A Story of Units
A Story of Units
Grade 4 – Module 5
Fraction Equivalence, Ordering, and Operations
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A Story of Units
Session Objectives
• Examination of the development of mathematical
understanding across the module with a focus on the
Concept Development within the lessons
• Introduction to mathematical models and
instructional strategies to support implementation of
A Story of Units
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A Story of Units
Agenda
Introduction to the Module
Concept Development
Module Review
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A Story of Units
Curriculum Overview of A Story of Units
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A Story of Units
Module Overview
• Read the Narrative, Standards, Lesson Objectives,
and Vocabulary.
• What is familiar to you and your students?
• What will be new learning?
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A Story of Units
Mid-Module Assessment
• Study this document, and the End Module
Assessment, prior to teaching the Lessons.
• Provides a snapshot of the concepts taught over the
next 21 Lessons.
• Scan the Mid-Module Assessment to
get an idea of what topic we are
about to cover.
Topics A - D
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A Story of Units
Agenda
Introduction to the Module
Concept Development
Module Review
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A Story of Units
Topic A: Decomposition and Fraction
Equivalence
• 6 lessons to show fraction equivalence
through decomposition of fractions
• Tools: number bond, folding paper
strips, tape diagram, area model
4.NF.3b, 4.NF.4a
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A Story of Units
Decomposition of Fractions
• Use a strip of paper to represent, for
example, thirds and sixths.
• Show how a non-unit fraction is the
sum of unit fractions.
• Extend learning to show that nonunxxit fractions can be expressed as
the sum of two or more fractions
(unit or non-unit).
Lesson 1, Concept Development
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A Story of Units
Decomposition of Fractions
• Extend learning to fractions representing
numbers greater than one.
• Use tape diagrams to represent fractional parts.
• Discover that there are multiple ways to
decompose fractions.
Lesson 1, Concept Development
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A Story of Units
Debrief: How is decomposing a non-unit fraction similar to
decomposing whole numbers in Kindergarten and Grade 1?
Lesson 2, Concept Development
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A Story of Units
Decomposition of Fractions
• Express a non-unit fraction
as a whole number times a
unit fraction.
Lesson 3, Concept Development
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A Story of Units
Decomposition of Fractions
• Decompose fractions
into sums of smaller
unit fractions.
• Draw and label to
show the equivalence
of fractions.
Lesson 4, Concept Development
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A Story of Units
Decomposition of Fractions
Lesson 5, Application Problem
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A Story of Units
Decomposition of Fractions
• Show fraction
equivalence using
an area model.
• Model equivalence by
drawing an area model.
Lesson 5, Concept Development
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A Story of Units
Decomposition of Fractions
• Create equivalent fractions by drawing an area
model and then dividing it into smaller parts.
Lesson 6, Concept Development
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A Story of Units
Decomposition of Fractions
Turn and Talk to those sitting at your table:
• How can decomposing fractions lead to better
understanding of the representation of non-unit
fractions and to fraction equivalence?
• How do number bonds, tape
diagrams, and area models aid in
the understanding of the
decomposition of fractions?
Topic A Debrief
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A Story of Units
Topic B: Fraction Equivalence Using
Multiplication and Division
• 5 lessons to show fraction equivalence
using multiplication and division.
• New Terms: numerator, denominator
• Tools: tape diagrams, area model,
number line
4.NF.1
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A Story of Units
Fraction Equivalence
Lesson 7, Application Problem
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A Story of Units
Fraction Equivalence
• Decompose the area model to show equivalence.
• Create equivalent fractions by multiplying the
numerator and denominator by the same number.
Lessons 7 and 8, Concept Development
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A Story of Units
Fraction Equivalence
Lesson 11, Application Problem
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A Story of Units
Fraction Equivalence
• Compose larger
fractional units.
• Discover that the
numerator and
denominator can be
divided by the same
number to model
equivalence.
Lesson 9, Concept Development
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A Story of Units
Fraction Equivalence
• Use knowledge of
common factors to show
equivalent fractions.
• Show equivalence using
the greatest common
factor.
Lesson 10, Concept Development
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A Story of Units
Fraction Equivalence
• Use a number line to show equivalence.
Lesson 11, Concept Development
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A Story of Units
Fraction Equivalence
Turn and Talk to those sitting at your table:
• How does the modeling used within this Topic help
students to conceptually understand fractional
equivalence?
Topic B
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A Story of Units
Topic C: Fraction Comparison
• 4 lessons to show comparison of
fractions using benchmark numbers
and common units
• New Term: benchmarks
• Tools: number line, tape diagrams,
area models, number bonds
4.NF.2
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A Story of Units
Fraction Comparison
• Think about
fractions in relation
to 0, ½ , and 1, first
in isolation and
then in fraction
comparison.
Lesson 12, Concept Development
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A Story of Units
Fraction Comparison
• Compare two fractions greater than one using
benchmarks.
Lesson 13, Concept Development
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A Story of Units
Fraction Addition and Subtraction
• Subtraction of a fraction from a
whole using different methods
Lesson 17, Concept Development
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A Story of Units
Fraction Comparison
• Common numerators
• Numerators with related units
Lesson 14, Concept Development
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A Story of Units
Fraction Comparison
• Denominators with
related units
Lesson 14, Concept Development
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A Story of Units
Fraction Comparison
•
•
•
Denominators with
unrelated units
Use of the area model to
create common units for
comparison
Use of number bonds to
‘pull out’ a whole and then
compare the parts
Lesson 15, Concept Development
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A Story of Units
Fraction Comparison
Turn and Talk:
• What methods of fraction comparison are familiar?
• What methods of fraction comparison are new?
• How can each of the methods be used to help
students as they compare fractions?
Topic C
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A Story of Units
Topic D: Fraction Addition and Subtraction
• 6 lessons to show addition and
subtraction of fractions with an
emphasis on common units
• New Terms: common denominators,
mixed units
• Tools: number line, tape diagram,
number bond
4.NF.3a, 4.NF.4d
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A Story of Units
Fraction Addition and Subtraction
• Unit language is
used to relate
addition and
subtraction of
fractions to
addition and
subtraction of
whole numbers.
Lesson 16, Concept Development
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A Story of Units
Fraction Addition and Subtraction
• Conversion from a
fraction greater
than one to a mixed
number using a
number bond
Lesson 16, Concept Development
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A Story of Units
Fraction Addition and Subtraction
Students work in
groups to find
multiple ways of
solving
Lesson 18, Concept Development
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A Story of Units
Fraction Addition and Subtraction
Lesson 19, Application Problem
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A Story of Units
Fraction Addition and Subtraction
•
•
Students use multiple solution strategies to solve.
Use RDW to solve. Try solving 2 ways.
Lesson 19, Concept Development
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A Story of Units
Fraction Addition and Subtraction
• Adding with related units
Lesson 20, Concept Development
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A Story of Units
Fraction Addition and Subtraction
Lesson 21, Concept Development
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A Story of Units
Fraction Addition and Subtraction
Turn and Talk:
• How is subtracting fractions similar to subtracting
whole numbers?
• Discuss the progression and the importance of the
number bond model across grades.
Topic D
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A Story of Units
Topic E: Extending Fraction Equivalence to
Fractions Greater Than One
• 7 lessons working with fractions
greater than one to rename, order,
and compare.
• New Terms: line plot
• Tools: number line, tape diagram,
number bond, line plot
4.NF.1, 4.NF.2, 4.NF.3
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A Story of Units
Extending Fraction Equivalence to Fractions >1
Lesson 22, Concept Development
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A Story of Units
Extending Fraction Equivalence to Fractions >1
Lesson 23, Concept Development
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A Story of Units
Extending Fraction Equivalence to Fractions >1
• Rename fractions as
mixed units using
decomposition
• Convert using
multiplication
Lesson 24, Concept Development
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A Story of Units
Extending Fraction Equivalence to Fractions >1
Use RDW to solve:
Lesson 25, Application Problem
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A Story of Units
Extending Fraction Equivalence to Fractions >1
• Convert a mixed number to a fraction greater
than one.
Lesson 25, Concept Development
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A Story of Units
Extending Fraction Equivalence to Fractions >1
• Compare fractions greater than one by reasoning
using benchmark numbers.
Lesson 26, Concept Development
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A Story of Units
Extending Fraction Equivalence to Fractions >1
• Comparison with related
denominators
• Comparison with unrelated
denominators
•
Lesson 27, Concept Development
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A Story of Units
Extending Fraction Equivalence to Fractions >1
Plot fractions on a line plot and then
use the line plot to help solve word
problems.
Lesson 28, Concept Development
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A Story of Units
Extending Fraction Equivalence to Fractions >1
Turn and Talk:
• How does the content of this Topic build on the
learning of Topics A-D? How does the content of A-D
help students realize success with this Topic?
• Which of the strategies from this
module have you previously taught?
Topic E
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A Story of Units
Topic F: Addition and Subtraction of
Fractions by Decomposition
• 6 lessons working with fractions
greater than one to estimate and find
sums and differences
• Tools: number line, number bond
4.NF.3c, 4.NF.3d, 4.MD.4
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
• Rounding is used to
establish an estimated
sum or difference.
Lesson 29, Concept Development
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
• Use different strategies to add fractions
Lesson 30, Concept Development
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
Lesson 30, Concept Development
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
Addition of mixed numbers using different strategies
Lesson 31, Concept Development
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
Subtraction of a fraction from a mixed number using
different strategies
Lesson 32, Concept Development
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
Subtraction of a fraction from a mixed number using
different strategies
Lesson 32, Concept Development
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
Subtract a mixed number from a mixed number using
multiple strategies
Lesson 33, Concept Development
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
Subtract with mixed numbers
Lesson 34, Concept Development
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A Story of Units
Addition & Subtraction of Fractions by Decomposition
Turn and Talk:
• What from this Topic was new learning?
• What advantages can you see in introducing multiple
strategies to students?
Topic F
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A Story of Units
Topic G:
Repeated Addition of Fractions as Multiplication
• 6 lessons focusing on addition of
fractions as multiplication
• Tools: number line, tape diagram,
number bond, line plot
4.NF.4abc, 4.MD.4
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A Story of Units
Repeated Addition of Fractions as Multiplication
Represent the multiplication of n times a/b as (n×a)/b
using the associative property and visual models.
Lesson 35, Concept Development
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A Story of Units
Repeated Addition of Fractions as Multiplication
Patti needs 6 pieces of yarn that are each 3 eighths
meter long. How much yarn does Patti need?
Lesson 36, Concept Development
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A Story of Units
Repeated Addition of Fractions as Multiplication
Lesson 37, Application Problem
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A Story of Units
Repeated Addition of Fractions as Multiplication
Lesson 37 and 38, Concept Development
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A Story of Units
Repeated Addition of Fractions as Multiplication
Solve multiplicative comparison word problems.
Lesson 39, Concept Development
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A Story of Units
Repeated Addition of Fractions as Multiplication
Solve word problems involving the multiplication of a whole
number and a fraction including those involving line plots.
Lesson 40, Concept Development
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A Story of Units
Repeated Addition of Fractions as Multiplication
Turn and Talk:
• How do the strategies taught within this Topic help
students to be successful when solving problems
involving a whole number times a mixed number?
Topic G
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A Story of Units
Topic H: Exploration
Find and use a pattern to calculate the sum of all fractional parts
between zero and one.
Students work together to solve and
critique the thinking of others.
4.OA.5
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A Story of Units
Complete the End-of-Module Assessment.
• Now with all of the mathematical knowledge and
understanding of the models, review the
assessment.
• You may work alone or with your table to discuss
challenges or successes students and teachers may
face.
• How do the Topics in this Module help to prepare
students for success?
End-of-Module Assessment
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A Story of Units
Agenda
Introduction to the Module
Concept Development
Module Review
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A Story of Units
Biggest Takeaway
Turn and Talk:
•
•
What questions were answered for you?
What new questions have surfaced?
© 2012 Common Core, Inc. All rights reserved. commoncore.org
A Story of Units
Key Points
• The learning of fractions follows the same instructional
sequence as the learning of whole numbers.
• Fractional units behave just as all other units do, and can be
manipulated like whole numbers.
• Decomposing fractions strengthens the part-whole
relationship.
• Fraction equivalence and comparison are supported by visual
models.
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