Fractions in Grades 3 -5:
One Arithmetic for Whole Numbers and Fractions
Sequence of Sessions
Overarching Objectives of this December 2014 Network Team Institute
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The sessions for Grades K-5 emphasize the coherence of the curriculum as a tool that enables teachers to identify, practice, and use appropriate
instruction moves and scaffolds. By examination of the sequence of concepts through which fraction operations are introduced and developed in
A Story of Units, educators teaching and supporting these grade levels will understand how the concepts and skills taught at each grade lead students
toward a profound understanding of fraction operations.
High-Level Purpose of this Session
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The purpose of this session is to extend K-2 understanding of addition and subtraction and Grade 3's understanding of the unit fraction to Grade 4 and
5's addition and subtraction of fractions. These connections will be explicit, studied within the context of the curriculum, and opportunities to practice
instruction of these concepts will be embedded within the session.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 series exploring coherence across the grades. Module Focus sessions also support implementation of the curriculum by closely
examining each module in A Story of Units.
Key Points
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In Grade 3, when formal introduction to fraction starts, we extend previous understandings by using unit language to describe fractions before we
introduce fraction notation.
We can add and subtract with fractions by adding or subtracting like units just like with whole numbers.
The use of models with whole numbers extends to fractions: number path, number bond, number line, tape diagrams, array, and area model.
Understanding of number lines and comparison of whole numbers helps build an understanding of comparison of fractions.
Understanding of part/whole relationships extends to fractions.
We came to understand whole numbers and their operations using concrete, pictorial, and abstract representations. We can do the same with
fractions.
We can decompose fractions just as we can decompose whole numbers by breaking them down into smaller parts.
We can rename fractions just as we can rename whole numbers.
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Solution strategies of addition (Make a 10, Make the Next Ten, Make a Simpler Problem) and subtraction (Take from 10, Take from the Next Ten, Take
from 100) with whole numbers builds a foundation for adding and subtracting with fractions (Make the Next Whole, Add to the Next Whole and then
Add More, Take from 1).
We can use the number line and ‘arrow way’ to add and subtract with fractions just as we can use them with whole numbers.
Our understanding of multiplication and division of whole numbers as repeated addition and repeated subtraction extends to fractions.
The associative, commutative, and distributive properties apply to fractions just as they do to whole numbers.
We use our understanding of factors and multiples when finding equivalent fractions.
We can think of division of fractions as partitive division and measurement division just as we do when dividing whole numbers.
Session Outcomes
What do we want participants to be able to do as a result 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. (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
Introduction
7 min
Explores the idea that units are
more than just numbers.
Prepared Resources
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Fractions Grades 3 – 5 PPT
Fractions Grades 3 – 5
Facilitator Preparation
Review Fractions Grades 3 – 5
PPT and Facilitator Guide
Facilitator Guide
Foundations to
Fractions
10 min
Explores how fractions are formed
using equal parts and the
foundational standards that are
introduced in Grades K – 2.
Fractions: Grade 3
52 min
Explores the Grade 3 standards
and begins the official
introduction to fractions.
Fractions: Grade 4
133 min
Explores fractions in Grade 4
using methods such as
decomposition in order to add,
subtract, multiply, and divide
fractions.
Fractions: Grade 5
95 min
Explores the Grade 5 standards
and focus on multiplying and
dividing fractions.
3 min
Concludes with a look at the
Curriculum Map to illustrate the
flow of whole number to fractional
operation modules.
Conclusion
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Fractions Grades 3 – 5 PPT
Fractions Grades 3 – 5
Facilitator Guide
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Fractions Grades 3 – 5 PPT
Fractions Grades 3 – 5
Facilitator Guide
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Fractions Grades 3 – 5 PPT
Fractions Grades 3 – 5
Facilitator Guide
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Fractions Grades 3 – 5 PPT
Fractions Grades 3 – 5
Facilitator Guide
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Fractions Grades 3 – 5 PPT
Fractions Grades 3 – 5
Facilitator Guide
Session Roadmap
Section: Introduction
Time: 7 minutes
In this section, you will be introduced to the ideas of units being
more than just numbers.
Materials used include:
• Fractions Grades 3 – 5 PPT
• Fractions Grades 3 – 5 Facilitator Guide
Time
7 min
Slide # Slide #/ Pic of Slide
1.
Script/ Activity directions
Slide 1: Title Page
7 minutes
3 minutes:
Ask participants to find a partner not at their tables. Once they are in pairs,
ask partner A to share their memories of learning fractions in school. After 1
minute, give a signal for the other partner to do likewise, and then return to
their original seats, bidding farewell to their new friends.
4 minutes:
Call attention to the name of the PK-5 curriculum as “The Story of Units.” Ask
participants to pair share as many units as they can think of.
Prompts: What would some measurement units be? Units in kindergarten?
What units might be new in Grade 1? 2? 3? 4? 5? (See the last pages of the
packet for a summation for your own reference, but do not direct the
participants at this time.)
As more ideas come forward either from the prompts or from teachers’ prior
knowledge, describe the units as “what we are counting” and the numbers as
“adjectives describing units.” Think, “How many, of what.”
(Please see the back of the Modified Problem Set for a list of some units
participants might state.)
**THIS SESSION IS DESIGNED TO BE 300 MINUTES.**
GROUP
Section: Foundations to Fraction Operations
Time: 10 minutes
In this section, you will explore how fractions are formed using
equal parts and the foundational standards that are introduced in
Grades K – 2.
Materials used include:
• Fractions Grades 3 – 5 PPT
• Fractions Grades 3 – 5 Facilitator Guide
Time
10 min
Slide # Slide #/ Pic of Slide
2.
Script/ Activity directions
Slide 2: Equal Parts
GRADES K, 1, 2 (See the standards pages in the back of the participants
hand-out to read the G1-2 standards.)
3 minutes to discuss with partners.
3 minutes to discuss as a whole group.
• Grade 3 starts the formal exploration of fractions.
• What do students know before grade 3?
• From Grade 1:
• Rectangles, circles, etc.. (Grade K)
• Halves, fourths, quarters, half of, fourth of, quarter of
(1.G.3)
• Partitioning into halves and quarters (1.G.3)
• Half-Hour (1.MD.3)
• From Grade 2:
• Halves, thirds, fourths (2.G.3)
• Partitioning into halves, thirds, and fourths (equal
shares) and describing the shares (2.G.3)
• Describing whole as 2 halves, 3 thirds, or 4 fourths
(2.G.3)
• Equal shares of the same whole need not have the same
shape. (2.G.3)
• If we were to partition this rectangle into equal parts, what might it
look like? (Have participants partition this rectangle into
quarters in 3 or 4 ways.)
• What vocabulary might be difficult for students?
• Halves, thirds (Why not “twoths” and threeths”?)
• Quarter of (Why is a quarter of a whole 1 fourth, a quarter of a
dollar 25 cents, and a quarter of an hour 15 minutes?)
(Click) Advance animation.
GROUP
Are these equal parts?
• Yes, each shows 1 fourth. (Equal shares need not be the same
shape. 2.G.3)
Give participants time to look at the Participant Handouts for Grades
1 and 2. (Includes G1-M5 Topic Overview, G1-M5-L5; G2-M8 Topic
Overview, G2-M8-L10, and the Standards.)
• What prior learning do students need to be successful with each
Problem Set?
• G1-M5-L9 – half, quarter, equal parts, larger, smaller, identify
shaded part as half or quarter, shapes
• G2-M8-L10 – halves, thirds, fourths, partitioning into fractional
parts, shading to show given fraction, 3 thirds and 4 fourths as
a whole.
• What prior learning will students be bringing to Grade 3 from these
modules?
• (Review discussion from 1st question on slide.)
• How can understanding the prior learning help to close gaps in
learning in G3, G4, and G5?
• This is the prior learning we expect students to have coming
into G3.
• Close gaps by incorporating these concepts/vocabulary into lessons.
•
Section: Grade 3
Time: 52 minutes
In this section, you will explore the Grade 3 standards and
introduction to fractions.
Materials used include:
• Fractions Grades 3 – 5 PPT
• Fractions Grades 3 – 5 Facilitator Guide
Time
Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
20 min
3.
Slide 3: Duck and Rectangular Array
GRADE 3
3.NF.1: Understand 1/b as the quantity formed by 1 part when a whole is
partitioned into b equal parts; understand a fraction a/b as the quantity
formed by a parts of size 1/b.
Ask participants:
1 minute with partners
Describe the number of ducks in unit form. 1 duck
How would you describe the orange section of the rectangle? 1 eighth
(Model unit language on the document camera: “1 eighth.”)
3 minutes with whole group
How does unit form help students link what they know about whole
numbers to fractions?
• Just like we can talk about ducks, dogs, cats, ones, tens,
hundreds, we can also talk about halves, thirds, fourths, etc….
• The difference is just the units. Fractions work in the same way
as whole numbers.
• Since early grades, students have been able to add like units. If
we have like units, we can add or subtract.
Participant Handout: (Note: Fractions with denominators of 2, 3, 4, 6,
& 8 for 3rd Grade.)
• Grade 3, Module Overview Lesson Objectives Chart
• Identify places where prior learning from G1 and G2 is
required.
• Topic A: Equal parts, identifying and describing shares
• Topic B: Partitioning, equal parts, identifying
fractional parts, number bonds (part/whole
relationship)
• Topic C: Compare (larger/smaller), fractional parts in
a whole
• Topic D: Number lines, comparison (<,>,=) using
number line
• Topic E: Number lines, equal parts
• Topic F: Number lines, comparison
• Identify the sequence of learning. How does it move from
simple to complex?
• Topic A: Similar to Grade 2 work – identifying fractions
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(concrete, pictorial, abstract); extends beyond halves,
thirds, fourths
• Topic B: Introduces ‘unit fraction’ - identifying and
building
• Topic C: Unit fraction – comparing – part/whole
relationship
• Topic D: Extends to number line – representing
fractions (placement and comparison)
• Topic E: Still number line, but fractional equivalency 
Leads to work with other models (number bond,
fraction strips)
• Topic F: Comparing fractions – different strategies
• Where do you see repetition?
• Interwoven throughout – “one step back, two steps
forward”.
Grade 3, Module 5, Lessons 1 and 5 Problem Set.
• Why do both lessons stress the unit fraction? Why is the unit
fraction important?
• Thinking of fractions in unit form ( 1 fourth, 1 third,
etc.) leads to introduction of unit fraction 1/b; relates
fractions back to whole numbers and to the idea of the
unit.
• Unit fractions tie fractions back to whole numbers and
are the basic building blocks of whole numbers.
• In kindergarten students learned to decompose 5 into
“hidden partners.” We are doing the same thing now
with fractions: 3/8 = 2/8 + 1/8, etc.
• How does this sequence between both lessons support
concrete to pictorial to abstract understanding?
• Lesson 1 uses concrete models (fraction strips and
beakers) to show fractional parts; familiar halves,
thirds, and fourths.
• Lesson 5 uses pictorial models; students determine the
fractional part numerically.
• We do many lessons before we introduce fraction
notation.
• What concrete experiences must students experience to
understand the unit fraction? Compare those experiences to
the experiences Grades PK-2 students have with units of ones
and tens.
• Idea of unit; manipulation of concrete units (Pattern
blocks)
Concrete experience with whole numbers – similar – units  unit form
5 min
4.
Slide 4: Renaming 320 as 32 tens (from Grade 2 learning)
• Using unit form, what do you see on the left?
• 3 hundreds, 2 tens
• Say that in standard form.
• 320
• What did we do going from left to right?
• We renamed 320 as 32 tens.
• Why might we do this?
• To express an equivalent amount or to rename so that we have
like units. Once we have like units, we can add or subtract.
• How is the term “renaming” like “equivalent?”
• Expressing the same amount in a different way.
• Describe other examples of how units have been manipulated.
Place value chart  ones, tens, hundreds decomposed for other units or
composed to other units through addition, subtraction, multiplication, and
division.
10 min
5.
Slide 5: Multiple Models
3.NF.2: Understand a fraction as a number on the number line; represent
fractions on a number line diagram.
a. Represent a fraction 1/b on a number line diagram by defining the
interval from 0 to 1 as the whole and partitioning it into b equal
parts. Recognize that each part has size 1/b and that the endpoint
of the part based at 0 locates the number 1/b on the number line.
b. Represent a fraction a/b on a number line diagram by marking off a
lengths 1/b from 0. Recognize that the resulting interval has size
a/b and that its endpoint locates the number a/b on the number
line.
Ask participants to discuss the multiple models students use to explore and
understand fractions.
How are these models used in earlier grades?
• Unit form – to build a foundation for understanding numbers,
place value, and the four operations (G2) Before grade 2
students use the “Say Ten Way.”
• Number bonds – to show part/whole relationships (introduced
in Kindergarten)
• Tape diagram (here, fraction strip) – To pictorially represent a
given quantity (G1)
• Number Path – Now the path is divided into fractional parts.
(Kindergarten)
• Number line – In G2 students make their own rulers, which is
the introduction to the number line. Students use it to show
the relationship between numbers; addition and subtraction of
numbers (G2)
• Which of these models might be the most sophisticated?
• The number line is used in fractions in a new way. Until now
we have only shown whole numbers on the number line. The
other models support this new complexity of the number line.
Have participants complete Grade 3, Module 5, Lessons 14 & 16.
Partners discuss, then whole group discusses:
What is the complexity between L14 and L16?
• L14 decomposes 1 into parts using number bonds, fraction
strips, and number lines. L16 decomposes whole numbers
greater than one into parts using a number line.
Why do we want students to think about fractions beyond 1?
• Fractions represent how whole numbers can be decomposed
into smaller parts. Just as we can decompose, for example, a
given number of hundreds and tens into smaller units, we can
decompose whole numbers (ones units) into smaller parts as
well  eventually, this relates not only to fractions greater
than one but also to decimal numbers.
If we only examined the distance between zero and 1, students would not see
that any place on the number line can be named. Perhaps, students could
surmise that fractions are only numbers between zero and one.
•
7 min
6.
Slide 6: ¾ = 6/8
3.NF.3: Explain equivalence of fractions in special cases, and compare
fractions by reasoning about their size.
a. Understand two fractions as equivalent (equal) if they are the same
size, or the same point on a number line.
b. Recognize and generate simple equivalent fractions, e.g., ½ = 2/4,
4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a
visual fraction model.
c. Express whole numbers as fractions, and recognize fractions that
are equivalent to whole numbers. Examples: Express 3 in the form
of 3 = 3/1; recognize that 6/1 = 6; locate 4/4 and 1 at the same point
of a number line diagram.
Have participants complete G3-M5, Lesson 22. (In the CD of this lesson,
students use fraction strips and the number line, along with area models to
show/prove equivalency. Here, on the Problem Set, work is more
abstract.)
Obviously, this lesson explores equivalent fractions. Analyze these
problems.
• What adds to the complexity of these problems?
• Support of equivalency using other models has been taken
away.
• Although equivalent amounts are shaded, they don’t appear to
be the same.
• What concrete experiences might students need to be exposed with
prior to this Problem Set?
• Pattern blocks – composing different shapes
Paper unit squares – manipulating size and shape of them – to ‘build’ what is
seen here and to prove equivalency. If students struggle to see the
equivalence pictorially, they can cut paper squares to prove equivalency.
10 min
7.
Slide 7: 4 as the Number of units/numerator
3.NF.3.d: Compare two fractions with the same numerator or the same
denominator by reasoning about their size. Recognize that comparisons
are valid only when the two fractions refer to the same whole. Record the
results of comparisons with the symbols >, =, <, and justify the conclusions,
e.g., by using a visual fraction model.
How does this G2-M2 visual help students to understand the size of units?
(Relate centimeters to meters, ones to tens, fifths to thirds.)
• Units represent different things. Here, the quantity is ‘4’ of
each. The question is ‘4’ of what unit. 4 centimeters means
something different than 4 meters, etc.. To compare, we must
look at the unit and also at the quantity. We need to think,
“How many of what?” If the “of what” is not identical, than the
“How many” is insignificant in comparing quantities.
Give participants time to complete G3-M5, Lesson 28.
Then ask:
• If we know that the units are of different sizes, but that the quantity
of each is the same, we can use the quantity to help us compare (e.g.,
4 meters > 4 cm). In other words, the how many is the same, but the
of what is different.
• How can we build on this understanding?
• With fractions, we can use the numerators to compare
fractions (e.g., thirds are greater than fifths, therefore 2 thirds
are greater than 2 fifths, assuming that the two fractions refer
to the same whole). How can understanding this help us
compare 2/3 and 4/5? Write these 2 fractions under the
document camera.
• If we know that 2/3 is equivalent to 4/6, we are then
comparing 4/6 to 4/5. Sixths is a smaller unit than fifths.
Therefore, 4/6 < 4/5  2/3 < 4/5. (The visual for this is on
the problem set of G3-M5-L28.)
• What other ways could you compare 2/3 and 4/5 using mental
math?
• We can think of the units of thirds and fifths. 2/3 is 1 third less
than 1. 4/5 is 1 fifth less than 1. Since thirds are greater than
fifths, 2/3 will be further from 1 and, therefore, be less than
4/5.
Although not a Grade 3 standard, one could find a common denominator:
2/3 = 10/15 and 4/5 = 12/15; therefore, 4/5 > 2/3.
Section: Grade 4
Time: 133 minutes
In this section, you will explore fractions in Grade 4 using methods
such as decomposition in order to add, subtract, multiply, and
divide fractions.
Materials used include:
• Fractions Grades 3 – 5 PPT
• Fractions Grades 3 – 5 Facilitator Guide
5 min
8.
Slide 8: Decomposing a group of Ducks & Chickens
GRADE 4
Since kindergarten, students are learning that we can decompose any number
into parts. The concept of part and whole starts early, and we continue to
explore this throughout the Story of Units.
How does this prior learning support work with fractions?
Just as we can decompose whole numbers into smaller parts, we can also
decompose fractions into smaller parts.
10 min
9.
Slide 9: Decomposing ½
4.NF.3: Understand a fraction a/b with a>1 as a sum of fractions 1/b.
a. Understand addition and subtraction of fractions as joining and separating
parts referring to the same whole.
b. Decompose a fraction into a sum of fractions with the same
denominator in more than one way, recording each decomposition by
an equation. Justify decompositions, e.g., by using a visual fraction
model. Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 2/8 + 1/8.
4.NF.4: Apply and extend previous understandings of multiplication to
multiply a fraction by a whole number.
a. Understand a fraction a/b as a multiple of 1/b. For example, use a visual
fraction model to represent 5/4 as the product 5 x (1/4), recording the
conclusion by the equation 5/4 = 5 x (1/4).
How is this number bond like the one with the ducks and chickens?
• We are looking at the part/whole relationship.
How is it different?
• The units are fractions instead of whole numbers. Instead of
representing whole numbers as two or more parts of a whole, we
express a fraction less than 1 as two or more smaller parts.
Have participants study the G4-M5 Module Overview. Reminder that Grade 4 is
limited to denominators of 2, 3, 4, 5, 6, 8, 10, and 12.
• What prior understanding do G4 students need to be successful
in this module?
• Topic A: Decomposition, unit fractions, equivalence, tape
diagrams, area model, part/whole relationships
• Topic B: Area model, number lines, whole number
multiplication and division, equivalence
• Topic C: Representation of fractions on a number line,
finding equivalent fractions
• Topic D: Whole number addition and subtraction
• Topic E: Decomposition of fractions, use of visual models,
line plots
• Topic F: Decomposition, whole number addition and
subtraction
• Topic G: Whole number multiplication, associative and
distributive property, multiplicative comparison, line plots
• Topic H: Patterns, whole number addition
• How does the first half of the module support the second half of
the module?
• Students use what they learned in the first half of the
module to build their understanding of fractions in the
second half of the module. They apply their understanding
of fractions less than 1 (first half) to fractions greater than
1 (second half).
Have participants complete G4-M5, Lessons 4 & 6.
• Why use a tape diagram and an area model to decompose
fractions? What multiple models do we use at lower grades to
decompose whole numbers?
• It allows us to visually see the decomposition and how the
shaded part stays the same; it’s the size of the units and
the quantity of the unit that is changing.
• At lower grades, we use 5- and 10-group cards, Hide-Zero
cards, place value disks and charts, Number paths,
Rekenrek, number bonds, and tape diagrams.
How does the sophistication of recording increase? What does
this prepare students for?
• L4 – The problems increase in complexity. The first
problems are shown in their entirely and then the other
problems are scaffolded, requiring the students to do more
and more work. It moves from decomposition of unit
fractions to decomposition of non-unit fractions.
L6 – We decompose horizontally to show fraction equivalence. This prepares
students to find equivalent fractions in a more abstract manner, recognizing
that, for example, when we find the equivalence of 2/3 and 4/6, we are doubling
the quantity of units and also making the size of the units half the size (thus
doubling).
•
20 min
10.
Slide 10: 10/12
4.NF.1: Explain why a fraction a/b is equivalent to a fraction (n x a)/n x b) by
using visual fraction models, with attention to how the number and the size of
the parts differ even though the two fractions themselves are the same size.
Use this principle to recognize and generate equivalent fractions.
Read the Teach Using Feedback Protocol on the last page in the packet.
Let participants know they will be expected to teach this strategy to a peer in
moments.
SUPER MODEL: (Refer to Grade 4, Module 5, Lesson 10.)
• Ask participants to draw an area model that represents 10/12.
• If we want to compose an equivalent fraction with smaller units, what
can we do?
• We can make groups of 2. If we do, we don’t have any remaining
groups.
• Ask participants to show this on the area model.
• There should be 5 groups of 2 units shaded. There should be a
sixth, unshaded group.
• Write the equivalent fraction.
• 5/6.
• Let’s consider the unit fractions of 1/12 and 1/6. What do you notice
about the denominators?
• 6 is a factor of 12.
• What do you notice about the numerators of 5 and 10?
• 5 is a factor of 10.
List the factors of 10 and 12.
• 10: 1, 2, 5, 10
• 12: 1, 2, 3, 4, 6, 12
• Are there any common factors?
• Yes, 1 and 2.
• How does 2 relate to the area model you drew?
• We made equal groups of 2.
• Model how this can be written as a division problem.
DELIBERATE PRACTICE: G4-M5-L10 Problem Set 1(b) (9/12) and 1(c )
(6/10)
• Review the protocol. Find a partner.
• Facilitator uses a stopwatch to monitor the time and lead the group.
• Remind Partner A not to respond to Partner B’s feedback. This is not
time to discuss. Accept their feedback and your possible errors, looking
forward to your improvement.
Have the participants complete G4-M5, Lesson 10.
• What is the complexity in Problem 2?
• When completing 2(b), we could compose to 4/6 or 2/3.
2(a) and 2(b) have the same fractional amount shaded in each. We can look at
the relationship of the (a) and (b).
•
16 min
11.
Slide 11: 25 + 14 & 400 – 30
4.NF.1: Explain why a fraction a/b is equivalent to a fraction (n x a)/n x b) by
using visual fraction models, with attention to how the number and the size of
the parts differ even though the two fractions themselves are the same size.
Use this principle to recognize and generate equivalent fractions.
4.NF.3: Understand a fraction a/b with a>1 as a sum of fractions 1/b.
Ask participants to answer this question: Using mental math how could you
solve the problem 25 + 14?
• (20 + 10) + (5 + 4) = 30 + 9
• (25 + 10 + 4)
• (14+ 20 + 5)
• (5 + 4 + 20 + 10)
• Other ways are also possible.
In pairs, ask participants to answer this question. Then discuss it as a whole
group.
What is the big idea that students must know in order to add these two
numbers?
We can add like units; ones with ones and tens with tens.
We can add on one unit at a time; add the ten and then add the
ones.
• We can decompose either (or both) of the numbers in order to
add more efficiently.
Model: 25 + 14 using number bonds. This is how we help young
students visualize the decomposition that eventually leads to mental
math.
•
•
•
Model on the document camera Make a Ten using 9 + 8. Decompose 8
as 1 + 7.
• Model 37 + 46. How would you add using mental math and the Make
the Next Ten strategy? Show using number bonds and the equation
below.
• (37 + 3) + 43 = 83 (or equivalent).
(CLICK) Advance the animation and repeat the questions.
• We can add to make the next ten and then add the remainder of
the second addend to find the total.
Model on the document camera 400 – 30, and model 52 – 26, 17 – 9.
• 400 – 30
• Take from 1 hundred using number bonds: 100 – 30= 70; 70 +
300 = 370
• Unit form: 40 tens – 3 tens = 37 tens = 370
• 52 – 26
• Take from the next ten using number bonds: 30 – 26 = 4; 4 +22 =
26
• 52 – 26 = 56 – 30 = 26 (Compensation – added 4 to the whole &
the part or the minuend & subtrahend)
• 17 - 9
• Take from 10: 10 – 9 =1; 1 + 7 = 8
• Compensation: 17 – 9 = 17 – 10 + 1 = 8
Ask participants to complete G4-M5, Lesson 22.
• How are the strategies Take from Ten, Take from the Next Ten, or Take
from a Hundred like the strategy practiced in Lesson 22?
• 1(c-d) – We are taking from 1. Isn’t this like what they learned in
earlier grades?
•
• 3(a-b) – We are taking from 1.
We make an easier problem by decomposing the minuend.
7 min
12.
Slide 12: Number Path (G1-M1) and Number Line (G2-M3)
4.NF.3b: Decompose a fraction into a sum of fractions with the same
denominator in more than one way, recording each decomposition by an
equation. Justify decompositions, e.g., by using a visual fraction model.
Examples: 3/8 = 1/8 + 1/8 + 1/8; 3/8 = 1/8 + 2/8; 2 1/8 = 1 + 1 + 1/8 = 8/8 +
8/8 + 1/8.
Discuss at tables what this model is and how it could be used.
• It’s a number path.
Write 2 number sentences to match. Draw a number bond to support the partwhole relationship.
• 6+3=9
• 9–6=3
CLICK.
Discuss at tables what this model is and how it could be used.
• It’s a number line.
• We can use it to show how we can add two numbers, focusing on
the units.
• We can use it to show how we can solve a subtraction problem by
adding on to find the difference, again focusing on units.
• We can use it to decompose a given number.
Write 2 number sentences to match. Draw a number bond to support the partwhole relationship.
• 776 + 124 = 900
• 900 – 776 = 124
Have participants complete G4-M5-L24.
Pair-Share:
• How would you describe what you did in this Problem Set?
• We renamed a fraction as a mixed number by pulling out whole
numbers.
• How is this different than the way we used to simplify fractions?
• We can visually see that they are the same as opposed to “dividing
the numerator by the denominator and expressing the remainder
as a fraction” which is the way that we likely learned.
• How could this same idea be expressed as a number bond?
1(a) shows the solution using a number bond to express 11/3 as
9/3 + 2/3. 9/3 = 3. 3 + 2/3 = 3 2/3.
• How can whole number work with number line models (including the
arrow way) support upcoming work with fractions?
• We can use the same model to rename a “fraction greater than 1”
into a mixed number. (This is the answer participants should say
after studying G4-M5-L24.)
We can use the same strategies that we used with whole numbers to add and
subtract with fractions.
•
10 min
13.
Slide 13: 23 + 71; 84 – 21; 84 – 19
4.NF.3c: Add and subtract mixed numbers with like denominators, e.g., by
replacing each mixed number with an equivalent fraction, and/or by using
properties of operations and the relationship between addition and
subtraction.
• What’s going on here?
• These are Grade 2 examples of the arrow way to add and
subtract.
• Adding to make the next ten; adding on a unit of ten; subtracting
a unit of ten; subtracting one from a ten.
• What model is similar to the arrow way?
• The open number line, or the number line.
Give participants time to discuss what they see.
• How would you generalize this strategy?
• Making a simpler problem by adding and subtracting the ones
and tens in two separate steps (units of ten and units of one) or
by subtracting an easier number and then adding back the
difference.
• What do students need to know to be able to do these strategies?
• Expressing a number as the sum of its parts (units)
• Adding tens, adding ones.
• Subtracting tens, subtracting ones.
Have participants analyze G4-M5, Lessons 30 and 33.
• How is the Arrow Way used for fractions?
• In the same way as whole numbers. We show how we can make
the next whole, add to the next whole and then add on, subtract
by adding on, or subtract by decomposing, or simply adding or
subtracting like units.
How does the sequence scaffold in L30?
• In Problem 1, students are adding like units with no regrouping
1(a) and then adding like units, making the next one (1b).
• In Problem 2, students are thinking about how much more they
need to get to the next one.
In Problem 3, students are decomposing the second addend in order to make the
next one and then are adding on.
•
20 min
14.
Slide 14: Rectangular Array & Area Model: Using the area model to Add
and Subtract Fractions with unrelated units (G5-M3).
5.NF.1: Add and subtract fractions with unlike denominators (including
mixed numbers) by replacing given fractions with equivalent fractions in such
a way as to produce an equivalent sum or difference of fractions with like
denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general,
a/b + c/d = (ad + bc)/bd.)
• What model precedes the rectangular array?
• The array is used in kindergarten to help students organize
objects in order to decide, “How many?” Ten frames are one
example of arrays used in many grade levels.
• The rectangular array is used in Grade 2, Module 6 to introduce
multiplication and division to students.
Review the Teach Using Feedback Protocol on the last page in the packet
(used earlier).
• Let participants know they will be expected to teach the modeled
strategy to a peer.
• Remind them to switch being Partner A and B so each person gets a
turn to play “teacher”.
SUPER MODEL Round 1, G5-M3-L3:
• Use area models to solve #1a: ½ + 1/3 and ½ + 2/3.
DELIBERATE PRACTICE Round 1: G5-M3-L3 Problem Set #1e ¾ + 1/5.
• Review the protocol. Find a partner.
• Facilitator uses a stopwatch to monitor the time and lead the group.
• Remind Partner A not to respond to Partner B’s feedback. This is not
time to discuss. Accept their feedback and your possible errors, looking
forward to your improvement.
SUPER MODEL Round 2: (G5, M3, Lesson 12).
• Use area models to solve #1a: 3 1/5 – 2 ¼ .
DELIBERATE PRACTICE Round 2: G5-M5-L12 Problem Set #1b (4 2/5 – 3 ¾ ).
• Review the protocol. Find a partner.
• Facilitator uses a stopwatch to monitor the time and lead the group.
• Remind Partner A not to respond to Partner B’s feedback. This is not
time to discuss. Accept their feedback and your possible errors, looking
forward to your improvement.
Upon completion, briefly examine the Module Overview for G5-M3 in the
Participant Handout. Look for comparisons in the sequence of learning from
G4-M5 and how G4-M5 sets students up for success in G5-M3.
• G5 -Topic A – Equivalent Fractions – All the standards in Topic A
are Grade 4 standards.
• G5 -Topic B – Making Like Units Pictorially • G5 -Topic C – Making Like Units Numerically –
• G5 -Topic D – Further Applications –
• G4 – Fraction equivalence (Topics A,B,&E) sets students up for TA
and TB of G5;
• G4 – Finding common units (Topic C) – sets students up for
finding common units (equivalence in G5);
• G4 – Fraction addition and subtraction (Topics D&F);
• All Topics – Relate to G5 – Further Applications
NOTE: Although the sequence jumped from G4 to G5 fraction addition and
subtraction, on the next slide we will return to G4 fraction multiplication and
continue to G5 fraction multiplication and division.
10 min
15.
Slide 15: 6 cm
4.NF.4: Apply & extend previous understanding of multiplication to multiply a
fraction by a whole number.
5 minutes
3 minutes:
The purpose of this component is to clarify the meaning of a/b as a multiple of
1/b and to also show the relationship between measurement units and fractional
units in regard to their behavior with the operations. (SEE G4-M5-Lesson 35 for
further explanation of using unit form to multiply a whole number and a fraction.)
Ask the participants:
True or false?
6 cm = 6 x 1 cm
•Ask the participants to apply this same thinking to 6/5. (6/5 = 6 x 1/5)
• 6 fifths = (6 x 1) fifth = 6 x (1 fifth) = 6 x 1/5
•Have participants review G4-M5, Lesson 35.
2 minutes
Debrief experience and be sure to bring out the easing of the students into
fractional notation and what a stumbling block that can be without using unit
form first. Call out the use of the associative property.
This is the associative property used to decompose a number into its unit parts. 6
x 1 cm is really showing the repeated addition of 1 cm + 1 cm + 1 cm + 1 cm + 1
cm + 1 cm. In turn 6 x 1/5 is showing the decomposition of 6/5 into its fractional
units: 1/5 + 1/5 + 1/5 + 1/5 + 1/5 + 1/5.
We know this is true from early on in our counting experiences: 3 crackers is
composed of 1 cracker + 1 cracker + 1 cracker. And in Grade 3 we see that as 3
crackers = 3 x 1 cracker.
Units change, but the mathematics, or the arithmetic, remains constant and true.
15 min
16.
Slide 16: 3 x 2; 3 x 2 cm
4.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction
by a whole number.
The purpose of this sequence of 3 problems is to show the relationship between whole
number multiplication with various units, multiplication of measurement units, and
multiplication of fractional units.
• Ask participants to consider the meaning of the factors 3 and 2 with their
partner.
• 3 groups of 2
• 3 twos
• 3 lengths of 2 cm or 3 times as much a 2 cm
• Share out whole group about their meaning, labeling the Grade 3 significance for
now as the first factor indicating the number of groups, the second factor
meaning the size of the group. What is the unit? (twos)
• Ask participants to tell their partner a number sentence for the array of ducks.
• 3x2=6
• Share out whole group: 3 times 2 ducks. What units are we using now? (ducks)
• In the Participant Handout, follow the directions for making a tape diagram to
solve for 3 x 2 cm. (Problem 1)
• Share out whole group the significance of equal groups in the tape diagram.
What is the unit? (centimeters)
• In the Participant Handout, follow the directions for making a tape diagram to
solve for 3 x 2 fifths. (Problem 2)
Share out whole group about how the previous examples shed light on successfully
answering a whole number times a fraction problem. How is using unit form
advantageous with fractions? Can you rename 6/5 as 1 1/5 using a number bond?
• Using unit form helps us to relate what we know about whole
numbers to fractions.
• 6/5 = 5/5 + 1/5; 5/5 = 1; 1 + 1/5 = 1 1/5
Model how the associative property works to solve this problem using unit or numerical
form:
• 3 x 2 fifths = 3 x (2 x 1 fifth) = (3 x 2) x 1 fifth = 6 x 1 fifth or (6 x 1) fifth = 6/5
• 3 x 2/5 = 3 x (2 x 1/5) = (3 x 2) x 1/5= 6 x 1/5 or (6 x 1)/5 = 6/5
(Perhaps have participants try one on their own.)
20 min
17.
Slide 17: Array of 3 x 6, 3 x 42; 3 x 4m 60cm
4.NF.4: Apply and extend previous understandings of multiplication to multiply a fraction
by a whole number.
4.NF.4c: Solve word problems involving multiplication of a fraction by a whole numbers,
e.g., by using visual fraction models and equations to represent the problem.
• Write a multiplication problem for the array. (3 x 6)
• Use the distributive property, guided by the shading of the array, to write 2
multiplication problems that equal 3 x 6.
• [(3 x 5) + (3 x 1)]
• Use unit form to describe the array.
• (3 fives + 3 ones)
• Solve for 3 x 42 using the distributive property, separating the tens and ones.
Write this as a multiplication equation and in unit form.
• [(3 x 40) + (3 x 2)] = 126
• OR [(3 x 4 tens) + (3 x 2 ones)] = 12 tens + 6
ones = 126
•What other times have we used the distributive property to solve
multiplication problems?
• We also use the distributive property when we divide.
• The models we use to show this are the Rekenrek with
basic facts, the area models and number bonds.
3 minutes:
• In the Participant Handout, have teachers draw a tape diagram representing 3 x
4 m 60 cm (Problem 3). (See Lessons 37 and 38 of G4-M5.)
• Model after they do, showing 3 x 4 meter 60 cm tapes first not distributed to be
adjacent and then redistributed to be adjacent.
4 minutes
• Ask them, would this same model apply to multiply 3 x 4 3/5?
• In the Participant Handout, have teachers draw a tape diagram representing 3 x
4 3/5 (Problem 4).
• As you see is necessary, release, guide or show the work with the model and
numerically. 3 x (4 + 3/5) and its equivalence to 3 x 4 + 3 x 3/5 including the
bond of 9/5 as 5/5 and 4/5.
4 minutes: Deliberate Practice
Re-read protocol for Deliberate Practice and prepare to practice Problem 1 from G4
Lesson 37. 1 minute prep time, 1 minute teach time, 30 seconds feedback, 1 minute
teach, 30 seconds feedback.
5 minutes:
Conclude by having participants solve the problems in the Participant Handout from G4
Lesson 39.
Whole group shares out Problem Set analysis and experience.
NOTE: We will be returning to this idea in grade 5, when we solve a fraction times a
whole number by using division.
Section: Grade 5
Time: 95 minutes
In this section, you will explores the Grade 5 standards and focus on Materials used include:
multiplying and dividing fractions.
• Fractions Grades 3 – 5 PPT
• Fractions Grades 3 – 5 Facilitator Guide
18 min
18.
Slide 18: 6 ÷ 2
10 minutes
Have participants discuss the meaning of the divisor and of the unknown.
Review the distinction between measurement and partitive division first with 6
divided by 2 and then with 4 divided by ½ as pictured below.
Construct the work collaboratively and interactively with the participants, using
both direct, guided and partner sharing processes to clarify the content.
Different groups will need different kinds of support. Have participants generate
word problems corresponding to each type.
Have participants review the Module Overview for G5-M4 in the participants’
packet.
17 min
19.
Slide 19: 3 Divided by 2
5.NF.3: Interpret a fraction as division of the numerator by the denominator
(a/b = a divided by b).
Ask participants to draw a picture of this fraction problem. Hopefully there will
be two interpretations.
Model on the document camera. Show 3 same colored post-it notes to
represent the dividend. Under the post-its draw 2 circles to show the number of
groups.
•Think of this as 3 crackers divided by two people. What do we know?
What is unknown?
• We know the group size, but not the size of the group.
(Mention to participants that we will return to this idea
of group size unknown.)
•Slide one full post-it note into each group. What do we do with the other
piece?
• We cut it in two equal pieces, and place a piece in each
group.
•What is the answer? What is the size of the group?
• 1½
Show 3 different colored post-it notes to represent the whole. Again, draw the 2
groups.
•We need to know how much in each group. This time, however, we
should consider that the post-it notes are 3 crackers of different
flavors. Both people would like to taste each cracker. How does this
change our picture?
• Each piece is divided into 2. The answer is 3/2 rather
than 1 ½.
•Show this by dividing each post-it and moving it into the group.
Have participants practice 2 divided by 3, and 3 divided by 4.
•If we do enough of these problems, what will students notice?
• That we can interpret a division problem as a fraction.
Model unit form on the document camera:
•Let’s go back to 3 divided by 2. What’s my divisor?
• 2.
•Let’s state the whole (the dividend) as halves. How many halves in 3?
• 6 halves.
•What is 6 halves divided by 2? How many is 6 divided by 2? What’s the
unit?
• 6 divided by 2 is 3, and the unit is halves. Our answer is
3 halves.
Model the standard algorithm for 3 divided by 2.
Have participants complete G5-M4, Lessons 2 & 3 in the packet.
10 min
20.
Slide 20: 1/3 of 18
5.NF.4: Apply and extend previous understanding of multiplication to multiply a
fraction or whole number by a fraction.
a. Interpret the product of (a/b ) x q as a parts of a partition of q into b equal
parts; equivalently, as the result of a sequence of operations a x q divided by b.
Ask participants to draw a tape diagram to match this problem.
•When did we see a fraction times a whole number before now?
• In Grade 4.
•How did we interpret a fraction times a whole number?
• We used the multiplication symbol, and we used
repeated addition. So this problem would have been
interpreted as 1/3 + 1/3 + 1/3 …… (eighteen times). We
would have used the associative property to think “1
times 18 thirds.”
Model on the tape diagram on the document camera. Write: 3 units = 18; 1
unit = 18 divided by 3 = 18/3 or 6.
•How does this thinking evolve from grade 4?
• In grade 4 we thought, “1 times 18 thirds.” Now we
think, “1 times 18 divided by 3 or 18/3.”
Have participants complete G5-M4-L7 in the packet.
•How does Lesson 7 build on Lesson 3?
• Lesson 3 taught us to think of a division problem as a
fraction. Now we can use this learning to solve
multiplication of a fraction.
L3 teaches us that 18÷3 can be interpreted as a fraction, 18/3. So when I think
about solving 1/3 of 18, I can interpret the multiplication problem as 1 times
18/3, which is 18/3, the fraction, or the division problem. So now I can easily
rename the fraction greater than 1 as 6. It’s showing how multiplication and
division of fractions is related.
15 min
21.
Slide 21: 2 x 20
5.NF.4: Apply and extend previous understanding of multiplication to multiply a
fraction or whole number by a fraction.
a. Interpret the product of (a/b ) x q as a parts of a partition of q into b equal
parts; equivalently, as the result of a sequence of operations a x q divided by b.
3 minutes
In Grade 3, this means 2 units of 20 or 2 twenties.
In Grade 4, this can mean 20 times as much as 2.
2 + 2 + 2 + 2 + 2… twenty times.
We can use multiplicative comparison language or think of the first factor as the
unit, the second factor being the number of units.
As shown in G5-M4-L7:
F: What does 2/5 x 20 mean in Grade 4 fraction work?
P: 2/5 + 2/5 + 2/5 …. Twenty times.
F: In Grade 5, this can now also mean 2 fifths of 20.
2/5 of 20 is represented as 2/5 x 20.
Model the problem.
4 minutes
What if the unit is a fraction?
1 half of 2 fifths, for example.
Super Model: Use the document camera to model this problem using the area
model. Explain to the participants that we are working in the first grid of the
area model.
Also model this in unit language. ½ of 2 fifths. How many? 1. Of what? Fifths.
Do other examples as necessary such as 1/3 x 3/4 and 2/3 x ¾.
• 1/3 of ¾ Think 1 third of 3 is 1. What’s the unit?
Fourths. What’s the answer? 1 fourth or 1/4.
• 2/3 of ¾ Think: 2 thirds of 3 is 2. What’s the unit?
Fourths. What’s the answer? 2 fourths or ½.
• Often when we solve in unit form, the answer is in a
more simple form that if we did the multiplication.
5 minutes
Have participants deliberately practice G5-M4-L14, using examples (a) and (b).
15 min
22.
Slide 22: 1 ÷ 2
5.NF.7: Apply and extend previous understanding of division to divide unit
fractions by whole numbers and whole numbers by unit fractions.
•In pairs, have participants write a story problem that matches 1 divided
by 2. Share some examples, and as you do, ask participants to label
the stories as either missing group size ( or missing number of groups.
•Consider 3 divided by 1/3.
•Write a story problem that matches this problem. Determine if the story
problem is partitive or measurement. Solve by using a tape diagram
or an area model.
• I have 3 apples. If I want to give a third of an apple to
each friend, how many friends will get a piece?
(Measurement – Number of Groups Unknown)
• I apple = 3 thirds. Three apples equals 9
thirds. The answer is 9 friends.
• I have 3 apples. This is a third of what I need to make
applesauce. How many apples do I need? (Partitive Group Size Unknown)
• I need 9 apples.
•Model 3 divided by 1/3 using a tape diagram.
• Consider 1/3 divided by 3.
•Write a story problem that matches this problem. Determine if the story
problem is partitive or measurement. Draw a tape diagram or area
model to solve.
• I have a third of a pan of lasagna. I want to share it
equally among 3 friends. How much (of the whole pan)
will each friend get? (Partitive - Group size Unknown)
• 1 third divided by 3 is 1 ninth.
• I can also think in unit form. 1 third is
the same as 3 ninths. 3 ninths divided by
3 is 1 ninth.
• I have a third of a pan of lasagna. I need 3 pans for a
big family dinner. How much of the whole amount do I
have? (Measurement – Number of Groups unknown)
• I have 1 ninth of what I need.
•Model 1/3 divided by 3 using a tape diagram.
5 minutes:
Have participants do G5, M4, L25 & 26.
3 minutes
Analyze Problem Set and experience.
15 min
23.
Slide 23: Rectangular Array and Area Model Revisited
5.NF.5: Interpret a fraction as scaling (resizing).
• What if our area model now has lengths that weren’t whole numbers?
Can we find the area? Ask participants to find the area of a rectangle
that is 2 ¾units long by 1 ½ units wide. Draw a model and show your
work.
• After participants have had an opportunity to work, Super model
this problem on the document camera.
Participants should Deliberately Practice G5-M5-L11, Problems 1(b)
and 1(c).
5 min
24.
Have groups discuss at their tables and then share out whole group.
Discussion may include, but is not limited to, the following:
•In Grade 3, when formal introduction to fraction starts, we extend
previous understandings by using unit language to describe fractions
before we introduce fraction notation.
•We can add and subtract with fractions by adding or subtracting like
units just like with whole numbers.
•The use of models with whole numbers extends to fractions: number
path, number bond, number line, tape diagrams, array, and area
model.
•Understanding of number lines and comparison of whole numbers helps
build an understanding of comparison of fractions.
•Understanding of part/whole relationships extends to fractions.
•We came to understand whole numbers and their operations using
concrete, pictorial, and abstract representations. We can do the same
with fractions.
•We can decompose fractions just as we can decompose whole numbers
by breaking them down into smaller parts.
•We can rename fractions just as we can rename whole numbers.
•Solution strategies of addition (Make a 10, Make the Next Ten, Make a
Simpler Problem) and subtraction (Take from 10, Take from the Next
Ten, Take from 100) with whole numbers builds a foundation for
adding and subtracting with fractions (Make the Next Whole, Add to
the Next Whole and then Add More, Take from 1).
•We can use the number line and ‘arrow way’ to add and subtract with
fractions just as we can use them with whole numbers.
•Our understanding of multiplication and division of whole numbers as
repeated addition and repeated subtraction extends to fractions.
•The associative, commutative, and distributive properties apply to
fractions just as they do to whole numbers.
•We use our understanding of factors and multiples when finding
equivalent fractions.
We can think of division of fractions as partitive division and measurement
division just as we do when dividing whole numbers.
Section: Conclusion: Curriculum Overview
Time: 3 minutes
In this section, you will use the Curriculum Overview Map to show
the flow from whole number to fraction operation modules.
Materials used include:
• Fractions Grades 3 – 5 PPT
• Fractions Grades 3 – 5 Facilitator Guide
Time
3 min
Slide # Slide #/ Pic of Slide
Script/ Activity directions
25.
GROUP
Slide 25: Story of Units Curriculum Overview Map
2 minutes
Close the session by having participants notice the flow from whole number
(yellow) to fraction operation modules (pink).
Thank them and encourage them to use ONE ARITHMETIC for both whole
numbers, measurement units, and fractions and thus empower students to do
the same.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Turnkey Materials Provided
•
•
•
Fractions Grades 3 – 5 PPT
Fractions Grades 3 – 5 Facilitator Guide
Fractions Grades 3 – 5 Participants’ Handout
Additional Suggested Resources
Active learning
Turn and talk
•
•
•
How to Implement A Story of Units
A Story of Units Year Long Curriculum Overview
A Story of Units CCLS Checklist