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?

How will we know that they are able to do this?

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.

Participants will be able to articulate the key points listed above.

Session Overview

Section Time Overview Prepared Resources Facilitator Preparation

Introduction to Module

10 minsEstablish 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 minsExamine and discuss the assessments in Grade K Module 4.

Grade K Module 4 Sample Assessment Video

Review assessments, rubric, and sample studen work.

Session Roadmap

Section: Grade K Module 4 Time: 4 hours 15 minutes

TimeSlide #

Slide #/ Pic of Slide Script/ Activity directions GROUP

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.

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, 5-

groups, 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 Overview2. Look at the objective chart. Video/demonstration lesson will be

on L13. You will have an opportunity to try key lessons and share

concepts from each topic4. 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 boardsNote: 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 10How 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 13How 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 real-world 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 DHow 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 real-world 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 DWhy 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 multi-digit 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.

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.

10 18. Demonstration and Debrief of Topic FHow 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 GHow 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 HHow 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 Active learning Turn and talk

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