Holyoke Public Schools: Grade 5 Curriculum Map
Unit Four: Multiplication and Division of Fractions & Decimal Fractions (~35 days)
Overview
Grade 5’s Module 4 extends student understanding of fraction operations to multiplication and division of both fractions and decimal
fractions. Work proceeds from interpretation of line plots which include fractional measurements to interpreting fractions as division and
reasoning about finding fractions of sets through fraction by whole number multiplication. The module proceeds to fraction by fraction
multiplication in both fraction and decimal forms. An understanding of multiplication as scaling and multiplication by n/n as multiplication by
1 allows students to reason about products and convert fractions to decimals and vice versa. Students are introduced to the work of division
with fractions and decimal fractions. Division cases are limited to division of whole numbers by unit fractions and unit fractions by whole
numbers. Decimal fraction divisors are introduced and equivalent fraction and place value thinking allow student to reason about the size of
quotients, calculate quotients and sensibly place decimals in quotients. Throughout the module students are asked to reason about these
important concepts by interpreting numerical expressions which include fraction and decimal operations and by persevering in solving realworld, multistep problems which include all fraction operations supported by the use of tape diagrams. (From EngageNY)
Focus Standards
 5.NF.4
a.
Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
Interpret the product of (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence
of operations a × q ÷ b. For example, use a visual fraction model to show (2/3 × 4 = 8/3, and create a story context for this
equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
 5.NF.7
Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers
by unit fractions. (Students capable of multiplying fractions can generally develop strategies to divide fractions by reasoning
about the relationship between multiplication and division. However, division of a fraction by a fraction is not a requirement at
this grade level.)
a.
Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story
context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication
and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.
b.
Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context
for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and
*These instructional maps were created and developed using information from Engage NY, Shelby County Schools, Saugus Public Schools,
Somerville Public Schools, and www.mdk12.org.
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Holyoke Public Schools: Grade 5 Curriculum Map
division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.
c.
Solve real world problems involving division of unit fractions by non‐zero whole numbers and division of whole numbers
by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much
chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of
raisins?
Foundational Standards
Write and interpret numerical expressions.
 5.OA.1
Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
 5.OA.2
Write simple expressions that record calculations with numbers, and interpret numerical expressions without
evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 +7). Recognize that 3 × (18932 +
921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Perform operations with multi-digit whole numbers and with decimals to hundredths.
 5.NBT.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based
on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a
written method and explain the reasoning used.
Apply and extend previous understandings of multiplication and division to multiply and divide fractions.
 5.NF.3 Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of
whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to
represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that
when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50‐pound sack of
rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
 5.NF.5 Interpret multiplication as scaling (resizing), by:
 Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated
multiplication.
 Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number
*These instructional maps were created and developed using information from Engage NY, Shelby County Schools, Saugus Public Schools,
Somerville Public Schools, and www.mdk12.org.
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Holyoke Public Schools: Grade 5 Curriculum Map
(recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a
fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b =
(n×a)/(n×b) to the effect of multiplying a/b by 1.
 5.NF.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or
equations to represent the problem.
Convert like measurement units within a given measurement system.
 5.MD.1
Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm
to 0.05 m), and use these conversions in solving multi-step, real world problems.
Represent and interpret data.
 5.MD.2
Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on
fractions for this grade to solve problems involving information presented in line plots. For example, given different
measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the
beakers were redistributed equally.
Critical Concepts
 Order of Operations
Use Order of Operations to evaluate expressions
Write numerical expressions
 Multiplication of Fractions
Relate decimal multiplication to fraction multiplication
Solve fraction multiplication problems using models
Compare the size of the factors in a fraction multiplication problem to the product
Multiply fractions
Multiply mixed numbers
Write word problems involving the multiplication of fractions.
 Division of Fractions
Understand fractions as division
Solve fraction division problems using models
Compare the size of the factors in a fraction multiplication problem to the product
Divide fractions by whole numbers
*These instructional maps were created and developed using information from Engage NY, Shelby County Schools, Saugus Public Schools,
Somerville Public Schools, and www.mdk12.org.
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Holyoke Public Schools: Grade 5 Curriculum Map
Divide whole numbers by fractions
Write word problems involving division of fractions.
 Measurement with Fractions
Measure and compare fractional amounts
Use line plots
Student Objectives
I can…
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Use the Order of Operations to evaluate numerical expressions. 1.11
Evaluate numerical expressions with parentheses, brackets, and braces 1.12
Write numerical expressions 1.10
Relate decimal and fraction multiplication (EngageNY – Lesson 17, 18 – Unit 4)
Interpret fractions as division. 8.3
Model to find the fractional part of a group. 7.1
Model the product of a fraction and a whole number. 7.2
Multiply fractions and whole numbers. 7.3
Multiply fractions using models. 7.4
Related the size of the product compared to the size of one factor when multiplying fractions. 7.5
Multiply fractions. 7.6
Relate the size of the product to the factors when multiplying fractions greater than one. 7.8
Solve fraction multiplication problems using the strategy, “guess, check, and revise.” 7.10
Solve and create fraction word problems involving addition, subtraction, and multiplication (EngageNY – Lessons 11,12 – Unit 4)
Divide a whole number by a fractions and divide a fraction by a whole number. 8.1
Solve fraction division problems by using the strategy, “draw a diagram.” 8.2
Interpret fractions as division and solve whole-number division problems that result in a fraction or mixed number. 8.3
Divide a whole number by a fraction and divide a fraction by a whole number. 8.4
Represent division by drawing diagrams and writing story problems and equations. 8.5
Measure and compare pencil lengths to the nearest ½, ¼, and 1/8 of an inch and analyze the data through line plots (EngageNY –
Lesson 1 – Unit 4)
See GoMath Resources for student practice (Chapter.Lesson)
*These instructional maps were created and developed using information from Engage NY, Shelby County Schools, Saugus Public Schools,
Somerville Public Schools, and www.mdk12.org.
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Holyoke Public Schools: Grade 5 Curriculum Map
Essential Questions
In what order must operations be evaluated to find the solution to a problem?
In what order must operations be evaluated to find the solution when there are multiple parentheses?
How can you use a numerical expression to describe a situation?
How are decimal and fraction multiplication related?
How does a fraction represent division?
How can I find a fractional part of a group?
How do I use a model to show the product of a fraction and a whole number?
How do I find the product of a fraction and a whole number without using a model?
How do I use the area model to show the product of two fractions?
How does the size of the product compare to the size of one factor when multiplying fractions?
How do I multiply fractions?
How does the size of the product compare to the size of one factor when multiplying fractions greater than one?
How do I divide a whole number by a fraction and divide a fraction by a whole number?
How can the strategy “draw a diagram” help you solve division problems by writing a multiplication sentence?
How does a fraction represent division?
How can I divide fractions by solving a related multiplication sentence?
How can I use diagrams, equations, and story problems to represent division?
How can I measure and compare different fractional lengths?
Vocabulary
Decimal divisor, Simplify, Conversion factor, Commutative property, Decimal fraction, Denominator, Distribute, Divide, Equation, Equivalent
fraction, Expression, Factors; Conversion Factor, Fractional Unit, Numerator, Parentheses, Tape Diagram
Misconceptions
o Expressions are evaluated from left to right without consideration of the type of operation.
o Students do not use the correct order of operations.
o When writing expressions…students do not place parentheses correctly.
o Students reverse the dividend and divisor when writing a division problem to represent fractions as division.
*These instructional maps were created and developed using information from Engage NY, Shelby County Schools, Saugus Public Schools,
Somerville Public Schools, and www.mdk12.org.
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Holyoke Public Schools: Grade 5 Curriculum Map
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Students may confuse the numerator and denominator when making equal groups.
Students may multiply the whole-number factor and the denominator instead of the numerator.
Students may incorrectly find a fraction of the whole when modeling the first factor.
There may be confusion as to which factor is being compared to the product.
Students may confuse whether multiplying by fractions less than 1 or greater than 1 result in a product greater than the
factor.
Students may add the scaling factor to the side instead of multiplying it.
Recalling that multiplication and division are inverse operations, students divide a fraction by a whole number by
multiplying the fraction by the whole number.
Students reverse the dividend and the divisor when writing a division problem to represent a real-world situation.
When dividing fractions, students may transpose numbers.
Resources
GoMath
See Chapters 1, 7, 8
Engage NY
https://www.engageny.org/resource/grade-5-mathematics
Unpacked Standards
http://www.ncpublicschools.org/docs/acre/standards/common-core-tools/unpacking/math/5th.pdf
Model Lessons by Standard
http://www.mdk12.org/instruction/curriculum/mathematics/index.html
Performance Tasks:
http://3-5cctask.ncdpi.wikispaces.net/5.NBT.1-5.NBT4
*These instructional maps were created and developed using information from Engage NY, Shelby County Schools, Saugus Public Schools,
Somerville Public Schools, and www.mdk12.org.
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