Multiplication and Division with Whole Number and Fractions
in Grades 3-5: Concepts, Skills, and Problem Solving
Sequence of Sessions
Overarching Objectives of this October 2014 Network Team Institute

These grade-band 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 multiplication and division 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
level lead students toward a profound understanding of multiplication and division.
High-Level Purpose of this Session


Participants will draw connections between the progression documents and the careful sequence of mathematical concepts that develop within each
grade, thereby enabling participants to enact cross- grade coherence in their classrooms and support their colleagues to do the same.
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


Addition and Subtraction: Concepts, Skills, and Problem Solving
Grade-level Module Focus Sessions
Key Points



Knowing the progression across the grade band prepares teachers to more effectively scaffold by knowing what comes before and after a particular
concept on the “ladder” of complexity.
Knowing the progression across the grade provides insight into how each concepts contributes to students’ overall conceptual understanding.
Knowing the progression across the grade band illustrates the power of coherent/consistent teaching of models, strategies, and approaches.
Session Outcomes
What do we want participants to be able to do as a result of this
session?


Participants will draw connections between the progression documents
and the careful sequence of mathematical concepts that develop within
each grade, thereby enabling participants to enact cross- grade
coherence in their classrooms and support their colleagues to do the
same.
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
Introduction
Grade 3
Grade 4
Grade 5
Time
Overview
•
20 min
Introduces using the coherence of
the curriculum as a tool for
teachers.
Examines the topics in Grade 3
that are the beginning stages for
multiplication and division.
•
Examines the expansion of
multiplication and division into
larger numbers in Grade 4.
•
Examines Grade 5 topics including
exponential notation and decimal
multiplication and division as well
as fractions.
•
90 min
100 min
100 min
Prepared Resources
•
•
•
•
Facilitator Preparation
Major Work of the Grade
Band: Grades 3-5 PPT
Facilitator Guide
Review Grade 2 Module 6 and
Grade 3 Module 1
Major Work of the Grade
Band: Grades 3-5 PPT
Facilitator Guide
Review Grade 3 Module
Overviews
Major Work of the Grade
Band: Grades 3-5 PPT
Facilitator Guide
Review Grade 4 Module
Overviews
Major Work of the Grade
Band: Grades 3-5 PPT
Facilitator Guide
Review Grade 5 Module
Overviews
Closing
21 min
Sums up how the use of coherent
models and strategies across
grade levels enables teachers to
support students at their current
level of understanding and helps
them advance to more abstract
thinking and challenging concepts.
•
•
Major Work of the Grade
Band: Grades 3-5 PPT
Facilitator Guide
Session Roadmap
Section: Introduction
Time: 20 minutes
In this section, you will begin an overview of the examination of the Materials used include:
sequence of concepts through which multiplication and division are
 Major Work of the Grade Band: Grades 3-5 PPT
introduced and developed in A Story of Units. You will be
 Major Work of the Grade Band: Grades 3-5 Facilitator
introduced to this session which emphasizes coherence of the
Guide
curriculum as a tool that enables teachers to identify, practice, and
use appropriate instruction moves and scaffolds.
Time Slide #
5 min
1.
Slide #/ Pic of Slide
Script/ Activity directions
This full-day session is intended to support a school-wide implementation of A
Story of Units. This session will examine the sequence of concepts through
which multiplication and division 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 level lead students towards
a profound understanding of multiplication and division. The session
emphasizes coherence of the curriculum as a tool that enables teachers to
identify, practice, and use appropriate instructional moves and scaffolds.
GROUP
2 min
2.
As we explore the progression of multiplication and division, you will
understand how the concepts and lessons at your grade level make critical
connections within the grade band.
Knowing the progression across the grade band:
• Helps you scaffold by knowing what comes before and after your
place on the “ladder” of complexity.
• Gives you insight into how the concepts you teach contribute to
students’ overall conceptual understanding.
• Helps you see the power in coherent/consistent teaching of models,
strategies, and approaches.
2 min
3.
This is the curriculum map for A Story of Units. It shows the overall
sequence of the 5-8 modules at each grade level, PK-5.
CLICK. Today we are focusing on multiplication and division in Grades 3-5.
CLICK. Our work today comes from the highlighted modules.
General Points:
The colors within the map show the Domains of the Common Core
Standards.
• Yellow – Numbers
• Blue – Geometry
• Green – Number and Geometry, Measurement
• Pink – Fractions
8 min
4.
G2-M6 lays the foundation for multiplication and division in G3. The
sequence on the screen shows how the instruction in this module gradually
grows in complexity.
Participants work with a partner/small group to first analyze A and B from
left to right, and then to compare A and B.
Note to presenter: Tape diagram may appear most abstract/out of place.
Images are organized by G2-M6 Topics:
• G2-M6-TA introduces equal groups concretely (not shown), pictorially,
•
then abstractly as tape diagrams.
G2-M6-TB moves from scattered configurations to arrays. Concrete
(not shown), to pictorial, to abstract. Tiles are most abstract because
objects/units are much more difficult to interpret without space in
between.
After participants analyze, facilitate a whole group discussion.
Possibly highlight some of the following:
G2-M6-TA Equal Groups Sequence:
• Begin making equal groups from a given number of objects using
concrete materials.
• Move to pictorial representations of equal groups shown here
(circle a group of 5 stars, add 5 more, then add 5 more).
• Relate drawings to repeated addition and write number sentences.
• Find sums by adding on each time, or by using
doubles/simplifications. (Skip-counting as a strategy is taught in
G3.)
• Draw abstract tape diagrams that show the number in each group as
a new unit.
• Begin understanding that any unit may be counted, e.g., 3 dogs, 3
tens, or even 3 fives.
G2-M6-TB Array Sequence:
• Familiarity with arrays goes back to Kindergarten (rekenrek, 10
frames, etc.)
• Organize equal groups into arrays (concrete, not shown).
• Understand that either a row or column is the unit being counted.
• Compose arrays one row or column at a time, and write repeated
addition sentences.
• Continue to find sums by adding on each time. (Skip-counting as a
strategy is taught in G3.)
• Represent arrays pictorially, distinguish rows/columns by
separating equal groups horizontally/vertically (Shown here: 3
rows of 5 or 5 columns of 3.)
• Next use tiles to make arrays, pushing them together as a
foundational step toward work with area in G3.
General Points:
• Concrete experiences in this module set students up to begin at the
pictorial level in G3-M1.
• G2 focuses on manipulation of place value units (1, 10, 100). G3
focuses on manipulation of numbers 1-10 as units. This module
bridges those understandings.
3 min
5.
The Progressions break strategies for multiplication and division into 3
levels that move from simple to complex.
(In the OA Progression, see appendix pg. 36-38 for a comprehensive look.)
Computation Strategies Sequence:
Level 1:
• Using the array as an example, a Level 1 strategy would be to count
dots one by one to find the total.
• Hopefully mastered in G2. G3 spends almost no time here.
• Could be used for remediation.
Level 2 and 3 strategies require an understanding of the unit and how to
manipulate it.
Level 2:
• G3-M1 starts at Level 2.
• Using the array, a Level 2 strategy would be to count the rows as
units.
CLICK. Count: 1 three, 2 threes, 3 threes, 4 threes, 5 threes, 6 threes, 7
threes.
• Seeing rows as units, students working at this Level could also skipcount. (3, 6, 9, etc.)
• L2 skip-counting strategies are used throughout G3-M1 and G3-M3.
Level 3:
• These are properties of operations strategies in G3. (Commutative,
Distributive, Associative.)
• Using the array, a Level 3 strategy would be looking at groups as
units.
CLICK. For example, using the Distributive Property to simplify 7 threes by
thinking of it as 5 threes and 2 threes.
• Level 3 strategies are introduced halfway through M1 and grow
slowly over M1 and M3.
• A strong foundation in models (e.g. array) is necessary so students
can use them as tools to access the properties.
General Points:
• The Progressions can be found at the University of Arizona website
and were a guiding document in the development of the Common
Core State Standards.
Section: Grade 3
Time: 90 minutes
In this section, you will examine the topics in Grade 3 that are the
beginning stages for multiplication and division.
Materials used include:
 Major Work of the Grade Band: Grades 3-5 PPT
 Major Work of the Grade Band: Grades 3-5 Facilitator
Guide
 Major Work of the Grade Band: Grades 3-5 Problem Set
Time Slide # Slide #/ Pic of Slide
0 min
6.
Script/ Activity directions
GROUP
10 min
7.
The models on the screen show the sequence of complexity in the first
lessons on multiplication.
Ask participants to analyze the sequence. You might ask them to notice how it
compares to Grade 2, or compare the way that each model shows 4, 3, and 12.
Teaching Sequence:
• Work with equal groups pictures, relate to repeated addition and
multiplication sentences.
• Re-discover that an array is a more efficient organization of an equal
groups picture.
• Formalize understanding that the array shows 1 equal group as 1
row. (Columns come halfway through the module.)
• Distinguish between number of groups and size of groups (meaning
of factors) as they count rows and how many in 1 row to write
multiplication facts.
• Array is a critical model for exploring multiplication, division, and the
arithmetic properties, so it’s highly emphasized.
• Arrays are in M1 & M3. They become an “old friend” that makes new
concepts/connections between concepts accessible.
• Number bond is familiar from earlier grades, but the part-whole
relationships it shows are harder to identify in terms of
multiplication. That’s what makes it the most abstract of these
models. (Possibly point out how 4 is shown.)
CLICK. Throughout these initial lessons, students use both unit form and
standard form to compose number sentences that represent the models.
(Unit form – the concept and/or the term – may be new to participants. If
necessary, practice counting by units of 3 with these models.)
Participants try Problem 1 on the Problem Set.
Arthur has 4 boxes of chocolates. Each box has 6 chocolates inside.
How many chocolates does Arthur have altogether?
Possibly debrief by having them compare/analyze their 3 models with a
partner, or bring a participant’s work to the document camera and analyze as
a whole group.
3 min
8.
The models on the screen show the sequence of complexity in the first
lessons on division, which immediately follow the sequence on the previous
slide. Multiplication and division are taught together in M1 & M3 so
students perceive the relationship between them from the beginning.
Ways this relationship is made evident:
• Understand division as an unknown factor problem: unknown in
division is size of groups or number of groups.
(Notice title of slide – “composing” units to divide.)
• Use the same models for division as for multiplication: equal groups
pictures, arrays (not shown), number bonds, and count-bys to
understand the meaning of the unknown in division as size of groups
or number of groups.
• Emphasis on conceptually understanding division and learning to
interpret problems by writing division expressions.
Review with participants:
• In each example on the screen, groups can also be referred to as
units: 7 groups of 2, or 7 units of 2, which is the same as 7 twos.
10 min
9.
The goal of this slide is to help participants recognize the difference
between measurement and partitive division.
Ask participants to analyze the problems to determine the difference between
them. (Unknowns are different.)
Help participants name/understand each type of problem based on their
observations:
CLICK
• Problem A is an example of measurement division because the number
of groups is unknown.
• Problem B is an example of partitive division because the size of each
group is unknown.
Background for presenter:
• Students are introduced to both types of division in M1. They are
both used throughout M3.
• G3 does not use the vocabulary ‘partitive’ and ‘measurement’ with
students.
• Partitive division tends to be easier; students usually have a good
understanding of “fair share”.
Possibly have participants discuss the 2 ways of modeling shown.
You might ask:
• Can either model be used for either type of division? (Yes.)
• Can you think of other ways to model each type of division? (M1 uses
tape diagrams)
• Write equations to represent each problem. (Possible practice: Write
unknown factor multiplication sentences to understand the meaning of
the unknown, then rewrite as division where the quotient is the
unknown factor.)
Direct participants to solve Problem 2 on the Problem Set.
A. Two students equally share 8 crackers. How many crackers does
each student get?
B. There are 8 crackers. Each student gets 2. How many students get
crackers?
Standards Connection:
(CLICK for star) 3.OA.3: Use division within 100 to solve word
problems involving equal groups.
5 min
10.
The Commutative Property of multiplication is the first Level 3 strategy to
be introduced.
By now students:
• Have built background with both multiplication and division.
• Are well on their way toward understanding the meaning of factors.
• Are comfortable with L2 strategies (skip-counting), and models
(array, number bond).
Guide participants through the following steps to demonstrate how the
Commutative Property is introduced.
1. Turn your board so it’s vertical. Draw an array that shows 5 threes. How
many rows of 3 did you draw? (Draw, then answer ‘5 rows’.)
2. How many columns of 5 did you draw? (3 columns.)
3. Write an equation for the array where the number of rows is first. Don’t
solve yet. (Write 5 × 3 = ____.)
4. Rotate your board so it’s horizontal. How many rows of 5 do you have now?
(3 rows.)
5. How many columns of 3? (5 columns.)
6. Write an equation for the array where the number of rows is first. Don’t
solve yet. (Write 3 × 5 = ____.)
7. Explain to a partner using the words columns and rows why your equation
changed.
8. Will 5 × 3 and 3 × 5 have the same total? Use the array; tell your partner
how you know.
CLICK. Ask participants to tell a partner how they know the equation is true.
Background for presenter:
•
•
•
•
Commutativity enables students to write multiplication equations
with number of groups OR size of groups first.
As with other properties, lessons are sprinkled throughout M1-3 to
allow practice before adding complexity.
Commutativity is a strategy to multiply (rather than thinking every
fact must newly be memorized).
Per G3 standards, students are not required to know the properties
by name. We do call the Commutative Property by name, but not the
others.
Standards Connection:
(CLICK for star) 3.OA.5: Apply properties of operations to multiply.
5 min
11.
Distributive Property
• Introduced through the array (familiar friend)
• First with multiplication and then with division
• Learn this property as a strategy for approaching unknown
problems.
Ask participants to do the problem with you on their boards. Guide them
through the following steps:
1. Draw an 8 by 4 array (CLICK to show). How many fours does
the array show? (8 fours.)
2. Break apart 8 fours into 2 smaller parts: make number bonds to
show pairs with a sum of 8. (1/7, 2/6, 5/3, 4/4.)
3. Go with 5/3, add the unit “fours” to 8, 5, and 3 in the number
bond. CLICK to show bond.
4. Draw a line to show array breaking apart to make 5 fours and 3
fours. Shade 5 fours. CLICK.
5. Write 5 fours and 3 fours as equations next to the parts of the
array they represent. CLICK.
6. CLICK. Tell a partner how these equations show your work
with the array.
Background for presenter:
• G3 refers to the Distributive Property as the “break apart and
•
•
•
distribute strategy” with students.
Focus on using five facts as a place to break apart arrays. Fives facts
are “easy.” Also, the Progressions call out the 5 + n pattern as a
strategy for multiplying.
Although 5 + n is the focus, the efficiency of using doubles (4/4 in this
case) is also practiced.
Number bond is just another perspective on the same idea.
Call attention to the final equation (5 + 3) x 4. This notation is not emphasized
in these lessons, but is shown to lay a foundation for the Associative Property.
Standards Connection:
(CLICK for star) 3.OA.5: Apply properties of operations to multiply.
8 min
12.
Using the Distributive Property to solve division is similar to
multiplication:
• Break apart an array
• Solve 2 smaller problems
• Add the quotients of these problems together to solve the original
Ask participants to study the work on this slide and compare the steps to those
they took to solve the multiplication problem. Facilitate a discussion about
their observations.
Background for presenter:
• The 5 + n strategy is used to break apart the dividend in the bond (28
breaks into 20 and 8 because 5 x 4 = 20).
• Students might just know the original facts from both the
multiplication and division examples. That’s okay; we use them to
get at the properties of arithmetic and show relationships between
various basic facts. The goal is to build understanding that they can
eventually apply to facts they don’t know.
Direct participants to solve Problem 3 on the Problem Set.
Use the Distributive Property to solve 32 4 = _______ .
Standards Connection:
(CLICK for star) 3.OA.5: Apply properties of operations to multiply.
10 min
13.
Associative Property of Multiplication
• Introduced through the array (familiar friend)
• Learn this property as a possible strategy for difficult problems
(involving factors beyond 10) like 16 x 2.
Guide participants through the following steps to understand the Associative
Property.
1. 16 is the tricky part. Let’s simplify. On your board, write factor
pairs that have a product of 16.
2. Let’s rewrite 16 as 8 x 2. (CLICK)
3. The problem isn’t easier yet. We know from significant prior
work that we can shift the parentheses in this situation without
affecting the answer. Let’s try moving them to simplify. (CLICK)
4. The array shows how we regrouped the numbers to show 8
groups of 2 x 2.
5. How did that make the problem friendlier? (Now it’s just 8 x 4!)
(CLICK)
6. Those are factors we know how to work with. 8 x 4 =? (32.)
(CLICK)
7. The array is there to help make it visually clear to students that
16 x 2 = 8 x 4.
Background for Presenter:
• This is almost exactly how the Associative Property is taught in the
lesson.
• In the lesson, students rewrite 16 as 4 x 4 too and see that it doesn’t
really help simplify (still 4 x 12).
• Not much emphasis placed on the Associative Property: just a
strategy for toolboxes.
• Prior work includes a lesson on parentheses (without order of ops
language) and practice seeing the outcome of how moving
parentheses changes/doesn’t change outcome with different
operations.
You might ask how work with the Distributive Property (including notation
from multiplication slide) helps create the foundational knowledge necessary
to access this property.
Direct participants to solve Problem 4 on the Problem Set., or use as a “we do.”
Use the Associative Property to solve 3 × 12 = _____.
Standards Connection:
(CLICK for star) 3.OA.5: Apply properties of operations to multiply.
4 min
14.
Arithmetic patterns:
• Become important strategies for problem solving with multiplication.
• Conceptually understanding patterns moves them from “tricks” to
tools.
• Nines facts are a vehicle for exploring patterns in G3 because nines
facts are replete with patterns.
Background for Presenter:
This slide shows using the 9 = 10 – 1 pattern:
• Distributive Property: Rather than break up 9 into smaller parts, we
subtract a unit from 10 since 9 is so close to 10.
• 9 = 10 – 1 put simply: use a tens fact to help solve a nines fact.
• Important because it lays the foundation for other patterns that
emerge with units of 9 later.
Ask participants to turn to Problem 5 on the Problem Set:
Guide them through the following process.
1. What’s easier to solve, 9 × 4 or 10 × 4? (10 x 4.)
2. How many fours in 10 × 4? (10 fours.)
3. Your tape diagram shows 10 fours. What is the value of 1 unit on your
tape diagram? (Four). Label each unit.
4. How many fours in 9 × 4? (9 fours.)
5. Change your tape diagram to show 9 fours. (Change.) What change
did you make? (e.g., crossed off a four.)
6. 9 fours equals 10 fours minus… (1 four.)
7. Write an equation to show that. (E.g., 9 × 4 = (10 × 4) – (1 × 4) OR 9
fours = 10 fours – 1 four.)
8. Rewrite your equation using the products of 10 × 4 (or 10 fours) and 1
× 4 (or 1 four). (Write 9 × 4 = 40 – 4.)
9. What is 40 – 4? (36.)
CLICK to show solution.
General Points:
5 min
15.
The nines finger strategy is based on the 9 = 10 – 1 pattern. Before they
are introduced to the finger strategy, students model how it works with
arrays.
Talk participants through the following sequence:
CLICK. To solve 3 x 9, draw a 3 x 10 array.
CLICK. Using 9 = 10 – 1, students realize that 3 nines = 3 tens – 3 ones.
CLICK. Cross off 3 ones from the 3 tens, which leaves 2 tens 7 ones.
(Specifically crossed off as shown so the array clearly pictures 2 tens and 7
ones like fingers do.)
CLICK. 3 nines = 3 tens – 3 ones can be written as 3 x 9 = 30 – 3.
CLICK. That gives us 3 x 9 = 27.
CLICK. With conceptual understanding of 9 = 10 – 1 in place, the nines finger
strategy is introduced.
Possible partner share: How does the 9 = 10 – 1 strategy relate to the nines
finger strategy?
Standards Connection:
(CLICK for star) 3.OA.9: Identify and explain patterns in the
multiplication table.
5 min
16.
Toward preparing students for Grade 4, they explore place value as a
strategy for multiplying with multiples of 10. They begin with concrete
experience.
Provide participants with place value disks. *This may be their first experience
with them. If so, review the tool.*
“We do” using the document camera (record equations and unit language):
• Model 2 x 3 with place value disks as 2 rows of 3. Write ‘2 x 3 ones.’
• Our array shows 2 x 3 ones, true? (True.)
• What is the total value of 2 x 3 ones? (6 ones.) Record ‘2 x 3 ones = 6
ones’.
• Let’s change our units from ones to tens (change disks).
• What happens to our equations? (unit changes from ones to tens, and
total value changes from 6 to 60.)
• We multiplied 2 x 3 ones by ten, so now we’ve got 2 x 30 = 60, or 2 x
3 tens = 60.
CLICK. This work is transferred to the place value chart. Students
pictorially model 2 × 3 in the ones place. Then they locate the same basic
fact in the tens column. They see that when multiplied by 10, the digit shifts
one place value to the left.
Standards Connection:
(CLICK for star) 3.NBT.3: Multiply one-digit whole numbers by
multiples of 10.
2 min
17.
Module 4:
• Area is introduced.
• Relate area to understanding of multiplication.
Sequence of progression to the area model:
This image shows how the array evolves to become the area model as
students’ understanding of area grows over the course of the module.
• Concrete/Pictorial materials:
• Columns and rows
•
•
•
Grid with discrete objects in each box:
• Students often don’t perceive the space in each box as the unit
at first and mistakenly count lines.
• The object in each box is a scaffold to avoid that.
Grid with no objects:
• Students use inch and centimeter tiles to cover shapes.
• Understand the importance of no gaps or overlaps for area.
• Students make use of grid paper as well.
Empty square:
• This is what we call the area model.
• A critical module because it’s where the area model is
established.
• Folks with pacing problems, please do not gloss over/skip M4
because the model is used extensively in G4 and G5 as a model
for multiplication.
Standards Connection:
(CLICK for star) 3.MD.7: Relate area to multiplication.
2 min
18.
Background for presenter:
• Area model is used for both multiplication and division.
• Given side lengths, students multiply to find the area.
• Given area and one side length, students find the other side length
either by solving an unknown factor multiplication equation or a
division equation.
• G3 area work is purely conceptual: the “formula” for area is not
formally taught until G4.
Standards Connection:
(CLICK for star)
• 3.OA.4: Find the unknown number in a multiplication or division
equation.
• 3.OA.6: Understand division as an unknown factor problem.
3 min
19.
Ask participants to study the work on screen. Which property is evidenced and
how?
• Distributive Property is evidenced. (CLICK to confirm Distributive
Property/show slide title.)
• Side length 8 is decomposed into 5 and 3.
• Smaller side lengths are used to find the areas of the smaller
rectangles.
• Areas of smaller rectangles are added to find the total area of the large
rectangle.
Application of this Level 3 strategy prepares students for work they do in G4
with multiplication and the area model.
Standards Connection:
(CLICK for star) 3.OA.7: Fluently multiply and divide within 100.
10-15
min
20.
•
•
•
•
•
Give participants about 5 minutes to review the Instructional
Practice guide and familiarize themselves with the Core Action
Indicators.
Examine the Grade 3 Sample Work.
CLICK to reveal reflection points to be used alongside a discussion
about the sample work.
Allow 5-10 minutes to reflect and discuss.
Participants should use their graphic organizers to assist in
identifying scaffolds for instruction.
Section: Grade 4
Time: 100 minutes
In this section, you will examine the expansion of multiplication and Materials used include:
division into larger numbers, including hundreds and thousands, in
 Major Work of the Grade Band: Grades 3-5 PPT
Grade 4.
 Major Work of the Grade Band: Grades 3-5 Facilitator
Guide
 Major Work of the Grade Band: Grades 3-5 Problem Set
Time Slide # Slide #/ Pic of Slide
0 min
21.
8 min
22.
Script/ Activity directions
Background for presenter:
• Connection to the place value disk and chip model from Grade 3 with
multiplying by multiples of 10. (e.g. 3 tens times 2)
• Grade 4 extends multiplication into the hundreds and thousands to
multiply by 4 digit numbers.
• Grade 4 extends multiplication past multiplying number of groups or
number in each group, thinking about multiplicative comparison.
• “Times as many as” and multiplicative comparison set the stage for
Grade 5 scaling and Grade 6 proportional reasoning.
MODEL. 10 times as much as 1 one.
MODEL. Have participants follow along: 10 times as much as 3 ones.
CLICK. 3 tens is 10 times as much as 3 ones.
CLICK. To show multiplication of 100, first we break 100 into 10 times 10.
CLICK. Tell your partner a statement using multiplicative language for this
image. (300 is 100 times as much as 3 ones.)
GROUP
Participants draw to show 1,000 x 3 = 3,000 and write a statement using
multiplicative comparison language. (3 thousands is 1,000 times as much as 3
ones.)
CLICK. Using place value disks, this image shows 3 times 4 ones is 12 ones.
CLICK. It is recorded here numerically and in unit form.
CLICK. Tell your partner the multiplication sentence for this image. (3 times
4 tens is 12 tens)
CLICK. Now we are multiplying by multiples of 10. (4 tens)
CLICK. Record the numerical multiplication sentence in unit form that
represents this image. Show me your boards.
CLICK. Check your work.
Standards Connection:
(CLICK for star)
• 4.NBT.1: A digit in the ones place represents 10 times what it
represents in the place to its right.
• 4.OA.1: Interpret a multiplication equation as comparison.
12 min
23.
Background for presenter:
• Students use the distributive property, place value disks, partial
products, standard algorithm, and area model to multiple 2, 3, and 4
digit numbers times a single digit.
• Start concretely using place value disks.
• Move pictorially to the chip model.
• Align the steps of the algorithm directly to the work occurring
in the place value chart.
• Fluency with the standard algorithm for multiplication is not
expected until Grade 5 – 5.NBT.5.
• Introduced to the algorithm, supported by place value
strategies as stated in 4.NBT.4, to prepare for Grade 5
multiplication.
• Not assessed in Grade 4 on using the algorithm for
multiplication.
Switch to the document camera. MODEL 4 x 23 using place value disks and a
place value chart with ones and tens drawn on a personal white board.
• Display 2 tens 3 ones.
• Display 4 copies of 2 tens 3 ones to show multiplication times 4.
• Set up 4 x 23 vertically.
• Point to problem. 4 times 3 ones is 12 ones. Record 12 ones as a
partial product.
• Point to the ones column in the place value chart. 4 times 3 ones
is 12 ones. Record 12 ones under the ones column.
• Point to problem. 4 times 2 tens is 8 tens. Record 8 tens (80
ones) as a partial product.
• Point to the tens column in the place value chart. 4 times 2 tens
is 8 tens. Record 8 tens under the ones column.
• Regroup 10 ones as 1 ten.
• Rename the number of units in the place value chart as 9 tens 2
ones.
• Add the partial products as 92.
CLICK. Participants should analyze 4 x 34 and notice such items as:
• 16 ones required regrouping of 10 ones as 1 ten.
• The tens now have 13 tens. 10 tens are renamed as 1 hundred.
• 12 tens 16 ones is the same as 1 hundred 3 tens 6 ones.
CLICK. Participants should complete Problem 6 on the Problem Set: 3 x 424
10 min
24.
MODEL. Guide participants in drawing an area model for 3 x 424.
Teaching sequence:
• Decompose units – 424 = 400 + 20 + 4
• Draw a rectangle with a width of 3 and a length of 400 + 20 + 4
• Multiply each composite rectangle. Focus on unit language (e.g. 3
times 4 hundreds is 12 hundreds)
• That multiplication is the distributive property in action
(3x400)+(3x20)+(3x4)
• Each partial product in the composite rectangles can be related to the
partial products when solved using a vertical problem.
CLICK. Image of problem just modeled displayed.
CLICK. Participants should complete Problem 7 on the Problem Set:
7 x 534
8 min
25.
Switch to document camera. Model 6 x 612 using the algorithm. Record
regroupings “on the line”.
CLICK. Participants should analyze 5 x 2,374 and notice such items as:
• 5 times 4 ones is 20 ones. 20 ones is the same as 2 tens 0 ones. So
we record 2 tens “on the line” and a 0 in the ones column below
the line. The 2 is recorded before the 0 because this is the
number twenty, not the number “zero-two”. The proximity of
the 2 and 0 remind us of the number 20.
• Using unit language to multiply supports the place value of the
algorithm.
• Renaming 37 tens as 3 hundreds 7 tens helps in knowing where
to place the 3 and the 7.
CLICK. Participants should complete Problem 8 on the Problem Set:
4 x 8,618
General Points:
• This is the standard algorithm for multiplication used in a Story of
Units, introduced in G4-M3-L9.
Standards Connection:
(CLICK for star) 4.NBT.5: Multiply a whole number up to 4 digits by
one-digit whole number.
5 min
26.
CLICK. Participants should complete Problem 9 on the Problem Set:
There are 12 students in PE class separated into 4 equal teams. How
many students are on each team? Draw an array to model the problem.
Then:
On your personal white board, revise your problem so that there are 13
students in PE class separated into 4 equal teams. How many students are
on each team?
Compare the two models and problems.
CLICK. Compare and contrast these 3 models.
Participants should analyze the models and notice such items as:
• Images represent varying levels of pictorial understanding for
representing multiplication.
• The far right image is most abstract, connecting to G3’s area model.
• A remainder is represented by not belonging to an equal group,
completing an array, nor completing the length of a rectangle.
CLICK. Let’s discuss how to record remainders. This is an invalid equation
because both sides of the equal sign must have equivalent values.
CLICK. 7 divided by 2 also equals 3 remainder 1. Therefore we cannot write
“3-R-1” after the equal sign because we would be equating that 13÷4 is
equivalent to 7÷2, which we know it is not. 13 fourths is not the same as 7
halves. Or 3 ¼ is not equal to 3 ½ .
CLICK. Also writing “13÷4=3R1” is invalid. The R is invalid in an equation.
Therefore we write our answers as statements (The quotient is 3 and the
remainder is 1.). We can write “3-R-1” in a vertical problem because there is
no equal sign.
10 min
27.
The division algorithm is introduced and supported alongside the place
value chart model in G4-M3-L17.
Switch to document camera.
Model. 4÷3 using place value disks.
Note the remaining 1 one cannot be further subdivided.
Model. Have participants following along with 42 ÷ 3 using place value disks.
Note the remaining 1 ten can be further subdivided; rename 1 ten as 10 ones.
Model. Have participants following along with 44 ÷ 3 using the chip model.
Note the remaining 1 ten can be further subdivided; rename 1 ten as 10 ones.
Note the remaining 2 ones cannot be further subdivided.
CLICK. Participants should analyze the 3 problems they just solved here on the
slide as the chip model and notice such items as:
• 4 is crossed off one by one, where 42 and 44 are crossed off
quickly, showing level of understanding for the model and facts.
• Remainders in the ones are circled.
• Each group of subdivided disks represents the quotient.
• Tens can be further subdivided because 1 ten is the same as 10
ones. G4 students do not yet know 1 one is equal to 10 tenths.
• The steps of the algorithm (divide, multiple, subtract, bring
down) are better explained and modeled using place value
understanding instead of memorizing the 4 steps or acronyms
or songs.
CLICK. Students are asked to check their work using multiplication and
addition.
CLICK. Relating the part/whole relationship of the quotient and remainder
to the dividend can be shown using a number bond.
(If needed, allow participants to try 58 ÷ 3 or 1,534 ÷ 4.)
28.
1,344 is the area and 6 is the width. Students solve for the length by thinking
6 times how many thousands is about 1 thousand? Zero thousands. So 6
times how many hundreds is about 13 hundreds? 6 times 2 hundreds is 12
hundreds, which is almost 13 hundreds.
(CLICK)
A new area model drawn below shows how one might decompose the area,
showing the 12 hundreds and a remaining 144. Then we can ask 6 times
how many tens is about 14 tens? 2 tens. That leaves a remaining 24 ones. 6
times 4 ones is 24 ones.
A number bond represents the 3 parts to the area model, and the number
sentence below shows the distributive property as we divided up the total
area into smaller parts.
Remember how this is similar to how we decomposed 1 division problem
into 2 division problems in Grade 3 using the distributive property, such as
48 ÷ 3 as 30÷3 and 18÷3.
Remember the area model breaks down for remainders, so the remaining
part of an area would get tacked on as 1 square unit to the end of the model,
as we showed a few slides back.
(Solve 1,214 ÷ 4 with participants.)
Standards Connection:
(CLICK for star) 4.NBT.6: Find whole number quotients and
remainders with up to 4-digit dividends and 1-digit divisors.
4 min
29.
Switch gears back to multiplication. How does this type of problem not
relate to the multiplication and division that were just practiced? (1-digit
factors or divisors.)
• Using what they know about 4 times 2 tens, students reason about
solving 4 tens times 2 tens.
• Instead of just teaching the trick of the 2 zeros, students learn about
tens times tens using place value understanding.
CLICK. Have participants analyze the 2 place value charts. They should notice:
• Top image: 20 was recorded as 2 times 10. 40 is represented in
the tens. 4 tens is doubled. Then multiplied by 10.
• Bottom image: 40 was recorded as 4 times 10. 20 is represented
in the tens. 2 tens is quadrupled. Then multiplied by 10.
• The place value chart provides reason for why a ten times a ten
is a hundred.
• All properties of multiplication were used to complete these
samples.
CLICK. Visually through the area model, students see 1 ten times 1 ten is 1
hundred. Just as with the area model, soon students remove the inner grid
and reason numerically that 4 tens times 2 tens is 8 hundreds.
4 min
30.
Draw an area model for 3 times 25. CLICK.
Revise your area model for 30 times 25. CLICK.
Notice the solution is 10 times as much because we multiplied by 10.
Note the unit language that drives the understanding of the multiplication.
As students can already solve for a problem using the area model or written
notation for a 2-digit by 1-digit number such as 3x25, here we ramp it up a
notch, adding a complex step of multiplying by a multiple of 10.
1 min
31.
Students record the 2 partial products under the problem.
Focus is on products of multiples of 10.
Connecting to the written notation is the step prior to solving for 4 partial
products. Students are already comfortable recording an area model and a
written notation for a single-digit times 2-digit number.
3 min
32.
Background for presenter:
• To multiply a 2-digit by 2-digit number, students use unit language
(which supports place value) and the area model, and alongside
record the steps for the algorithm.
To solve the area model as it relates to the step of the algorithm, solve right
to left, top to bottom in the area model.
To start in the algorithm, we would start in the ones: 3 times 1 one. CLICK.
Record the partial product. Notice the distributive property at work,
recorded below the area model.
3 times 3 tens is 9 tens. CLICK. Record the partial product.
2 tens times 1 is 2 tens. CLICK. Record the partial product.
2 tens times 3 tens is 6 hundreds. CLICK. Record the partial product.
Add all partial products. CLICK.
2 min
33.
Soon we start to see the multiplication of 43 and 67 as 3 times 67 and 40
times 67, moving us to the recording of 2 partial products, more formally as
the steps of the algorithm.
8 min
34.
Switch to document camera.
Model. 46 x 63 as participants follow along.
CLICK. Display of problem just solved.
(Solve 48 x 74 with participants if needed.)
CLICK. Participants should complete Problem 10 on the Problem Set:
84 x 73 using an area model, partial products, and the standard
algorithm.
Standards Connection:
(CLICK for star) 4.NBT.5: Multiply two 2-digit numbers.
3 min
35.
Let’s think about how we can apply what we know about multiplication of
whole numbers for multiplication of fractions.
CLICK. I know that 1 apple + 1 apple + 1 apple = 3 apples.
CLICK. So, 1 one + 1 one + 1 one = 3 ones.
CLICK. Therefore, 1 fourth + 1 fourth + 1 fourth = 3 fourths.
CLICK. We can say there are 3 copies of 1 fourth.
Background for presenter:
Similar to repeated addition for whole numbers is multiplication, just
applied to fractions or different units (e.g. ones, tens, fourths, fifths).
Students have come to the table with
• understanding unit and non-unit fractions, their size and location on
a number line
• ability to add fractions and mixed numbers
• Decompose non-unit fractions in various ways
• Comparing fractions with like and unlike denominators
3 min
36.
Background for presenter:
Tape diagram
• 1 whole is represented by the entire length.
• 1 whole is decomposed into 4 equal sized parts or units.
• 3 units out of 4 are shaded representing 1/4+1/4+1/4=3/4.
CLICK.
• Knowledge of repeated addition from whole numbers applies to
fractions.
• Only the units have changes (say from twos, threes, and fours, to
fourths, fifths, and eighths)
• The associative property allows flexibility in renaming 3 times (1
fourth) as (3 times 1) fourth.
• Writing in unit form supports the numerical work to the right.
8 min
37.
Background for presenter:
• Sequence of multiplication of fractions begins with fractions less than
1.
• Relates to learning addition with numbers 0-10 and multiplication of
whole numbers with easy factors of 2, 3, and 4.
• The tape diagram (first use in presentation) represents 4 copies of
3/5.
CLICK. Fractions can be renamed in unit form (e.g. 3 fifths) to apply the
associative property for multiplication.
CLICK. To verify the answer, each unit or copy of 3/5 in the tape diagram
could be partitioned into three 1 fifths.
fifths or 12 fifths total.
This image shows 4 copies of 3-1
CLICK. Images of 12 fifths disappears.
(This can easily be applied to a word problem context. Sally’s ribbon is 3
fifths meters long. Marcus’s ribbon is 4 times as long as Sally’s ribbon. How
long is Marcus’s ribbon?)
CLICK. Participants should complete Problem 11 on the Problem Set:
5 x 11/12
8 min
38.
Alternatively, students can use the Distributive Property to multiply a
whole number times a fraction. This tape diagram shows this pictorially and
works really great with mixed number multiplication. I start with 3 and 1
fifth.
CLICK. I represent 2 copies because I am multiplying times 2. I can see how
I have 2 copies of 3 and 2 copies of 1 fifth.
CLICK. I can multiply those separately, multiplying the whole number 3,
and then multiplying the fraction 1 fifth, and then add them back together.
This last image shows how I distributed the parts of my tape diagram to
more clearly shows this.
CLICK. Participants should complete Problem 12 of the Problem Set:
3 x (4 2/3)
Standards Connection:
(CLICK for star) 4.NF.4: Multiply a fraction by a whole number.
10-15
min
39.
•
•
•
•
•
Section: Grade 5
Give participants about 5 minutes to review the Instructional
Practice guide and familiarize themselves with the Core Action
Indicators.
Examine the Grade 4 Sample Work.
CLICK to reveal reflection points to be used alongside a discussion
about the sample work.
Allow 5-10 minutes to reflect and discuss.
Participants should use their graphic organizers to assist in
identifying scaffolds for instruction.
Time: 100 minutes
In this section, you will examine Grade 5 topics including
Materials used include:
exponential notation and decimal multiplication and division as
 Major Work of the Grade Band: Grades 3-5 PPT
well as fractions.
 Major Work of the Grade Band: Grades 3-5 Facilitator Guide
 Major Work of the Grade Band: Grades 3-5 Problem Set
Time Slide # Slide #/ Pic of Slide
0 min
40.
Script/ Activity directions
GROUP
2 min
41.
In G5, students continue to notice place value patterns and see how
adjacent place value units are related.
Encourage participants to analyze this slide and notice that exponents are an
added complexity.
• Here students are introduced to exponential notation where they
learn to recognize that multiplying by 10 to the 3rd power is
equivalent to multiplying 10 x 10 x 10.
• Each multiplication by 10 results in the movement of the DIGIT, one
place to the left.
General Points:
This work very closely mirrors the math on the place value chart from
grades 3 and 4.
2 min
42.
Students also use exponential notation when showing division by a power of
ten.
Background for presenter:
Previously, students learned that when moving to the left on the place value
chart, the units become 10 times greater than those to the right. Here,
1
students recognize that places to the right on the place value chart are 10 the
size of the place to the immediate left.
• It is very important for students to realize, and for teachers to
reiterate, that the digits are doing the shifting.
• As you can see, the digit 7 in the hundreds place shifts 3 places to the
right when divided by 10 to the 3rd power.
• The DECIMAL POINT DOES NOT SHIFT! The decimal is ALWAYS
between the ones place and the tenths place. It does not move.
Encourage participants to see the additional complexity on this slide. (In G5,
the only added complexity on the place value chart is the addition of a new
place value called, the “thousandths” place.)
3 min
43.
After students multiply decimals by powers of 10, we move to the
multiplication of multi-digit decimal numbers by one-digit whole
numbers. As always, though the complexity of content may be changing,
the models remain the same.
In this example, we’re using the chip model and a place value chart to show
the multiplication and connecting it to the standard algorithm.
Invite participants to analyze the image and make sense of what is happening.
Participants should share with their neighbor what they understand about the
work shown.
Encourage participants to identify how this model would be different if it were
showing 423 x 4. (The ONLY difference would be the names of the places in the
PV chart–proof that multiplying decimals is just like multiplying whole
numbers.)
2 min
44.
The area model, another old friend from previous grades, can be used to
support learning with decimal multiplication.
Give participants time to analyze and make sense of this area model.
Encourage participants to recognize and appreciate the unit language shown.
4 x 4 is always 16. The unit being counted here is tenths, so 4 x 4 tenths is
16 tenths, etc.
8 min
45.
TIME ALLOTTED FOR THIS SLIDE:
• 2 min analysis
• 3 min modeling
• 3 min Problem Set
8 minutes
Students also divide a decimal dividend using a place value chart and the
standard algorithm.
Participants take a moment to analyze this work and make sense of what is
happening in this model. (The initial complexity of this problem is that the
ones must be changed for 10 tenths before sharing/dividing.)
Have participants discuss with a neighbor how they might use unit language
and place value reasoning to solve this division problem using both the chip
model and the written method.
Switch to Document Camera.
Model. 1.401 ÷ 3
(If necessary, check G5-M1-L14 CD to model appropriate language. It
begins like this.)
• Let’s begin by dividing or sharing our largest units. Can we share 1
one with 3 groups?  No, we must change 1 one for 10 tenths.
(Draw change using chip model.)
• 10 tenths plus the original 4 tenths in our whole makes… 14 tenths.
• Can we share 14 tenths with 3 groups? How many tenths in each
group?  4 tenths. (Record 4 tenths in both groups using chip model
and record 4 tenths in algorithm.)
• When we shared 4 tenths with 3 groups, how many tenths were
shared?  12 tenths. (Record in algorithm.)
• How many tenths remain?  2 tenths.
• Can we share 2 tenths with 3 groups?  No, we must change 2 tenths
for 20 hundredths, etc.
CLICK. Participants should complete Problem 13 of the Problem Set:
Use the place value chart and algorithm to solve 3.445 ÷ 5 = _______.
2 min
46.
Multi-digit multiplication continues in G5 as students multiply up to 4digit numbers by 3-digit numbers, shown here.
Participants analyze and explain to a neighbor what they understand about
the area model on the left and its connection to the algorithm on the right.
Note: Though this is a 4-digit by 3-digit problem, the area model is a 3-by-3
model due to the zero in the tens place of the first factor.
3 min
47.
Here, students multiply decimal values by multi-digit numbers.
Give participants time to analyze and again explain to a neighbor what they
understand about the model and the accompanying algorithm.
Encourage participants to discuss how the model/algorithm are so similar to
the model/algorithm used for whole number multiplication in G4.
Background for presenter:
Participants often ask why the width 43 is decomposed (in the area model)
with the ones above and the tens below. This is just a convention, and
because area is additive in nature, it could have been done another way
(with 4 tens above and 3 ones beneath). However, decomposing the length
and width in this way is more similar to what will be done when recording
in the written method (e.g., we begin by multiplying ones by ones and move
from right to left, and then we multiply tens times ones and record beneath
the first partial product).
9 min
48.
TIME ALLOTTED FOR THIS SLIDE:
• 4 min model
• 5 min Problem Set
9 minutes
Dividing by 2-digit divisors is new to Grade 5 students. Students round
the divisor in order to know where to put the first digit in the quotient of the
written method. Notice how unit language helps students estimate.
Use Document Camera to Model: (Use the vignette from M2-L22 as an
example of one way that unit language and place value reasoning could be
used to solve.)
• Let’s begin by sharing/dividing our largest units. Can we
divide 5 hundreds by 17?  Not without regrouping.
• We can rename/decompose 5 hundreds as/into 50 tens, plus
the 9 tens in our whole gives us 59 tens. We can divide 59
tens into 17 groups (or groups of 17). How can we estimate to
divide 59 tens by 17?  60 tens ÷ 20 = 3 tens.
• Record 3 tens in the quotient. 3 tens times 17 is?  51 tens.
• How many tens remain?  8 tens.
• Can we divide 8 tens by 17?  Not without regrouping.
• We need to decompose these 8 tens into 80 ones, and since
there are zero ones in our whole (point to 0 in dividend), we
have 80 ones divided by 17. How can we estimate to divide 80
by 17?  80 ones ÷ 20 = 4 ones.
• Record 4 ones in the quotient. What is 17 times 4 ones?  68
ones.
• How many ones remain?  12 ones.
• Can we make another group of 17?  No.
• What is our quotient?  34
• What is 34 units of 17, plus 12 ones?  590!
CLICK: To show solution.
CLICK: Participants should complete Problem 14 with a partner. (Partner A
uses unit language to explain each step of the algorithm in Problem (a).
Partner B does the same for Problem (b).)
PS shows 2 completed problems: (Directions) The following problems
have been completed for you. With a partner, take turns to explain
each step of the algorithm using unit language.
This problem should be modeled step-by-step so that participants can see how
we estimate, first using unit language. It is probably easier to calculate how
many groups of 20 are in 60 tens, compared with deciding how many groups
of 20 are in 600. Once we decide to start with 3 tens, the digit “3” is placed in
the quotient in the tens place.
5 min
49.
The focus now becomes fractions and decimals.
In the second lesson of Module 4, student begin to realize that fractions are
really just another way to write a division expression. They realize that
every fraction they’ve ever seen can be interpreted as a division problem
and every division problem they’ve ever solved could be expressed as a
fraction.
MODEL: Participants should think of each post-it note as one cracker.
Participants will model each scenario concretely, presenter clicks to animate
an image that matches their concrete work.
Place 2 crackers on your board. Work with a partner to show how you would
share these crackers with 2 people.  (Participants work.)  Write the
division sentence that matches the work you’ve done.  2 ÷ 2 = 1.
CLICK: to show first image and explain.
Now, put 1 cracker on your white board. Talk with your neighbor about what
you’ll do to share this 1 cracker with 2 people  (Participants share.)  Raise
your hand if your partner had a good idea. Share it.  (1 participant shares
the idea of cutting/splitting the cracker.)
Work with a partner to show how you would share this cracker with 2 people.
 (Participants work.)  Write the division sentence that matches the work
1
you’ve done.  1 ÷ 2 = 2.
CLICK: to show second image.
Invite participants to explain the unit language division sentence at bottom of
image.
Put a new whole cracker on your board. Talk with a neighbor about what
you’ll do to share this 1 cracker with 3 people.  (Participants share.) 
Work with a neighbor to show how you would share this cracker with 3
people.  (Participants work.)  Write the division sentences that match
your work. Write one sentence using numerals and a second sentence using
unit language.
CLICK: to show 3rd image.
Invite participants to share about the pattern they notice in the 2 numeral
division sentences they see in the 2nd and 3rd images  (The numerator is the
number of crackers being shared and the denominator is the number of people
being shared with, or the numerator is the whole and the denominator is the
number of groups.)
3 min
50.
(Continued from previous slide.)
Now put 3 crackers on your board. Talk with a neighbor about what you’ll do
to share these 3 cracker with 2 people.  (Participants share.)  Work with a
neighbor to show how you would share these crackers with 2 people. 
(Participants work.)  Write a division sentence that matches your work. 
(Participants work.)
CLICK: to show image. (Some, if not all, participants will have modeled the
sharing in this manner.)
Invite participants to share about what happened to the pattern that we saw
3
earlier.  Though 3 ÷ 2 does equal 2, the pattern (for the time being) has
disappeared because most people showed that if you have 3 crackers shared
by 2 people, each person can get 1 whole cracker and only the 3rd cracker
needs to be halved.
Ask participants to consider the following scenario: What if the 3 crackers
were each a different flavor, and each person wanted to taste each flavor?
Would this method of sharing still work? Tell your neighbor why or why not.
 This method would NOT work. Each cracker would need to be halved before
sharing.
2 min
51.
Invite participants to share about what this image shows.
Would this method for sharing work if each cracker were a different flavor?
Explain. (Call on participants to share their ideas.)
Invite participants to share about what they notice is happening again in the
division sentence. 
3
The pattern returns! 3 ÷ 2 = 2. The numerator again shows the number of
crackers (or the whole) and the denominator shows the number of groups.
6 min
52.
TIME ALLOTTED FOR THIS SLIDE:
• 3 min analysis
• 3 min Problem Set
6 minutes
Students spend 2 lessons interpreting fractions as division within the
context of sharing crackers equally. On the second day, they begin to make
the connection to the standard algorithm and expressing the remainder as a
fraction, which is new for 5th graders. The pictorial model supports this
complexity perfectly.
Invite participants to discuss with their neighbor what the image is showing.
How are the two models (drawn beneath each arrow) similar and different?
Explain. (Call on participants to share.)
CLICK: to display image of algorithm.
• Let’s see how the work in the algorithm matches the pictorial model
shown at left.
• How many whole crackers did we begin with?  4 (point to 4 in
algorithm).
• How many people are sharing the crackers?  3 (point to 3 in
algorithm).
How many whole crackers does each person get?  1 (point to 1 in
algorithm).
When we shared 1 cracker with 3 people, how many whole crackers
are shared?  3 crackers (point to 3 in algorithm).
How many crackers remained?  1 (point to 1 remaining cracker in
algorithm).
In order to share that remaining cracker, what was done?  It was
split into thirds.  How many of those thirds did each person get?  1
third (point to 1/3 in algorithm).
So if 4 crackers are equally shared by 3 people, how many crackers
1
does each person get?  1 and 3 crackers (point to quotient in
algorithm).
•
•
•
•
•
CLICK: Participants should complete Problem 15 in Problem Set.
Solve 5 ÷ 2 = _______ using a pictorial model and the standard
algorithm.
2 min
53.
From there, students begin to multiply fractions by whole numbers, but we
1
1
begin with the context of fraction of a set. So 4 x 12 is interpreted as 4 of 12.
Invite participants to explain how this model can be used to explain or support
1
an understanding of 4 of 12. (Call on participants to explain.)
3
What other problems could we solve using this model, or a similar one?  4 of
4
1
1
2
3
12  4 of 12  2 of 12  3 of 12 (by drawing 2 horizontal lines)  3 of 12  3
of 12.
4 min
54.
We can also use tape diagrams to show fraction of a set.
2
2
2
Invite participants to explain what this model is showing. (6 copies of 3  3 + 3
2
2
2
2
+ 3 + 3 + 3 + 3 . This work is just like the work done in G4.)
CLICK: Compare these tape diagrams. Share with a neighbor what you
understand about this pictorial model.
Call on participants to share.
• The tape at right shows fraction of a set.  The tape at right
shows a whole of 6 being partitioned into 3 units (or thirds)
and the question mark identifies that we are finding the value
2
of 2 of those units (or 3 of 6).
• If 6 is divided into three equal units (or thirds) each unit (or
2
third) has a value of 2. Therefore 2 units (or 3 of 6) are equal
to 4.
8 min
55.
TIME ALLOTTED FOR THIS SLIDE:
• 4 min model
• 1 min analysis
• 3 min Problem Set
8 minutes
We then move to multiplying fractions by fractions. We begin by
multiplying unit fractions by unit fractions.
1
1
Switch to Document Camera: Let’s solve 2 x 3 together. (Model for
participants.)
1
• It might be helpful to initially interpret this as 1 half of 1 third, or 2 of
1
•
•
•
1
. (E.g. I have 3 a pan of crispy rice treats left and I’d like to eat half of
3
them.)
Draw a fraction model and label the whole as 1.
Use vertical lines to partition the model into thirds. Shade 1 of them
and label the fraction that’s shaded.
If I’d like to eat half of this third, what must I do to it? Turn and share.
•
•
•
•
•
 You must cut it in half.
Let’s use horizontal lines to partition this third into 2 equal parts, or
halves. (Initially use a solid line to only partition the one third into
2 equal parts. See notes below.)
1
Shade the 1 half that I’ll eat and label the fraction that is shaded, 2.
Great! So now I can see which portion I’ll be eating. I can locate it on
this model. But I don’t yet know how to name this portion or this
fraction. What must I do?  You have to partition the rest of the
fraction model to show equal parts or equal units.
Use a dashed line to partition the remaining model in half.
Now I can name this fraction. What is the name of this fraction? What
1
is 1 half of 1 third?  1 sixth  6.
Background for presenter:
• After a little practice students learn to draw a solid line and a
dashed line at the same time. We only do it as two separate
steps initially.
• The second shading should only shade the portion that we are
1
finding. Some other models shade 3 of the whole first, then
1
shade 2 of the whole next (and then name the portion that is
1
1
double-shaded). But we are ONLY finding 2 of 3, so we only
shade that portion.
1
1
CLICK: to show image of 4 x 3. Share with your partner what you understand
about this model.
CLICK: Participants should complete Problem 16 in their Problem Set.
𝟏
𝟏
Solve 𝟐 of 𝟒 = ___. Draw an area model to show your thinking.
5 min
56.
TIME ALLOTTED FOR THIS SLIDE:
• 2 min model
• 3 min Problem Set
5 minutes
We then increase the complexity by multiplying unit fractions by nonunit fractions and finally multiply non-unit fraction by non-unit
fractions.
Switch to Document Camera: Let’s multiply 2/3 x 2/3 together. (Model with
participants.)
CLICK: to show completed solution.
CLICK: Participants should complete Problem 17 in their Problem Set.
𝟐
𝟑
Solve 𝟑 × 𝟒 = ____. Draw an area model to show your thinking.
Background for presenter:
We spend 3 lessons using models to show a fraction times a fraction.
Students will easily recognize the pattern that is happening: we can just
multiply the numerators and the denominators. Until students can articulate
why 1 third times 1 third results in 1 ninth, have them continue to use the
models to solve. However once they are able to articulate a clear
understanding of what is happening, the models are no longer necessary for
that student.
1 min
57.
We also multiply decimals by decimals using fraction notation.
Explain to your partner how the pictorial model at right supports what the
stick figure is thinking.
How is the fraction model at right similar and different from the fraction
models we just did?  Here students are only required to shade and label the
model, not actually draw a model showing hundredths. (It would otherwise be
too sloppy and time-consuming.)
3 min
58.
We also multiply larger decimals by larger decimals using fraction
notation.
Analyze the work shown here and explain to a neighbor what you see.
2 min
59.
Background for presenter:
Once students are comfortable multiplying fractions, they are then ready to
find the area of rectangles with fractional side lengths. But in order to
progress to a mixed number times a mixed number, we can start with a
simpler problem.
For this exercise, students use what we call “mystery rectangles” and patty
paper (or something like it). (Patty paper is translucent paper used to
separate hamburger patties. They’re sold in boxes of 1000 very cheaply,
and are typically 5.5 inches by 5.5 inches.) Patty paper does not have to be
used, but since we are finding area it is important to use units that are
square in shape, since area is measured in square units. Post-it notes would
also work but they are far more expensive. The translucent nature of patty
paper is also nice, so that students can still see the shape that they are
covering.
Students begin by finding the area of their mystery rectangle by seeing how
many whole units (or whole pieces of patty paper) it takes to completely
cover the rectangle without any spaces between square units, and without
any overlap.
CLICK. In this case it takes 6 square units to cover this rectangle so we say
it has an area of 6 square units.
CLICK. From this concrete experience, students can draw an area model to
represent what they just did, to find that a rectangle with a length of 3 units
and width of 2 units has an area of 6 square units. This can be done by
showing each of the six units.
CLICK. Or more abstractly, just using a multiplication expression to find the
area of 6 square units.
2 min
60.
Then we increase the complexity.
Explain to a neighbor how this rectangle is different from the one you just saw.
(Invite participants to share.)
CLICK: Analyze these 2 pictorial models and discuss how they are similar
and different. (Invite participants to share.)
2 min
61.
In the next lesson, students use a ruler to find the side lengths of a rectangle.
1
1
Here we have a rectangle measuring 2 2 in x 1 4 in.
CLICK:
Take a moment to analyze the work shown here to find the area.
Share with you neighbors what the work is showing.  Students are using the
Distributive Property to solve.
How is this work with the Distributive Property similar and different from the
work they’ve done previously?  It’s the same, except now we’re multiplying
with fractions and whole numbers.
5 min
62.
The final few lessons of Module 4 are dedicated to division with unit
fractions. The G5 standard limits fractional division to the use of just unit
fractions, so again we’ll be looking at fractions with 1 as the denominator.
As is usually the case, when discovering a new concept we want to provide
students with a context. So here, we have a word problem that states that
1
Jenny has 2 pounds of pecans. If she puts 3 pound in each bag, how many
bags can she make?
Just as with whole numbers there are two ways to interpret any division
problem. As we learned earlier there is measurement division and partitive
division, or more simply, number of groups unknown or group size
1
unknown. In this case, the context tells us size of the groups (they’re 3 pound
groups) and we are trying to find the number of groups (or how many thirds
are in 2). Since it’s a number of groups unknown problem this is the
measurement division interpretation.
Switch to Document Camera (Model):
We can use a tape diagram to help support our solution. And since we’re being
asked to find the number of thirds in 2, it might be helpful for us to first think
of how many thirds are in 1.
6 min
63.
TIME ALLOTTED FOR THIS SLIDE:
• 3 min analysis
• 3 min Problem Set
6 minutes
Participants read this problem and discuss how it’s different from the problem
we just solved. Participants should then discuss how the tape diagram drawn
to support the solution will be different.
CLICK: Analyze the solution and discuss what role the number line serves in
supporting the correct response.
CLICK: Participants should complete Problem 18 in the Problem Set.
𝟏
Jenny has 2 pounds of pecans. If she puts 𝟒 pound in each bag, how
many bags can she make?
8 min
64.
TIME ALLOTTED FOR THIS SLIDE:
• 1 min analysis
• 4 min model
• 3 min Problem Set
8 minutes
Analyze this context, and see how it’s related to the previous problem. Discuss
with a neighbor how this problem is similar and different. (Invite participants
1
to share.)  It’s a very similar problem (it’s 2 ÷ 2 )  It’s the partitive division
1
interpretation.  We know the number of groups (Jenny has 2 of a group),
what we don’t know is the size of the group.
Switch to Document Camera (Model the solution using a tape diagram.)
Again, we can use a tape diagram to help support our solution. In this model,
we show the 2 pounds of pecans being equal to half the amount needed.
Therefore, if each of 2 units is equal to 2 lbs, then the whole must be 4 lbs. So,
Jenny will need 4 pounds of pecans to make her pies.
• How many pounds of pecans does Jenny have?  2 lbs. (Draw a
tape and label it ‘2’.)
• These 2 lb represent what?  It’s one half of what she needs.
• So if 2 is one half, how many additional units of 2 are in the
whole?  One more unit of 2.  If 2 is half then she still needs
the other half to make a whole. (Draw another unit of 2 to the
right of original tape.)
• If 2 is half, what is the whole? (Draw a bracket and a question
mark above tape to indicate unknown.)  I can see that there
are 2 units of 2 there, so 4 is the whole.  If 2 is half, 4 is the
whole.
1
• Right, 2 divided by 2 = 4. Jenny will need 4 pounds of pecans.
CLICK: to show solution
CLICK: Participants should solve Problem 19 in Problem Set.
𝟏
Jenny has 2 pounds of pecans. If this is 𝟑 the amount she needs, how
many pounds will she need?
5 min
65.
In prior lessons, students divided a whole number by a unit fraction. Now
students move into dividing a unit fraction by a whole number.
Students are provided with a context to solve through. Here, Nolan gives
1
some pans of brownies to his 3 friends to share equally. If he has 2 a pan of
brownies, how much of a pan will each friend get?
Model (“We do”: complete with participants.)
After completing tape diagram and problem, encourage participants to see
1
the similarities between dividing by 3 and multiplying by 3.
In other words, when working with these models (both the fraction model and
the tape diagram) the process for showing division by 3 is just like the process
1
for showing multiplication by 3.
In both cases, you partition the unit into thirds. Said differently, if you have a
pan of lasagna (or even a portion of a pan) and you want to share it equally
with 3 friends, what will you do? You would do the same as if you wanted to
1
take a third (or multiply it by 3) of the pan: you would cut the lasagna into 3
equal pieces.
1
Therefore, multiplying by 3 and dividing by 3 result in the same value. (Which
is the reason why teachers encourage students to “invert and multiply” or
“keep, change, flip,” which we do not do in Eureka Math, but it’s interesting to
note why it works.)
CLICK: to show solution.
10-15
min
66.
•
•
•
•
•
Section: Closing
Give participants about 5 minutes to review the Instructional
Practice guide and familiarize themselves with the Core Action
Indicators.
Examine the Grade 5 Sample Work.
CLICK to reveal reflection points to be used alongside a discussion
about the sample work.
Allow 5-10 minutes to reflect and discuss.
Participants should use their graphic organizers to assist in
identifying scaffolds for instruction.
Time: 21 minutes
In this section, you will reflect upon how the use of coherent models Materials used include:
and strategies across grade levels enables teachers to support
 Major Work of the Grade Band: Grades 3-5 PPT
students at their current level of understanding and helps them
 Major Work of the Grade Band: Grades 3-5 Facilitator
advance to more abstract thinking and challenging concepts and
Guide
take a brief look into how these lessons will apply moving forward
to the middle grades.
Time Slide # Slide #/ Pic of Slide
0 min
67.
Script/ Activity directions
GROUP
1 min
68.
In Grade 6 Module 2, students divide fractions by fractions through visual
models, such as, tape diagrams, area models, and number line diagrams.
CLICK: This example shows how a tape diagram, which is used in grades 3-5
to divide, is used to divide 8 ninths by 2 ninths.
Notice how powerful and simple the use of unit language makes this
problem. 8 divided by 2 is ALWAYS 4. A ninth divided by a ninth is one (just
like a ten divided by a ten is 1, or 12 divided by 12 is 1). Therefore 8 ninths
divided by 2 ninths is 4!
10 min
69.
The use of coherent models and strategies across grade levels enables
teachers to support students at their current level of understanding and
help them advance to more abstract thinking and challenging concepts.
Because of the consistent reliance on the ten-structure and the continuous
practice with identifying a unit, students’ conceptual understanding will
rapidly and solidly develop. Students not only use Grades 3-5 coherence,
but fall back upon coherent learning in K-2 which develops into more
advanced 3-5 strategies.
Discuss these considerations at your table. Identify examples from today
that fit with each bullet point.
• What is a familiar model to a G3, G4 or G5 student learning
multiplication and division?
• What is a familiar strategy to a G3, G4 or G5 student learning
multiplication and division?
• What is a familiar property or concept to a G3, G4 or G5 student
learning multiplication and division?
• What is a concrete, pictorial, abstract approach to a G3, G4 or G5
student learning multiplication and division?
• What is a simple to complex approach to a G3, G4 or G5 student
learning multiplication and division?
5 min
70.
•
•
•
•
How does this quote summarize the road ahead for mathematics
education?
How does this coherence of multiplication and division allow
teachers and students to better their practice and understanding of
mathematics?
How do I get there? How do I start to see the improvement?
How do I get my educators there? How do I convince them of this
improvement over time?
Allow 5-10 minutes to discuss.
5 min
71.
Use this time to reflect personally, with a partner, and as a group on today’s
session.
Turnkey Materials Provided




Multiplication and Division with Whole Number and Fractions in Grades 3-5: Concepts, Skills, and Problem SolvingPPT
Facilitator Guide
Templates and Graphic Organizers
Instruction Practice Guide – Coaching – Summary of Core Actions
Additional Suggested Resources



How to Implement A Story of Units
A Story of Units Year Long Curriculum Overview
A Story of Units CCLS Checklist