize and interpret mathematical information provided orally, visually, or in written form within the given context.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.3. Explain, display, or justify mathematical ideas and arguments using precise mathematical language in written or oral communications.
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
• 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 doubles always results in an even sum, regardless of whether the addends are even or odd.
• Double plus/minus 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

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
• 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.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
2.4B Add up to four two-digit numbers and subtract two-digit numbers using mental strategies and algorithms based on knowledge of place value and properties of operations.

UP TO FOUR TWO-DIGIT NUMBERS USING MENTAL STRATEGIES AND ALGORITHMS BASED ON KNOWLEDGE OF PLACE VALUE AND PROPERTIES OF OPERATIONS

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)
• Sums of up to four two-digit whole numbers
• With and without regrouping
• Mental strategies based on place value
• Application of basic facts within each place value
• Compatible numbers
• Composition/decomposition of numbers to form friendly numbers (compatible numbers)
• Algorithms based on place value
• With and without regrouping
• Partial sums
• Addition of numbers in expanded form
• Partial sums recorded vertically
• Standard algorithm
• Properties of operations
• Addends may be added in any order to produce the same sum.
• Addends may be decomposed and grouped in any order to produce the same sum.
• Relationships between addition using mental strategies, algorithms, and properties of operations to addition using concrete models
• Relationships between addition using mental strategies, algorithms, and properties of operations to addition using open number lines

Subtract

TWO-DIGIT NUMBERS USING MENTAL STRATEGIES AND ALGORITHMS BASED ON KNOWLEDGE OF PLACE VALUE AND PROPERTIES OF OPERATIONS

Including, but not limited to:

• Whole numbers (0 – 1,000)
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}
• Subtraction
• 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
• Difference of two-digit whole numbers
• With and without regrouping
• Mental strategies based on place value
• Application of basic facts within each place value
• Composition/decomposition of numbers to form friendly numbers (compatible numbers)
• Algorithms based on place value
• With and without regrouping
• Partial differences
• Subtraction of numbers in expanded form
• Standard algorithm
• Properties of operations
• Minuend and/or subtrahend may be decomposed to produce friendly numbers.
• Relationships between subtraction using mental strategies, algorithms, and properties of operations to subtraction using concrete models
• Relationships between subtraction using mental strategies, algorithms, and properties of operations to subtraction using open number lines

Note(s):

• Grade 1 explained strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentences.
• Grade 2 will solve one-step and multi-step word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms.
• Grade 3 will solve with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction.
• 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.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
2.4C Solve one-step and multi-step word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms.

Solve

ONE-STEP AND MULTI-STEP WORD PROBLEMS INVOLVING ADDITION AND SUBTRACTION WITHIN 1,000 USING A VARIETY OF STRATEGIES BASED ON PLACE VALUE, INCLUDING ALGORITHMS

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}
• Mathematical and real-world problem situations
• One-step and multi-step problems
• 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
• Sums of up to four two-digit whole numbers
• Sums of two three-digit whole numbers
• 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
• Differences of two- or three-digit whole numbers
• With or without regrouping
• Strategies based on place value and properties of operations in mathematical and real-world problem situations
• With or without concrete models
• With or without pictorial models or open number lines
• One-step and multi-step problems
• Algorithms based on place value in mathematical and real-world problem situations
• Partial sums
• Addition of numbers in expanded form
• Partial sums recorded vertically
• Partial differences
• Subtraction of numbers in expanded form
• Standard subtraction algorithm
• One-step and multi-step problems

Note(s):

• Grade 1 explained strategies used to solve addition and subtraction problems up to 20 using spoken words, objects, pictorial models, and number sentences.
• Grade 2 will solve one-step and multi-step word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms.
• Grade 3 will solve with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction.
• 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.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
2.4D Generate and solve problem situations for a given mathematical number sentence involving addition and subtraction of whole numbers within 1,000.

Generate, Solve

PROBLEM SITUATIONS FOR A GIVEN MATHEMATICAL NUMBER SENTENCE INVOLVING ADDITION AND SUBTRACTION OF WHOLE NUMBERS WITHIN 1,000

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
• Sums of up to four two-digit whole numbers
• Sums of two three-digit whole numbers
• 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
• Differences of two- or three-digit whole numbers
• With or without regrouping
• 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
• Unknown in any position
• Generate and solve mathematical and real-world problem situations when given an addition number sentence.
• One-step problems
• Appropriate mathematical language
• Connection between information in the problem and problem type
• Joining action result unknown
• Joining action change unknown
• Joining action start unknown
• Part-part-whole whole unknown
• Part-part-whole part unknown
• Additive comparison compare quantity (larger quantity) unknown
• Additive comparison referent (smaller quantity) unknown
• Generate and solve problem mathematical and real-world situations when given a subtraction number sentence
• One-step problems
• Appropriate mathematical language
• Connection between information in the problem and problem type
• Subtraction situations
• Separating action result unknown
• Separating action change unknown
• Separating action start unknown
• Part-part-whole part unknown
• Additive comparison referent (smaller quantity) unknown
• Generate and solve problem mathematical and real-world situations when given a multi-operation number sentence
• Multi-step problems
• Appropriate mathematical language

Note(s):

• Grade 1 generated and solved problem situations when given a number sentence involving addition or subtraction of numbers within 20.
• Grade 2 will solve one-step and multi-step word problems involving addition and subtraction within 1,000 using a variety of strategies based on place value, including algorithms.
• Grade 3 will solve with fluency one-step and two-step problems involving addition and subtraction within 1,000 using strategies based on place value, properties of operations, and the relationship between addition and subtraction.
• 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.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
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:
2.7C 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 = __
• 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 = __
• 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.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• VIII.A. Communication and Representation – Language, terms, and symbols of mathematics
• VIII.A.1. Use mathematical symbols, terminology, and notation to represent given and unknown information in a problem.
• VIII.C. Communication and Representation – Presentation and representation of mathematical work
• VIII.C.1. Communicate mathematical ideas, reasoning, and their implications using symbols, diagrams, models, graphs, and words.
2.11 Personal financial literacy. The student applies mathematical process standards to manage one's financial resources effectively for lifetime financial security. The student is expected to:
2.11A Calculate how money saved can accumulate into a larger amount over time.

Calculate

HOW MONEY SAVED CAN ACCUMULATE INTO A LARGER AMOUNT OVER TIME

Including, but not limited to:

• Saving – setting aside money earned or received for future use
• Saving can result in an increase of money over time.
• Money may be saved in a bank account, piggy bank, etc.
• Calculate money saved over time.
• Relationship between saving money and addition
• Saving money is equivalent to adding money to a bank account, piggy bank, etc.
• Adding money to a bank account or piggy bank will result in a larger amount of money.

Note(s):

• Grade 1 distinguished between spending and saving.
• Grade 3 will list reasons to save and explain the benefit of a savings plan, including for college.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Financial Literacy
• TxCCRS:
• I.A. Numeric Reasoning – Number representations and operations
• I.A.2. Perform computations with rational and irrational numbers.
• IX.B. Connections – Connections of mathematics to nature, real-world situations, and everyday life
• IX.B.1. Use multiple representations to demonstrate links between mathematical and real-world situations.
2.11C Distinguish between a deposit and a withdrawal.

Distinguish

BETWEEN A DEPOSIT AND A WITHDRAWAL

Including, but not limited to:

• Money may be stored in a bank account.
• Checking account usually used for frequent transactions
• Savings account usually used for less frequent transactions or for earning interest
• Terminology for bank transactions
• Deposit – money put into an account
• Withdrawal – money taken out of an account
• Subtract from previous balance
• Balance – the amount of money that is in a bank account after a deposit or withdrawal
• New total after adding or subtracting
• Distinguish between a deposit and a withdrawal in mathematical and real-world problem situations.

Note(s):