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 Instructional Focus DocumentGrade 5 Mathematics
 TITLE : Unit 11: Making Connections SUGGESTED DURATION : 10 days

#### Unit Overview

Introduction
This unit bundles student expectations that address examining problem situations involving whole numbers, fractions, and decimals, discerning the necessary operations needed to solve the problems, and verifying and/or justifying the solutions. 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 4, students represented and solved addition and subtraction of fractions with equal denominators using objects and pictorial models that build to the number line and properties of operations and evaluated the reasonableness of sums and differences of fractions using benchmark fractions 0, , and 1, referring to the same whole. In Grade 5 Units 01 – 07, students explored extending whole number operations to include using algebraic reasoning, extended place value understandings to the thousandths place, and divided and multiplied decimals to the hundredths place. Students added and subtracted fractions with unequal denominators, multiplied a fraction by a whole number, and divided a unit fraction by a whole number and whole number by a unit fraction.

During this Unit
Students revisit representing and solving problems involving addition and subtraction of fractions with unequal denominators referring to the same whole using objects and pictorial models and properties of operations. Students also solve addition and subtraction situations involving positive rational numbers. Students revisit representing and solving problems involving the multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models. Students continue multiplication as they solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship of decimal multiplication to the multiplication of whole numbers. Students revisit division as they solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm. They divide whole numbers by unit fractions and unit fractions by whole numbers. Students continue to simplify numerical expressions that do not involve exponents, including up to two levels of grouping.

After this Unit
In Grade 6, students will be expected to multiply and divide positive rational numbers fluently. Students will determine, with and without computation, whether a quantity is increased or decreased when multiplied by a fraction, including values greater than or less than one. They will also generate equivalent numerical expressions and determine if two expressions are equivalent through the use of properties of operations and prime factorization.

In Grade 5, solving for products of decimals to the hundredths, solving for quotients of decimals, adding and subtracting positive rational numbers fluently, and dividing whole numbers by unit fractions and unit fractions by whole numbers are identified as STAAR Readiness Standards 5.3E, 5.3G, 5.3K, and 5.3L. Representing and solving multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models is identified as STAAR Supporting Standard 5.3I. All of these standards are subsumed within the Grade 5 STAAR Reporting Category 2: Computations and Algebraic Relationships and the Grade 5 Texas Response to Curriculum Focal Points (TxRCFP): Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals. Simplifying numerical expressions is identified as STAAR Readiness Standard 5.4F and subsumed within the Grade 5 STAAR Reporting Category 1: Numerical Representations and Relationships and Grade 5 Focal Point: Understanding and generating expressions and equations to solve problems (TxRCFP). This is furthering the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, VIII. Problem Solving and Reasoning, IX. Communication and Representation, X. Connections.

Research
According to Van de Walle, Karp, and Bay-Williams (2004) in regards for the need to connect fractions and decimals, “Linking the ideas of fractions to decimals can be extremely useful, both from a pedagogical view as well as a practical, social view.” (pg. 280). He goes on to state that “decimal numbers are simply another way of writing fractions. Both notations have value. Maximum flexibility is gained by understanding how the two symbol systems are related.” (p. 280). Fosnot and Dolk (2002) add, “Once children develop a sense of landmark fractions like , , and and know the decimal equivalents, using them interchangeably is a powerful strategy. For example, 75 × 80 can easily be solved by thinking of the problem as × 80. You only have to remember to compensate for the decimal answer (multiply 60 by 100, because you treated 75 as )” (p.125).

Fosnot, C., & Dolk, M. (2002). Young mathematicians at work: constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from http://www.thecb.state.tx.us/index.cfm?objectid=E21AB9B0-2633-11E8-BC500050560100A9
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from https://www.texasgateway.org/resource/txrcfp-texas-response-curriculum-focal-points-k-8-mathematics-revised-2013
Van de Walle, J., Karp, K., & Bay-Williams, J. (2004). Elementary and middle school mathematics: Teaching developmentally. Boston, MA: Pearson Education, Inc.

 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 positive rational numbers; multiplication of whole numbers and decimals with products through the hundredths; multiplication of whole numbers and fractions; division of decimals by whole numbers with quotients through the hundredths; division of whole numbers and unit fractions; numerical expressions up to two levels of grouping that do not include exponents).
• 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 …
• objects
• area models
• other pictorial models
… aid in problem solving?
• What patterns and relationships can be found within and between the words, unknown variable(s), grouping symbols, expressions, and equations used to represent a problem situation?
• How does understanding …
• relationships within and between operations
• properties of operations
• place value
• relationships between addition and subtraction involving whole numbers and addition and subtraction involving decimals
• relationships between addition and subtraction involving fractions with like denominators and addition and subtraction involving fractions with unlike denominators
• relationships between multiplication with whole numbers and multiplication with decimals
• relationships between multiplication with whole numbers and multiplication with fractions
• relationships between division with whole numbers and division with decimals
• relationships between division with whole numbers and division with unit fractions
… 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 involving positive rational numbers?
• What strategies can be used to determine …
• the product
• an unknown factor
… in a multiplication situation involving …
• whole numbers?
• whole numbers and decimals?
• whole numbers and fractions?
• What strategies can be used to determine …
• the quotient
• an unknown
… in a division situation involving …
• whole numbers?
• whole numbers and decimals?
• whole numbers and unit fractions?
• Why is it important to understand when and how to use standard algorithms?
• Why is it important to be able to add and subtract positive rational numbers fluently?
• What relationships exist between …
• addition and subtraction?
• addition and multiplication?
• multiplication and division?
• subtraction and division?
• operations with whole numbers and operations with decimals?
• operations with whole numbers and operations with fractions?
• What generalizations can be made about order of operations and its purpose?
• How can …
• properties of numbers and operations
• order of operations
… be used to simplify an expression?
• 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?
• When using multiplication to solve a problem situation, why can the order of the factors be changed?
• When using division to solve a problem situation, why can the order of the dividend and divisor not be changed?
• Operational understandings lead to generalizations that aid in determining the reasonableness of solutions (addition and subtraction of positive rational numbers; multiplication of whole numbers and decimals with products through the hundredths; multiplication of whole numbers and fractions; division of decimals by whole numbers with quotients through the hundredths; division of whole numbers and unit fractions; numerical expressions up to two levels of grouping that do not include exponents).
• When adding two non-zero positive rational numbers, why is the sum always greater than each of the addends?
• When subtracting two non-zero positive rational numbers with the minuend greater than the subtrahend, why is the difference always less than the minuend?
• When multiplying two non-zero positive rational numbers greater than one, why is the product always greater than each of the factors?
• When multiplying two non-zero positive rational numbers with one of the factors greater than one, why is the product always greater than the smallest factor?
• When multiplying two non-zero positive rational numbers with both factors less than one, why is the product always less than the smallest factor?
• When dividing two non-zero positive rational numbers with the dividend greater than the divisor, why is the quotient always greater than one?
• When dividing two non-zero positive rational numbers with the dividend less than the divisor, why is the quotient always greater than zero and less than one?
• Number and Operations
• Number
• Positive rational numbers
• Operations
• Subtraction
• Multiplication
• Division
• Properties of Operations
• Relationships and Generalizations
• Operational
• Equivalence
• Solution Strategies and Algorithms
• Algebraic Reasoning
• Equivalence
• Order of Operations
• Representations
• Concrete models
• Pictorial models
• Expressions
• Equations
• Associated Mathematical Processes
• Application
• Problem Solving Model
• Tools and Techniques
• Communication
• Representations
• Relationships
• Justification
 Assessment information provided within the TEKS Resource System are examples that may, or may not, be used by your child’s teacher. In accordance with section 26.006 (2) of the Texas Education Code, "A parent is entitled to review each test administered to the parent’s child after the test is administered." For more information regarding assessments administered to your child, please visit with your child’s teacher.

#### MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS

Underdeveloped Concepts:

• Some students may have a procedural understanding of the standard algorithms for addition, subtraction, multiplication, or division while lacking conceptual understanding of the operations.
• Some students may think that when subtracting mixed numbers, the whole numbers and fractions can be subtracted separately rather than thinking about the fraction as part of the numbers (e.g., For 2 – 1, students may think 2 – 1 = 1 and = , so 2 – 1 = 1).
• Some students may think the dividend always goes on the left side of a division sentence rather than understanding where to place the dividend and divisor based on the symbol being used in long division.
• Some students may be able to perform a symbolic procedure for decimal multiplication or division with limited understanding of the multiplication or division concepts involved.
• Some students may think that when you multiply with fractions, the product is always larger as in whole numbers rather than seeing the reverse is true for quantities less than one.
• Some students may think that when you divide with fractions, the quotient is always smaller as in whole numbers rather than seeing the reverse is true for quantities less than one.
• Some students may not realize that the properties of numbers apply to fractions (e.g., associative, commutative, and distributive properties).
• Some students may not understand that “a third” of something is the same as dividing by three (e.g., of 30 = 30 ÷ 3).
• Some students may think that dividing "by one half” (e.g., 6 ÷ = 12) is the same as dividing "in half” (e.g., 6 ÷ 2 = 3).

#### 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
• Counting (natural) numbers – the set of positive numbers that begins at one and increases by increments of one each time {1, 2, 3, ..., n}
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• 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)
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• 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)
• Factor – a number multiplied by another number to find a product
• Fluency – efficient application of procedures with accuracy
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Grouping symbols – symbols to show a group of terms and/or expressions within a mathematical expression
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Least common denominator (LCD) – the least common multiple of the denominators of two or more fractions
• Mixed number – a number that is composed of a whole number and a fraction
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Order of operations – the rules of which calculations are performed first when simplifying an expression
• Positive rational numbers – the set of numbers that can be expressed as a , where a and b are counting (natural) numbers
• Product – the total when two or more factors are multiplied
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Strip diagram – a linear model used to illustrate number relationships
• Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number
• Whole numbers – the set of counting (natural) numbers and zero {0, 1, 2, 3, ..., n}

Related Vocabulary:

 Area model Brackets Decimal grid Difference Eighths Elevenths Equivalent Fifths Fourths Halves Hundredths Magnitude Multiple Ninths Number line Parentheses Partial quotients Position Ratio table Remainder Sevenths Simplify fractions Sixths Standard algorithm Sum Tenths Thirds Thousandths Twelfths Whole
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 5 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.
5.1 Mathematical process standards. The student uses mathematical processes to acquire and demonstrate mathematical understanding. The student is expected to:
5.1A 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 an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• X. Connections
5.1B 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 an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VIII. Problem Solving and Reasoning
5.1C 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 an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• VIII. Problem Solving and Reasoning
5.1D 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 an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.1E 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 an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.1F 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 an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• X. Connections
5.1G 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 an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• Understanding and generating expressions and equations to solve problems
• Representing and solving problems with perimeter, area, and volume
• Organizing, representing, and interpreting sets of data
• TxCCRS:
• IX. Communication and Representation
5.3 Number and operations. The student applies mathematical process standards to develop and use strategies and methods for positive rational number computations in order to solve problems with efficiency and accuracy. The student is expected to:
5.3E Solve for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers.

Solve

FOR PRODUCTS OF DECIMALS TO THE HUNDREDTHS, INCLUDING SITUATIONS INVOLVING MONEY, USING STRATEGIES BASED ON PLACE-VALUE UNDERSTANDINGS, PROPERTIES OF OPERATIONS, AND THE RELATIONSHIP TO THE MULTIPLICATION OF WHOLE NUMBERS

Including, but not limited to:

• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products of decimals limited to three-digit factors by two-digit factors with products to the hundredths
• Multiply tenths by tenths (e.g., 0.3 × 0.7 = 0.21, 1.2 × 1.2 = 1.44, 14.3 × 1.3 = 18.59, etc.)
• Multiply tenths by hundredths or vice versa (e.g., 0.5 × 0.12 = 0.06, 1.4 × 0.15 = 0.21, 21.4 × 0.45 = 9.63, etc.)
• Multiply tenths by thousandths or vice versa (e.g., 0.4 × 0.125 = 0.05, 0.125 × 8.4 = 1.05, etc.)
• Multiply whole numbers by tenths, hundredths, and thousandths or vice versa (e.g., 3 × 1.3 = 3.9, 42 × 7.45 = 312.9, 7.02 × 78 = 547.56, 6 × 0.125 = 0.75, etc.)
• Multiplying by a lesser factor results in lesser products.
• Connections between whole number multiplication and decimal multiplication
• Base-10 place value system
• A number system using ten digits 0 – 9
• Relationships between places are based on multiples of 10.
• Moving left across the places, the values are 10 times the position to the right.
• Moving right across the places, the values are one-tenth the value of the place to the left.
• Place value relationships to determine products
• 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)
• Recognition of multiplication in mathematical and real-world problem situations
• Strategies for multiplication
• Distributive property for partial products
• Doubling and halving
• Relate multiplication (associative property) to numerical notation
• Ratio tables
• Equation(s) to reflect solution process

Note(s):

• Grade 5 introduces solving for products of decimals to the hundredths, including situations involving money, using strategies based on place-value understandings, properties of operations, and the relationship to the multiplication of whole numbers.
• Grade 6 will multiply and divide positive rational numbers fluently.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• TxCCRS:
• I. Numeric Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
• X. Connections
5.3G Solve for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm.

Solve

FOR QUOTIENTS OF DECIMALS TO THE HUNDREDTHS, UP TO FOUR-DIGIT DIVIDENDS AND TWO-DIGIT WHOLE NUMBER DIVISORS, USING STRATEGIES AND ALGORITHMS, INCLUDING THE STANDARD ALGORITHM

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}
• Decimals (positive decimals less than one and greater than one to the tenths and hundredths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Division
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• Quotients of decimals limited to four-digit dividends and two-digit whole number divisors, with quotients to the hundredths
• Dividend to the tenths and whole number divisor (e.g., 1.2 ÷ 24 = 0.05, 358.8 ÷ 23 = 15.6, 721.7 ÷ 14 = 51.55, etc.)
• Dividend to the hundredths and whole number divisor (e.g., 8.68 ÷ 4 = 2.17, 8.25 ÷ 15 = 0.55, 62.76 ÷ 12 = 5.23, etc.)
• Whole number dividends and whole number divisors that yield quotients to the hundredths (e.g., 3 ÷ 4 = 0.75, 10 ÷ 8 = 1.25, 1000 ÷ 16 = 62.5, etc.)
• Relationships between multiplication and division to help in solution process
• a ÷ b = c, so b × c = a
• Connections between division of whole numbers and division with decimals
• Decimal quotients will have the same digits as whole number quotients when the number of digits in the dividend and number of digits in the divisor of both the decimal problem and whole number problem are the same.
• Base-10 place value system
• A number system using ten digits 0 – 9
• Relationships between places are based on multiples of 10.
• Moving left across the places, the values are 10 times the position to the right.
• Moving right across the places, the values are one-tenth the value of the place to the left.
• Place value relationships to determine quotients
• Recognition of division in mathematical and real-world problem situations
• Division structures
• Partitive division
• Total amount known
• Number of groups known
• Size or measure of each group unknown
• Quotative division (also known as Measurement division)
• Total amount known
• Size or measure of each group known
• Number of groups unknown
• Decomposing division problems into partial quotients
• Standard algorithm using the distributive method
• Record steps that relate to the algorithm used including distributing the value in the quotient according to place value.
• Standard algorithm
• Remainder dependent upon the mathematical and real-world problem situation
• Various ways to record remainder
• Ignore the remainder
• Add one to the quotient
• Remainder is written as a decimal
• Remainder is the answer
• Conversion of remainder into smaller units
• Equation(s) to reflect solution process

Note(s):

• Grade 5 introduces solving for quotients of decimals to the hundredths, up to four-digit dividends and two-digit whole number divisors, using strategies and algorithms, including the standard algorithm.
• Grade 6 will multiply and divide decimals fluently.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• TxCCRS:
• I. Numeric Reasoning
• VIII. Problem Solving and Reasoning
• IX. Communication and Representation
5.3I Represent and solve multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models.
Supporting Standard

Represent, Solve

MULTIPLICATION OF A WHOLE NUMBER AND A FRACTION THAT REFERS TO THE SAME WHOLE USING OBJECTS AND PICTORIAL MODELS, INCLUDING AREA MODELS

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}
• Fractions (positive proper, improper, or mixed numbers)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction –a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Multiplication
• Product – the total when two or more factors are multiplied
• Factor – a number multiplied by another number to find a product
• Products limited to a whole number and a fraction that refers to the same whole
• Fractional relationships
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Referring to the same whole
• Fractions are relationships, and the size or the amount of the whole matters.
• Equivalent fractions to simplify solutions
• Recognition of multiplication in mathematical and real-world problem situations
• Concrete objects and pictorial models
• Pattern blocks and other shapes
• Skip counting
• Fraction bars
• Number lines
• Area models
• Strip diagrams
• Strip diagram – a linear model used to illustrate number relationships
• Equation(s) to reflect solution process

Note(s):

• Grade 5 introduces representing and solving multiplication of a whole number and a fraction that refers to the same whole using objects and pictorial models, including area models.
• Grade 6 will multiply and divide positive rational numbers fluently.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• TxCCRS:
• I. Numeric Reasoning
• IX. Communication and Representation
5.3K Add and subtract positive rational numbers fluently.

POSITIVE RATIONAL NUMBERS FLUENTLY

Including, but not limited to:

• Positive rational numbers – the set of numbers that can be expressed as a fraction , where a and b are counting (natural) numbers
• Various forms of positive rational 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}
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Fractions (positive proper, improper, or mixed numbers with equal or unequal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number
• Fluency – efficient application of procedures with accuracy
• Sums of whole numbers
• Sums of decimals up to the thousandths
• Sums of fractions with equal and unequal denominators
• Subtraction
• Differences of whole numbers
• Differences of decimals with values limited to the thousandths
• Differences of fractions with equal and unequal denominators
• Fractional relationships
• Equivalent fractions to determine common denominator prior to adding or subtracting fractions
• Least common denominator (LCD) – the least common multiple of the denominators of two or more fractions
• Equivalent fractions to simplify solutions
• Recognition of addition and/or subtraction in mathematical and real-world problem situations

Note(s):

• Grade 4 evaluated the reasonableness of sums and differences of fractions using benchmark fractions 0, and 1 referring to the same whole.
• Grade 7 will add, subtract, multiply, and divide rational numbers fluently.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• TxCCRS:
• I. Numeric Reasoning
• X. Communication and Representation
5.3L Divide whole numbers by unit fractions and unit fractions by whole numbers.

Divide

WHOLE NUMBERS BY UNIT FRACTIONS AND UNIT FRACTIONS BY 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}
• Fractions (positive unit fractions)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number
• Division
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• Quotients of fractions where dividend and divisors are limited to whole numbers by unit fractions and unit fractions by whole numbers
• Fractional relationships
• Relationship between the whole and the part
• Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
• Denominator – the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
• Referring to the same whole
• Fractions are relationships, and the size or the amount of the whole matters.
• Recognition of division in mathematical and real-world problem situations
• Division structures
• Partitive division
• Total amount known
• Number of groups known
• Size or measure of each group unknown
• Quotative division (also known as Measurement division)
• Total amount known
• Size or measure of each group known
• Number of groups unknown
• Division strategies
• Partitive
• Ratio tables

Note(s):

• Grade 5 introduces dividing whole numbers by unit fractions and unit fractions by whole numbers.
• Grade 6 will multiply and divide positive rational numbers fluently.
• Various mathematical process standards will be applied to this student expectation as appropriate.
• TxRCFP:
• Developing an understanding of and fluency with addition, subtraction, multiplication, and division of fractions and decimals
• TxCCRS:
• I. Numeric Reasoning
• IX. Communication and Representation
5.4 Algebraic reasoning. The student applies mathematical process standards to develop concepts of expressions and equations. The student is expected to:
5.4F Simplify numerical expressions that do not involve exponents, including up to two levels of grouping.

Simplify

NUMERICAL EXPRESSIONS THAT DO NOT INVOLVE EXPONENTS, INCLUDING UP TO TWO LEVELS OF GROUPING

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}
• Decimals (positive decimals less than one and greater than one to the tenths, hundredths, and thousandths)
• Decimal number – a number in the base-10 place value system used to represent a quantity that may include part of a whole and is recorded with a decimal point separating the whole from the part
• Fractions (positive proper, improper, or mixed numbers with equal or unequal denominators)
• Fraction – a number in the form where a and b are whole numbers and b is not equal to zero. A fraction can be used to name part of an object, part of a set of objects, to compare two quantities, or to represent division.
• Proper fraction – a number in the form where a and b are whole numbers and a < b where b is not equal to zero
• Improper fraction – a number in the form where a and b are whole numbers and a > b where b is not equal to zero
• Mixed number – a number that is composed of a whole number and a fraction
• Unit fraction – a fraction in the form representing the quantity formed by one part of a whole that has been partitioned into b equal parts where b is a non-zero whole number
• Sums of whole numbers
• Sums of decimals up to the thousandths
• Sums of fractions with equal and unequal denominators
• Subtraction
• Differences of whole numbers
• Differences of decimals with values limited to the thousandths
• Differences of fractions with equal and unequal denominators
• 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 three-digit factors by two-digit factors
• Products of decimals limited to three-digit factors by two-digit factors with products to the hundredths
• Multiply tenths by tenths (e.g., 0.3 × 0.7 = 0.21, 1.2 × 1.2 = 1.44, 14.3 × 1.3 = 18.59, etc.)
• Multiply tenths by hundredths or vice versa (e.g., 0.5 × 0.12 = 0.06, 1.4 × 0.15 = 0.21, 21.4 × 0.45 = 9.63, etc.)
• Multiply tenths by thousandths or vice versa (e.g., 0.4 × 0.125 = 0.05, 0.125 × 8.4 = 1.05, etc.)
• Multiply whole numbers by tenths, hundredths, and thousandths or vice versa (e.g., 3 × 1.3 = 3.9, 42 × 7.45 = 312.9, 7.02 × 78 = 547.56, 6 × 0.125 = 0.75, etc.)
• Products of fractions where factors are limited to a fraction and a whole number
• Division
• Quotient – the size or measure of each group or the number of groups when the dividend is divided by the divisor
• Dividend – the number that is being divided
• Divisor – the number the dividend is being divided by
• Whole numbers with quotients up to four-digit dividends and two-digit divisors
• Quotients of decimals limited to four-digit dividends and two-digit whole number divisors, with quotients to the hundredths
• Dividend to the tenths and whole number divisor (e.g., 1.2 ÷ 24 = 0.05, 358.8 ÷ 23 = 15.6, 721.7 ÷ 14 = 51.55, etc.)
• Dividend to the hundredths and whole number divisor (e.g., 8.68 ÷ 4 = 2.17, 8.25 ÷ 15 = 0.55, 62.76 ÷ 12 = 5.23, etc.)
• Whole number dividends and whole number divisors that yield quotients to the hundredths (e.g., 3 ÷ 4 = 0.75, 10 ÷ 8 = 1.25, 1000 ÷ 16 = 62.5, etc.)
• Quotients of fractions where dividend and divisors are limited to whole numbers by unit fractions and unit fractions by whole numbers
• 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)
• Numerical expressions without exponents
• Grouping symbols – symbols to show a group of terms and/or expressions within a mathematical expression
• Parentheses ( )
• Brackets [ ]
• Up to two levels of grouping
• Grouping symbols within grouping symbols
• Two sets of grouping symbols
• Order of operations – the rules of which calculations are performed first when simplifying an expression
• Parentheses/brackets: simplify expressions inside parentheses or brackets in order from left to right
• Multiplication/division: simplify expressions involving multiplication and/or division in order from left to right
• Various indicators of multiplication include ×, •, or grouping symbols without a multiplication symbol.
• Addition/subtraction: simplify expressions involving addition and/or subtraction in order from left to right
• Recognition of expressions in real-world problem situations

Note(s):