INSTRUCTIONAL FOCUS DOCUMENT
Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
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UNIT OVERVIEW
This unit bundles student expectations that address representing and solving multiplication and division problems involving a whole number and a fraction. 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 solved with fluency multi-step problems involving addition and subtraction of fractions with equal denominators, as well as multiplication
and division of whole numbers, including interpreting remainders. In Grade 5 Units 01 – 06, students added and subtracted positive rational numbers fluently, inclusive of whole
numbers, decimals, and fractions with unequal denominators, and found products and quotients of decimals through the hundredths.
During this unit, students represent a product of whole number and a fraction referring to the same whole using concrete and pictorial models, including area models and strip
diagrams. Students continue to estimate solutions in mathematical and real-world problems. Concrete and pictorial models, including area models and strip diagrams, are also
used to represent the division of a whole number by a unit fraction and the division of a unit fraction by a whole number. Students also continue to simplify numerical
expressions that involve all operations with whole numbers, decimals, and fractions, respectively. This unit concludes the formal study of operations with whole numbers,
fractions, and decimals in Grade 5.
After this unit, in Units 08 – 10, students will add, subtract, multiply, and/or divide whole numbers, decimals, and fractions to solve problems in geometry, measurement,
algebra, and data analysis. In Grade 6, students will be expected to multiply and divide positive rational numbers fluently. Students will also generate equivalent numerical
expressions using order of operations, including whole number exponents and prime factorization.
In Grade 5, using estimation to determine solutions, representing and solving multiplication of a whole number and a fraction, representing division of a unit fraction by a whole
number and the division of a whole number by a unit fraction are identified as STAAR Supporting Standards 5.3A, 5.3I, and 5.3J, and dividing whole numbers by unit fractions
and unit fractions by whole numbers is identified as STAAR Readiness Standard 5.3L. 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 part of the Grade 5
STAAR Reporting Category 1: Numerical Representations and Relationships as well as the Grade 5 Focal Point: Understanding and generating expressions and equations to
solve problems (TxRCFP). This unit is supporting the development of the Texas College and Career Readiness Standards (TxCCRS): I. Numeric Reasoning, VIII. Problem
Solving and Reasoning, and IX. Communication and Representation.
According to Van de Walle, Karp, and Bay­Williams (2004) “a firm understanding of fractions is the most critical foundation for fraction computation” (p. 254). The National
Research Council (2001) supports this by stating, “Instructional practices that tend toward premature abstraction and extensive symbolic manipulation lead students to have
severe difficulty in representing rational numbers with standard written symbols and using the symbols appropriately” (p. 234). Furthermore, “One helpful approach is to embed
the calculation in a realistic problem. Students can then use the context to connect their previous work with whole numbers to the new situations with rational numbers”
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
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(National Research Council, 2001, p. 239). “If you have spent adequate time with your students exploring multiplication of fractions… the traditional multiplication algorithm will
be relatively simple to develop. Shift from contextual problems to a straight computation” (Van de Walle, Karp & Bay­Williams, 2004, p. 272).
National Research Council. (2001). Adding It Up: Helping children learn mathematics. Washington, DC: National Academy Press.
Texas Education Agency & Texas Higher Education Coordinating Board. (2009). Texas college and career readiness standards. Retrieved from
http://www.thecb.state.tx.us/collegereadiness/crs.pdf
Texas Education Agency. (2013). Texas response to curriculum focal points for kindergarten through grade 8 mathematics. Retrieved from
http://projectsharetexas.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.
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
Numeric Reasoning
Estimation is a strategy that can be used to approximate and
reasonable solutions.
Mathematics Grade 5 Unit 07 PA 01
Provide concrete models for students to select, as
needed.
Analyze the problem situation(s) described below.
Organize and record your work for each of the following
tasks. Using precise mathematical language, justify
and explain each solution process.
1) The Lady Bird Johnson Wildflower Center is an
educational and research facility that serves to
conserve wildflowers and increase the use of native
plants. The center will buy bulk bags of wildflower seed
to divide into small packets and distribute to children at
Division
Expressions
Fractions
Multiple Representations
Multiplication
Order of Operations
Simplify
Whole Numbers
Algebraic Reasoning
Estimation
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UNIT UNDERSTANDINGS
What estimation strategies can be used to help solve
computation problems?
How do estimated solutions compare with the actual solution?
The product of a whole number and a fraction can be represented using
various models.
What is the process for multiplying a whole number and a
fraction?
What types of concrete and/or pictorial models are used to
multiply a whole number and a fraction?
page 2 of 29 INSTRUCTIONAL FOCUS DOCUMENT
Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
PERFORMANCE ASSESSMENT(S)
an event. Each bulk Bluebonnet seed bag contains
of a pound of seeds. Each bulk Rockrose bags
contains
pounds of seeds. The center will buy 14
Bluebonnet bags and 7 Rockrose bags.
a) Write an expression that can be used to
represent the exact number of pounds of seed the
Wildflower Center will purchase.
b) Estimate to determine how many pounds of
seed the Wildflower Center will purchase.
c) Use concrete and/or pictorial models to
determine the exact number of pounds of seed
the Wildflower Center will purchase.
d) Describe the relationship between the number
of pounds of Bluebonnet seed and the number of
pounds of Rockrose seed that will be ordered.
Standard(s): 5.1A , 5.1B , 5.1C , 5.1D , 5.1E , 5.1F ,
5.1G , 5.3A , 5.3I ELPS.c.1A , ELPS.c.2D ,
ELPS.c.3D , ELPS.c.3H , ELPS.c.4E , ELPS.c.4H ,
ELPS.c.5B , ELPS.c.5F , ELPS.c.5G
OVERARCHING CONCEPTS
UNIT CONCEPTS
Solve
Associated Mathematical Processes
Application
Problem Solving Model
Tools and Techniques
Communication
Representations
Relationships
Justification
Numeric Reasoning
xxx
UNIT UNDERSTANDINGS
How can an area model be used to multiply a whole number and
a fraction?
How is multiplying a fraction by a whole number similar and
different to multiplying whole numbers?
Estimation is a strategy that can be used to approximate and
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
Mathematics Grade 5 Unit 07 PA 02
Provide concrete models for students to select, as
needed.
Analyze the problem situation(s) described below.
Organize and record your work for each of the following
tasks. Using precise mathematical language, justify
and explain each solution process.
1) The Lady Bird Johnson Wildflower Center is an
educational and research facility that serves to
conserve wildflowers and increase the use of native
plants. The center purchased bulk Purple Coneflower
seeds and Pink Cosmo seeds to divide into small
packets and distribute to children at an event. They
received a total of 12 pounds of Purple Coneflower
seeds and 8 pounds of Pink Cosmo seeds. The
Wildflower Center would like to put
of a pound of
Purple Coneflower seeds into one set of small packets
and
of a pound of Pink Cosmo seeds into another
set of small packets.
a) Write a numerical expression that can be used
to represent the number of packets the Wildflower
Center will be able to fill with Purple Coneflower
seeds.
b) Write a numerical expression that can be used
to represent the number of packets the Wildflower
xxx
UNIT UNDERSTANDINGS
reasonable solutions.
Division
Fractions
Multiple Representations
Multiplication
Whole Numbers
Algebraic Reasoning
Estimation
Expressions
Order of Operations
Simplify
Solve
Associated Mathematical Processes
Application
Communication
Justification
Problem Solving Model
Relationships
Representations
Tools and Techniques
What estimation strategies can be used to help solve
computation problems?
How do estimated solutions compare with the actual solution?
The quotient of a unit fraction by a whole number or a whole number by
a unit fraction can be represented using various models.
What is the process for dividing a unit fraction by a whole
number?
What is the process for dividing a whole number by a unit
fraction?
What types of concrete and/or pictorial models can be used to
divide a whole number by a unit fraction or a unit fraction by a
whole number fraction?
How can an area model be used to divide a whole number by a
unit fraction or a unit fraction by a whole number fraction?
How is dividing a whole number by a unit fraction or a unit
fraction by a whole number fraction similar and different to
dividing whole numbers?
Numerical expressions can be simplified using the order of operations.
What is the purpose of the order of operations?
What is the order of operations?
What is the process for simplifying a numeric expression?
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
OVERARCHING CONCEPTS
UNIT CONCEPTS
PERFORMANCE ASSESSMENT(S)
SUGGESTED DURATION :
UNIT UNDERSTANDINGS
Center will be able to fill with Pink Cosmo seeds.
c) Use concrete and/or pictorial models to
determine the exact number of Purple Coneflower
seed packets the Wildflower Center will create.
d) Simplify the expression used to represent the
number of packets the Wildflower Center will be
able to fill with Pink Cosmo seeds.
e) Describe the relationship between the number
of Purple Coneflower seed packets and Pink
Cosmo seed packets the Wildflower Center will
create.
2) The Lady Bird Johnson Wildflower Center has three
demonstration garden beds where they showcase the
Purple Coneflowers and Pink Cosmos. They use
pound bag of Purple Coneflower seeds and
pound
bag of Pink Cosmo seeds evenly between the three
demonstration beds.
a) Write a numerical expression that can be used
to represent how many pounds of Purple
Coneflower seeds that will be planted in each
demonstration bed.
b) Write a numerical expression that can be used
to represent how many pounds of Pink Cosmo
seeds that will be planted in each demonstration
bed.
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INSTRUCTIONAL FOCUS DOCUMENT
Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
PERFORMANCE ASSESSMENT(S)
OVERARCHING CONCEPTS
UNIT CONCEPTS
SUGGESTED DURATION :
xxx
UNIT UNDERSTANDINGS
c) Estimate if there will be more pounds of Purple
Coneflower or Pink Cosmo seeds planted in each
demonstration bed.
d) Use concrete and/or pictorial models to
determine the exact number of pounds of Purple
Coneflower seeds that will be planted in each
demonstration bed.
e) Simplify the expression used to represent how
many pounds of Pink Cosmo seeds that will be
planted in each demonstration bed.
Standard(s): 5.1A , 5.1B , 5.1C , 5.1D , 5.1E , 5.1F ,
5.1G , 5.3A , 5.3J , 5.3L , 5.4F ELPS.c.1A ,
ELPS.c.2D , ELPS.c.3D , ELPS.c.3H , ELPS.c.4E ,
ELPS.c.4H , ELPS.c.5B , ELPS.c.5F , ELPS.c.5G
MISCONCEPTIONS / UNDERDEVELOPED CONCEPTS
Misconceptions:
Some students may think that they will need to find a common denominator when multiplying or dividing rational numbers.
Some students may think that when dividing a fraction by a whole number, only the denominator is divided, rather than considering the fraction as representing a quantity
(e.g.,
).
Some students may multiply both the numerator and denominator by the whole number (e.g.,
quantity (e.g.,
), rather than considering the fraction as representing a
).
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TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
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Some students may think that when you multiply with fractions, the product is always larger, 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.
Underdeveloped Concepts:
Some students may not recognize the unreasonableness of their solution because they do not have a firm grasp of the meaning of a fraction.
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 “one­half” (e.g., 6 ÷ = 12) is the same as dividing in “half” (e.g., 6 ÷ 2 = 3).
Some students may not use benchmarks fractions to estimate solutions because they think of fractions as only part-whole relationships instead of thinking of fractions as
quantity.
Some students may think that
be smaller than
is always bigger than
ignoring the whole, rather than thinking of fractions quantities related to the whole (e.g.,
of a small pizza may
of an extra-large pizza).
UNIT VOCABULARY
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}
Denominator– the part of a fraction written below the fraction bar that tells the total number of equal parts in a whole or set
Dividend – the number that is being divided
Divisor – the number the dividend is being divided by
Estimation – reasoning to determine an approximate value
Factor – a number multiplied by another number to find a product
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.
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TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
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Front-end method – a type of estimation focusing first on the largest place value in each of the numbers to be computed and then determining if the next smallest place
value(s) when grouped should be considered or ignored
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
Numerator – the part of a fraction written above the fraction bar that tells the number of fractional parts specified or being considered
Numeric expression – a mathematical phrase, with no equal sign, that contains numbers and an operator(s)
Order of operations – the rules of which calculations are performed first when simplifying an expression
Parentheses and brackets – symbols to show a group of terms and/or expressions within a mathematical expression
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
Rounding – a type of estimation with specific rules for determining the closest value
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:
Eighths
Elevenths
Equivalent
Fifths
Fourths
Fraction bar
Halves
Multiple
Ninths
Sevenths
Sixths
Tenths
Thirds
Twelfths
Whole
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SYSTEM RESOURCES
SUGGESTED DURATION :
xxx
OTHER RESOURCES
Mathematics Grade 5 Implementation TAG Tool
STAAR Grade 5 Mathematics Enhanced Blueprint
Grade 5 TEKS Introduction and Focal Points with Aligned Standards
Texas Education Agency:
Texas Education Agency - Revised Mathematics TEKS: Side-by-Side TEKS
Comparison
Project Share Gateway at www.projectsharetexas.org (click on the
Resource Index; search for Side by Side)
Texas Education Agency - Revised Mathematics TEKS: Vertical Alignment Charts
Project Share Gateway at www.projectsharetexas.org (click on the
Resource Index; search for Vertical Alignment)
Texas Higher Education Coordinating Board and Texas Education Agency:
Texas College and Career Readiness Standards (beginning on page 59 or a11)
TEKS#
SE#
TEKS
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.
UNIT LEVEL SPECIFICITY
Apply
MATHEMATICS TO PROBLEMS ARISING IN EVERYDAY LIFE, SOCIETY, AND THE WORKPLACE
Note(s):
The mathematical process standards may be applied to all content standards as appropriate.
TxRCFP:
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TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
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.
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
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
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TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
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TEKS#
SE#
5.1C
TEKS
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.
UNIT LEVEL SPECIFICITY
Select
TOOLS, INCLUDING REAL OBJECTS, MANIPULATIVES, PAPER AND PENCIL, AND
TECHNOLOGY AS APPROPRIATE, TO SOLVE PROBLEMS
Select
TECHNIQUES, INCLUDING MENTAL MATH, ESTIMATION, AND NUMBER SENSE AS
APPROPRIATE, TO SOLVE PROBLEMS
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.
Communicate
MATHEMATICAL IDEAS, REASONING, AND THEIR IMPLICATIONS USING MULTIPLE
REPRESENTATIONS, INCLUDING SYMBOLS, DIAGRAMS, AND LANGUAGE AS APPROPRIATE
Note(s):
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TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
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.
Create, Use
REPRESENTATIONS TO ORGANIZE, RECORD, AND COMMUNICATE 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.
Analyze
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TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
MATHEMATICAL RELATIONSHIPS TO CONNECT AND COMMUNICATE 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:
X. Connections
5.1G
Display, explain, and justify mathematical ideas
and arguments using precise mathematical
language in written or oral communication.
Display, Explain, Justify
MATHEMATICAL IDEAS AND ARGUMENTS USING PRECISE MATHEMATICAL LANGUAGE IN
WRITTEN OR ORAL COMMUNICATION
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:
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TEKS#
TEKS
SE#
UNIT LEVEL SPECIFICITY
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:
5.3A
Estimate to determine solutions to mathematical
and real-world problems involving addition,
subtraction, multiplication, or division.
Supporting Standard
Estimate To Determine
SOLUTIONS TO MATHEMATICAL AND REAL-WORLD PROBLEMS INVOLVING ADDITION,
SUBTRACTION, MULTIPLICATION, OR DIVISION
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 (less than and greater than one to the tenths, hundredths, and thousandths)
Decimal number – a number in the base­10 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 (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.
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Grade 5 Mathematics
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TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
Proper fraction – a number in the form 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 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
Addition
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 x 0.7 = 0.21, 1.2 x 1.2 = 1.44, 14.3 x 1.3 =
18.59, etc.)
Multiply tenths by hundredths or vice versa (e.g., 0.5 x 0.12 = 0.06, 1.4 x 0.15 =
0.21, 21.4 x 0.45 = 9.63, etc.)
Multiply tenths by thousandths or vice versa (e.g., 0.4 x 0.125 = 0.05, 0.125 x 8.4
= 1.05, etc.)
Multiply whole numbers by tenths, hundredths, and thousandths or vice versa
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
(e.g., 3 x 1.3 = 3.9, 42 x 7.45 = 312.9, 7.02 x 78 = 547.56, 6 x 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 (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
Estimation strategies for solving mathematical and real-world problem situations
Estimation – reasoning to determine an approximate value
Vocabulary indicating estimation 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.
Front­end method – a type of estimation focusing first on the largest place value in each
of the numbers to be computed and then determining if the next smallest place value(s)
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
when grouped should be considered or ignored (compensation)
Rounding – a type of estimation with specific rules for determining the closest value
Round numbers to a common place then compute.
Keep one value the same and round the other value to estimate solutions.
Compatible numbers – numbers that are slightly adjusted to create groups of numbers
that are easy to compute mentally
Multiple operations with various forms of numbers
Note(s):
Grade Level(s):
Grade 4 added and subtracted whole numbers and decimals to the hundredths place
using the standard algorithm.
Grade 5 estimates solutions with whole numbers, fractions, and decimals.
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.
Represent, Solve
MULTIPLICATION OF A WHOLE NUMBER AND A FRACTION THAT REFERS TO THE SAME
WHOLE USING OBJECTS AND PICTORIAL MODELS, INCLUDING AREA MODELS
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
TEKS
SE#
Supporting Standard
UNIT LEVEL SPECIFICITY
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 (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
Fraction 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.
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
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 Level(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
5.3J
Represent division of a unit fraction by a whole
number and the division of a whole number by a
Represent
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
TEKS
SE#
unit fraction such as 1/3 ÷ 7 and 7 ÷ 1/3 using
objects and pictorial models, including area
models.
Supporting Standard
UNIT LEVEL SPECIFICITY
DIVISION OF A UNIT FRACTION BY A WHOLE NUMBER AND THE DIVISION OF A WHOLE
NUMBER BY A UNIT FRACTION 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 (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 limited to a unit fraction by whole number and a whole number by a unit
fraction
Fraction 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
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page 20 of 29 INSTRUCTIONAL FOCUS DOCUMENT
Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
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.
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
Concrete objects and pictorial models
Strip diagrams
Strip diagram – a linear model used to illustrate number relationships
Clocks
Number lines
Adapted area models (e.g., brownie pan method)
Note(s):
Grade Level(s):
Grade 5 introduces representing division of a unit fraction by a whole number and the
division of a whole number by a unit fraction such as ÷ 7 and 7 ÷ 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.
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page 21 of 29 INSTRUCTIONAL FOCUS DOCUMENT
Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
TEKS
SE#
UNIT LEVEL SPECIFICITY
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
5.3L
Divide whole numbers by unit fractions and unit
fractions by whole numbers.
Divide
Readiness Standard
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 (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
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page 22 of 29 INSTRUCTIONAL FOCUS DOCUMENT
Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
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
Fraction 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.
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 Level(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
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page 23 of 29 INSTRUCTIONAL FOCUS DOCUMENT
Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
SUGGESTED DURATION :
xxx
TEKS#
SE#
TEKS
UNIT LEVEL SPECIFICITY
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
ELPS#
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
The English Language Proficiency Standards (ELPS), as required by 19 Texas Administrative Code, Chapter 74, Subchapter A, §74.4, outline English language
proficiency level descriptors and student expectations for English language learners (ELLs). School districts are required to implement ELPS as an integral part of
each subject in the required curriculum.
School districts shall provide instruction in the knowledge and skills of the foundation and enrichment curriculum in a manner that is linguistically accommodated
commensurate with the student’s levels of English language proficiency to ensure that the student learns the knowledge and skills in the required curriculum.
School districts shall provide content-based instruction including the cross-curricular second language acquisition essential knowledge and skills in subsection (c) of the
ELPS in a manner that is linguistically accommodated to help the student acquire English language proficiency.
http://ritter.tea.state.tx.us/rules/tac/chapter074/ch074a.html#74.4
Choose appropriate ELPS to support instruction.
ELPS.c.1
The ELL uses language learning strategies to develop an awareness of his or her own learning processes in all content areas. In order for the ELL to
meet grade-level learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically
accommodated (communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. The student is
expected to:
ELPS.c.1A
use prior knowledge and experiences to understand meanings in English
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Grade 5 Mathematics
TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
ELPS#
SUGGESTED DURATION :
xxx
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
ELPS.c.1B
monitor oral and written language production and employ self-corrective techniques or other resources
ELPS.c.1C
use strategic learning techniques such as concept mapping, drawing, memorizing, comparing, contrasting, and reviewing to acquire basic and
grade-level vocabulary
ELPS.c.1D
speak using learning strategies such as requesting assistance, employing non-verbal cues, and using synonyms and circumlocution (conveying
ideas by defining or describing when exact English words are not known)
ELPS.c.1E
internalize new basic and academic language by using and reusing it in meaningful ways in speaking and writing activities that build concept
and language attainment
ELPS.c.1F
use accessible language and learn new and essential language in the process
ELPS.c.1G
demonstrate an increasing ability to distinguish between formal and informal English and an increasing knowledge of when to use each one
commensurate with grade-level learning expectations
ELPS.c.1H
develop and expand repertoire of learning strategies such as reasoning inductively or deductively, looking for patterns in language, and
analyzing sayings and expressions commensurate with grade-level learning expectations.
ELPS.c.2
The ELL listens to a variety of speakers including teachers, peers, and electronic media to gain an increasing level of comprehension of newly
acquired language in all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced high stage of English language
acquisition in listening. In order for the ELL to meet grade-level learning expectations across the foundation and enrichment curriculum, all
instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded) commensurate with the student's
level of English language proficiency. The student is expected to:
ELPS.c.2A
distinguish sounds and intonation patterns of English with increasing ease
ELPS.c.2B
recognize elements of the English sound system in newly acquired vocabulary such as long and short vowels, silent letters, and consonant
clusters
ELPS.c.2C
learn new language structures, expressions, and basic and academic vocabulary heard during classroom instruction and interactions
ELPS.c.2D
monitor understanding of spoken language during classroom instruction and interactions and seek clarification as needed
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ELPS#
SUGGESTED DURATION :
xxx
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
ELPS.c.2E
use visual, contextual, and linguistic support to enhance and confirm understanding of increasingly complex and elaborated spoken language
ELPS.c.2F
listen to and derive meaning from a variety of media such as audio tape, video, DVD, and CD ROM to build and reinforce concept and language
attainment
ELPS.c.2G
understand the general meaning, main points, and important details of spoken language ranging from situations in which topics, language, and
contexts are familiar to unfamiliar
ELPS.c.2H
understand implicit ideas and information in increasingly complex spoken language commensurate with grade-level learning expectations
ELPS.c.2I
demonstrate listening comprehension of increasingly complex spoken English by following directions, retelling or summarizing spoken
messages, responding to questions and requests, collaborating with peers, and taking notes commensurate with content and grade-level needs.
ELPS.c.3
The ELL speaks in a variety of modes for a variety of purposes with an awareness of different language registers (formal/informal) using vocabulary
with increasing fluency and accuracy in language arts and all content areas. ELLs may be at the beginning, intermediate, advanced, or advanced
high stage of English language acquisition in speaking. In order for the ELL to meet grade-level learning expectations across the foundation and
enrichment curriculum, all instruction delivered in English must be linguistically accommodated (communicated, sequenced, and scaffolded)
commensurate with the student's level of English language proficiency. The student is expected to:
ELPS.c.3A
practice producing sounds of newly acquired vocabulary such as long and short vowels, silent letters, and consonant clusters to pronounce
English words in a manner that is increasingly comprehensible
ELPS.c.3B
expand and internalize initial English vocabulary by learning and using high-frequency English words necessary for identifying and describing
people, places, and objects, by retelling simple stories and basic information represented or supported by pictures, and by learning and using
routine language needed for classroom communication
ELPS.c.3C
speak using a variety of grammatical structures, sentence lengths, sentence types, and connecting words with increasing accuracy and ease as
more English is acquired
ELPS.c.3D
speak using grade-level content area vocabulary in context to internalize new English words and build academic language proficiency
ELPS.c.3E
share information in cooperative learning interactions
ELPS.c.3F
ask and give information ranging from using a very limited bank of high-frequency, high-need, concrete vocabulary, including key words and
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TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
ELPS#
SUGGESTED DURATION :
xxx
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
expressions needed for basic communication in academic and social contexts, to using abstract and content-based vocabulary during extended
speaking assignments
ELPS.c.3G
express opinions, ideas, and feelings ranging from communicating single words and short phrases to participating in extended discussions on a
variety of social and grade-appropriate academic topics
ELPS.c.3H
narrate, describe, and explain with increasing specificity and detail as more English is acquired
ELPS.c.3I
adapt spoken language appropriately for formal and informal purposes
ELPS.c.3J
respond orally to information presented in a wide variety of print, electronic, audio, and visual media to build and reinforce concept and
language attainment.
ELPS.c.4
The ELL reads a variety of texts for a variety of purposes with an increasing level of comprehension in all content areas. ELLs may be at the
beginning, intermediate, advanced, or advanced high stage of English language acquisition in reading. In order for the ELL to meet grade-level
learning expectations across the foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated
(communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1,
certain of these student expectations apply to text read aloud for students not yet at the stage of decoding written text. The student is expected to:
ELPS.c.4A
learn relationships between sounds and letters of the English language and decode (sound out) words using a combination of skills such as
recognizing sound-letter relationships and identifying cognates, affixes, roots, and base words
ELPS.c.4B
recognize directionality of English reading such as left to right and top to bottom
ELPS.c.4C
develop basic sight vocabulary, derive meaning of environmental print, and comprehend English vocabulary and language structures used
routinely in written classroom materials
ELPS.c.4D
use prereading supports such as graphic organizers, illustrations, and pretaught topic-related vocabulary and other prereading activities to
enhance comprehension of written text
ELPS.c.4E
read linguistically accommodated content area material with a decreasing need for linguistic accommodations as more English is learned
ELPS.c.4F
use visual and contextual support and support from peers and teachers to read grade-appropriate content area text, enhance and confirm
understanding, and develop vocabulary, grasp of language structures, and background knowledge needed to comprehend increasingly
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TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
ELPS#
SUGGESTED DURATION :
xxx
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
challenging language
ELPS.c.4G
demonstrate comprehension of increasingly complex English by participating in shared reading, retelling or summarizing material, responding
to questions, and taking notes commensurate with content area and grade level needs
ELPS.c.4H
read silently with increasing ease and comprehension for longer periods
ELPS.c.4I
demonstrate English comprehension and expand reading skills by employing basic reading skills such as demonstrating understanding of
supporting ideas and details in text and graphic sources, summarizing text, and distinguishing main ideas from details commensurate with
content area needs
ELPS.c.4J
demonstrate English comprehension and expand reading skills by employing inferential skills such as predicting, making connections between
ideas, drawing inferences and conclusions from text and graphic sources, and finding supporting text evidence commensurate with content
area needs
ELPS.c.4K
demonstrate English comprehension and expand reading skills by employing analytical skills such as evaluating written information and
performing critical analyses commensurate with content area and grade-level needs.
ELPS.c.5
The ELL writes in a variety of forms with increasing accuracy to effectively address a specific purpose and audience in all content areas. ELLs may
be at the beginning, intermediate, advanced, or advanced high stage of English language acquisition in writing. In order for the ELL to meet gradelevel learning expectations across foundation and enrichment curriculum, all instruction delivered in English must be linguistically accommodated
(communicated, sequenced, and scaffolded) commensurate with the student's level of English language proficiency. For Kindergarten and Grade 1,
certain of these student expectations do not apply until the student has reached the stage of generating original written text using a standard writing
system. The student is expected to:
ELPS.c.5A
learn relationships between sounds and letters of the English language to represent sounds when writing in English
ELPS.c.5B
write using newly acquired basic vocabulary and content-based grade-level vocabulary
ELPS.c.5C
spell familiar English words with increasing accuracy, and employ English spelling patterns and rules with increasing accuracy as more English
is acquired
ELPS.c.5D
edit writing for standard grammar and usage, including subject-verb agreement, pronoun agreement, and appropriate verb tenses
commensurate with grade-level expectations as more English is acquired
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TITLE : 5M_U7_ Multiplication and Division of Whole Numbers with a Fraction_Oct2014 - ABILENE ISD
ELPS#
SUGGESTED DURATION :
xxx
SUBSECTION C: CROSS-CURRICULAR SECOND LANGUAGE ACQUISITION ESSENTIAL KNOWLEDGE AND SKILLS.
ELPS.c.5E
employ increasingly complex grammatical structures in content area writing commensurate with grade-level expectations, such as:
ELPS.c.5F
write using a variety of grade-appropriate sentence lengths, patterns, and connecting words to combine phrases, clauses, and sentences in
increasingly accurate ways as more English is acquired
ELPS.c.5G
narrate, describe, and explain with increasing specificity and detail to fulfill content area writing needs as more English is acquired.
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