Modeling Word Problems with Tape Diagrams:
Addition and Subtraction
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 the 12 subtraction and addition word problem situations, categorizing prior to drawing.
o Become more familiar with the situation types designated for their grade level and see how those situations manifest in multi-step problems at
later grades, thereby gaining a deeper understanding of the ways in which foundational work laid in the early grades is critical for success
through middle school, especially in the arena of comparison problems.
Related Learning Experiences



This topic is extended in a companion session, Modeling Word Problems with Tape Diagrams: Multiplication and Division.
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 additional and
subtraction problem types.
How will we know that they are able to do this?
Participants will be able to articulate the key points listed above.


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.
Session Overview
Section
Time
Overview
Prepared Resources
•
Introduction
11 min
Introduces the key ideas in this
session.
•
•
Problem Solving
Processes
31 min
Explores Polya’s Problem Solving
Process and the Read-Draw-Write
(RDW) process and apply them to
specific problems.
•
•
Modes of
Instructional
Delivery
31 min
Explores modes of instructional
delivery and view them in action.
•
Modeling Word Problems
with Tape Diagrams:
Addition and Subtraction
PPT
Modeling Word Problems
with Tape Diagrams
:Addition and Subtraction
Facilitator Guide
Modeling Word Problems
with Tape Diagrams:
Addition and Subtraction
PPT
Modeling Word Problems
with Tape Diagrams
:Addition and Subtraction
Facilitator Guide
Modeling Word Problems
with Tape Diagrams:
Addition and Subtraction
PPT
Modeling Word Problems
with Tape Diagrams
:Addition and Subtraction
Facilitator Guide
Facilitator Preparation
Review Modeling Word Problems
with Tape Diagrams :Addition
and Subtraction PPT and
Facilitator Guide
Review Modeling Word Problems
with Tape Diagrams :Addition
and Subtraction PPT and
Facilitator Guide
Review Modeling Word Problems
with Tape Diagrams :Addition
and Subtraction PPT and
Facilitator Guide
•
Following Protocol D:
Digging Into a
71 min
Vignette
Explores Protocol D and use all of
the protocols discussed so far to
solve problems.
•
•
Deliberate Practice
34 min
Explores the Deliberate Practice
Protocol to solve difficult
problems and use all of the
protocols discussed to help solve
challenging problems.
•
Modeling Word Problems
with Tape Diagrams:
Addition and Subtraction
PPT
Modeling Word Problems
with Tape Diagrams
:Addition and Subtraction
Facilitator Guide
Modeling Word Problems
with Tape Diagrams:
Addition and Subtraction
PPT
Modeling Word Problems
with Tape Diagrams
:Addition and Subtraction
Facilitator Guide
Review Modeling Word Problems
with Tape Diagrams :Addition
and Subtraction PPT and
Facilitator Guide
Review Modeling Word Problems
with Tape Diagrams :Addition
and Subtraction PPT and
Facilitator Guide
Session Roadmap
Section: Introduction
Time: 11 minutes
In this section, you will be introduced to the key ideas in this
section.
Materials used include:
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction PPT
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Facilitator Guide
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Participant Handout
Time
Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
4 min
1.
(PH – Page 2)
PS
Participants work on solving one challenge problem.
(Prompt participants to solve the “Leonard’s comic books” problem. If they
finish early, challenge them to try the “Reading Club” problem.)
Inform participants that the solutions for each of the problems will be in the
answer packet they will receive at the end of the day.
𝟏
Leonard spent 𝟒 of his money on a sandwich. He spent 𝟐 times as much
𝟑
on a gift for his brother as on some comic books. He had 𝟖 of his money
left. What fraction of his money did he spend on comic books? (G5 M3
L7)
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 Multiplication and Division Problem Types
(Situations) can be found on Appendix B in the back of the Participant
Handout. **
As time allows, consider using Protocol C at various times throughout
the session.
A NOTE REGARDING APPENDIX D – STAGES OF MODEL DRAWING:
At an ideal time within the session, turn attention to Appendix D in 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?
WG
5 min
3.
(PH – Page 3)
ITT 
1. Participants read and analyze the highlighted situation types
PS 
silently. (1 minute)
WG
2. Participants jot down a word problem on page 4 that corresponds to
each of the 6 problem situations. (1 minute)
3. Participants stand and find a partner from a different table to share
and analyze each other’s problems. Change partners with each
different problem type. (3 minutes total)
Facilitator Considerations - Possible Whole Group Discussion Questions:
• Which problems were the easiest to create?
• Which problems were the hardest to create?
Which problems do you think the students will struggle solving the most?
Section: Problem Solving Processes
Time: 31 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 :Addition and
problems.
Subtraction PPT
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Facilitator Guide
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Participant Handout
Time
10 min
Slide # Slide #/ Pic of Slide
4.
Script/ Activity directions
GROUP
(PH – Pages 5-6)
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.
WG 
PS
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. ***
Notes on Pedagogy:
The RDW process 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. Reread the second sentence. Draw and
label, etc. Consider the following example:
1
Leonard spent 4 of his money on a sandwich. He spent 2 times as much on a
3
gift for his brother as on some comic books. He had 8 of his money left. What
fraction of his money did he spend on comic books?
Read:
1
Leonard spent 4 of his money on a sandwich.
Draw: (see slide note)
Read:
He spent 2 times as much on a gift for his brother as on some comic books.
Draw: (see slide note)
Read:
3
He had 8 of his money left.
Draw: (see slide note)
Read:
What fraction of his money did he spend on comic books?
Draw and write: (see slide note)
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.
7 min
5.
(PH – Page 7-9, Pages 45-46)
WG 
Set 1
ITT 
1. Facilitator super-models the first problem using the RDW process using PS
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.
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 the referents.
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.
4. Go over Appendix D: Stages of Model Drawing. Then have participants
analyze on PH Pages 8-9.
7 min
6.
(PH – Page 10)
Set 2
1. Facilitator models the first problem using the RDW process using the
“Modeling with Interactive Questioning” method.
NOTE: It is up to the presenter on how often they model questions. It may
not be necessary to model the first problem of every set.
2. Participants complete the Problem Solving Protocol A.
3. Classify the problem type and label the referents.
WG 
ITT 
PS
7 min
7.
(PH – Page 11)
Set 3
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 the referents.
WG 
ITT 
PS
Facilitator Considerations: Possible Whole Group Discussion Questions
• What are some road blocks students will face?
• How can you help through these road blocks?
Section: Modes of Instructional Delivery
Time: 31 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 :Addition and
Subtraction PPT
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Facilitator Guide
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Participant Handout
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
GROUP
2 min
(PH – Pages 12 -13)
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.
1. Show the video and ask participants to pay attention to the mode of delivery
WG 
being used.
PS 
2. Participants determine which type of delivery is being modeled in the video.
WG
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?
7 min
10.
(PH – Page 14)
Set 4
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 the referents.
WG 
ITT 
PS
7 min
11.
(PH – Page 15)
Set 5
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 the referents.
Facilitator Considerations: Possible Whole Group Discussion Questions
• What are some road blocks students will face?
• How can you help through these road blocks?
WG 
ITT 
PS
7 min
12.
(PH – Page 16)
Set 6
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 the referents.
Facilitator Considerations: Possible Whole Group Discussion Questions
• What are some road blocks students will face?
• How can you help through these road blocks?
WG 
ITT 
PS
Section: Following Protocol D: Digging Into a Vignette
Time: 73 minutes
In this section, you will explores Protocol D and use all of the
protocols discussed so far to solve problems.
Materials used include:
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction PPT
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Facilitator Guide
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Participant Handout
Time
10 min
Slide # Slide #/ Pic of Slide
13.
Script/ Activity directions
GROUP
(PH – Pages 17-22)
Follow Protocol D: Digging Into a Vignette - “Question the Author” for G1M4-Lesson 20.
PS 
WG
8 min
14.
(PH – Pages 23-24)
Show distribute student work.
Participants compare and contrast the student work.
PS
Facilitator Considerations:
Encourage participants to discuss how they would help the struggling
student after examining their work.
7 min
15.
(PH – Page 25)
Set 7
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 the referents.
10 min
16.
(PH – Pages 26-27)
ITT PS
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.
NOTE: The actual word problems given in the curriculum for all models
throughout the session are presented in Appendix C 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?
WG 
ITT 
PS
7 min
17.
(PH – Page 28)
Set 8
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 the referents.
WG 
ITT 
PS
7 min
18.
(PH – Page 29)
1. Participants silently examine each model and write a “start
unknown” 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 C of the Participant
Handout.
Facilitator Considerations:
If time allows, direct participants to Appendix C to see the word problems
presented in the curriculum for these models.
Possible Whole Group Discussion Questions:
• How were your and your partners’ problems similar/different?
• Is one problem more difficult than another?
10 min
19.
(PH – Pages 30-32)
1. Participants read and analyze the highlighted situation types
silently. (1 minute)
2. Participants jot down word problems on Page 27 that correspond
with the highlighted problem types (1 minute)
3. Participants stand and find a partner from a different table to share
and analyze each other’s problems. Change partners with each
different problem type. (3 minutes total)
4. Participants read the Notes on Pedagogy silently.
5. When signaled, participants will discuss their thoughts about the
ITT 
PS 
WG
process of the RDW modeled here and how these models are
different than the ones presented so far in this presentation.
6. Discuss as a whole group. Pay attention to the similarities and
differences between the earlier models and the comparison model.
Facilitator Considerations - Possible Whole Group Discussion Questions:
• How are these different than the problem types we discussed and
practiced earlier in the day?
7 min
20.
(PH – Page 33)
Set 9
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label the referents.
WG 
ITT 
PS
7 min
21.
(PH – Page 34)
Set 10
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label the referents.
WG 
ITT 
PS
Section: Deliberate Practice
Time: 34 minutes
In this section, you will explore the Deliberate Practice Protocol to
solve difficult problems and use all of the protocols discussed to
help solve challenging problems.
Materials used include:
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction PPT
• Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Facilitator Guide
•
Modeling Word Problems with Tape Diagrams :Addition and
Subtraction Participant Handout
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
7 min
(PH – Page 35)
PS
Out of Seat Sharing
1. Participants choose one problem in either Set 9 or Set 10 they found
difficult to model and then find a partner from another table.
2. One partner shares the model drawn for the problem they chose and
why they believe this problem was difficult for them.
3. Partners follow the Deliberate Practice Protocol with the difficult
problem.
4. Repeat steps 2 and 3 for the problem the other partner chose. If
partners chose the same problem, the second partner should pick
another challenging problem.
22.
GROUP
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.
7 min
23.
(PH – Page 36)
Set 11
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label the referents.
WG 
ITT 
PS
7 min
24.
(PH – Page 37)
Set 12
1. Facilitator models the first problem using the RDW process.
2. Participants complete the Problem Solving Protocol.
3. Classify the problem type and label the referents.
WG 
ITT 
PS
5 min
25.
(PH – Page 38)
1. Participants silently examine each model and write a “comparison”
word problem that can be solved using the model.
2. Signal participants to discuss their word problems with a partner.
ITT PS
Facilitator Considerations:
If time allows, direct participants to Appendix C to see the word problems
presented in the curriculum for these models.
Possible Whole Group Discussion Questions
Which set and group would you problem fall into?
5 min
26.
(PH –Page 39)
• Revisit the two opening problems.
• Use “Share It” cards to have participants share their thinking.
WG
3 min
27.
How does knowledge of the various problem situations support educators
to be more effective in empowering every student to be a better problem
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 E of the Participant Handout.
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
MP.2
MP.3
MP.4
MP.5
MP.6
MP.7
MP.8
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
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)
ITT 
WG
•
•
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
•
•
•
Modeling Word Problems with Tape Diagrams: Addition and Subtraction PPT
Modeling Word Problems with Tape Diagrams: Addition and Subtraction Facilitator Guide
Modeling Word Problems with Tape Diagrams: Addition and Subtraction Participant Handout
Additional Suggested Resources
•
•
•
How to Implement A Story of Units
A Story of Units Year Long Curriculum Overview
A Story of Units CCLS Checklist
Turn and talk