Module Focus: Grade 4 – Module 3
Sequence of Sessions
Overarching Objectives of this November 2013 Network Team Institute

Module Focus sessions for K-5 will follow the sequence of the Concept Development component of the specified modules, using this narrative as a tool
for achieving deep understanding of mathematical concepts. Relevant examples of Fluency, Application, and Student Debrief will be highlighted in
order to examine the ways in which these elements contribute to and enhance conceptual understanding.
High-Level Purpose of this Session




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.
Instructional supports. Participants will be prepared to utilize models appropriately in promoting conceptual understanding throughout A Story of
Units.
Related Learning Experiences
●
This session is part of a sequence of Module Focus sessions examining the Grade 4 curriculum, A Story of Units.
Key Points
•
Number disks and area models are used heavily throughout the module to support the algorithms.
•
The multiplication and division algorithms are not expected fluencies in Grade 4.
•
Unit language and place value understanding drives the experience of the algorithms.
•
Keep a balance of rigor by addressing each component of a lesson.
•
Honor and respect the objectives.
•
Find a balance between success and mastery.
Session Outcomes
What do we want participants to be able to do as a result of this
session?





Focus. Participants will be able to identify the major work of each grade
using the Curriculum Overview document as a resource in preparation
for teaching these modules.
Coherence: P-5. Participants will draw connections between the
progression documents and the careful sequence of mathematical
concepts that develop within each module, thereby enabling participants
to enact cross- grade coherence in their classrooms and support their
colleagues to do the same . (Specific progression document to be
determined as appropriate for each grade level and module being
presented.)
Standards alignment. Participants will be able to articulate how the
topics and lessons promote mastery of the focus standards and how the
module addresses the major work of the grade in order to fully
implement the curriculum.
Implementation. Participants will be prepared to implement the
modules and to make appropriate instructional choices to meet the needs
of their students while maintaining the balance of rigor that is built into
the curriculum.
Instructional supports. Participants will be prepared to utilize models
appropriately in promoting conceptual understanding throughout A
Story of Units.
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 to the
Module
8 minutes
Overview of the instructional
focus of Grade 1 Module 3.


Grade 4 Module 3
Grade 4 Module 3 PPT
110
minutes
Examination of the development
of mathematical understanding
across the module using a focus on
Concept Development within the
lessons.


Grade 4 Module 3
Grade 4 Module 3 PPT
Concept
Development
Prepared Resources
Facilitator Preparation
Module Review
9 minutes
Articulate the key points of this
session.


Grade 4 Module 3
Grade 4 Module 3 PPT
Session Roadmap
Section: Introduction to the Module
Time: 8 minutes
[8 minutes] In this section, you will…
Materials used include: none
Provide an overview of the instructional focus of Grade 4 Module 3.
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
1
1.
NOTE THAT THIS SESSION IS DESIGNED TO BE 127 MINUTES IN LENGTH.
To cut down to 120 minutes, shorten Slide 22 by 7 minutes, giving just a
brief overview of Lesson 26 and not having participants work through
the lesson in its entirety.
Welcome! In this Module Focus Session, we will examine the second half
of lessons of Grade 4 – Module 3.
1
2.
Our objectives for this session are to:
• Examination of the development of mathematical
understanding across the module using a focus on Concept
Development within the lessons.
• Introduction to mathematical models and instructional
strategies to support implementation of A Story of Units.
GROUP
0
3.
We will begin by exploring the module overview to understand the purpose
of this module. Then we will dig in to the math of the module. We’ll lead
you through the teaching sequence, one concept at a time. Along the way,
we’ll also examine the other lesson components and how they function in
collaboration with the concept development. Finally, we’ll take a look back
at the module, reflecting on all the parts as one cohesive whole.
Let’s get started with the module overview.
0
4.
The third module in Grade 4 is Multi-Digit Multiplication and Division. The
module includes 38 lessons and is allotted 43 instructional days. Our focus
today will be Topics E-H.
3
5.
Spend about 2 minutes looking at the Topic Titles for Topics A-D and the
Objectives for the first half of lessons (up to Mid-Module Assessment) (Page
x of the Module Overview). How do Modules 1 and 2 connect to Module 3?
Gain a general understanding for where the module has taken students thus
far.
• What conceptual understanding have they learned? Multiplicative
Comparison (times as much); Multiplication by 10, 100, and 1000;
Single-Digit by Multi-Digit (up to 4-digit) Multiplication; Application
of new learning in the form of word problems;
• What models have been used? Diagrams (area and perimeter);
number disks and place value charts (to establish place value when
multiplying by units of 10, 100, and 1000); arrays (for pictorial
representation of place value understanding); area model (to model
multiplication  goes back to concept of finding area);
• Why might the module begin and end with multiplication? To bring
coherence to the lessons within the module; multiplication and
division at the beginning of the module build place value
understanding and understanding of the area model that is needed
for 2-digit×2-digit multiplication
• Why does the module begin with area and perimeter? To provide
context for multiplicative comparison; to build understanding of the
process of multiplication and division using the area model; to
provide context for word problems
Discuss importance in understanding the flow and pace of the module,
keeping the end in mind, so we don’t get “stuck” on a lesson.
Finding success in lessons, not complete mastery. Mastery will be
accomplished through each lesson, as they build upon themselves.
3
6.
Spend about 2 minutes looking at the Topic Titles for Topics E-H and the
Objectives for the second half of lessons (Pages xi and xii of the Module
Overview). Gain a general understanding for where the Module will take the
students.
• How does the first half of the Module connect to the second half? The
first half builds the foundation for the division and multiplication
work of the second half (place value, area model).
• What conceptual understanding will they learn? Division with
remainders; Divisibility; Division of Thousands, Hundreds, Tens, and
Ones; 2-digit by 2-digit Multiplication;
• What models will be used? Array/area models; number disks; long
division algorithm; place value chart; partial products (two- and
four-); standard algorithm for 2-digit by 2-digit multiplication;
• What is important to note regarding the standard algorithms for
multiplication and division? Students are not assessed on the
standard algorithms for multiplication or division in Grade 4. The
algorithm for multiplication is a Grade 5 standard. The algorithm for
division is a Grade 6 standard.
Include discussion of the following:
• Importance in understanding the flow and pace of the module,
keeping the end in mind; don’t get “stuck” on a lesson.
• Finding success in lessons, not complete mastery. Mastery will be
accomplished through each lesson, as they build upon themselves.
Section: Concept Development
Time: 111 minutes
[111 minutes] In this section, you will…
Materials used include:personal whiteboards and participant
problem set
Examine of the development of mathematical understanding across the
module using a focus on Concept Development within the lessons.
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
1
7.
We will examine the Concept Development of the Second Half of this
module by focusing on the progression and coherence of the lessons within
and across each topic.
We will examine the models, structure, and language used and the
movement from concrete to pictorial to abstract.
We will go step by step, lesson by lesson, primarily modeling the Concept
Developments, but also including some Application Problems, Debriefs, and
Fluencies.
You will have many opportunities to be the “teacher” at your table. Try to
vary who that is throughout the presentation so that everyone has the
chance to be comfortable , not only practicing the mathematics, but also
presenting the sequence of the lessons to the “students”.
3
8.
Take 2 minutes to read Topic Opener E. Read for the standards addressed
and the content students will be expected to have understanding of for
moving on.
Standard:
Content: “Division of tens and ones with successive remainders”
• Division types (groups unknown, group size unknown)
• Interpreting remainders
• Division with arrays and the area model
• Division algorithm as it relates to work with number disks and area
GROUP
models
3
9.
Complete first part with participants:
10 ÷ 2
11 ÷ 2
8÷4
9÷4
Let participants practice the rest. Keep in mind that, at this point, the word
‘remainder’ is not used. The idea is for students to gain the conceptual
understanding of what is happening. ‘Remainder’ will be introduced in this
lesson.
3
10.
We start the Topic by connecting to the G3 models of number bonds and
arrays.
Discuss connection to G3 models using number bonds and arrays to find
‘groups of’.
• Model array using 13 ÷6. Have participants complete 17 ÷3 on their
‘Problem Set’ (#1).
• Model number bond using 13 ÷6. Have participants complete #2 on
their Problem Set.
• Show how the array can be used to show a transition to a tape
diagram. Model how to box the array and to show the remainder.
Have participants complete #3 on their Problem Set.
** Lesson 14 connects and advances the array to a tape diagram in order to
model word problems.
5
11.
Application problem asked students to draw an array to solve for 38 ÷ 4.
Show the model of an array first (groups of 4)  transition to an area
model, first using graph paper so students can see that they are similar 
transition to a rectangle, being mindful of proportions.
Introduce the language of “The quotient is 9. The remainder is 2.”
Have participants complete #4 on the Problem Set.
Discuss Debrief questions as a whole group or in tables.
• What does the quotient represent in the area model? The side length
• When does the area model present a challenge in representing
division problems? When there is a remainder
• How is the whole represented in the area model? By the rectangle
and square units
• The quotient represents a side length. The remainder consists of
square units. Why? There are not enough units to make another
group.
6
12.
Deliver the script: Model for participants how to deliver the Concept
Development without reading from the script word for word; Remind them
to pay attention to the progression
3
13.
Lesson 17’s complexity is to regroup in the tens when there is a remaining
number after division in the tens column. Problems are contrasted using
remainders in the ones compared to remainders in the tens. We can keep
dividing when there is a remainder in the tens.
Model for participants the two examples on this slide and then have them
work to complete #7 of the Problem Set.
4
14.
Lesson 18 provides students with the opportunity to practice dividing twodigit numbers by a single-digit number, regrouping in the tens and having a
quotient with a remainder. Students check their work by multiplying.
Students visualize the process and work to complete the algorithm without
a model.
Show participants the sample problem on the slide and talk them through
finding the solution.
4
15.
In Lesson 20, we look at the area model and use the area model to represent
division problems. Students will see that there are many ways to break
apart a division problem into pieces that make it easier to solve.
In the example, 48 ÷4, students see that we can break the 48 into the more
manageable pieces of 40 and 8 or of 20, 20, and 8. The area models
represent how we break apart the problem each time. Students will further
discover that they could break 48 down to 24 and 24 or even 12, 12, 12, and
12. Each example has the consistent side length of 4.
Explain this process to participants and then have them practice with #8 of
the Problem Set.
4
16.
We can also build from part to whole. We do this by starting with the side
length of 4 and then building up to the area of 96. We first give tens to our
side length. Can we give one ten?  Yes! It’s 40.  Can we give two tens to
our side length of 4?  Yes! That would be 80.  Can we give three tens?
 No! That would be 120.  Draw one length of 20.  We’ve used 80
square units. Write that in the area model. How many are left?  16. 
Let’s give ones to our side length. Can we give one one?  Yes! That would
be 4.  Can we give two ones?  Yes! That would be 8.  How many can
we give?  4 because 4 fours is sixteen and that’s how many ones there are.
 We have a side length of 20 + 4. That equals 24.
Explain and model for participants. Further show how the solution using
the area model can then relate to the algorithm for division. Participants
will complete #9 of the Problem Set.
4
17.
Show participants how to solve by building an area model part to whole.
Participants can then complete #10 of the Problem Set.
3
18.
Take 2 minutes to read Topic Opener F. Read for the standards addressed
and the content students will be expected to have understanding of for
moving on.
Standard:
Content:
Find factor pairs
Define prime and composite
Test for factors and observe patterns
Determine whether a number is a multiple of another
Explore properties of prime and composite
Why is this Topic placed between multiplication and division in the
module?
Knowledge of factors and multiples helps students to solve multiplication
and division problems in a more efficient manner; they are able to
determine unknown side lengths and to break apart problems more easily.
3
19.
Lesson 22s objective is to find factor pairs. Students use their knowledge of
the basic facts for multiplication in order to identify factor pairs. Further,
they learn the definitions of prime and composite and identify numbers as
one or the other.
In Lesson 23, students test for factors and observe patterns. For example,
they use division to test for factors and use the associative property as
represented below. Explain both methods to participants and then have
them answer the questions found on the slide.
(Click to advance each question taken from the script.)
3
20.
Lesson 24s objective is to determine whether a whole number is a multiple
of another number. Students use division and the associative property (as
demonstrated below) to determine this.
In Lesson 25, students work to complete the Sieve of Eratosthenes to
identify the prime numbers to 100. This is a guided lesson. If time allows,
have participants complete #11 of the Problem Set.
3
21.
Take 2 minutes to read Topic Opener G. Read for the standards addressed
and the content students will be expected to have understanding of for
moving on.
Standards:
Content:
Use unit language to divide multiples of 10, 100, and 1,000 by singledigit numbers (12 thousands ÷4 = 3 thousands).
• Represent division of up to four-digits with number disks and
numerically with and without remainders and requiring
decomposing remainders up to three times.
• Solve division problems with a zero in the dividend or in the
quotient.
Why is division of larger dividends separated in this module?
The work with factors and multiples is helpful when dividing with larger
dividends. Work with larger dividends entails the same process as with
smaller dividends. It is an extension of that learning.
•
8
22.
Hand out copies of Lesson 26.
Give 5 minutes to read the lesson front to back.
Mark 2 ah-has relating to concepts, content, or other.
Give 2-3 minute for participants to complete the Problem Set. Discuss the
importance of completing the Problem Set prior to the Concept
Development.
Plan how to present this lesson in a shorter amount of time.
Share strategies at tables and a few out loud.
4
23.
Lesson 27 is an extension of the work done in Topic E in Lessons 16-18.
Working with numbers disks to divide will now be familiar to students. The
difference here is that this process is repeated given the larger dividend.
Walk the participants through the division problem using the script. If time
allows, have participants complete #12 of the Problem Set.
4
24.
Lessons 28 and 29 build on the work of Lesson 27. Students will divide
with larger numbers (up to 4-digits) using the place value chart and the
standard algorithm for division side-by-side. Students will see that division
with larger numbers follows the same process as that with smaller
numbers.
Have participants practice solving and walking through a script using unit
language to solve Problem #13 on the Problem Set (The same problem that
appears on this slide).
Debrief: You have practiced division with 2, 3, and 4 digit numbers. Discuss
what you think would be true for dividing a number with a greater number
of digits.
4
25.
At this point, participants have had much practice with hearing, seeing, and
delivering unit language when explaining the process of division. Challenge
participants to work with a partner to solve one of the problems on the
slide. Each partner will solve one problem and then will explain to his/her
4partner how to teach about a zero using unit language. (Problem Set #14)
4
4
26.
Lesson 31 guides students through word problems using tape diagrams to
model. The model allows students to consider about the type of division
and what the quotient means. The example shown in the slide models a
‘number of groups unknown’ problem.
The tape diagram shows how the problem can be modeled. Notice the
whole of 292 is used to label the length of the tape diagram. The size of the
group is modeled on the left side of the tape diagram. The ‘?’ in the
remaining section of the tape diagram indicates that we don’t know the
number of groups. We can divide to show how many groups of 4 there will
be.
(Have participants quickly sketch a tape diagram and solve. Show result.)
Participants may then solve #15 of the Problem Set. (The tape diagram and
solution are modeled below.)
4
27.
Larger divisors can stump some students who aren’t familiar with their
division or multiplication facts. Students have been provided several
lessons prior to this one to gain understanding of the method of the
algorithm. Now they can be successful using these larger, and often more
challenging, divisors.
Participants should note that there are multiple ways to solve a problem.
Two ways to solve the problem (as displayed on the slide) are outlined
below. It is wise to call out and celebrate alternative solutions that are
mathematically sound. Remind participants that the division algorithm is
not an expected fluency until Grade 6. In Grade 4, the process is introduced
to prepare students for division work in Grade 5.
5
28.
This lesson builds on the Concept Development of Topic E’s Lesson 20. In
this lesson, students extend their learning of the connection of the area
model to division to include larger numbers. Students will first build
lengths of 1 hundred instead of lengths of ten. Students will see that there is
more than one way to break apart a number.
Discuss with participants both methods for decomposing the whole using
number bonds and the area model. Stress that this work relates directly to
earlier lessons and G3 division.
Model 672 ÷3 to find the
length of the unknown side. (See example to the right.)
If time allows, have participants complete #16 of the
Problem Set.
3
29.
Take 2 minutes to read Topic Opener H. Read for the standards addressed
and the content students will be expected to have understanding of for
moving on.
Standard:
Content:
• Multiply 2-digit multiples of 10 by two-digit numbers using a place
value chart and an area model.
• Multiply 2-digit by 2-digit numbers using four partial products and
the standard algorithm for two-digit by two-digit multiplication.
How have students prepared for this type of multiplication?
Students have multiplied single-digit by multi-digit numbers. Students have
modeled multiplication using place value disks, the area model, and partial
products. This multiplication is simply an extension of a process with
which students are already familiar.
How are the students prepared to use the area model?
Students have used the area model throughout the module to both multiply
and divide. Earlier lessons have given a solid foundation of work with the
area model to allow for a smooth transition to 2-digit by 2-digit
multiplication.
2
30.
This fluency activity is included within this module to show that fluencies
can be used to anticipate future lessons in order to ready students for
success. As we are near the end of Module 3, fluencies anticipating the
work of future modules begin to appear.
What is this fluency preparing students for? Geometry and Fractions
Why is it placed here? Anticipation of upcoming Lessons/Topics/Modules
How could this fluency be revised/adapted/postponed?
• If it is not new learning, it is not a fluency and should not be treated
as such.
•
•
•
3
31.
To revise: If students are not familiar with the geometric terms,
revise to those that do have
familiarity.
To adapt: Focus on only the geometry or only the fractions, but not
both.
To postpone: Set priorities. If another fluency within the lesson is
more applicable to the lesson
or to future lessons,
postpone this one and choose another instead.
The Application Problem for Lesson 34 leads directly into the lesson where
students discover they are prepared to solve 40 times 22 because they
know it to be the same as 4 times 10 times 22.
Have participants read the Application Problem and then study the
diagram. Point out that the area model shows 40×22 but that they can look
at it as (4 × 10) ×22. The idea is to show students that they are able to solve
more difficult problems by using what they already know as strategies.
They see that in the area model. They also model it on place value charts.
Deliver the script as follows:
3
32.
In this lesson, students apply what they learned in Lesson 34 and extend it
to the next level. To show 60 as one side length and 34 as the other, the 34
is shown as 30 + 4. Students are able to solve for each of the parts as the
multiplication of units is something with which they are familiar. The
expression is written in numeric form and in unit language within the
model.
Participants should study the model and then look to see how it is then
written in the algorithm as two partial products.
Debrief: When recording the partial products, must we start with the
smallest product? Why or why not?
We can start with either of the two partial products. The sum of the two
will be the same no matter which order is used. The smaller of the two is
written first in this lesson in order to allow for a smoother transition to 2-
digit by 2-digit multiplication.
Participants may complete #17 of the Problem Set.
5
33.
Lesson 36 builds to the next level in preparing students to multiply 2-digit
by 2-digit numbers. Instead of multiplying a multiple of ten by a 2-digit
number, students multiply two 2-digit numbers which are not multiples of
ten.
Show participants how the area model is set up for this type of problem.
Within each quadrant, an expression is written and solved. We now have 4partial products instead of two. (Students could argue that there are 4partial products in the previous examples and that their products are
simply zero.)
Point out how the partial products are written in vertical format and then
are added to determine the area of the large rectangle (the product of 23
and 31). The distributive property can also be used to show the breakdown
of each number into partial products.
Walk participants step by step through the area model, recording also
within the algorithm.
Participants complete #18 of the Problem Set.
4
34.
Lesson 37 transitions from 4-partial products to 2-partial products,
bringing students one step closer to the standard algorithm for
multiplication. Using a visual of the area model, students will see that
(6×30) + (6×5) is the same as 6×35. Students know how to calculate 6×35
as it was discovered in Topic C. Similarly, students will see that the same
concept applies to the bottom portion of the area model.
Walk students step-by-step through the example shown on the slide and
then have them work to complete #19 of the Problem Set. Participants
should first draw the model representing four partial products and then
shade the area model to show how two partial products are also
represented. Encourage participants to use the example on this slide when
completing #19.
2
35.
The module culminates in Lesson 38 where students see the final link from
2 partial products to solving 2-digit by 2-digit multiplication using the area
model.
Explain to participants the model that is shown on the slide. Students now
draw just 2 partial products instead of four. An expression is written within
each to show multiplication by ones and by tens. The partial products are
then linked to the algorithm for multiplication. Initially, no regroupings are
needed. Students are then faced with problems where regrouping becomes
necessary. (See next slide.)
Please note to participants that using hide zero cards can allow students to
‘see’ in the numbers in the algorithm.
Use of 2 partial products encourages unit language as we multiply.  We
are not multiplying by ‘2’ each time, rather we are multiplying by ‘2’ and
then by ‘2 tens’.
4
36.
In this final problem, regrouping is necessary. Note that the regrouping is
completed on the line as it has been in earlier lessons throughout this
module.
Show participants the regrouping within the 2nd partial product as shown
above. The regrouping is completed as if there were a ‘line’ above the 2nd
number. The number that is regrouped is added immediately following the
multiplication by the next place value (as is typically done within the
algorithm).
Have participants solve both problems (#20 on the Problem Set).
Section: Module Review
Time: 9 minutes
[9 minutes] In this section, you will…
Materials used include:
Faciliate as participants articulate the key points of this session and
clarify as needed.
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
0
37.
The Concept Development portion of the module is now complete.
5
38.
Allow participants a few minutes to look at the End-of-Module Assessment.
If time allows, participants may work to complete the assessment. If time
does not allow, encourage a brief discussion regarding the assessment and
how the work within the module prepares students for success.
How did Module 1 and Module 2 prepare you for Module 3?
Place value understanding, work with addition and subtraction, work with
various models, and application of what was learned helped to prepare for
Module 3.
Take two minutes to turn and talk with others at your table. During this
session, what information was particularly helpful and/or insightful? What
new questions do you have?
Allow 2 minutes for participants to turn and talk. Bring the group to order
and advance to the next slide.
3
39.
Take two minutes to turn and talk with others at your table. During this
session, what information was particularly helpful and/or insightful? What
new questions do you have?
Allow 2 minutes for participants to turn and talk. Bring the group to order
and advance to the next slide.
1
40.
Review the key points of the session as outlined above.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Turnkey Materials Provided
●
Grade 4 Module 3 PPT
Additional Suggested Resources


How to Implement A Story of Units
A Story of Units Year Long Curriculum Overview
Active learning
Turn and talk