New York State Common Core
P-5
GRADE
Network Team Institute
Rigor Breakdown:
Conceptual Understanding
for Grades 3-5
Sequence of Sessions
Overarching Objectives of this November 2012 Network Team Institute






Participants will understand the features of A Story of Units and how the curriculum is aligned with
the CCLS, Learning Progressions, Instructional Shifts, Publishers’ Criteria, PARCC, Major Content
Emphases, and the Pre-Post Math Standards.
Participants will understand how to use the How to Implement A Story of Units and A Story of Units:
A Curriculum Overview for Grades P-5 documents and be prepared to share these documents with
others.
Participants will explore G2—M3 and G5—M3 and be prepared to train others to teach these
modules.
Participants will extend understanding of the structure of these modules to other modules in A Story
of Units.
Participants will analyze assessment tasks associated with G2—M3 and G5—M3.
Participants will examine evidence of all three components of rigor in A Story of Units and be
prepared to promote balanced instruction.
High-Level Purpose of this Session





Understand the conceptual understanding component of rigor called for in the Standards, as defined
by the Instructional Shifts and the Publishers’ Criteria
Examine various activities that promote conceptual understanding in A Story of Units
Compare and contrast conceptual understanding activities and analyze the impact and advantages of
each
Highlight Standards of Mathematical Practice in the conceptual understanding activities of A Story of
Units
Recognize the balance and intensity of all three components of rigor in A Story of Units
Related Learning Experiences

This session, preceded by an overview of rigor, is the first of a series of three Rigor Breakdown
sessions. While this session will focus on the conceptual understanding component of rigor, the next
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding for Grades 3-5
7/13/17
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NYS COMMON CORE MATHEMATICS PD

Rigor Breakdown: Conceptual Understanding
two sessions will focus on the other two components: procedural skill and fluency, and application.
This series of three sessions on rigor will be followed by a collaborative session during which
participants will lay out the structure and foundation needed for lesson planning within a module.
Key Points



Conceptual understanding can be promoted in a variety of ways including use of concrete and
pictorial models, conceptual questioning, and writing/speaking about understanding.
Conceptual understanding represents a major part of the instructional vision that shapes A Story of
Units. Frequent, rich opportunities to help students achieve conceptual understanding are woven
throughout A Story of Units.
These opportunities for conceptual development are often also opportunities to nurture the
Standards for Mathematical Practice. The Standards for Mathematical Practice are embedded into all
components of rigor and are not separate from the content instruction.
Session Outcomes
What do we want participants to be able to do as a result of this session?


Understand the importance of conceptual understanding
Recognize conceptual understanding activities in A Story of Units
How will we know that they are able to do this?


Participants will share their grasp of conceptual understanding and its importance with colleagues,
providing examples of how it occurs in A Story of Units.
Participants and their colleagues will successfully implement conceptual understanding activities as
described in A Story of Units.
Session Overview
Section
Opening
Conceptual
Understanding –
Concrete and
Pictorial Models
Time
0:00-0:05
(5 min)
0:05-0:25
(20 min)
Overview
 Link to previous sessions; frame the
session, referencing the agenda
 Examine conceptual understanding
as defined in the Shifts and
Publishers’ Criteria
 Examine how A Story of Units
develops conceptual understanding
 Review the flow and objectives of
this session
 Watch video clip of a teacher
(Example 1) engaging with students
to developing conceptual
understanding using concrete or
pictorial models
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Prepared Resources
Facilitator Preparation
 Session PowerPoint
 Review session notes and
PowerPoint presentation
 How to Implement A Story of Units
(can be used as a reference for the  Download the CCLS for easy
Standards of Mathematical
reference to the Standards of
Practice)
Mathematical Practice
 Download Crosswalk of
Instructional Shifts for easy
reference as needed
 Download Publishers’ Criteria for
easy reference as needed
 Video Clip: Fractions as Division –
 Review and download video file as
Concrete to Pictorial
a back-up in case there is a problem
with the embedded video
Rigor Breakdown: Conceptual Understanding
7/13/17
2
NYS COMMON CORE MATHEMATICS PD
Conceptual
Understanding –
Conceptual
Questioning
Conceptual
Understanding –
Writing and
Speaking About
Understanding
0:25-0:40
(15 min)
0:40-0:50
(10 min)
Closing
0:50-1:00
(10 min)
TOTAL TIME
60 min
Rigor Breakdown: Conceptual Understanding
 Analyze the impact and advantages
of the conceptual understanding
work in Example 1
 Identify evidence of Standards for
Mathematical Practice in Example 1
 Recall that Mathematical Practices
are not a separate activity
 Engage in Example 2 of work
developing conceptual
understanding through conceptual
questioning
 Identify evidence of Standards for
Mathematical Practice in Example 2
 Compare and contrast examples 1
and 2, analyzing the impact and
advantages of each
 Engage in Example 3 of developing
conceptual understanding through
writing and speaking about
understanding
 Compare and contrast conceptual
understanding activities of the
three examples, analyzing the
impact and benefits of each
 Summarize key points
 Reflect on next steps
 Reflect on the role of conceptual
understanding in balanced
instruction
 Video Clip: 10 Cakes Shared
Among 4
 Review and download video file as
a back-up in case there is a problem
with the embedded video
 Video Clip: Washing Machine
Problem
 G5—M3—B—L10 Worksheet
 Review and download video file as
a back-up in case there is a problem
with the embedded video
• How to Implement A Story of Units
Session Roadmap
Opening
Time: 0:00-0:05
[5 minutes] In this section, you will…
 Link to previous sessions; frame the session, referencing the agenda
 Examine conceptual understanding as defined in the Shifts and Publishers’ Criteria
 Examine how A Story of Units develops conceptual understanding
 Review the flow and objectives of this session
Materials used include:


Session PowerPoint
How to Implement A Story of Units
(SLIDE 1) During the last session we focused on rigor and its three different components from a holistic point of view.
You had an opportunity to take a look at rigor from an assessment and problem solving approach by analyzing the
PARCC Type I, II, and III Tasks as well as assessment tasks and rubrics from A story of Units. We saw how the three
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
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NYS COMMON CORE MATHEMATICS PD
Rigor Breakdown: Conceptual Understanding
components of rigor are embedded into the assessment tasks.
(SLIDE 2) In the next three sessions we will carefully examine each of the three components of rigor: conceptual
understanding, procedural skill and fluency, and application by experiencing and analyzing specific examples of each
component.
(SLIDE 3) In this first rigor breakdown session we will focus on the conceptual understanding component of rigor.
(SLIDE 4) The objectives of this session are to:





Understand the conceptual understanding component of rigor called for in the Standards, as defined by the
Instructional Shifts and the Publishers’ Criteria
Examine various activities that promote conceptual understanding in A Story of Units
Compare and contrast conceptual understanding activities and analyze the impact and advantages of each
Highlight Standards of Mathematical Practice in the conceptual understanding activities of A Story of Units
Recognize the balance and intensity of all three components of rigor in A Story of Units
(SLIDE 5) Let’s begin with referencing the description of conceptual understanding provided by the Crosswalk of
Common Core Instructional Shifts: Mathematics. It explains that for deep understanding to occur, “Teachers teach
more than ‘how to get the answer’ and instead support students’ ability to access concepts from a number of
perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students
demonstrate deep conceptual understanding of core math concepts by applying them to new situations as well as
writing and speaking about their understanding.”
(SLIDE 6) According to page 5 of the Publishers’ Criteria, “The word ‘understand’ is used in the Standards to set explicit
expectations for conceptual understanding…” Pages 9-10 of the Publishers’ Criteria state with respect to understanding
that, “Materials amply feature high-quality conceptual problems and questions that can serve as fertile conversationstarters in a classroom if students are unable to answer them…”
(SLIDE 7) The Publishers’ Criteria goes on to explain that, “…This includes brief conceptual problems with low
computational difficulty (e.g., ‘Find a number greater than 1/5 and less than 1/4’); brief conceptual questions (e.g., ‘If
the divisor does not change and the dividend increases, what happens to the quotient?’); and problems that involve
identifying correspondences across different mathematical representation of quantitative relationships.”
(SLIDE 8) In this session, we will examine three forms of conceptual understanding work by watching video clips and
engaging in sample activities of conceptual understanding from the module.
Conceptual Understanding – Concrete and Pictorial Models
Time: 0:05-0:25
[20 minutes] In this section, you will…


Watch video clip of a teacher (Example 1) engaging with students to developing conceptual understanding
using concrete or pictorial models
Analyze the impact and advantages of the conceptual understanding work in Example 1
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
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NYS COMMON CORE MATHEMATICS PD
Rigor Breakdown: Conceptual Understanding
 Identify evidence of Standards for Mathematical Practice in Example 1
 Recall that Mathematical Practices are not a separate activity
Materials used include:

Video Clip: Fractions as Division – Concrete to Pictorial
(CLICK TO ADVANCE ANIMATION ON SLIDE 8) Let’s begin with developing conceptual understanding through the use of
concrete and pictorial models.
(SLIDE 9) We’re going to watch a video clip of a 5th grade class engaged in learning about the relationship between
fractions and quotients. After the video you’ll have the opportunity to discuss the following:



If you teach this concept in your grade level, compare and contrast the example with how you develop this
same concept in your classroom today.
Analyze the impacts and advantages of developing conceptual understanding as demonstrated.
Identify one or more mathematical practices that are being employed in this example.
The video shows Anita Meadows, a teacher at Saint Edward’s School in Vero Beach, Florida. Her
students have are just beginning to look at the concept of fractions as quotients. Let’s watch how Anita
begins her lesson with a concrete experience and progresses into pictorial representations.
(SLIDE 10 – Play Video Clip: Fractions as Division – Concrete to Pictorial)
(SLIDE 11) Take a moment to consolidate your thoughts on the questions posed before the video.
Then you will have an opportunity to discuss your thoughts.
(Allow 2 minutes for independent reflection.)
Turn and talk with a partner at your table your response to these questions.
(Allow 2 minutes for turn and talk, and then facilitate a discussion.)
(SLIDE 12) According to page 8 of the CCSSM, “The Standards for Mathematical Practice describe ways in which
developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter
as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years.”
Recall that the Mathematical Practices are to be consciously integrated into daily instruction, but are not a separate
activity that takes away from content instruction.
(SLIDE 13) Let’s review some of the benefits associated with using concrete and pictorial materials to develop
conceptual understanding.


(CLICK TO ADVANCE FIRST BULLET ON SLIDE 13) Concrete materials give students an experiential
understanding of a given operation or concept.
(CLICK TO ADVANCE SECOND BULLET ON SLIDE 13) When we progress from a concrete model to a pictorial
representation, students are in charge of creating their own visual representations. This new responsibility
offers more flexibility in their representation, and in turn, often leads to erroneous representations. It may
take a bit of practice for students to correctly produce a visual model of what they have just experienced
concretely. Talking about what they have drawn often encourages them to analyze and correct any
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
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NYS COMMON CORE MATHEMATICS PD

Rigor Breakdown: Conceptual Understanding
misrepresentations.
(CLICK TO ADVANCE THIRD BULLET ON SLIDE 13) If we skip the concrete and pictorial representations, concepts
and operations become disconnected from their actual meaning, thus robbing students of understanding when
and how to apply the concepts and operations they are learning about.
Conceptual Understanding – Conceptual Questioning
Time: 0:25-0:40
[15 minutes] In this section, you will…
 Engage in Example 2 of work developing conceptual understanding through conceptual questioning
 Identify evidence of Standards for mathematical practice in Example 2
 Compare and contrast examples 1 and 2, analyzing the impact and advantages of each
Materials used include:

Video Clip: 10 Cakes Shared Among 4
(SLIDE 14) Throughout this session, we aim to provide varied samples of conceptual understanding work in A Story of
Units. Know that there is no one set way to promote deep understanding. This example uses conceptual questions to
build understanding.
(SLIDE 15) The example comes from later in this same module as students are learning to add and subtract fractions
with unlike denominators using visual fraction models. This time we will be engaging in the lesson ourselves. In this
short exchange, I will play the role of the teacher, and you will play the role of the students. While you may certainly
recognize concrete and pictorial models being used here, we would like to identify an example of conceptual
questioning within the lesson. So, here are the prompts for reflection that we will discuss after the exercise.


Identify conceptual questioning in the example.
Identify a mathematical practice that was employed in this example.
Exercise: 4 minutes from G5—M3—B—L3
Earlier in this lesson students dialoged with their teacher in a manner similar to this:
T: What is 3 chairs + 2 chairs?
S: 5 chairs.
T: What is 3 dogs + 2 dogs?
S: 5 dogs.
T: What is 3 sevenths + 2 sevenths?
S: 5 sevenths.
T: What is 3 dogs + 2 cats? (Write” 3 dogs + 2 cats”)
S: We can’t add dogs and cats?
T: Why is that? Turn and talk to your partner.
S: (Students share.)
T: I heard Michael tell his partner that children and adults are not the same unit. We would need to
have common units before we add. What do dogs and cats have in common?
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
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NYS COMMON CORE MATHEMATICS PD
Rigor Breakdown: Conceptual Understanding
S: They are animals.
T: Let’s add animals, not dogs and cats. Say the addition sentence with animals.
S: 3 animals + 2 animals = 5 animals.
Problem 2
𝟏 𝟏
+
𝟑 𝟐
In problem 2 you can have students fold a paper again to transition into drawing, or start directly
with drawing. This is a simple problem involving two unit fractions, like problem 1. The primary
purpose is to reinforce understanding of what is happening to the units within a very simple
context. Problem 3 moves on to address a unit fraction plus a non-unit fraction.
T: Do our units get bigger or smaller when we create like units? Talk to your partner.
S: “There are more units.”  “The units get smaller because it is the same amount of space but more
parts.”  “The units are definitely getting smaller. There are more, yes. But, they are smaller.”
T: Let’s draw a diagram to help solve the problem and see if you are right.
T: Did the half become 3 smaller units and each third become 2 smaller units?
S: Yes!
T: Tell me the addition sentence.
S: 2 sixths + 3 sixths = 5 sixths.
1
3
1
2
3
5
+2=6+6=6
Take two minutes now to reflect on the prompts. Then you will have time to discuss your thoughts
with a partner at your table. (Allow 2 minutes for independent reflection.)


Identify conceptual questioning in the example.
Identify a mathematical practice that was employed in this example.
Turn and talk with a partner at your table about your reflections. (Allow 2 minutes for turn and talk,
and then facilitate a discussion.)
(SLIDE 16) Next we have a short video example of conceptual questioning. As you watch this video clip, here are topics
for reflection:


Identify conceptual questioning in the example.
Compare and contrast the impacts and advantages of conceptual questioning with the example of concrete
and pictorial models.
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
7
NYS COMMON CORE MATHEMATICS PD
Rigor Breakdown: Conceptual Understanding
After the video, you will have a chance to share your thoughts with a partner at your table.
Anita Meadows teaches 5th graders at Saint Edward’s School in Vero Beach, Florida. In this video the
students are solving a word problem that asks them to find how much cake is in each share when 10
cakes are divided equally into 4 shares.
(SLIDE 17 – Play Video Clip: 10 Cakes Shared Equally Among 4)
(SLIDE 18) Turn and talk with a partner at your table about your reflections of the video. (Allow 2
minutes for turn and talk, and then facilitate a discussion.)
(SLIDE 19) Let’s review some of the key points of conceptual questioning.



(CLICK TO ADVANCE FIRST BULLET ON SLIDE 19) Conceptual questioning gets students to think well beyond
getting the right answer. So often students get confirmation that their answer is right and all thinking and
learning stops at that point.
(CLICK TO ADVANCE SECOND BULLET ON SLIDE 19) A series of questioning may begin with a yes/no question,
but in all cases, it should require more… possibly asking them to find a non-example, or find how a change in
one quantity would affect the answer.
(CLICK TO ADVANCE THIRD BULLET ON SLIDE 19) Questions should be inserted throughout the lesson insisting
that students push forward to reflect on subtleties, implications, insights, and tools that might be applied to
other problems. Confirmation of an answer can be delayed until after such reflections, or even left open for the
students to decide. When we question students in this way, we, as teachers, get a much clearer picture of the
students’ current level of understanding. Here are some examples.
(CLICK TO ADVANCE FINAL EXAMPLE) The Publishers’ Criteria gives this simple example of conceptual questioning:
“Can you think of a number between 1/4 and 1/5?”
One additional note on questioning students: invariably we will ask a few yes/no questions in our classroom, keep in
mind that such questions can be worth asking if our body language and tone do not give away the desired response.
Your students should get used to having to build confidence and a defense to their positions by teachers questioning
them even when they have given a correct response.
(Note to the presenter: Some teachers may feel uncomfortable with the idea of students proving each other right or
wrong as is suggested in the sample questions. If so, encourage discussion of the benefits of establishing a classroom
environment where students feel comfortable defending their own positions and are not afraid of saying something
that they later decide is wrong. Done well, this environment truly promotes understanding and confidence.)
Conceptual Understanding – Writing and Speaking About Understanding
Time: 0:40-0:50
[10 minutes] In this section, you will…

Engage in Example 3 of developing conceptual understanding through writing and speaking about
understanding
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
8
NYS COMMON CORE MATHEMATICS PD
Rigor Breakdown: Conceptual Understanding

Compare and contrast conceptual understanding activities of the three examples, analyzing the impact and
benefits of each
Materials used include:


Video Clip: Washing Machine Problem
G5—M3—B—L10 Worksheet
(SLIDE 20) The previous activities emphasized the use of conceptual questions to build deep understanding. Now let’s
examine opportunities for students to write and speak about their understandings.
(SLIDE 21) Recall that the Crosswalk of Common Core Instructional Shifts: Mathematics describes deep understanding
in this way:
“Teachers teach more than ‘how to get the answer’ and instead support students’ ability to access concepts
from a number of perspectives so that students are able to see math as more than a set of mnemonics or
discrete procedures. Students demonstrate deep conceptual understanding of core math concepts by applying
them to new situations as well as writing and speaking about their understanding.”
Notice the final phrase in this definition. Think back to the video clip we watched of students learning about the
relationship between fractions and quotients. Can you recall students speaking about their understanding in the clip?
(SLIDE 22) Here is another short clip showing students speaking about their understanding of a problem they are
working on. As you watch the video consider the following question:

How does the speaking about understanding evident in the video compare with what occurs in your
classroom/school/district today?
(SLIDE 23) Here is the problem that the students have been given: 2/5 of the washing machines in a store were sold at
a total price of $3840. If 18 washing machines remained in the store. What was the cost of one washing machine?
The video clip is of Anita Meadows and her students at Saint Edward’s School as they discuss ways to
solve this problem. (SLIDE 24 – Play Video Clip: Washing Machine Problem.)
(SLIDE 25) Take a moment to consolidate your thoughts on the questions posed before the video.
Then you will have an opportunity to discuss your thoughts.
(Allow 2 minutes for independent reflection.)
Turn and talk with a partner at your partner about how the speaking about understanding seen in the
video compares with what occurs in your classroom/school/district today?
(Allow 2 minutes for turn and talk, and then facilitate a discussion.)
(SLIDE 26) Now let’s look for more evidence of how A Story of Units implements the ‘writing and
speaking about their understanding’ piece of conceptual understanding as you complete the worksheet
that accompanies Lesson 10 from this module.
As you work, reflect on the following question:
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
9
NYS COMMON CORE MATHEMATICS PD

Rigor Breakdown: Conceptual Understanding
What evidence of the “writing and speaking about their understanding” requirement did you
see as you completed the worksheet from G5—M3—B—L10.
After the activity, you will have a chance to share your thoughts with a partner at your table.
When you have finished, turn and talk with a partner at your table about where you saw evidence of
students being asked to write and speak about their understanding. (Allow 2 minutes for turn and talk,
and then facilitate a discussion.)
(SLIDE 27) Let’s review some key points about how speaking and writing about understanding can occur in the
classroom:



(CLICK TO ADVANCE FIRST BULLET ON SLIDE 27) Speaking about understanding can occur among students
debating a problem, or with a teacher questioning students individually or as a group.
(CLICK TO ADVANCE SECOND BULLET ON SLIDE 27) Writing about understanding can occur at the board, on
worksheets, on homework, or in student journals.
(CLICK TO ADVANCE 3rd BULLET) Both speaking and writing are valuable ways to consolidate learning and reveal
students’ current level of understanding. Daily exit tickets, provided with each lesson, are designed to give
teachers immediate information about individual students’ understandings on a regular basis so that they can
use that information to tailor daily instruction to meet students’ needs.
(SLIDE 28) We have examined three examples of how conceptual understanding is fostered in A Story of Units:
 Use of concrete and pictorial models
 Conceptual questioning
 Writing and speaking about understanding
Take two minutes now to make notes comparing and contrasting all three approaches, analyzing the
impact and advantages of each. Then you will have an opportunity to discuss your thoughts.
(Allow 2 minutes for independent reflection.)
Turn and talk with a partner at your table about your reflections.
(Allow 2 minutes for turn and talk, and then facilitate a discussion.)
Closing
Time: 0:50-1:00
[10 minutes] In this section, you will…
 Summarize key points
 Reflect on next steps
 Reflect on the role of conceptual understanding in balanced instruction
Materials used include:

How to Implement A Story of Units
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
10
NYS COMMON CORE MATHEMATICS PD
Rigor Breakdown: Conceptual Understanding
(SLIDE 29) Let’s review the key points of this session:



Conceptual understanding can be promoted in a variety of ways including use of concrete and pictorial models,
conceptual questioning, as well as writing and speaking about understanding.
Conceptual understanding represents a major part of the instructional vision that shapes A Story of Units.
Frequent, rich opportunities to help students achieve conceptual understanding are woven throughout A Story
of Units.
These opportunities for conceptual understanding are often also opportunities to nurture the Standards for
Mathematical Practice. The Standards for Mathematical Practice are embedded into all components of rigor
and are not separate from the content instruction.
(SLIDE 30) Take a moment to reflect on these next steps. Then you will have an opportunity to discuss
your thoughts.


How can you increase students’ deep understanding of the concepts you will be covering when you
return to your classroom/school?
How you can help your colleagues implement the conceptual understanding component of rigor
Turn and talk with a partner at your table about how you will to begin to implement these strategies for
promoting conceptual understanding.
(Allow 2 minutes for turn and talk, and then facilitate a discussion.)
(SLIDE 31) Recall that conceptual understanding is but one of three components of rigor. The Instructional Shifts
require that, “Students are practicing and understanding. There is more than a balance between these two things in
the classroom – both are occurring with intensity. Teachers create opportunities for students to participate in ‘drills’
and make use of those skills through extended application of math concepts…”
(SLIDE 32) So we see that the standards call for equal intensity and balance among the three components. On page 5
of the Publishers’ Criteria we are told that, “To help students meet the expectations of the Standards, educators will
need to pursue, with equal intensity, three aspects of rigor in the major work of each grade: conceptual understanding,
procedural skill and fluency, and applications.” On page 9, the Publishers’ Criteria tells us that, in respect to rigor,
“Materials and tools reflect the balances in the Standards…” The variety of strategies we have explored that promote
conceptual understanding is an important part of balanced, rigorous instruction.
(SLIDE 33) The time spent on each component of rigor in daily lessons varies and is guided by the rigor emphasized in
the standard(s) addressed in any given lesson. To clearly communicate the carefully balanced and rigorous instruction
of A Story of Units, the overview of each module includes a Distribution of Instructional Minutes diagram. This is an
example. It shows, at a glance, the time devoted to each component of rigor in the module as a whole, as well as in any
given lesson of the module. Look at Lesson 4 of this example module. Note that this lesson begins with 15 minutes of
fluency work, followed by 20 minutes of concept development and 15 minutes of problem-solving applications. As
usual, the lesson ends with a 10 minute student debrief.
(SLIDE 34) To further promote balanced rigorous instruction, every lesson begins with a pie graph illustrating the
Suggested Lesson Structure. Note that this information is from Lesson 4 of the Distribution of Instructional Minutes
diagram that we just discussed. The actual order of Fluency Practice, Concept Development, Application Problems, and
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
11
NYS COMMON CORE MATHEMATICS PD
Rigor Breakdown: Conceptual Understanding
Student Debrief in each lesson is determined by the content and pedagogy of the lesson. While the time spent on
conceptual development varies among lessons depending on the standards being addressed and the nature of that
particular lesson content, conceptual development is commonly a large portion of the total lesson time. The Student
Debrief usually wraps up the ideas learned for the day and is often a powerful component of the lesson structure. It is a
daily opportunity to promote discussion about concepts.
In closing, let’s look at where this information can be found in the How to Implement A Story of Units document. Turn
to the section explaining The Common Core Approach to Lesson Structure on page 21. Note that information about the
Distribution of Instructional Minutes and the Suggested Lesson Structure is provided.
Turn to page 23 and read the section on Concept Development. (Allow 2 minutes for participants to read
independently, and then facilitate a discussion.)
Turnkey Materials Provided






PowerPoint Presentation
How to Implement A Story of Units
G5—M3—B—L10 Worksheet (provided in G5—M3)
Video Clip: Fractions as Division – Concrete to Pictorial
Video Clip: 10 Cakes Shared Among 4
Video Clip: Washing Machine Problem
Additional Suggested Resources



CCLS – http://www.p12.nysed.gov/ciai/common_core_standards/pdfdocs/nysp12cclsmath.pdf
Instructional Shifts – http://schools.nyc.gov/NR/rdonlyres/9375E046-3913-4AF5-9FE3D21BAE8FEE8D/0/CommonCoreInstructionalShifts_Mathematics.pdf
Publishers’ Criteria – http://www.corestandards.org/assets/Math_Publishers_Criteria_K8_Summer%202012_FINAL.pdf
Session Notes:
Date:
© 2012 Common Core, Inc. All rights reserved. commoncore.org
Rigor Breakdown: Conceptual Understanding
7/13/17
12