Modeling Word Problems with Tape Diagrams:
Multiplication and Division
Sequence of Sessions
Overarching Objectives of this March 2015 Network Team Institute

Participants will be able to identify, practice, and use best instructional moves and scaffolds for chosen common core standards.
High-Level Purpose of this Session

This session provides an opportunity to understand more deeply the coherence of mathematics K-7 and suggests a pedagogy allowing students to make
intuitive drawings followed by teachers’ guidance with the tape diagram. Throughout the session, participants will:
o Analyze 7 multiplication and division word problem situations, “equal groups” and “multiplicative comparison with greater and lesser
unknowns,” categorizing prior to drawing.
o Become more familiar with situation types and how those situations manifest in multi-step problems at later grades, thereby gaining a deeper
understanding of the ways in which the foundational work of grades : Multiplication and Division contributes to success through middle school
and into algebra.
o Develop mathematically as they work on the upper grade problems, making important connections in particular to ratio and algebra.
Related Learning Experiences



This topic is extended in a companion session, Modeling Word Problems with Tape Diagrams: Addition and Subtraction.
This session is part of a series supporting implementation of A Story of Units. Sessions on Crafting Teaching Sequences for Extended Interventions
will also develop teachers’ ability to customize the curriculum to meet the specific needs of their students.
Module Focus sessions also support implementation of the curriculum by closely examining each module in A Story of Units.
Key Points
•
•
•
•
Participants will learn about Polya’s Problem Solving Process and compare it to the RDW (Read-Draw-Write) Process.
Participants will discuss obstacles students may have with each of these processes.
Participants will explore different modes of instructional delivery and their effectiveness.
Participants will work challenging problems using the Deliberate Practice protocol.
Session Outcomes
What do we want participants to be able to do as a result of this
session?



Participants will be able to distinguish the various multiplication and
division problem types.
Participants will be able to describe the relationship between simple
problem types in early grades and multi-step problem types that are
found in upper grades.
Participants will be able to support students in use of the tape diagram
using the RDW process.
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.
Session Overview
Section
Time
Overview
Prepared Resources
•
Introduction
11 min
Introduces the key ideas in this
session.
•
•
Problem Solving
Processes
Modes of
Instructional
Delivery
35 min
Explores Polya’s Problem Solving
Process and the Read-Draw-Write
(RDW) process and apply them to
specific problems.
73 min
Explores modes of instructional
delivery and view them in action.
•
•
Modeling Word Problems
With Tape Diagrams:
Multiplication and Division
PPT
Modeling Word Problems
With Tape Diagrams:
Multiplication and Division
Facilitator Guide
Modeling Word Problems
With Tape Diagrams:
Multiplication and Division
PPT
Modeling Word Problems
with Tape Diagrams :
Multiplication and Division
Facilitator Guide
Facilitator Preparation
Review Modeling Word Problems
with Tape Diagrams :
Multiplication and Division PPT
and Facilitator Guide
Review Modeling Word Problems
with Tape Diagrams :
Multiplication and Division PPT
and Facilitator Guide
Modeling Word Problems Review Modeling Word Problems
With Tape Diagrams:
with Tape Diagrams :
Multiplication and Division Multiplication and Division PPT
•
•
Protocol D: Precise
Descriptions of
Terminology
34 min
Explores Protocol D and use all of
the protocols discussed so far to
solve problems.
•
•
Problem Analysis
27
Explores using the different
models and protocols from the
previous sections to solve difficult
problems.
•
PPT
and Facilitator Guide
Modeling Word Problems
with Tape Diagrams :
Multiplication and Division
Facilitator Guide
Modeling Word Problems
With Tape Diagrams:
Multiplication and Division
PPT
Modeling Word Problems
with Tape Diagrams :
Multiplication and Division
Facilitator Guide
Modeling Word Problems
With Tape Diagrams:
Multiplication and Division
PPT
Modeling Word Problems
with Tape Diagrams :
Multiplication and Division
Facilitator Guide
Review Modeling Word Problems
with Tape Diagrams :
Multiplication and Division PPT
and Facilitator Guide
Review Modeling Word Problems
with Tape Diagrams :
Multiplication and Division PPT
and Facilitator Guide
Session Roadmap
Section: Introduction
Time: 11 minutes
In this section, you will be introduced to …
Materials used include:
• Modeling Word Problems with Tape Diagrams : Multiplication
and Division PPT
• Modeling Word Problems with Tape Diagrams : Multiplication
and Division Facilitator Guide
• Modeling Word Problems with Tape Diagrams : Multiplication
and Division Participant Handout
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
4 min
(PH – Page 2)
PS
1.
Participants work on solving one challenge problem.
(Prompt participants to solve the “Camp Okey-Fun-Okey” problem. If they
finish early, challenge them to try the other problem.)
Model the tape diagram. Inform participants that the solutions for each of
the problems will be in the answer packet they will receive at the end of
the session.
Last summer, at Camp Okey-Fun-Okey, the ratio of the number of
boy campers to the number of girl campers was 𝟖:𝟕. If there were a
total of 195 campers, how many boy campers were there? How many
girl campers? (G6-M1-L5)
2 min
2.
Total Session Time: 180 Minutes
Introduce yourself and the professional development session to
participants.
Codes within PPT:
(PS = Partner Share; ITT = Independent Think Time; WG = Whole
Group; PH = Participant Handout)
** Descriptions of Addition and Subtraction Problem Types
(Situations) can be found on Page 46 of the Participant Handout. **
As time allows, consider using Protocol C (PH – Page 43) at various
times throughout the session.
WG
A NOTE REGARDING APPENDIX C – STAGES OF MODEL DRAWING:
At an ideal time within the session, turn attention to Appendix C (pp
48-49 of the Participant Handout). Give participants time to read
and analyze the document. Encourage them to pay close attention to
the 6 Stages of Model Drawing. Engage participants in a discussion.
What are the new complexities at each stage? How does the
narrative on page 48 contribute to an understanding of tape
diagrams? How can this document help inform teachers?
5 min
3.
(PH – Page 3)
ITT  PS 
1. Participants read and analyze the highlighted situation types
WG
silently. (2 minutes)
2. Participants stand and find a partner from a different table to
create an A x B = ____ for one minute. Change partners and create a C ÷
A = ___ situation, and finally change to create a C ÷ B = ___ situation.
Encourage participants to use grade-appropriate measurement, whole
number, or fractional units. (3 minutes total)
Facilitator Considerations - Whole group: share examples and have all
identify the situation type.
1. Which situation types do we use most frequently?
Section: Problem Solving Processes
Time: 35 minutes
In this section, you will explore Polya’s Problem Solving Process and Materials used include:
the Read-Draw-Write (RDW) process and apply them to specific
• Modeling Word Problems with Tape Diagrams : Multiplication
problems.
and Division PPT
• Modeling Word Problems with Tape Diagrams : Multiplication
and Division Facilitator Guide
• Modeling Word Problems with Tape Diagrams : Multiplication
and Division Participant Handout
Time Slide # Slide #/ Pic of Slide
5 min
4.
Script/ Activity directions
1. Provide some history of Polya’s Problem Solving Process.
2. Participants compare Polya’s process to the RDW process.
3. Discuss obstacles students may have with each of these
processes.
4. Share thoughts about why Eureka Math has chosen to use the
RDW process.
5. Participants read through the Pedagogy Notes on page 5 of the
Participant Handout and discuss in groups how this would look
in their classroom.
NOTE: Facilitator will use RDW during “super-modeling” of problems
throughout the PD.
*** RDW has been chosen because the Draw (and Label) process helps
students to make sense of the problem (understand). Students then
write and solve an equation. ***
GROUP
WG  PS
Notes on Pedagogy:
When using the RDW process, it often involves moving back and forth
between reading and drawing. Students might first read the problem
entirely then reread the first sentence. Draw and label. Read the second
sentence. Draw and label. Consider the following example:
Mr. Peterson bought a case (24 boxes) of fruit juice. One-third of the
drinks were grape and two-thirds were cranberry. How many boxes of
cranberry drinks did Mr. Peterson buy? (G5 M2 L10)
Read:
Mr. Peterson bought a case (24 boxes) of fruit juice.
Draw:
Read:
One-third of the drinks were grape.
Draw:
Read:
Two-thirds were cranberry.
Draw:
Read:
How many boxes of cranberry drinks did Mr. Peterson buy?
Write:
Polya’s process culminates with “look back.” The RDW process, on the
other hand, culminates with a statement and a labeled drawing, an
illustration of the story. The statement puts the answer back into
context. Does the statement make sense? Does it correspond correctly
to the drawing? Does the drawing tell the story? This is MP.2 in action,
“reasoning abstractly and quantitatively.” The drawing precipitates the
reasoning. The student does not figure out the problem and then draw
but rather decodes the relationships through the drawing. The abstract
numbers are manipulated in the calculation and restored as quantities
in the statement.
10
min
5.
(PH – Pages 6-7)
Set 1, Group A (Unknown product – whole numbers (#4 has
fractions))
1. Facilitator super-models the first problem using the RDW
process using the “Modeling with Interactive Questioning”
method.
NOTE: Solve numerically below the tape diagram so the right and left
handed people can see the model while they work. Follow this
suggestion throughout the day.
2. Participants complete the Problem Solving Protocol A (PH –
Page 43).
NOTE: Reasoning for Step 3 in the protocol: When we, as educators,
model a problem, it is important we present a solution whose
relationship to the problem situation is understandable by the most
students.
3. Classify the problem type and label with the variables, identifying
the meaning of the factors.
A X B = C. Label the factors.
Facilitator Considerations:
Criteria for a good model – have a
piece of chart paper up and we add
criteria of a good model as we go through the day.
WG 
ITT 
PS
10
min
6.
(PH – Page 8)
WG  ITT
Set 1, Group B (Unknown product – fractions)
 PS
1. Facilitator models the first problem using the RDW process using
the “Guided Practice” method.
2. Participants complete the Problem Solving Protocol A.
3. Classify the problem type and label with the variables,
identifying the meaning of the factors.
A X B = C. Label the
factors.
Facilitator Considerations:
Possible Whole Group Discussion Questions:
1. What are some road blocks students will face?
2. How can you help them through these road blocks?
10
min
7.
(PH – Page 9)
Set 1, Group C (Unknown product – fraction of a whole)
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol A.
3. Classify the problem type and label with the variables, identifying
the meaning of the factors.
A X B = C. Label the factors.
WG  ITT
 PS
Facilitator Considerations:
Possible Whole Group Discussion Questions:
1. What is different about this group compared to the previous two
groups?
2. What type of problem was the hardest to model?
Section: Modes of Instructional Delivery
Time: 73 minutes
In this section, you will explore modes of instructional delivery
and view them in action.
Materials used include:
• Modeling Word Problems with Tape Diagrams : Multiplication and
Division PPT
•
•
Modeling Word Problems with Tape Diagrams : Multiplication and
Division Facilitator Guide
Modeling Word Problems with Tape Diagrams : Multiplication and
Division Participant Handout
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
GROUP
2 min
(PH – Page 10)
Participants read the different modes of instructional delivery along with the
paragraph that precedes it.
ITT
8.
Modes of Instructional Delivery
A teacher also needs to make a decision about the mode of delivery of instruction
each day as it is not dictated in the curriculum at all. Will students be encouraged to
use a specific model to reason about the relationships within the problem (e.g. an
array or tape diagram) or will any math drawing that makes sense be encouraged?
Will this be a better problem to use a step-by-step guided approach because of new
complexities, or will the students work independently and then share out their
strategies? Will the students work independently or in pairs? In cooperative groups
with a protocol or solo and then share with a partner? The chart below lays out three
modes of delivery of instruction. There are many gradations within and between
each one.
8 min
9.
(PH – Page 10) (Set 1, Group A, #3)
WG 
1. Show the video and ask participants to pay attention to the mode of delivery PS 
being used.
WG
2. Participants determine which type of delivery is being modeled in the video.
3. Participants discuss what they would hear, see, and experience with the other
two modes of delivery.
4. Participants discuss strengths and weaknesses of each mode of instructional
delivery and when each might be the right choice.
Facilitator Considerations:
Possible Whole Group Discussion Questions:
1. How would the other two modes look/sound different than the one we saw in
the video?
2. During instruction, when did the teacher release the students to work on
their own? What affect did that have on the students' work?
3. If the students had worked independently, how would the student work have
been different?
4. If the teacher had controlled the guided instruction more, how would the
work be different?
5. If the teacher had modeled interactively, how would the students' experience
be different?
6. When is it best to control more? Less?
10
min
10.
(PH – Page 11)
1. Direct participants to the Deliberate Practice Protocol B and have them
silently read the protocol.
2. Participants decide who will be the teacher and then follow the protocol for
the indicated problem (Set 1,Group A, #3).
Three boxes weighing 128 pounds each and one box weighing 254 pounds
were loaded onto the back of an empty truck. A crate of apples was then loaded
onto the same truck. If the total weight loaded onto the truck was 2,000
pounds, how much did the crate of apples weigh? (G4 M3 L13)
** Refer to Deliberate Practice Stems found on Page 44 of the Participant Handout.**
PS
10
min
11.
(PH – Pages 12-15)
WG 
Set 2, Group A
(Problems 1-4 only) (Group size unknown – partitive – whole ITT 
numbers)
PS
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label with the variables, identifying the
meaning of the factors.
A X B = C. Label the factors.
Facilitator Considerations:
Possible Whole Group Discussion Questions:
1. What type of division is being modeled in these problems?
2. Are these problems more/less difficult than problems in Set 1? Why?
3. Where may students get stuck when modeling these types of problems?
15
min
12.
(PH – Pages 16-17)
Set 2, Group B (Group size unknown – partitive – fractions (“whole” as 1))
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label with the variables, identifying the
meaning of the factors.
A X B = C. Label the factors.
WG 
ITT 
PS
8 min
13.
(PH – Pages 18-19)
1. Participants silently examine each model and write a word problem that can
be solved using the model.
2. Signal participants to discuss their word problems with a partner.
ITT PS
NOTE: The actual word problems given in the curriculum for all models throughout
the session are presented in Appendix B of the Participant Handout (Page 47).
Facilitator Considerations:
Possible Whole Group Discussion Questions:
1. How were your and your partners’ problems similar/different?
2. Is one problem more difficult than another?
10
min
14.
(PH – Pages 20-21)
Set 3, Group A (Number of groups unknown – measurement – whole
numbers)
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label with the variables, identifying the
meaning of the factors.
A X B = C. Label the factors.
WG 
ITT 
PS
10
min
15.
(PH – Page 22-23)
Set 3, Group B (Number of groups unknown – measurement – fractions)
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label with the variables, identifying the
meaning of the factors.
A X B = C. Label the factors.
WG 
ITT 
PS
Section: Protocol D: Precise Descriptions of Terminology Time: 34 minutes
In this section, you will explore Protocol D and use all of the
protocols discussed so far to solve problems.
Materials used include:
• Modeling Word Problems with Tape Diagrams : Multiplication and
Division PPT
• Modeling Word Problems with Tape Diagrams : Multiplication and
Division Facilitator Guide
• Modeling Word Problems with Tape Diagrams : Multiplication and
Division Participant Handout
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
GROUP
4 min
(PH – Page 24)
Protocol D: Precise Descriptions of Terminology
1. Participants individually derive a description of these terms based on their
past experiences.
2. Participants share description with partners and revise as necessary.
3. Participants share their descriptions with the whole group.
4. Participants are guided to the glossary, providing the precise descriptions
from the curriculum.
Glossary (PH – Page 45)
ITT 
PS
16.
Measurement Division: The whole and the size or measure of one group is known,
but the number of groups needs to be determined.
Partitive Division: The whole and the number of groups being created are known,
but the size of the groups needs to be determined.
10
min
17.
(PH – Page 25)
1. Distribute student work.
2. Participants compare and contrast the student work.
3. Participants discuss how they would help the struggling student after
examining their work.
ITT 
PS
5 min
18.
(PH – Page 26 and 27)
ITT 
1. Participants read and analyze the highlighted problem types silently. (1
PS 
minute)
WG
2. Participants stand and find a partner from a different table. For 1 minute,
using grade-appropriate measurement, whole number, or fractional units,
they create a ‘Compare, Larger Unknown’ situation.
3. Participants change partners. For 1 minute, they create a ‘Compare, Smaller
Unknown A > 1’ situation.
4. Participants change partners. For 1 minute, they create a ‘Compare, Smaller
Unknown’ situation.
5. Participants change partners. For 1 minute, they create a ‘Compare, Larger
Unknown A < 1’ situation.
Facilitator Considerations:
Possible Whole Group Discussion Questions:
1. How are these different than the problem types we discussed and practiced
earlier?
15
min
19.
(PH – Pages 28-29)
WG 
ITT 
PS
Set 4, Group A (Compare – larger unknown)
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label with the variables, identifying the
meaning of the factors.
A X B = C. Label the factors.
Note: Set 4, Group B (Compare – smaller unknown – A<1) is on Page 30 of the PH.
It will not be used in this session. It remains in the handout for the participants to
reference at a later time.
Section: Problem Analysis
Time: 27 minutes
In this section, you will explore using the different models and
protocols from the previous sections to solve difficult problems.
Materials used include:
• Modeling Word Problems with Tape Diagrams : Multiplication
and Division PPT
• Modeling Word Problems with Tape Diagrams : Multiplication
and Division Facilitator Guide
• Modeling Word Problems with Tape Diagrams : Multiplication
and Division Participant Handout
Time
5 min
Slide # Slide #/ Pic of Slide
20.
Script/ Activity directions
GROUP
(PH – Page 31)
1. Participants choose one problem in Set 4 they found difficult to
model and then find a partner from another table.
2. Each partner shares the model drawn for the problems he/she
chose and why he/she believes this problem was difficult.
Out of
Seat
Sharing
Facilitator Considerations:
Participants share some questions that could help students. “Share It”
cards.
Survey the room to see who chose difficult problems from each group,
showing that one type of problem may be more difficult than another
group.
13 min
21.
(PH – Pages 32-34)
Set 5, Group A (Compare – smaller unknown - (A>1))
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label with the variables, identifying
the meaning of the factors.
A X B = C. Label the factors.
WG 
ITT 
PS
Note: Although not included within the content of this session, Set 5,
Group B (Compare – larger unknown - (A<1)) (PH - Pages 35-36) and Set
6, Group A (Ratios) (PH – Pages 38-40) are included within the Participant
Handout. Both sets remain in the handout for the participants to reference
at a later time. Also included within the Participant Handout, on pages 37
and 41, are models for which word problems can be written.
6 min
22.
(PH – Page 42)
WG
Revisit the Camp Okey-Fun-Okey problem. Have a brief conversation about
how the multiplicative comparison model bridges so beautifully into
student work with ratios.
Use “Share It” cards to have participants share their thinking.
Last summer, at Camp Okey-Fun-Okey, the ratio of the number of boy
campers to the number of girl campers was 𝟖:𝟕. If there were a total of
195 campers, how many boy campers were there? How many girl
campers? (G6-M1-L5)
3 min
23.
How does knowledge of the various problem situations support educators ITT 
to be more effective in empowering every student to be a better problem
WG
solver?
THE FOLLOWING VIDEO/ACTIVITY COULD BE INCORPORATED INTO ANY
PART OF THE SESSION. THE PURPOSE OF THE VIDEO IS TO SHOW HOW
THE RDW PROCESS ENCOURAGES DEEP THINKING AND CAN BE
EVIDENCED IN THE STANDARDS FOR MATHEMATICAL PRACTICE.
Prompt participants to turn to APPENDIX D of the Participant Handout
(Page 50). Show the video of the G2 student explaining his thinking.
Prompt participants to take note of the model drawing and of the
explanation that the student gives. Discuss how the student’s answers are
evidenced in the Standards for Mathematical Practice. (See activity below.)
Analysis of Evidence of the Mathematical Practice Standards within a
Grade 2 Classroom
Directions: Watch the following short video clip of the Grade 2 student’s
thinking about his work.
Discuss with your partner: What Standards of Mathematical Practice are
evidenced? How are they evidenced?
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
Discussion questions might include the following:
How is the student…
• decontextualizing and contextualizing? (MP.2)
• modeling with mathematics? (MP.4)
• using appropriate tools strategically? (MP.5)
• attending to precision? (MP.6)
• looking for and making use of structure? (MP.7)
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Active learning
Turnkey Materials Provided
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•
Modeling Word Problems with Tape Diagrams : Multiplication and Division PPT
Modeling Word Problems with Tape Diagrams : Multiplication and Division Facilitator Guide
Modeling Word Problems with Tape Diagrams : Multiplication and Division Participant Handout
Additional Suggested Resources
•
•
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How to Implement A Story of Units
A Story of Units Year Long Curriculum Overview
A Story of Units CCLS Checklist
Turn and talk