Module Focus: Grade K – Module 4
Sequence of Sessions
Overarching Objectives of this February 2014 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.
Standards alignment. Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module
addresses the major work of the grade in order to fully implement the curriculum.
Implementation. Participants will be prepared to implement the modules and to make appropriate instructional choices to meet the needs of their
students while maintaining the balance of rigor that is built into the curriculum.
Related Learning Experiences
●
This session is part of a sequence of Module Focus sessions examining the Grade K curriculum, A Story of Units.
Key Points



Numbers can be decomposed from wholes into parts in a variety of ways. These decompositions can be represented with
concrete, pictorial, or numerical number bonds.
Using number bonds and drawings, story situations can be mathematized and subsequently represented as addition or
subtraction equations.
In the context of number bonds, addition and subtraction sentences, students formalize their understanding of part/whole
relationships and begin to see the inverse relationship between addition and subtraction.
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.
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
Facilitator Preparation
Introduction to
Module
10 mins
Establish the instructional focus of
Grade K Module 4.


Grade K Module 4 PPT
Facilitator Guide
Review Grade K Module 4.
Concept
Development
215 mins
Examine the mathematical
concepts developed in Grade K
Module 4 through demonstration
lessons and fluency practice.


Grade K Module 4 PPT
Facilitator Guide
Review Grade K Module 4.
Module Assessments
30 mins
Examine and discuss the
assessments in Grade K Module 4.

Grade K Module 4 Sample Review assessments, rubric, and
sample studen work.
Assessment Video
Session Roadmap
Section: Grade K Module 4
Time: 4 hours 15 minutes
Time Slide Slide #/ Pic of Slide
#
Script/ Activity directions
1
1.
NOTE THAT THIS SESSION IS DESIGNED TO BE 4 HOURS and 15
MINUTES IN LENGTH.
Turnkey Materials Provided in Addition to PowerPoint:
• Grade K—Module 4 Selected Lessons Handout
• GK-M4- Module Overview Handout
Additional Suggested Resources:
• Operations and Algebraic Thinking Progression Document
• A Story of Units: A Curriculum Overview for Grades P-5
• How to Implement A Story of Units
Welcome! In this module focus session, we will examine Grade K –
Module 4.
1
2.
Our objectives for this session are to:
• Examine the development of mathematical
understanding across the module using a focus on the
lessons.
• Introduce mathematical models and instructional
strategies to support implementation of A Story of Units.
GROUP
1
3.
We will begin by briefly exploring the module overview to understand
the purpose of this module. Then we will dig in to the math of the
module, with the bulk of our time spent on lesson study. We’ll lead you
through the teaching sequence, one concept at a time. You will have the
opportunity to take part in live demonstrations and possibly even
facilitate a lesson. Finally, we’ll take a look back at the module,
reflecting on all the parts as one cohesive whole through the lens of
assessment.
Let’s get started with the module overview.
*Idea- ask for a volunteer who feels comfortable with our assessment
process. Have that volunteer administer the assessment to one of the
facilitators and have group use rubric to score (in afternoon).
3
4.
The fourth module in Kindergarten is Number Pairs, Addition and
Subtraction to 10. The module includes 41 lessons and is allotted 47
instructional days.
How many of you have had experience teaching/observing M1? M2?
M3? Based on that experience, what knowledge and experiences do
your students already have that prepare them for addition and
subtraction?
• Number core: number name sequence, 1:1 correspondence,
cardinality, numeral recognition
• Tracking a counting path
• Models: Rekenrek, counting the Math Way, number towers, 5groups, number path
• Hidden partners (decomposition/composition)
• Exposure to expressions and equations
• Comparing numbers
This module builds on understandings of number and operations
established in Modules1-3, formalizing their understanding of number
pairs using addition and subtraction operations. This module prepares
students for success as they move into Level 2 and 3 methods for
solving single-digit addition and subtraction problems.
5
5.
To become familiar with Module 4, read the narrative in the module
overview. We will not spend much time going through the other
components of the module overview or topic openers to leave as much
time as possible to exploring and practicing key mathematical concepts
within the lessons.
1. Read the narrative in the Module Overview
2. Look at the objective chart. Video/demonstration lesson will be
on L1
3. You will have an opportunity to try key lessons and share
concepts from each topic
4. The Operations and Algebraic Thinking Progressions Document
together with the progressions document for Number and Operations
in Base Ten will provide keen insight into the relationship between
standards between grades K-2. Throughout the session, we will connect
portions of these progressions documents to our learning.
1
6.
Now that you know the main objectives for this module, let’s dig into
the lessons.
12
7.
Demonstration of Lesson 1 Concept Development by facilitator.
While facilitating the CD, click to show image of birds on the next slide.
1
8.
Use this image as part of the Lesson 1 Concept Development
demonstration. Participants should decompose 5 birds into two groups: 2
chickens and 3 geese.
12
9.
In Topic A, students learn a critical new model- the number bond. They
saw a preview of the number bond in Module 1 when they worked with
hidden partners in Lesson 9, but this topic formalizes the model in
students minds. The number bond is a pictorial representation of the
part-part-whole relationship.
As you would expect, number bond work progresses from concrete to
abstract.
(Click to advance.) Students initially learn about the model by acting
out a story, as we just saw in Lesson 1.
(Click to advance.) Students then practice with manipulatives using a
number bond template.
(Click to advance.) They then begin to use pictorial representations of
objects in the number bonds.
(Click to advance.) Finally, they begin to use numerals in place of
objects and pictures.
Note that the number bond can be presented in any orientation.
Throughout Topic A, students see the number bond in all of its
orientations so that they do not develop a rigid understanding of the
model or how it is used.
Why do we spend so much time with the relationships between
numbers 1-5?
1. K.OA.5 Fluently add and subtract within 5. Students need to
understand the relationships among these numbers deeply
through extended practice to become fluency with addition and
subtraction within 5.
2. “Students work with small numbers first, though many
kindergarteners will enter school having learned parts of
the Kindergarten standards at home or at a preschool
program. Focusing attention on small groups in adding and
subtracting situations can help students move from
perceptual subitizing to conceptual subitizing in which they
see and say the addends and the total, e.g., ‘Two and one
make three.’” – OA Progression, page 8. Spending time on 1-5
establishes the part-part-whole relationship. It also allows an
understanding of composing and decomposing within a
manageable quantity.
3
10.
Instruct participants to take out their personal white boards. Facilitate
the following fluency activity from Lesson 7. It should take only 2 minutes
with adults.
Number Bond Flash (5 minutes)
Materials: (T) Magnetic shapes or dry erase markers (S) Personal white
boards
Note: This is a maintenance activity to support fluent understanding of
the relationships between numbers to 5 through number bonds.
T: (Show 3 red squares and 1 yellow square.) How many squares do I
have?
S: 4 squares.
T: How many are yellow?
S: 1.
T: How many are red?
S: 3.
T: 1 and 3 are the parts. 4 is the whole. Draw a number bond to tell
about my squares. Lift up your board when you are done.
S: (Write number bonds using drawings or numerals. Lift board to
signal completion.)
T: Nice job.
Repeat with (CLICK TO ADVANCE) 2 + 2,
(CLICK TO ADVANCE) 4 + 1,
(CLICK TO ADVANCE) 2 + 3. As students show mastery, stop naming
the parts and whole before they draw.
How is this activity designed so that students can self-differentiate?
(The student chooses whether to use drawings or numerals in the
number bond. Student chooses orientation of bond.)
How can you adjust the materials to provide additional differentiation?
(Slide number bond mat into personal white board, give students cubes
to create concrete number bonds, allow advanced students to create
number bonds for higher sums if they show mastery within 5.
12
11.
Assign each Topic to a table to demonstrate. (In cases where they are
presenting a CD, 1 participant “teacher” will teach to the rest of their
table.)
• TB, Lesson 10: Concept Development
• TC, Lesson 13: Concept Development and Debrief (questions
that do not relate to PS)
• TD, Lessons 22-24: Demonstrate subtraction strategies: break
off, hide, and cross of a part (Provide support for this group
early to make sure that they understand their task. They are to
provide examples of how students will use each of the three
strategies to subtract.)
• TE, Lesson 27: Concept Development and Debrief
• TF, Lessons 29-31: Describe the progression of addition word
problems in this topic. (Teachers will read the four lessons and
describe how the problems differ as the complexity increases,
giving examples from the text.)
• TG, Lessons 33-36: Describe the progression of subtraction
word problems in this topic. (Teachers will read the four lessons
and describe how the problems differ as the complexity
increases, giving examples from the text.)
• TH, Lesson 37:
1. Each table will have 10 minutes to prepare their demonstration.
If you are presenting a CD, one person should plan to be the
teacher while other participants are students. If you are
presenting on a series of problems from a topic, the group may
decide how to split up the presentation responsibilities.
2. Ideally, 1-2 participants from each group will skim the other
lessons in the topic to better understand the context.
3. As the demo lesson is being presented, the rest of the room
gathers around the demo teacher and table of “students”.
Note to facilitator: Depending on how many participants you have, you
might split the lessons between pairs of “teachers” to teach to the whole
room (if you have a small group) or you can split the components of each
lesson (if you have a large group).
10
12.
Demonstration and Debrief of LESSON 10
How will decomposing numbers less than or equal to 10 in more than
one way help students with Level 2 and 3 problem solving? (K.OA.3)
(Show 8 + 4. In order to engage in Level 2 counting on strategies,
students must understand that an addend is embedded in the total
(perceive addend simultaneously as an addend and as part of the total).
They are therefore able to omit the counting of one addend, 8 (show 8
fingers), and count on by the other addend, 9, 10, 11, 12. For level 3
strategies, they make use of this knowledge to convert to an easier
problem. Level 3 example: 8 + 4= 8 +(2 + 2)= 10 + 2 = 12)
The quotes that follow each lesson are excerpts taken from the
Operations and Algebraic Thinking progressions document. These
portions of the progressions document pertain to the mathematical
concepts each lesson is written from.
“Put Together/Take Apart situations with Both Addends Unknown play
an important role in Kindergarten because they allow students to
explore various compositions that make each number. (K.OA.3) This
will help student to build the Level 2 embedded number
representations used to solve more advance problem subtypes. As
students decompose a given number to find all of the partners that
compose the number, the teacher can record each decomposition with
an equation such as 5 = 4 +1, showing the total on the left and the two
addends on the right. Students can find patterns in all of the
decompositions of a given number and eventually summarize these
patterns for several numbers.” (p. 10, Operations and Algebraic
Thinking progressions)
10
13.
Demonstration and Debrief of LESSON 13
How do math drawings and number bonds support the use of number
sentences (equations) in this lesson?
(They help the students mathematize the situation and provide
pictorial references for writing equations. Drawings and number bonds
help students identify the referents within the number sentence (MP.2),
helping them to better understand the part-part-whole relationship in
equation form.)
The quotes that follow each lesson are excerpts taken from the
Operations and Algebraic Thinking and Number and Operations in Base
Ten progressions documents. These portions of the progressions
document pertain to the mathematical concepts each lesson is written
from.
“Students act out adding and subtracting situations by representing
quantities in the situation with objects, their fingers, and math
drawings (MP.5, K.OA.1). To do this, students must mathematize a realworld situation (MP.4), focusing on the quantities and their
relationships rather than non-mathematical aspects of the situation.
Situations can be acted out and/or presented with pictures or words.
Math drawings facilitate reflection and discussion because they remain
after the problem is solved. These concrete methods that show all of the
objects are called Level 1 methods.” (p. 8, Operations and Algebraic
Thinking progressions)
“Numerical expressions and recordings of computations, whether with
strategies or standard algorithms, afford opportunities for students to
contextualize, probing into the referent for the symbols involved
(MP.2). Representations such as bundled objects or math
drawings…and diagrams…afford the mathematical practice of
explaining correspondences among different representations (MP.1).”
(p. 4, Number and Operations in Base Ten)
NOTE: In Topic C, Lesson 16, students learn to use a mystery box to
represent the unknown in an equation. In GK, the unknown is often the
total. Using this strategy helps prepare students for missing addend
problems in G1.
10
14.
Demonstration and Debrief of Topic D
How do these concrete methods help Kindergarteners understand the
concept of subtraction?
(They help the students visualize and mathematize the situation and
provide pictorial references for writing equations. Drawings and
number bonds help students identify the referents within the number
sentence, helping them to better understand the part-part-whole
relationship in equation form. There is less flexibility in how
subtraction sentences are written, so identifying the referents and
applying them to the equations may be more difficult.)
The quotes that follow each lesson are excerpts taken from the
Operations and Algebraic Thinking progressions document. These
portions of the progressions document pertain to the mathematical
concepts each lesson is written from.
“Students act out adding and subtracting situations by representing
quantities in the situation with objects, their fingers, and math
drawings (MP.5, K.OA.1). To do this, students must mathematize a realworld situation (MP.4), focusing on the quantities and their
relationships rather than non-mathematical aspects of the situation.
Situations can be acted out and/or presented with pictures or words.
Math drawings facilitate reflection and discussion because they remain
after the problem is solved. These concrete methods that show all of the
objects are called Level 1 methods.” (p. 8, Operations and Algebraic
Thinking progressions)
10
15.
Demonstration and Debrief of Topic D
Why do Kindergarteners need to be able to find the number that makes
10 with any number from 1 to 9? (K.OA.4)
(This prepares students to compose and decompose 10, and later any
base ten unit. This is critical for use of standard algorithms beginning in
G2. Kindergarten teachers are setting an early basis for conceptual
understanding of addition and subtraction strategies and algorithms. It
also supports the Level 3 strategy of making a ten.)
The quotes that follow each lesson are excerpts taken from the Number
and Operations in Base Ten progressions document. These portions of the
progressions document pertain to the mathematical concepts each lesson
is written from.
“Standard algorithms for base-ten computations with the four
operations rely on decomposing numbers written in base-ten notation
into base-ten units. The properties of operations then allow any multidigit computation to be reduced to a collection of single-digit
computations. These single-digit computations sometime require the
composition or decomposition of a base-ten unit.” (p. 8, Operations and
Algebraic Thinking progressions)
16.
We introduce core fluency practice sets and sprints in Topic F. These
provide daily practice for review and master of the GK core fluency,
sums and differences with totals to 5.
The fluency practice sets come in a series that progresses from easy to
hard. The first time you administer the sets, all of your students will
start on Sheet A. Students continue to work on Sheet A until they are
able to complete all problems correctly in 96 seconds. Once a student
has mastered Sheet A, she moves on to Sheet B, and so on. In this way,
your students will continue to practice at their own level until they
have achieved mastery.
Encourage students who did not finish to take the sheet home and
continue to practice.
10
17.
The sprints come in four sets that progress from easy to hard. With the
sprint, you will need to choose one set of problems for all of your
students in order to continue the sprint correction pattern you’ve
already established for the class. Choose the set that best meets the
needs of most of your class, and remember that sprints are
differentiated so that every child in your class should be able to
complete at least the first few problems.
18.
Demonstration and Debrief of Topic F
How does the progression of word problems adhere to the OA
Progressions Table 2: Addition and subtraction situation by grade
level? (K.OA.2)
(The progression of problems first helps students mathematize the
situation by providing the parts and the total. Once the referents are
established, students are ready to try three types of addition word
problems: Add To/Result Unknown, Put Together/Total Unknown, and
Both Addends Unknown)
19.
The quotes that follow each lesson are excerpts taken from the
Operations and Algebraic Thinking progressions document. These
portions of the progressions document pertain to the mathematical
concepts each lesson is written from.
“Add To/Take From situations are action-oriented; they show changes
from an initial state to a final state. These situations are readily
modeled by equations because each aspect of the situation has a
representation as a number, operation (+ or -), or equal sign (here with
the meaning of “becomes,” rather than the more general “equals”).” (p.
8-9, Operations and Algebraic Thinking progressions)
“In Put Together/Take Apart situations, two quantities jointly
compose a third quantity (the total), or a quantity can be decomposed
into two quantities (the addends). This composition/decomposition
may be physical or conceptual. These situations are acted out with
objects initially and later children begin to move to conceptual mental
actions of shifting between seeing the addends and seeing the total
(e.g., seeing children or seeing boys and girls, or seeing red and green
apples or all the apples.)” (p. 10, Operations and Algebraic Thinking
progressions)
“Put Together/Take Apart situations with Both Addends Unknown
play an important role in Kindergarten because they allow students to
explore various compositions that make each number. (K.OA.3) This
will help students to build the level 2 embedded number
representations used to solve more advanced problem subtypes. As
students decompose a given number to find all of the partners that
compose the number, the teacher can record each decomposition with
an equation…” (p. 10, Operations and Algebraic Thinking progressions)
10
20.
Demonstration and Debrief of Topic G
How does the progression of word problems adhere to the OA
Progressions Table 2: Addition and subtraction situation by grade
level? (K.OA.2)
(The progression of problems first helps students mathematize the
situation by providing the parts and the total. Once the referents are
established, students are ready to try two types of subtraction word
problems: Take From/Result Unknown and Take Apart/Total
Unknown)
21.
The quotes that follow each lesson are excerpts taken from the
Operations and Algebraic Thinking progressions document. These
portions of the progressions document pertain to the mathematical
concepts each lesson is written from.
“Add To/Take From situations are action-oriented; they show changes
from an initial state to a final state. These situations are readily
modeled by equations because each aspect of the situation has a
representation as a number, operation (+ or -), or equal sign (here with
the meaning of “becomes,” rather than the more general “equals”).” (p.
8-9, Operations and Algebraic Thinking progressions)
“In Put Together/Take Apart situations, two quantities jointly
compose a third quantity (the total), or a quantity can be decomposed
into two quantities (the addends). This composition/decomposition
may be physical or conceptual. These situations are acted out with
objects initially and later children begin to move to conceptual mental
actions of shifting between seeing the addends and seeing the total
(e.g., seeing children or seeing boys and girls, or seeing red and green
apples or all the apples.)” (p. 10, Operations and Algebraic Thinking
progressions)
10
22.
Demonstration and Debrief of Topic H
How does working with the number path aid students’ understanding
of the additive identity? (Acting out situations of adding or subtracting
0 on the number path shows students in a concrete way how 0 acts as
an additive identity. Adding 0 to any number does not change the
original number!)
How does the number path assist students in understanding that
addition and subtraction are inverse operations? (They experience in a
physical demonstration that moving the same number of spaces
forward and backward on the number path will leave them where they
began; addition and subtraction are opposites.)
The quotes that follow each lesson are excerpts taken from the
Operations and Algebraic Thinking progressions document. These
portions of the progressions document pertain to the mathematical
concepts each lesson is written from.
“The relationship between addition and subtraction in the Add To/Take
From and the Put Together/Take Apart action situations is that of
reversibility of actions: an Add To situation undoes a Take From
situation and vice versa and a composition (Put Together) undoes a
decomposition (Take Apart) and vice versa.” (p. 8, Operations and
Algebraic Thinking progressions)
1
23.
So far, we have examined the mathematical concepts, curriculum
organization, and mathematical models critical to Module 4. Now let’s
examine how we assess student learning.
8
24.
Now that you know the focus of the module, let’s examine how students
will be assessed on their mastery of these skills and concepts. In
previous sessions, we talked about the value of 1:1 assessments and
strategized for their implementation. Today, we are going to evaluate a
Kindergartener as he goes through the Module 4 Mid-Module
Assessment.
Turn to Topic A in the assessment. How will these questions assess true
mathematical understanding?
As you watch the video, fill in the assessment sheet. When the video is
over, we will take some time to score the assessment using the rubric.
NOTE TO FACILITATOR: Play video.
6
25.
Allow participants 2 minutes to turn and talk about their observations of
the content and implementation of the assessment. Then have
participants share their notes on Topic A using the questions below. This
is will give you and the participants a common starting point for using
the rubric.
• Let’s start with the first question: (Questions to be determined
on review of the video clip)
3
26.
Now that we have gathered our data, we are ready to score using the
rubric. But first, what is the purpose of a rubric?
Possible answers:
• To assess the student understanding of the standards
• To provide context and language to discuss student work
• To plan next steps for future instruction
• To grade (This is not the intention of the writers.)
Take the next 2 minutes to score Topic A using the rubric.
NOTE TO FACILITATOR: Allow 2 minutes for participants to work
independently.
7
27.
Turn and talk with others at your table. Share your scores and ask them
to do the same. However, don’t get overly focused on the numeric score.
Instead, think about the following questions:
• What skills or understanding has the student mastered?
• Where do the student’s misunderstandings lie?
• What can you do to support this student towards reaching
mastery of this concept/standard?
NOTE TO FACILITATOR: Allow 4 minutes for participants to turn and
talk about the rubric and the student’s work. Then facilitate a discussion
in the remaining 3 minutes.
If participants don’t agree that the student earned a 4, please touch
briefly on the following talking points (again, try to help them focus on
the student’s mathematical understanding rather than the numerical
score):
• Question #1 She was able to correctly identify a matching shape
and describe why they matched.
• Question #2 She was able to choose the 3 triangles despite a
difficult distractor.
• Question #3 She talked about attributes of the two shapes,
number of sides and corners, angle size (pointy), and type of
sides. Didn’t discuss closed figures.
0
28.
3
29.
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.
2
30.
Let’s review some key points of this session.
• Numbers can be decomposed from wholes into parts in a variety
of ways. These decompositions can be represented with
concrete, pictorial, or numerical number bonds.
• Using number bonds and drawings, story situations can be
mathematized and subsequently represented as addition or
subtraction equations.
• In the context of number bonds, addition and subtraction
sentences, students formalize their understanding of part/whole
relationships and begin to see the inverse relationship between
addition and subtraction.
Use the following icons in the script to indicate different learning modes.
Video
Reflect on a prompt
Turnkey Materials Provided



Grade K Module 4 PPT
Grade K—Module 4 Selected Lessons Handout
GK-M4- Module Overview 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
Active learning
Turn and talk