Module Focus: Grade 4 – Module 5
Sequence of Sessions
Overarching Objectives of this February 2014 Network Team Institute

Module Focus sessions for K-5 will follow the sequence of the Concept Development component of the specified modules, using this narrative as a tool
for achieving deep understanding of mathematical concepts. Relevant examples of Fluency, Application, and Student Debrief will be highlighted in
order to examine the ways in which these elements contribute to and enhance conceptual understanding.
High-Level Purpose of this Session

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

Focus. Participants will be able to identify the major work of each grade using the Curriculum Overview document as a resource in preparation for
teaching these modules.
Coherence: P-5. Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that
develop within each module, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the
same.
Standards alignment. Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module
addresses the major work of the grade in order to fully implement the curriculum.
Implementation. Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their
students while maintaining the balance of rigor that is built into the curriculum.
Related Learning Experiences
●
This session is part of a sequence of Module Focus sessions examining the Grade 4 curriculum, A Story of Units.
Key Points

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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.
Session Outcomes
What do we want participants to be able to do as a result of this
session?




Focus. Participants will be able to identify the major work of each grade
using the Curriculum Overview document as a resource in preparation
for teaching these modules.
Coherence: P-5. Participants will draw connections between the
progression documents and the careful sequence of mathematical
concepts that develop within each module, thereby enabling participants
to enact cross- grade coherence in their classrooms and support their
colleagues to do the same . (Specific progression document to be
determined as appropriate for each grade level and module being
presented.)
Standards alignment. Participants will be able to articulate how the
topics and lessons promote mastery of the focus standards and how the
module addresses the major work of the grade in order to fully
implement the curriculum.
Implementation. Participants will be prepared to implement the
modules and to make appropriate instructional choices to meet the needs
of their students while maintaining the balance of rigor that is built into
the curriculum.
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.
Session Overview
Section
Time
Overview
Prepared Resources
Introduction to the
Module
20 mins
Establish the instructional focus of
Grade 4 Module 5.
 Grade 4 Module 5 PPT
 Facilitator Guide
Topic A Lessons
40 mins
Examine the lessons of Topic A.
 Grade 4 Module 5 PPT
 Grade 4 Module 5 Lesson
Excepts
Topic B Lessons
20 mins
Examine the lessons of Topic B.
 Grade 4 Module 5 PPT
 Grade 4 Module 5 Lesson
Excepts
Facilitator Preparation
Review Grade 4 Module 5.
Review Topic A.
Review Topic B.
Topic C Lessons
40 mins
Examine the lessons of Topic C.
 Grade 4 Module 5 PPT
 Grade 4 Module 5 Lesson
Excepts
Review Topic C.
Review Topic D.
Topic D Lessons
30 mins
Examine the lessons of Topic D.
 Grade 4 Module 5 PPT
 Grade 4 Module 5 Lesson
Excepts
Topic E Lessons
40 mins
Examine the lessons of Topic E.
 Grade 4 Module 5 PPT
 Grade 4 Module 5 Lesson
Excepts
Review Topic E.
Review Topic F.
Review Topic D.
Topic F Lessons
25 mins
Examine the lessons of Topic F.
 Grade 4 Module 5 PPT
 Grade 4 Module 5 Lesson
Excepts
Topic G Lessons
20 mins
Examine the lessons of Topic C.
 Grade 4 Module 5 PPT
 Grade 4 Module 5 Lesson
Excepts
20
Reiterate the key points of the
lessons and faciliatate a discussion
of the final exploratory lesson and
end-of-module assessment.
Summary / Closing
 Grade 4 Module 5 PPT
Review Topic H and End-ofModule Assessment.
Session Roadmap
Section: Grade 4 Module 5
Time: 255 minutes
[255 minutes] In this section, you will…
Materials used include:
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
1 min
Welcome! In this Module Focus Session, we will
examine Grade 4 – Module 5.
1.
GROUP
1 min
2.
Our objectives for this session are the following:
•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
1 min
3.
We will begin by exploring the Module Overview to
understand the purpose of this module, and then we
will dig into the math of the module. We’ll lead you
through the teaching sequence, paying close attention
to the mathematics that is taught within the Concept
Developments and to the progression of the
mathematics as students move through each of the
lessons. We’ll examine the other lesson components as
well and how they function in collaboration with the
Concept Development. Finally, we’ll take a look back at
the module, reflecting on all the parts as one cohesive
whole.
1 min
4.
The fifth module in Grade 4 is Fraction Equivalence,
Ordering, and Operations. This module includes 41
lessons and is allotted 45 instructional days.
This module builds on understandings established in
Module 5 of Grade 3, Fractions as Numbers on the
Number Line. It also incorporates understandings of
multiplication and division from Grade 3 and from
Module 3 of Grade 4. This module prepares students
for work in Module 6 where they will extend their
learning to express decimal fractions as decimals and to
add and subtract decimal fractions. Students will also
build the foundational knowledge necessary for success
with fractions in Grade 5.
10
min
5.
Allow 8 minutes for participants to read through the
Module Overview. Encourage participants to underline
familiar components. Highlight what is new or
unfamiliar for teachers and/or students.
This session will strive to cover many of the unfamiliar
components in the module including the models and
methods used to build a deeper conceptual
understanding of fractions. Further information about
fractions can be found in the Progressions. In order to
allot more time to the models of this module, we will
not be reading the Progressions.
3 min
6.
Studying this document prior to teaching the first half
of the module allows a teacher to see where he/she
needs to take the students. Sometimes teachers can get
stuck on a lesson where students may not be reaching
mastery. Remember to think what lies ahead and how
reaching success in each lesson, not necessarily 100%
mastery, will allow teachers to stay on pace, and allow
students to continue learning new concepts as they
strengthen the ones from the days before. This module
moves slowly and comprehensively across the fraction
standards.
Take a quick glance at the mid-module assessment to
see where we will be going as we study the
mathematics in the 1st half of the module.
1 min
7.
We will now take a look at the lessons, focusing on each
Concept Development. The goal is for you to get a
general sense of the module and of the progression of
concepts throughout.
1 min
8.
Topic A builds on student learning of unit fractions
from Grade 3. Students work with non-unit fractions,
decomposing them to find the unit fraction equivalence.
Students express fractions using addition and
multiplication. Paper folding activities work, at the
concrete level, to bridge Grade 3 knowledge of work
with unit fractions. Tape diagrams and the area model
are used to support learning and conceptual
understanding at the pictorial level. Students are able
to ‘see’ the equivalence rather than simply learning the
‘tricks’.
6 min
9.
Complete a paper folding activity to begin the slide:
1.Decompose 1 strip into thirds.
a) Draw a number bond to represent the whole
decomposed into thirds.
b) Ask for a number sentence. (1 = 1/3 + 1/3 + 1/3)
c) Place your finger between the 2nd and 3rd units. Ask for
a new number sentence. (1 = 2/3 + 1/3)
2.Decompose 1 strip into sixths, shade 5 sixths. (Repeat
with steps a-c.)
3.Decompose 2 strips into fourths, shade 7 fourths.
(Repeat with steps a-c.)
(Click to advance the slide.) Each bullet appears with a
corresponding image.
We begin the module with a direct reference to models
and understanding from Grade 3 - Module 5 where
students used paper strips and number bonds to
identify and compose fractions. Now we are
introducing them to the additive nature of fractions.
1 min
10.
As we saw with our fraction strip representing 1 ¾, we
can also decompose fractions greater than 1 whole.
Moving past the concrete stage of the paper folding
strips, we represent fractions pictorially as tape
diagrams, which was also done in Grade 3. Here we
introduce students to modeling fractions greater than 1
with tape diagrams. We also discover that there can be
multiple ways to decompose a fraction. We move from
the pictorial number bond to a more abstract
representation, horizontal addition number sentences.
3 min
11.
Lesson 2 is a continuation of Lesson 1. Here students
are called upon to find the multiple decompositions of a
given fraction.
(Click to advance the slide.)
Debrief: Think to yourself, how is decomposing a
fraction similar to the work done in Grades K and 1
with whole numbers?
Turn and talk. Will anyone share?
(GK-1 decompose whole numbers, first using counting
pieces, then using 5 or 10 frames, and then using number
bonds as they prepare to write addition and subtraction
number sentences.)
2 min
12.
Students can express non-unit fractions as the sum of
unit fractions, which they have been doing for two
lessons, but now they use multiplication. This is
familiar territory for them, as:
3 bananas = 1 banana + 1 banana + 1 banana = 3 x 1
banana, or
3 tens = 1 ten + 1 ten + 1 ten = 3 x 1 ten.
Easily, students make the connection to these prior
experiences and the visual of a tape diagram to see 3
fourths equals 3 times 1 fourth.
Students also use the distributive property to
decompose 5 times 1 third into 2 multiplication
equations, thus “pulling out one”. This concept of
“pulling out one” is prevalent in this module,
specifically the 2nd half of the lessons, and is connected
to pulling out a ten or hundred to decompose when
subtracting.
2 min
13.
Just as we can decompose 1 ten into 10 ones or 1
hundred into 10 tens, we can decompose a unit fraction,
such as 1 fifth, into smaller units. Again, working with
the tape diagram and number bonds keeps this work
visually supportive and accessible for students. Using
what we learned in the first 3 lessons, the unit or nonunit fractions can be decomposed using addition or
multiplication, as shown in the 2nd graphic.
4 min
14.
(Click to advance the slide.) AP Problem appears.
Solve this problem using the models and learning from
the first 4 lessons of this module. Compare your
answers with your table partners.
(Click to advance the slide.) AP sample answer appears.
Here, a student drew a tape diagram to represent the
loaf of bread cut into 6 slices. Each slice is decomposed
into 2 equal pieces creating twelfths. The equivalence of
2/6 to 4/12 can be shown using multiplication or
division or addition.
5 min
15.
(Click 2 TIMES to advance the slide.)
Lesson 5 is the first lesson in the module and the
curriculum where the area model is introduced to
decompose fractions into smaller units. The area
represents 1 whole and can be decomposed vertically,
as shown here, into fifths. A horizontal line further
decomposes the fifths in half, creating tenths. This is a
consistent model used in G4 to find equivalent fractions
and is used as well as in G5-M3 to add fractions with
related denominators.
(Click to advance the slide.) (Switch to document
camera.)
Model for participants how to show ½ = 5/10.
12
min
16.
Lesson 6 is a continuation of Lesson 5. Lesson 6 focuses
on decomposing non-unit fractions to find equivalent
fractions.
Call attention to the importance of labeling the whole.
The first two area models don’t show the whole, but, with
the fractions below it, one could determine that the first
image is ¾. Without the fraction labeling the area model,
the first image could also represent 3/2.
Instruct participants to complete 1-7 of the Problem Set.
Allow 10 minutes to do so.
Encourage participants to share their work with table
partners as opposed to providing an ‘answer key’. This
process would then mirror what occurs in a classroom
during a Debrief – students sharing work and critiquing
peers (MP3 and MP6).
4 min
17.
Allow 3 minutes for participants to discuss at tables.
Allow 1 minute for share out.
Sample answers:
•Decomposing fractions leads to better understanding of
the representation of non-unit fractions and to fraction
equivalence because they are able to use reasoning to
explain why two different fractions can represent the
same part of a whole.
1 min
18.
Topic B builds on student learning from Topic A. In
Topic B, students begin to generalize and to see that
multiplication and division can be used to create
equivalent fractions. The terms numerator and
denominator are introduced to the students in Lesson
7. The learning is supported throughout Topic B by use
of the area model and tape diagrams and, finally, with
the number line.
1 min
19.
The example shown here of student work for this
Application Problem is supported by student
understanding from Topic A. It allows students to see
that there is more than one way to decompose a
fraction. Here, each seventh is decomposed in half and
in thirds. Similarly, one could decompose sevenths into
fourths, fifths, tenths, or hundredths. That is not,
however, realistic to show in an area model. Let’s use
Topic B to discover a more efficient method using
multiplication and division.
3 min
20.
Lessons 7 and 8 develop the concept of finding
equivalent fractions, those with smaller units, using
multiplication. First, an area model is decomposed and
evaluated. Then students are introduced to the concept
that each unit was doubled or tripled and, thus, the
units selected were doubled or tripled. When we double
or triple whole numbers, we multiply. So here we
multiply the numerator by 2 or 3, and we multiply the
denominator by 2 or 3.
The last image shows how students must reason using
an area model or multiplication when a number
sentence may be untrue, such as ¾ being equivalent to
6/12.
4 min
21.
Although much of the module’s work is with abstract
numbers, all Application Problems put the learning into
a context for students. It is important to stress the realworld connections we can make with fractions. That
connection is also brought out in particular Debrief
questions.
Solve this Application Problem and discuss your
solutions with those at your table.
(Click to advance the slide.)
Solution 1 shows finding equivalent fractions using
multiplication as done here in Topic B. Solution 2 refers
back to both Grade 3 and Topic A of Grade 4 using
number bonds as a model.
Accept all reasonable answers and explanations, and
probe students to explain their thinking. This
encourages development of many mathematical
practices.
1 min
22.
Starting with non-unit fractions, students compose
larger fractional units using an area model. Relating this
work to decomposing fractions, they find division can
be used as an abstract method to find an equivalent
fraction.
1 min
23.
Students consider the numerator and the denominator
as a single number when composing. “What must be
done to the top number, must be done to the bottom
number.” Using what they learned about factors in
Module 3, they can quickly determine equivalent
fractions. Finding the greatest common factor of both
the numerator and the denominator helps us to
simplify fractions. Remember, the CCSS do not require
students to simplify fractions unless it aids in finding a
solution or in making sense of a problem. Here 8/12
can also be written as 2/3 or 4/6. Just as we learned in
younger grades that addition and subtraction are
related operations and are taught together, so must the
composition and decomposition of numbers be taught
in union, so as to build greater fluency in fractions.
7 min
24.
Another pictorial model used to show the equivalence
of fractions is the number line. The number line is
introduced by drawing it directly below a tape diagram
so that students may see the decomposition, or
composition, is the same.
Allow participants 5 minutes to complete #s 8-10 of the
Problem Set.
Encourage participants to share their work with table
partners as opposed to providing an ‘answer key’. This
process would then mirror what occurs in a classroom
during a Debrief – students sharing work and critiquing
peers (MP3 and MP6).
3 min
25.
Allow 2 minutes for participants to discuss at tables.
Allow 1 minute for share out.
Sample Answer:
•The models allow students to clearly SEE that although
equivalent fractions may “look” very different they
represent the same portion of the whole.
1 min
26.
The focus of Topic C is comparison of fractions. The
topic begins with comparing fractions using benchmark
numbers, such as 0, ½ and 1. Students then reason
about the comparison of fractions using common units.
Number lines and tape diagrams can be decomposed to
show or plot the related denominators for comparison.
Finally, students use the area model to show
equivalence for denominators that are not related.
3 min
27.
Present the following script from Problem 1 of Lesson 12
to engage participants.
2 min
28.
Practice with the benchmarks between 0 and 1 allows
students to compare larger fractions.
The first number line asks for students to recognize
fractions greater than one using whole numbers as
benchmarks. The second number line compares
fractions with the same whole number but different
fractional units.
Students can pull out the whole number from mixed
numbers or improper fractions using a number bond. If
the whole numbers are equal to each other, the
fractional parts can be compared, referring to their
Lesson 12 learning.
Here both fractions have a whole number of 1. It is
easiest, therefore, to simply compare the fractional
units of 3/8 and 4/6 which they learned in the previous
lesson.
3 min
29.
Subtraction with fractions can be complicated. Always
converting to an improper fraction is not always a
viable method, especially as we begin to work with
larger numbers. We show a variety of methods that
students can employ and apply to any numbers.
(Click to advance.)
Lesson 17 begins with subtracting from 1. Easily the
whole number can be renamed as a fraction, such as 5
fifths, and the students can subtract.
4 min
30.
Use the following script, adapted from Problem 1 of
Lesson 14, to introduce the next lesson.
T:
Which is greater, 1 apple or 3 apples?
S:
3 apples.
T:
Which is greater, 1 fourth or 3 fourths?
S:
3 fourths.
T:
What do you notice about these
statements? (Hold up white board with comparisons.)
S:
The units are the same.  It is easy to
compare because the units are the same.
T:
Which is greater, 1 fourth or 1 fifth?
S:
I can’t compare them. The units aren’t the
same.  1 fourth because fifths are
smaller fractional units than fourths.
T:
Which is greater, 2 fourths or 2 sixths?
S:
2 fourths.
T:
What do you notice about these
statements? (Hold up white board with
comparisons.)
S:
The numerators are the same.
(CLICK TO ADVANCE THE SLIDE.)
Here, we see a student comparing two fractions with
unlike denominators, but related numerators. 1 seventh
is greater than 1 twelfth, sevenths having a larger
fractional size than twelfths. 5/7 is, therefore, greater
than 5/12.
(CLICK TO ADVANCE THE SLIDE.)
On occasion, it is possible for students to compare
fractions with related numerators. Students can make
related numerators into like numerators by
decomposing or composing a fraction, as seen in the
second image. Although this method is valuable and
valid, it is not stressed and is not a major component of
this Topic.
5 min
31.
Comparing fractions with related denominators is the
subject of the remainder of Lesson 14. A student can
compare 3/5 to 7/10 easily when finding that fifths and
tenths are related. Decomposing fifths into tenths can
be done using a tape diagram or a number line. Once
both fractions have like units, the comparison is simple.
Students are armed with several methods for
comparing any two fractions and practice their
reasoning during the Concept Development and Debrief
of the lesson.
(Switch to document camera.)
With participants following along, model how to draw
tape diagrams and a number line to show the
comparison of 5/6 and 9/12.
After drawing ask:
1.How could I reason about these fractions without
drawing a model?
2.Could changing 9/12 to ¾ simplify this problem?
3.How could I move to the abstract level to solve
without drawing a model?
4.How could I back up and support students at a
concrete level?
16
min
32.
(Click to advance the slide.)
Compare ¾ to 4/5.
Can I use benchmarks? (Yes, but fourths and fifths are
tricky to see.)
Can I use common numerators? (Sure, but they aren’t
related so I would have to decompose both fractions.)
Can I use common denominators? (Sure, but they aren’t
related so I would have to decompose both fractions.)
A tape diagram or number line is not the place to be
comparing fractions such as these with unrelated
denominators or unrelated numerators. To make like
units, it is easiest to use the area model. Developed
further in Grade 5 to add fractions with unrelated
denominators, decomposing area models to make like
units for two fractions is introduced in Lesson 15.
(Click to advance the slide.)
First, an area model is drawn for each fraction with
vertical lines to decompose one area and horizontal
lines to decompose the other. Next, students overlap
each model with the other model’s lines. Fifths are
drawn on the ¾ model. Fourths are drawn on the 4/5
model. Each model now is composed of twentieths.
(Click to advance the slide.)
When numbers start to increase their value and we
must compare fractions greater than 1, number bonds
are used to pull out the whole and parts of improper
fractions, thus allowing students to see if the wholes
are the same, then the parts can be compared.
Note this lesson is heavily weighted with pictorial
representations to present the math. By the end of
Lesson 15, students are encouraged to begin comparing
fractions with unrelated denominators abstractly by
using multiplication to create like units for both
fractions in order to compare. A concrete
representation could use paper area models that can be
cut and rearranged for students to count the number of
units.
(Switch to document camera.)
Model for participants how to compare 2/3 and ¾.
Allow participants 9 minutes to complete Problem Set
#11- 17.
4 min
33.
Allow 3 minutes for participants to discuss at tables.
Allow 1 minute for share out.
Sample answers:
•Comparison of fractions with like numerators and
denominators, and the use of a tape diagram are familiar
concepts.
•Fraction comparison using benchmarks, area models,
and number lines are new.
•Each method shines a different light on the skill of
comparing fractions .Exposure to and practice with all of
the methods will enable to students to choose the one
most appropriate to use when they need to compare
fractions.
1 min
34.
The focus of Topic D is addition and subtraction of
fractions. Students work with fractions less than one
and fractions greater than one up to two. Addition and
subtraction of fractions begins with common units and
then extends to related units. Students apply what they
have learned to solve word problems involving the
addition and subtraction of fractions.
3 min
35.
(Click to advance the slide.)
The language of units is imperative to stress in the
younger grades so its importance can once again be
strengthened and applied when working with fractions.
T:
1 banana plus 2 bananas equals?
S:
3 bananas.
T:
1 banana plus 2 apples equals?
S:
I can’t solve. The units are not like.
T:
How can I make like units?
S:
Both are fruits. 1 piece of fruit plus 2 pieces of
fruit equals 3 pieces of fruit.
(Click to advance the slide.)
Lesson 16 begins with relating units used in previous
modules, like ones and meters, to the addition and
subtraction of fractions.
A number line is used to “slide” left or right to model
the addition and subtraction.
The new term ‘mixed units’ is introduced in this lesson,
allowing students to communicate precisely when
renaming fractions is necessary or possible. For
example, 9 fifths minus 3 fifths equals 6 fifths. A
number bond can show us that 6 fifths can be renamed
as 1 and 1 fifth by pulling out one.
1 min
36.
Again, the number bond can be used to rename the final
sum or difference. It is not, however, required that
students do so. When applying these sums and
differences to word problems, a context will be clearer
if a student reports they drank 1 and ¾ cups of milk
rather than 7/4 cups of milk which is an
unconventional way to report the sum based on a
context.
2 min
37.
In this lesson, students work together to show multiple
ways of adding and subtracting more than two
fractions. Collaboration and discussion as well as
critical analysis are all important parts of this lesson.
Particular attention to making one or pulling out one is
a focus as it will strengthen their fraction fluency for
the 2nd half of lessons which include larger numbers.
3 min
38.
In this lesson, students work together to show multiple
ways of adding and subtracting more than two
fractions. Collaboration and discussion as well as
critical analysis are all important parts of this lesson.
Particular attention to making one or pulling out one is
a focus as it will strengthen their fraction fluency for
the 2nd half of lessons which include larger numbers.
4 min
39.
Lesson 19 is a Word Problem Lesson wherein students
use the Problem Set of word problems to solve in class.
As students gain fluency with fractions, various solution
strategies will arise. Share different strategies, as one
may be more efficient than another strategy, or a
student may learn a new strategy they had yet to
consider.
(Click to advance the slide.)
Use Read, Draw, Write to solve. If you finish early, try
finding an alternative solution strategy.
(Click to advance the slide.)
Read the problem with participants.
Encourage them to first draw what they know. Hint that
a bar model would be acceptable.
Allow 2 minutes to work.
(Click to advance the slide.)
Show Solution 1. Discuss it is straightforward adding
and renaming the sum.
(Click to advance the slide.)
Shows Solution 2. Discuss it is pulling out one to make a
whole and a part before finding the sum. This solution
also renames to a larger unit, 3/5.
(Click to advance the slide.)
Students should always write a response so as to put
their answer back into context to check for a reasonable
response.
3 min
40.
A special note: the Grade 4 standards limit students to
adding fractions with like units. However, 4.NF.5 asks
students to add tenths and hundredths by converting
tenths to hundredths. This work is limited mostly to
work with decimal fractions and will be heavily covered
in Module 6. To prepare students for this work, and to
use the work of converting fractions to different units
learned in previous Topics, Lessons 20 and 21 extend
beyond Grade 4 expectations to add fractions with
related units. It is possible to skip these lessons, not
effecting the remainder of the module’s learning, if your
students are struggling with Grade level concepts.
However, challenge others with this work as you
remediate others. This concept is not assessed. It will
be revisited in Grade 5 Module 3.
(Click to advance the slide.)
Again, related units would be ones such as thirds,
sixths, and twelfths. Or halves, fourths, and eighths. Or
halves, fourths, and twelfths.
(Click to advance the slide.)
Beginning pictorially, students rename thirds as sixths,
which is previous grade level work, to add like units
using tape diagrams.
(Click to advance the slide.)
Pictorially, this time on a number line, students
decompose and rename sixths as twelfths, sliding along
the number line and recording numerically.
(Click to advance the slide.)
Abstractly, students can use multiplication to rename
fractions, such as fifths as tenths and, again, add
numerically.
12
min
41.
In Lesson 21, tape diagrams, number lines, and number
bonds are used to solve addition problems involving
fractions with related units whose sum is greater than
1.
The image on the left shows a tape diagram model for
adding fractions. The number bond is used to rename
the sum as a mixed number.
The image on the right shows the number line model
and, again, the number bond to assist in the renaming
of the sum.
Note that subtraction with related units is not taught
because subtraction of decimal fractions is not a Grade
4 standard.
Allow participants 10 minutes to complete Problem Set
#18 - 24.
4 min
42.
Allow 3 minutes for participants to discuss at tables.
Allow 1 minute for share out.
Sample Answers:
•The use of unit language in this module enables students
to relate subtraction of fractions to their work with
subtracting whole numbers. Students still see a part,
part, whole relationship, as with whole numbers,
modeled with the tape diagram.
•Because the number bond is such a familiar model for
students by Grade 4, using it with fractions is
comfortable transition for them. They easily see the
appropriateness of using this model to express these new
“part-part-whole” relationships. It shows more
connectedness of fractions and whole numbers.
1 min
43.
Now we have reached the 2nd half of the module. The 1st
half of the module kept all fractions within 2 - All sums,
all fractions decomposed, all wholes were never greater
than 2. It’s purposeful, just as in Kindergarten where
students aren’t expected to decompose 86 or 147 but
rather numbers 0 to 19. Now that students have spent
21 lessons building upon their Grade 3 experience with
fractions, they are prepared to decompose, find
equivalence, order, add, and subtract fractions with
greater values. This will strengthen their understanding
of fractions.
The focus of Topic E is to extend the learning from
Topics A-D and to relate it to fractions greater than 1.
At the end of the Topic, students apply what they have
learned about fractions to solve word problems
involving the addition and subtraction of fractions and
representation of the data on a line plot.
3 min
44.
The opening lesson of the 2nd half of the module has
students apply many concepts they are already familiar
working with, but now with larger numbers. Again, the
same models can be used, but as students strengthen
their fluency of fraction concepts, the work becomes
more and more abstract.
(Click to advance.)
Students add fractions to whole numbers and subtract
fractions from whole numbers using tape diagrams as
models, accompanied by a number sentence.
(Click to advance.)
Students are reminded, by writing fact family number
sentences, of how fractions and whole numbers behave
the same when adding and subtracting.
(Click to advance.)
A number line is an alternative model to the tape
diagram to show subtraction from a whole number.
A number bond also models for students that
subtracting a fraction less than one from a whole
number can simply be modeled as subtraction from one
and adding the remaining whole. The last image has a
student thinking about subtracting 5/12 from 12/12. 9
can be decomposed as 8 and 12/12. The remaining
whole is 8. The difference is 7/12. The answer is,
therefore, 8 7/12.
4 min
45.
Students must recall that multiplication is repeated
addition. Just as 2+2+2+2+2+2=12, so does 6x2=12.
Instead of adding 6 halves, we multiple 6 times 1 half.
The number line allows us to quickly see how we can
rename 6 halves as 3.
The associative property is then brought out to show
that 6 halves can actually be renamed as 3 x (2 halves).
This really helps to see that 6 times 1 half is 3.
(Click to advance.)
From here, students are ready to express a whole
number times a unit fraction as a mixed number.
Practice expressing 13 x 1/5 with participants.
Ask participants to brainstorm a context for this type of
problem.
(Takes 1/5 minute to run around his house, ran 13 times,
how long?)
(Making 1/5 pound burgers, 13 guests, how many pounds
to buy?)
2 min
46.
First, students use what they know about fraction
decomposition to rename improper fractions as wholes
and parts, shown with this number bond. 6 thirds is
renamed as 2 ones. The number line models the
addition of the wholes and parts to make 7/3, and
students find that 7/3 is equal to 2 1/3 on the number
line.
(Click TWICE to advance.)
Next, students simplify that method in conjunction with
what they learned in the previous lesson. Moving to the
abstract, students convert improper fractions to mixed
numbers using multiplication. This is the most efficient
method when working with larger values.
5 min
47.
Application Problems allow students to place this
fraction work within a context. Use RDW to solve this
problem.
Allow participants about 2 minutes to solve.
Share out solutions at tables or with the whole group.
Share student sample work by clicking to animate.
(Click to advance the slide.)
This student drew a number line showing the 13- 1
sixth mile laps. Decomposing 13 sixths as 12 sixths and
1 sixth, the students finds 2 1/6 is equivalent to 13/6.
(Click to advance the slide.)
Here a student shows how multiplication can be used to
find the same solution.
(Click to advance the slide.)
Finally, a statement is written to contextualize the
answer.
3 min
48.
Now students work in the opposite direction. Instead of
finding equivalent mixed numbers, the mixed number is
given and the fraction greater than one is solved for.
Students convert a mixed number to a fraction. First,
they use models. Next, using mental math instead, they
calculate without a model.
(Click to advance the slide.)
Working with the numbers from the Application
Problem, the Concept Development opens with
decomposing 2 1/6 into 12/6 and 1/6. Of course,
instead of the addition, one could associate the
numbers and use multiplication. A number line models
this work, counting by sixths to see 2 1/6 is equal to
13/6.
(Click to advance the slide.)
Soon, the work becomes more abstract, associating the
mixed number to find a fraction greater than one.
5 min
49.
Using their knowledge of benchmark fractions and
comparison and ordering of fractions from the 1st half
of lessons, students apply that work with larger-valued
fractions.
Work with a partner for 1 minute to solve this problem.
Don’t forget to work pictorially. Models not only
support the abstract work but provide reasoning for an
answer.
(Click to advance the slide.) Shows the word problem.
(After working…)
What are some strategies you and your partner used to
solve this problem?
(Renamed all fractions as fractions greater than
one.)
(Drew a number line and plotted points.)
(Renamed all fractions as eighths so I could work
with like units.)
(Thought about benchmarks of 0, ½, and 1.
Renamed all fractions as mixed numbers.)
(Renamed 3 6/8 as 3 ¾.)
(Click to advance the slide.)
Here is a student sample number line.
(Discuss the results and possible incomplete student
answer.)
6 min
50.
Comparing using tape diagrams for related
denominators or using the area model and creating
common denominators for unrelated denominators is
not new to students. But when faced with larger valued
fractions, mixed number or improper, students must
reason about the whole and parts of each number. If the
wholes are the same, the parts are compared using any
method they already learned. Students may also create
like denominators or like numerators using
multiplication.
(Click to advance the slide.)
Work with your partner to compare 7 3/5 to 7 4/6. Try
to find 2 solutions.
(Allow several minutes to work. Share the below
images by modeling the multiplication. Draw area
models for like denominators if helpful.)
7 min
51.
Students are introduced formally to line plots in Lesson
28. Here, students use the Problem Set as part of the
Concept Development. Students use measurement with
a ruler to precisely decompose a number line into
equal-sized units. A table with data is provided and
students plot points and answer questions related to
the line plot within a context.
Allow participants 10 minutes to complete Problem Set
#25- 31.
4 min
52.
Allow 3 minutes for participants to discuss at tables.
Allow 1 minute for share out.
Sample Answer:
•The content of this topic builds on the learning of Topics
A-D by giving students the opportunities to apply the
same models and strategies they used with fractions less
than one to fractions greater than one. Students are
successful because this opportunity to link new learning
to old.
1 min
53.
Topic F provides students with the opportunity to use
their understanding of fraction addition and
subtraction as they explore mixed number addition and
subtraction by decomposition. Students first estimate
the sums and differences and then work to find the
exact sums and differences using a variety of different
strategies. This Topic F closely mirrors Topic D from
the 1st half of lessons. Again, a variety of models are
used to bridge the work of the 1st half to this more
complex work with larger numbers.
2 min
54.
Students use number lines to roughly plot a rounded
point and add on. They round fractions to benchmark
numbers. They find that rounding fractions and
estimating the sums is usually quite accurate.
More care must, however, be taken when estimating a
difference.
In subtracting 4 8/9 minus 3 1/5, each fraction can be
rounded to the nearest whole because each is simply
one unit away from the whole and those units are
relatively small.
(Click to advance.)
3 ¾ minus 3 1/7, however, does not fall into the same
category. Here we show on the number line the actual
difference.
(Click to advance.)
Since each fraction is just 1 unit from the nearest whole,
if we round to the nearest whole, our estimated
difference actually increases, much beyond the actual
difference. By rounding one number up and another
number down, the estimate is not reasonable.
(Click to advance.)
Here we find that in rounding both numbers, still to
benchmarks but in the same direction of the number
line, the estimated difference will likely be much more
accurate and reasonable compared to the actual
difference. Students practice this reasoning with
several carefully chosen subtraction problems to
illustrate this fact.
How is this similar to whole number estimation of the
difference?
(Explain using this image.)
2 min
55.
Just as with addition of whole numbers, there are many
solution strategies in the addition of fractions. In
Kindergarten and Grade 1, we work on making 5 and
making 10 and then work to count on or make 10 by
decomposing and addend.
(Show example of how many more make 10. 7? (3.) Next,
7 + 5. Count on past 7- 8, 9, 10, 11, 12. Or decompose 5
into 3 and 2. 7 + 3 =10 and 10 +2 = 12.)
Counting in unit form, making 10, and counting on are
all whole number strategies to be applied to fraction
addition.
(Click to advance the slide.)
Unit Counting - This is similar to G4-M1 and the
addition algorithm. We add like units. The number line
models and supports this.
(Click to advance the slide.)
Making 10 - In fractions, we make a whole or the next
whole. For example, from 3 1/8, how many more
eighths make 4? (7 eighths.)
2 min
56.
Continuation from previous slide’s discussion…
(Click to advance the slide.)
Renaming units - 7 ones plus 5 ones is? (12 ones.) We
rename as 1 ten 2 ones. Here 5 fourths is renamed as
4/4 and ¼. 4/4 is 1 so the sum is 6 ¼.
(Click to advance the slide.)
Making 10 - We make 10 before adding by
decomposing an addend. 7 + 5 is rather 10 + 2. So by
decomposing ¾ to make a whole, we simplified the
addition.
(Click to advance the slide.)
Lastly, we can show counting on the “arrow way” which
is a strategy used in the lower grades and one used in
Module 2 with the addition of metric units to make the
next unit. Here we add 2/4 to make the next whole, 6.
1 min
57.
Here is where teachers may continue to see a difference
in the solution strategies that students use to solve.
When adding a mixed number to a mixed number, add
like units: ones with ones, fourths with fourths. The
length to which that is recorded is up to the mental
math a student may sustain. Some may need to record a
long chain of their work. Others may compute mostly in
their heads. Continue to probe students for reasoning
and understanding. Strategies for making one still apply
here, but we encourage students to add the ones first. If
needed, students may still choose to model with
number lines.
1 min
58.
Write on your white boards the following problem 34 –
13. Solve. (21.) Why was that so easy? (There were
enough of each unit to subtract.)
(Click to advance the slide.)
Are there enough fifths to subtract 3/5 from 3 4/5?
(Yes.) So the subtraction here is simple. A number line
can model this for us.
3 min
59.
On your personal boards solve 33-14. (19.) What was a
strategy used here to solve? (Rename 1 ten as 10 ones.)
(Click to advance the slide.)
Are there enough fourths to subtract? (No.) Just as we
did before, let’s decompose the subtrahend. 3/5 minus
3/5 is zero. Now the problem is 3-1/5 which we solved
before. We also show the counting up strategy to keep
in mind the relationship that subtraction and addition
have.
(Click to advance the slide.)
A different solution strategy used for when there are
not enough units is that we can decompose the
minuend, or the total, to pull out one. 3 1/5 becomes 2
1/5 plus 1. We learned how to subtract fractions from 1
in Topic D, so we have simplified this subtraction
problem. Again, we can add up as well.
There are many strategies. Remember that we don’t
need each student to be fluent with them all. Encourage
students to find a strategy that works. It is important
that they are exposed to many strategies. This helps
increase their fluency and conceptual understanding of
fractions. It also helps them to see the correlation
fractions and whole number addition and subtraction
have.
2 min
60.
Parallel to mixed number addition where we add like
units, in subtraction we subtract like units. We
encourage students to subtract the wholes first so as to
simplify the problem. Now they can use any of the
strategies learned in the previous lessons. Here are a
few. Look them over and discuss them with your
partner.
7 min
61.
This lesson is a continuation of Lesson 33 and, again,
shows another solution method. This time the focus is
on the decomposition of the whole to make more units
available to subtract. This aligns nicely to the
subtraction algorithm as we are renaming the whole
and making more of the smaller units.
Allow participants 10 minutes to complete Problem Set
#32- 35.
4 min
62.
Allow 3 minutes for participants to discuss at tables.
Allow 1 minute for share out.
Sample Answer:
•After being introduced to and working with multiple
strategies, students will have the freedom to use the
strategy that is most comfortable, or makes the most
sense, to them.
1 min
63.
Topic G builds on the concept of representing a fraction
as a whole number times a mixed number. Students
use both the associative and distributive properties to
express a representation of the multiplication. At the
end of the Topic, students solve word problems and,
once again, interpret a line plot and answer questions
regarding the points that are plotted.
2 min
64.
Students continue using the associative property to
represent fractions as multiplication, building upon
their Topic E experience.
To show how this number sentence can associate, first
students are given the example of 4 times 3
centimeters.
Write 4 x 3 centimeters = 4 x (3 centimeters) = (4 x 3)
centimeters = 12 centimeters.
By showing this in unit form, students can make the
connection to the numerical form, supported by the
area model visual.
4 min
65.
Lesson 36 further develops the concepts of Lesson 35
and places the problems into a context.
Let’s use RDW together to solve.
Solve problem with participants.
(Click to advance to show the solution.)
4 min
66.
Let’s try another. This time, I’ll give you 2 minutes to
get started. Then we will work together to finish the
problem. If you finish early, compare your solution
strategy with your partner’s.
Solve problem together, showing multiple solutions.
(Click to advance the slides to show the solution
strategies.)
3 min
67.
Lesson 38 advances in complexity as students are
multiplying mixed numbers times whole numbers. To
do so, the distributive property is used. As shown here,
it is introduced by rearranging a tape diagram. The top
tape shows 2 copies of 3 1/5. The bottom tape groups
the whole numbers and the fractions.
(Click to advance the slide.)
Next, students write out how to solve the problem. First
they see 2 copies of 3. We multiply 2 times 3. There are
also 2 copies of 1/5. So we multiply 2 times 1/5. The
products are added together to find the final answer.
(Click to advance the slide.)
As a scaffold and support of the pictorial drawings of
tape diagrams, students may write the expression as
repeated addition, decomposing each mixed number.
Providing a context to these abstract problems allows
students to contextualize the meaning of the numerical
work. They decontextualize to solve and then place
their answer back into context to check for the
reasonableness of their answer. As students progress
through fractions and deepen their experiences, they
further develop mathematical practices if allowed such
opportunities. Lessons 37 and 38 offer various word
problems to do just this.
4 min
68.
As we have discussed today, the learning of fractions
follows a similar path as the learning of whole numbers.
In the previous lessons, we thought of the
multiplication as copies. Remember, Patti needed 6
pieces of the same length of yarn. Now we bring it
further and apply the multiplication of a whole number
times a fraction as comparison using the familiar
phrase from Grade 4-Module 3 “times as many as”.
Let’s solve this problem together.
Use RDW to solve showing 2 solutions.
(Click to advance the slides when finished to show the
solutions.)
1 min
69.
Topic G culminates with data for students to create a
line plot and then answer several questions related to
the line plot. Students have a chance to practice
addition, subtraction, and multiplication of fractions.
Allow participants 10 minutes to complete Problem Set
#36- 39.
4 min
70.
Allow 3 minutes for participants to discuss at tables.
Allow 1 minute for share out.
Sample Answer:
•Students are successful with this topic because the
strategies taught allow them to apply what they’ve
learned earlier. They are able to decompose the mixed
number and then apply the distributive property, a
concept they are quite familiar with by this point.
2 min
71.
In this final lesson of the module, students explore
fractions even further. First they are given cards, each
card labeled with a fraction for each unit between 0 and
1. For example, 0 sixths, 1 sixth, 2 sixths…all the way to
6 sixths. They are asked to arrange the cards so that
they can find the sum of all the fractions in an efficient
manner. Students work together, critique each other,
and solve various other similar problems.
5 min
72.
1 min
73.
Allow participants a few minutes to look at the End-ofModule Assessment. If time allows, participants may
work to complete the assessment. If time does not
allow, encourage a brief discussion regarding the
assessment and how the work within the module
prepares students for success.
3 min
74.
Take two minutes to turn and talk with others at your
table. During this session, what information was
particularly helpful and/or insightful? What new
questions do you have?
Allow 2 minutes for participants to turn and talk. Bring
the group to order and advance to the next slide.
2 min
75.
Let’s review some key points of this session.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Turnkey Materials Provided
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Grade 4 Module 5 PPT
Grade 4 Module 5 Faciliator Guide
Grade 4 Module 5 Lesson Excerpts
Additional Suggested Resources
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How to Implement A Story of Units
A Story of Units Year Long Curriculum Overview
A Story of Units CCLS Checklist
Active learning
Turn and talk