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 Instructional Focus DocumentGrade 4 Mathematics
 TITLE : Unit 03: Multiplication of Whole Numbers SUGGESTED DURATION : 8 days

Unit Overview

Introduction
This unit bundles student expectations that require solving one-, two-, or multi-step problem situations, as well as estimating and solving multiplication problems using a variety of strategies, including the standard algorithm. 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 3, students represented one- and two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations. They represented and solved one- and two-step multiplication problems within 100 using arrays, strip diagrams, and equations. Students also determined the unknown whole number in a multiplication equation relating three whole numbers when the unknown is either a missing factor or product. Additionally, students represented multiplication facts by using a variety of approaches (e.g., repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting). They also recalled facts to multiply up to 10 by 10 with automaticity. Students used strategies (e.g., mental math, partial products, and the commutative, associative, and distributive properties) and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number.

During this Unit
Students determine products of a number by 10 or 100 using properties of operations (commutative and distributive properties) and place value understandings. Multiple representations of products of 2 two-digit numbers are used (e.g., arrays, area models, or equations), including representing the products for perfect squares through 15 by 15. Grade 4 students use strategies (e.g., mental math, partial products, and the commutative, associative, and distributive properties) and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. These understandings build the students’ capacity to fluently solve one-, two-, and multistep problems using combinations of addition, subtraction, and multiplication. They also represent problem situations using strip diagrams and equations with a letter standing for the unknown quantity. Students use rounding and compatible numbers to find estimates and justify reasonableness for solutions to problems involving these three operations.

After this Unit
In Units 05, 11, and 13, students will continue to apply multiplication skills as they use estimation along with knowledge of other operations to make connections across the curriculum. In Grade 5, students will extend their application of strategies for rounding numbers to decimals to the tenths or hundredths place. Students will also estimate to determine solutions to mathematical and real-world problems involving addition, subtraction, multiplication, or division. They will multiply, with fluency, a three-digit number by a two-digit number using the standard algorithm, in addition to representing and solving multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.

In Grade 4, solving with fluency one- and two-step problems involving multiplication is identified as STAAR Readiness Standard 4.4H, and representing multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity is addressed in STAAR Readiness Standard 4.5A. Determining products of a number and 10 or 100 using properties of operations and place value understandings, representing the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15, and using strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number are identified as STAAR Supporting Standards 4.4B, 4.4C, and 4.4D. STAAR Supporting Standard 4.4G addresses rounding to the nearest 10, 100, or 1,000 or using compatible numbers to estimate solutions involving whole numbers. These standards are all included under STAAR Reporting Category 2: Computations and Algebraic Relationships, and are further contained in the Grade 4 Texas Response to Curriculum Focal Points (TxRCFP): Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems. Rounding whole numbers to a given place value is identified as STAAR Supporting Standard 4.2D and is subsumed under the STAAR Reporting Category 1: Numerical Representations and Relationships. This standard is further classified under the Grade 4 Focal Point: Understanding decimals and addition and subtraction of decimals (TxRCFP). This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Reasoning, and X. Connections.

Research
According to the National Council of Teachers of Mathematics(2011) “Multiplication is one of two fundamental operations, along with addition, which can be defined so that it is an appropriate choice for representing and solving problems in many different situations” (p. 8). This publication further asserts that “the properties of multiplication and addition provide the mathematical foundation for understanding computational procedures for multiplication and division, including mental computation and estimation strategies, invented algorithms, and standard algorithms” (2011, p. 9). Ma (2010) states that “being able to calculate in multiple ways means that one has transcended the formality of the algorithm and reached the essence of the numerical operations—the underlying mathematical ideas and principles” (p. 112).

Ma, L. (2010). Knowing and teaching elementary mathematics. New York, NY: Routledge
National Council of Teachers of Mathematics. (2011). Developing essential understanding of multiplication and division grades 3 – 5. Reston, VA: National Council of Teachers of Mathematics, Inc.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013

 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)
• Estimation strategies can be used to mentally approximate solutions and determine reasonableness of solutions (multiplication of whole numbers).
• What strategies can be used to estimate solutions to problems?
• What are the similarities and differences between rounding numbers and using compatible numbers to estimate a solution?
• When might an estimated answer be preferable to an exact answer?
• How can an estimation aid in determining the reasonableness of an actual solution?
• When might one estimation strategy be more beneficial than another?
• Recognizing and understanding operational relationships in a variety of problem situations leads to efficient, accurate, and flexible representations and solution strategies (multiplication of whole numbers).
• 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 …
• arrays
• area models
• an equation(s) with a letter standing for the unknown
• strip diagrams or other pictorial models
… aid in problem solving?
• What patterns and relationships can be found within and between the words, pictorial models, and equations used to represent a problem situation?
• How does understanding …
• relationships within and between operations
• properties of operations
• place value
• partial products
• multiples of 10 or 100
… aid in determining an efficient strategy or representation to investigate problem situations?
• What strategies can be used to determine …
• the product
• an unknown factor
… in a multiplication situation?
• Why is it important to understand when and how to use standard algorithms?
• Why is it important to be able to perform operations with whole numbers fluently?
• What relationships exist between addition and multiplication?
• When using multiplication to solve a problem situation, why can the order of the factors be changed?
• Operational understandings lead to generalizations that aid in determining the reasonableness of solutions (multiplication of whole numbers).
• When multiplying two counting numbers greater than one, why is the product always greater than each of the factors?
• Number and Operations
• Estimation
• Rounding
• Compatible Numbers
• Number
• Counting (natural) numbers
• Whole numbers
• Operations
• Multiplication
• Problem Types
• Properties of Operations
• Relationships and Generalizations
• Operational
• Equivalence
• Solution Strategies and Algorithms
• Algebraic Reasoning
• Representations
• Concrete models
• Pictorial models
• Expressions
• Equations
• Patterns and Relationships
• Multiples
• Perfect squares
• 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 that multiplication by 10, 100, or 1,000 means “adding the appropriate number of zeroes” rather than thinking of the product as 10, 100, or 1,000 times greater.
• Some students may think of the area model simply as another procedure for solving multiplication problems rather than as a geometric representation of the distributive property.
• Some students may think that area models are not related to the standard algorithm for multiplication rather than realizing that area models are a visual representation of multiplication and can be used to show the partial products produced through standard algorithms.
• Some students may think that the most efficient way to break up an area model into chunks (distributive property) is to break it up by place value rather than thinking about the numbers and then determining the most efficient way to solve from a variety of strategies.
• Some students may think that the standard algorithm for multiplication is always the most efficient way to solve a multiplication problem rather than thinking about the numbers and then determining the most efficient way to solve from a variety of strategies.
• When using standard algorithm for multiplying, some students may forget to place a zero in the second partial product to hold the number of tens by simply using the digits rather than considering the place value of the product. Underdeveloped Concepts:

• Some students may lack fluency with basic multiplication facts.
• Some students may have a procedural understanding of the standard algorithms for multiplication while lacking conceptual understanding of the operation.

Unit Vocabulary

• Associative property of multiplication – if three or more factors are multiplied, they can be grouped in any order, and the product will remain the same; a × b × c = (a × b) × c = a × (b × c)
• Commutative property of multiplication – if the order of the factors are changed, the product will remain the same; a × b = c; therefore, b × a = c
• Compatible numbers – numbers that are slightly adjusted to create groups of numbers that are easy to compute mentally
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Distributive property of multiplication – if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together; a × (b + c) = (a × b) + (a × c)
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Estimation – reasoning to determine an approximate value
• Factor – a number multiplied by another number to find a product
• Fluency – efficient application of procedures with accuracy
• Product – the total when two or more factors are multiplied
• Rounding – a type of estimation with specific rules for determining the closest value
• Strip diagram – a linear model used to illustrate number relationships
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 About Approximately Area model Around Array Close Consecutive Estimate Halfway Little less Little more Magnitude Mental math Multiple Multiplication Number line Open number line Partial product Perfect square Place value Properties of operations Standard algorithm
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 Creator 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 4 Mathematics TEKS

TEKS# SE# Unit Level Taught Directly TEKS Unit Level Specificity

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.
• Student Expectations (TEKS) are labeled Readiness as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Supporting as identified by TEA of the assessed curriculum.
• Student Expectations (TEKS) are labeled Process standards as identified by TEA of the assessed curriculum.
• 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.

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.
4.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
4.1A Apply mathematics to problems arising in everyday life, society, and the workplace.
Process Standard

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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• X. Connections
4.1B 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.
Process Standard

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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VIII. Problem Solving and Reasoning
4.1C 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.
Process Standard

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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• VIII. Problem Solving and Reasoning
4.1D Communicate mathematical ideas, reasoning, and their implications using multiple representations, including symbols, diagrams, graphs, and language as appropriate.
Process Standard

Communicate

MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, GRAPHS, AND LANGUAGE AS APPROPRIATE

Including, but not limited to:

• Mathematical ideas, reasoning, and their implications
• Multiple representations, as appropriate
• Symbols
• Diagrams
• Graphs
• Language

Note(s):

• The mathematical process standards may be applied to all content standards as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• IX. Communication and Representation
4.1E Create and use representations to organize, record, and communicate mathematical ideas.
Process Standard

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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• IX. Communication and Representation
4.1F Analyze mathematical relationships to connect and communicate mathematical ideas.
Process Standard

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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• X. Connections
4.1G Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication.
Process Standard

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 fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• Measuring angles
• Understanding decimals and addition and subtraction of decimals
• Building foundations for addition and subtraction of fractions
• TxCCRS:
• IX. Communication and Representation
4.4 Number and operations. The student applies mathematical process standards to develop and use strategies and methods for whole number computations and decimal sums and differences in order to solve problems with efficiency and accuracy. The student is expected to:
4.4B Determine products of a number and 10 or 100 using properties of operations and place value understandings.
Supporting Standard

Determine

PRODUCTS OF A NUMBER AND 10 OR 100 USING PROPERTIES OF OPERATIONS AND PLACE VALUE UNDERSTANDINGS

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}
• Multiplication of whole numbers
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Knowledge of patterns in place value to solve multiplication involving multiples of 10 or 100 (e.g., 98 × 10; 98 × 100; 980 × 10; 980 × 100; 9,800 × 10; 9,800 × 100; etc.)
• Properties of operations
• Distributive property of multiplication – if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together
• a × (b + c) = (a × b) + (a × c)
• Multiplying a number by 10 is equal to multiplying each place value digit by 10.
• Multiplying a number by 100 is equal to multiplying each place value digit by 100.
• Commutative property of multiplication – if the order of the factors are changed, the product will remain the same
• a × b = c; therefore, b × a = c
• Place value understanding
• When multiplying a number by 10, the product is 10 times larger meaning that each digit in the number shifts 1 place value position to the left, leaving a zero in the ones place to show groups of tens.
• When multiplying a number by 100, the product is 100 times larger meaning that each digit in the number shifts 2 place value positions to the left, leaving zeros in the ones place and tens place to show groups of hundreds.

Note(s):

• Grade 3 represented multiplication facts by using a variety of approaches such as repeated addition, equal-sized groups, arrays, area models, equal jumps on a number line, and skip counting.
• Grade 3 recalled facts to multiply up to 10 by 10 with automaticity and recalled the corresponding division facts.
• Grade 5 will multiply with fluency a three-digit number by a two-digit number using the standard algorithm.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I. Numeric Reasoning
• IX. Communication and Representation
4.4C Represent the product of 2 two-digit numbers using arrays, area models, or equations, including perfect squares through 15 by 15.
Supporting Standard

Represent

THE PRODUCT OF 2 TWO-DIGIT NUMBERS USING ARRAYS, AREA MODELS, OR EQUATIONS, INCLUDING PERFECT SQUARES THROUGH 15 BY 15

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}
• Multiplication of whole numbers
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of two-digit factors by two-digit factors
• Arrays
• Arrangement of a set of objects in rows and columns
• Area models
• Arrangement of squares/rectangles in a grid format
• Connect the factors as the length and width, and the product as the area
• Equations
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Factor × factor = product
• Product = factor × factor
• Multiplication is commutative
• 14 × 18 = 252
• 18 × 14 = 252
• 252 = 14 × 18
• 252 = 18 × 14
• Perfect squares (through 15 × 15)
• Factors of a perfect square are the same
• Models of perfect squares result in a square
• Equations of perfect squares
• Factor × same factor = product
• Product = factor × same factor

Note(s):

• Grade 3 used strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number. Strategies may have included mental math, partial products, and the commutative, associative, and distributive properties.
• Grade 5 will multiply with fluency a three-digit number by a two-digit number using the standard algorithm.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I. Numeric Reasoning
• II.D. Algebraic Reasoning – Representations
• IX. Communication and Representation
4.4D Use strategies and algorithms, including the standard algorithm, to multiply up to a four-digit number by a one-digit number and to multiply a two-digit number by a two-digit number. Strategies may include mental math, partial products, and the commutative, associative, and distributive properties.
Supporting Standard

Use

STRATEGIES AND ALGORITHMS, INCLUDING THE STANDARD ALGORITHM, TO MULTIPLY UP TO A FOUR-DIGIT NUMBER BY A ONE-DIGIT NUMBER AND TO MULTIPLY A TWO-DIGIT NUMBER BY A TWO-DIGIT NUMBER. STRATEGIES MAY INCLUDE MENTAL MATH, PARTIAL PRODUCTS, AND THE COMMUTATIVE, ASSOCIATIVE, AND DISTRIBUTIVE PROPERTIES

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}
• Multiplication of whole numbers
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Strategies and algorithms for multiplication
• Mental math
• Reasoning strategies that may or may not require the aid of paper, pencil, or other tools
• Partial products
• Decomposing the factor(s) into smaller parts, multiplying the parts, and combining the intermittent parts
• Properties of operations
• Commutative property of multiplication – if the order of the factors are changed, the product will remain the same
• a × b = c; therefore, b × a = c
• Associative property of multiplication – if three or more factors are multiplied, they can be grouped in any order, and the product will remain the same
• a × b × c = (a × b) × c = a × (b × c)
• Distributive property of multiplication – if multiplying a number by a sum of numbers, the product will be the same as multiplying the number by each addend and then adding the products together
• a × (b + c) = (a × b) + (a × c)
• Standard algorithm
• Standardized steps or routines used in computation
• Connections between strategies and operations
• Equation(s) to reflect solution process

Note(s):

• Grade 3 used strategies and algorithms, including the standard algorithm, to multiply a two-digit number by a one-digit number. Strategies may have included mental math, partial products, and the commutative, associative, and distributive properties.
• Grade 5 will multiply with fluency a three-digit number by a two-digit number using the standard algorithm.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I. Numeric Reasoning
• II.D. Algebraic Reasoning – Representations
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
4.4G Round to the nearest 10, 100, or 1,000 or use compatible numbers to estimate solutions involving whole numbers.
Supporting Standard

Round

TO THE NEAREST 10, 100, OR 1,000 TO ESTIMATE SOLUTIONS INVOLVING WHOLE NUMBERS

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}
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Recognition of operations in mathematical and real-world problem situations
• Multi-step problems
• Estimation – reasoning to determine an approximate value
• Rounding – a type of estimation with specific rules for determining the closest value
• To the nearest 10; 100; or 1,000
• Number lines
• Proportionally scaled number lines (pre-determined intervals)
• Open number line (no marked intervals)
• Relative magnitude of a number describes the size of a number and its relationship to another number.
• Rounding to the nearest 10 on a number line
• Determine the two consecutive multiples of 10 that the number being rounded falls between.
• Begin with the value of the original tens place within the number and then identify the next highest value in the tens place.
• Determine the halfway point between the consecutive multiples of 10.
• Locate the position of the number being rounded on the number line.
• Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 10 on the number line.
• If the number being rounded is before the halfway point on the number line, round to the value of the original tens place.
• If the number being rounded is past the halfway point on the number line, round to the value of the next highest tens place.
• If the number being rounded is on the halfway point on the number line, round to the value of the next highest tens place.
• Rounding to the nearest 100 on a number line
• Determine the two consecutive multiples of 100 that the number being rounded falls between.
• Begin with the value of the original hundreds place within the number and then identify the next highest value in the hundreds place.
• Determine the halfway point between the consecutive multiples of 100.
• Locate the position of the number being rounded on the number line.
• Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 100 on the number line.
• If the number being rounded is before the halfway point on the number line, round to the value of the original hundreds place.
• If the number being rounded is past the halfway point on the number line, round to the value of the next highest hundreds place.
• If the number being rounded is on the halfway point on the number line, round to the value of the next highest hundreds place.
• Rounding to the nearest 1,000 on a number line
• Determine the two consecutive multiples of 1,000 that the number being rounded falls between.
• Begin with the value of the original thousands place within the number and then identify the next highest value in the thousands place.
• Determine the halfway point between the consecutive multiples of 1,000.
• Locate the position of the number being rounded on the number line.
• Determine if the number being rounded is before, past, or on the halfway point between the consecutive multiples of 1,000 on the number line.
• If the number being rounded is before the halfway point on the number line, round to the value of the original thousands place.
• If the number being rounded is past the halfway point on the number line, round to the value of the next highest thousands place.
• If the number being rounded is on the halfway point on the number line, round to the value of the next highest thousands place.
• Round a given number to the closest multiple of 10; 100; or 1,000 on a number line.
• Round a given number to the higher multiple of 10; 100; or 1,000 if it falls exactly halfway between the multiples on a number line.
• Round numbers to a common place then compute.
• If not designated, find the greatest common place value of all numbers in the problem to determine the place value to which you are rounding (e.g., round to the nearest 10 if only two-digit numbers are being considered in the problem; round to the nearest 100 if only three-digit numbers are being considered in the problem; round to the nearest 1,000 if only four-digit numbers are being considered; round to the nearest 10 if both two-digit and three-digit numbers are being considered in the problem; round to the nearest 100 if both three-digit and four-digit numbers are being considered; etc.).
• Vocabulary indicating estimation in mathematical and real-world problem situations (e.g., about, approximately, estimate, etc.)
• Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)
• Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
• Rounding numerically based on place value
• Find the place to which you are rounding.
Look at the digit of the next lowest place value, the digit to the right of which you are rounding.
If the digit in that place is less than 5, then the digit in the rounding place remains the same.
If the digit in that place is greater than or equal to 5, then the digit in the rounding place increases by 1.
The digit(s) to the right of the place of which you are rounding is replaced with “0”.
• Round numbers to a common place then compute.
• If not designated, find the greatest common place value of all numbers in the problem to determine the place value to which you are rounding (e.g., round to the nearest 10 if only two-digit numbers are being considered in the problem; round to the nearest 100 if only three-digit numbers are being considered in the problem; round to the nearest 1,000 if only four-digit numbers are being considered; round to the nearest 10 if both two-digit and three-digit numbers are being considered in the problem; round to the nearest 100 if both three-digit and four-digit numbers are being considered; etc.).
• Vocabulary indicating estimation in mathematical and real-world problem situations (e.g., about, approximately, estimate, etc.)
• Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.
• Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.

Use

COMPATIBLE NUMBERS TO ESTIMATE SOLUTIONS INVOLVING WHOLE NUMBERS

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}
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Recognition of operations in mathematical and real-world problem situations
• Multi-step problems
• Estimation – reasoning to determine an approximate value
• Compatible numbers – numbers that are slightly adjusted to create groups of numbers that are easy to compute mentally
• Determine compatible numbers then compute.
• Vocabulary indicating estimation in mathematical and real-world problem situations (e.g., about, approximately, estimate, etc.)
• Vocabulary descriptors of the effects of the adjustment on the estimation compared to the actual solution (e.g., about, close, little more/little less, around, approximately, estimated, etc.)
• Variation of the estimate from the actual solution is dependent upon the magnitude of the adjustment(s) of the actual numbers.

Note(s):

• Grade 3 rounded to the nearest 10 or 100 or use compatible numbers to estimate solutions to addition and subtraction problems.
• Grade 5 will round decimals to tenths or hundredths.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I. Numeric Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
4.4H

Solve with fluency one- and two-step problems involving multiplication and division, including interpreting remainders.

Solve

WITH FLUENCY ONE- AND TWO-STEP PROBLEMS INVOLVING MULTIPLICATION

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}
• Fluency – efficient application of procedures with accuracy
• Standard algorithms for the four operations
• Automatic recall of basic facts
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• One- and two-step problem situations
• One-step problems
• Recognition of multiplication in mathematical and real-world problem situations
• Two-step problems
• Two-step problems must have one-step in the problem that involves multiplication and/or divison; however, the other step in the problem can involve addition and/or subtraction.
• Recognition of multiplication in mathematical and real-world problem situations
• Equation(s) to reflect solution process

Note(s):

• Grade 4 introduces solving with fluency one- and two-step problems involving multiplication and division, including interpreting remainders.
• Grade 5 will multiply with fluency a three-digit number by a two-digit number using the standard algorithm.
• Grade 5 will solve with proficiency for quotients of up to a four-digit dividend by a two-digit divisor using strategies and the standard algorithm.
• Various mathematical process standards will be applied to this student expectation as appropriate
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I. Numeric Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
4.5 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:
4.5A

Represent multi-step problems involving the four operations with whole numbers using strip diagrams and equations with a letter standing for the unknown quantity.

Represent

MULTI-STEP PROBLEMS INVOLVING MULTIPLICATION WITH WHOLE NUMBERS USING STRIP DIAGRAMS AND EQUATIONS WITH A LETTER STANDING FOR THE UNKNOWN QUANTITY

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}
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of whole numbers up to two-digit factors by two-digit factors and up to four-digit factors by one-digit factors
• Representations of an unknown quantity in an equation
• Equation – a mathematical statement composed of algebraic and/or numeric expressions set equal to each other
• Any single letter to represent the unknown quantity (e.g., 24 – 8 = y, etc.)
• Equal sign at beginning or end and unknown in any position
• Multi-step problem situations involving the four operations in a variety of problem structures
• Recognition of multiplication in mathematical and real-world problem situations
• Representation of problem situations with strip diagrams and equations with a letter standing for the unknown
• Strip diagram – a linear model used to illustrate number relationships
• Relationship between quantities represented and problem situation

Note(s):

• Grade 3 represented one- and two-step problems involving addition and subtraction of whole numbers to 1,000 using pictorial models, number lines, and equations.
• Grade 3 represented and solved one- and two-step multiplication and division problems within 100 using arrays, strip diagrams, and equations.
• Grade 3 determined the unknown whole number in a multiplication or division equation relating three whole numbers when the unknown is either a missing factor or product.
• Grade 5 will represent and solve multi-step problems involving the four operations with whole numbers using equations with a letter standing for the unknown quantity.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing fluency with efficient use of the four arithmetic operations on whole numbers and using this knowledge to solve problems
• TxCCRS:
• I. Numeric Reasoning
• II.D. Algebraic Reasoning – Representations
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation 