The Number System: Crafting Teaching Sequences for
Extended Interventions for Grades 6-9
Sequence of Sessions
Overarching Objectives of this October 2014 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


Participants will experience the cycle of plan → teach → assess → analyze and will practice a planning protocol to support the development of extended
interventions for students in grades 6-9.
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 series exploring coherence across the grades and the development of short- and long-term interventions.
Module Focus sessions are also support implementation of the curriculum by closely examining each module in A Story of Ratios and
A Story of Functions.
Key Points



Math content is best taught as a coherent story from Kindergarten to Grade 12. This coherence is essential to both student success and the advancement
of the craft of teaching.
A teacher’s pedagogical content knowledge of the grade levels preceding and following his or her own impacts student success daily and is the primary
engine necessary to meet the needs of all students.
The cycle of plan → teach → assess → analyze is critical for development of successful interventions of all types (instant, short-term, and long-term).
Session Outcomes
What do we want participants to be able to do as a result of this
session?


Participants will experience the cycle of plan → teach → assess → analyze
and will practice a planning protocol to support the development of
extended interventions for students in grades 6-9.
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 and demonstrate
the ability to develop an extended teaching sequence for intervention.
Session Overview
Section
Time
Overview
Prepared Resources
●
Introduction
29 min
Introduces the concepts behind
teaching sequences and
interventions.
●
●
Short-term
Interventions
75 min
Explores ways to identify a need
for and craft short-term
interventions.
●
●
●
Extended
Interventions
100 min
Explores ways to identify a need
for and craft extended
interventions.
●
●
Crafting Extended
Teaching Sequences
58 min
Explores crafting an extended
teaching sequence.
●
Grades 6-9, The Number
System: Crafting Teaching
Sequences for Extended
Interventions PPT
Facilitators Guide
Grades 6-9, The Number
System: Crafting Teaching
Sequences for Extended
Interventions PPT
Facilitators Guide
Grade 8 Module 7 MidModule Assessment
Grades 6-9, The Number
System: Crafting Teaching
Sequences for Extended
Interventions PPT
Facilitators Guide
Grades 6-9, The Number
System: Crafting Teaching
Sequences for Extended
Interventions PPT
Facilitators Guide
Facilitator Preparation
●
●
Review facilitator guide
and curriculum overview
documents for A Story of
Ratios and A Story of
Functions.
Review and complete
Grade 8 Module 7 MidModule Assessment
Session Roadmap
Section: Introduction
Time: 29 minutes
In this section, you will be introduced to the concepts behind
teaching sequences and interventions.
Materials used include:
 Grades 6-9, The Number System: Crafting Teaching
Sequences for Extended Interventions PPT
 Grades 6-9, The Number System: Crafting Teaching
Sequences for Extended Interventions Facilitators Guide
Time Slide # Slide #/ Pic of Slide
3 min
Script/ Activity directions
1.
2.
One of the earliest estimates of pi can be found in the ancient Egyptian
document, The Rhind Mathematical Papyrus. Dated at around 1650 BC, this
document gave a rule for computing the area of a circular field.
For example, to find the area of a round field having diameter 9 khet, it
suggests:
Take away 1/9 of the diameter, in this case 1; the remainder is 8. Multiply 8
times 8; it makes 64. Therefore it contains 64 setat of land.
What value of pi is implied by this procedure?
GROUP
10 min
3.
Belief #1: Math content is best taught as a coherent story from Kindergarten
to Grade 12. This coherence is essential to both student success and the
advancement of the craft of teaching.
Consider a vision of elementary, middle and high schools joining hands to
provide a coherent educational experience from Pre-Kindergarten or
Kindergarten through Grade 12. The EngageNY math curriculum tells a
story, a trilogy, from A Story of Units (PK-5) to A Story of Ratios (6-8) to A
Story of Functions (9-12). The structure of this curriculum reflects a
commitment to a holistic journey, whereby skills build upon skills, themes
are interwoven and understandings develop out of previous
understandings. Because the curriculum spans from PK – 12, educators
move students through the grade levels in such a way that each teacher’s
contribution stems from what came before, and leads into the work of those
that follow. Teaching must be a collaborative, team endeavor. This is in the
best interest of the student and ultimately in the best interest of the teacher
as well.
Belief #2: A teacher’s pedagogical content knowledge of the grade levels
preceding and following his or her own impacts student success daily and is
the primary engine necessary to meet the needs of all students.
With a commitment to a coherent PK-12 experience, attention turns to PK12 schools, not necessarily in one physical location but committed to, for
example, a Grade 5 student graduating to partnered middle and high
schools. A master teacher is one who is very conscious of his/her role in the
continuum of a student’s journey, who has an ever-expanding knowledge of
pedagogical content knowledge beyond that of his/her own grade level.
That knowledge is the engine behind the responsive questioning and
delivery of instruction that can rectify a misunderstanding in a minute
rather than in an hour, in a simple demonstration rather than with remedial
workbooks or alternate lessons.
As participants engage in the protocol, circulate and listen. This might be a
very new paradigm, to truly think of teaching as a collaborative endeavor.
You might pepper conversations with questions such as:
• In your own experience learning math, did you experience
connnectedness and continuity?
• How do you want your students’ experience to be different than your
•
•
8 min
4.
own?
Do you think it important that an elementary curriculum transition
smoothly to middle school?
What has given the teaching of mathematics coherence over the
years?
Intuitive thought is the initial analysis, from that teachers plan their lesson.
The process occurs at the micro level, several times within a given lesson,
teachers begin to teach, ask a question or given a problem to assess
understanding and instantly analyze and adjust their plans going forward
accordingly.
What we’ll be exploring today, is the analysis that happens primarily on the
larger scale, perhaps after an exit ticket or mid or end of module assessment
is given.
5 min
5.
Instant – a single question, doesn’t detract from the lesson progress,
teachers review content only as far back as is needed, and as efficiently as
possible.
Short-term – note taken, I’m going to go home and plan out the actual
sequence of questions, and I’ll do it as fluency or 10 min segment tomorrow
Extended – I’ve found that the problem is deeper than a gap in this grade’s
knowledge, I’m going to go home and plan 4 to 10, 20-min chunks of
instruction
In each case, the teacher starts by analyzing, ‘What is the error?’, ‘Is the
source of the error clearly evident?’ ‘What questions could I ask to reveal
with certainty the source or cause of the error?’
3 min
6.
Section: Short-term Interventions
Review the process as outlined.
Time: 75 minutes
In this section, you will explore ways to identify a need for and craft Materials used include:
short-term interventions.
 Grades 6-9, The Number System: Crafting Teaching
Sequences for Extended Interventions PPT
 Grades 6-9, The Number System: Crafting Teaching
Sequences for Extended Interventions Facilitators Guide
 Grade 8 Module 7 Mid-Module Assessment
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
3 min
Participants will complete G8M7 MMA #1a and predict the challenges they
think students will face with this problem.
7.
GROUP
12 min
8.
8 minutes
Students will analyze the question using the MP protocol.
4 minutes
Share our analysis.
20 min
9.
Participants create a short term sequence of questions that identify with
clarity the source of the error in the original work, including questions that
reveal that the error cannot be addressed without a more extended
intervention.
20 min
10.
Participants review the sequence of the student outcomes of relevant
lessons from this module and reflect on / adjust their sequence of questions.
15 min
11.
Participants plan a 10-min intervention (something that would open up or
be a part of the next day’s lesson).
5 min
12.
Participants write new assessment questions modeled after the question(s)
for which the student error was seen.
Section: Extended Intervention
Time: 100 minutes
In this section, you will explore ways to identify a need for and craft Materials used include:
extended interventions.
 Grades 6-9, The Number System: Crafting Teaching
Sequences for Extended Interventions PPT
 Grades 6-9, The Number System: Crafting Teaching
Sequences for Extended Interventions Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
GROUP
10 min
13.
Share our extended intervention with poster.
20 min
14.
Explain the game.
The object is to find as many rows of 5 as possible, vertical, horizontal or
diagonal.
Let’s do a practice word.
• Draw their attention to the top left square. Invite everyone to search
for the words “multiplicative comparison” in G4 M3. (note the use of
quotation marks!!!)
• Guide the participants as necessary to use the PDF search feature on
their computer.
• Have them scan through the entire module to notice where it is used.
• Once they have scanned, they quickly write an example seen within
the module into the Bingo board box.
• Invite them to share: “Even just by scanning, what did you notice
about “multiplicative comparison” in G4 M3?”
• Let the participants know that we have chosen these terms carefully,
to be most helpful in their coming study.
1. When the clock starts, participants search the A Story of Units’ PDFs
in the grade levels designated for the vocabulary indicated. (Use the
curricular map and PDF search to locate terms. Be careful to search
with these precise terms! Use quotation marks.)
2. Once a term is located, quickly scan through the ENTIRE module to
get a feel for its frequency and the context in which it is used.
3. When time is called invite a team that has 5 (or 4) in a row to briefly
go through their terms and share what they noticed with each.
15 min
15.
Participants work in grade-level triads, and analyze student work from a
sample of their own grade level.
20 min
16.
Participants create a sequence of questions designed to assess the last point
of strength, they are careful to include questions that go back beyond this
grade level.
20 min
17.
Participants review the sequence of student outcomes related to the given
problem from the module it was taught in.
15 min
18.
Participants navigate through other modules and grade levels to analyze the
sequence of student outcomes that build to the current module’s outcomes.
Section: Crafting an Extended Teaching Sequence
Time: 58 minutes
In this section, you will explore crafting an extended teaching
sequence.
Materials used include:
 Grades 6-9, The Number System: Crafting Teaching
Sequences for Extended Interventions PPT
 Grades 6-9, The Number System: Crafting Teaching
Sequences for Extended Interventions Facilitators Guide
Time Slide # Slide #/ Pic of Slide
Script/ Activity directions
30 min
Participants craft a series of four to 10 lessons, each about 20 min in length,
that could be delivered over a couple of weeks as an intervention for gaps in
student knowledge.
19.
GROUP
15 min
20.
Participants share and critique their extended teaching sequences with
other triads at their same grade level.
Participants make a poster outlining their extended teaching sequence.
Participants share their posters with triads from other grade levels.
3 min
21.
Participants create second chance assessments related to the student error
that prompted their extended teaching sequence.
10 min
22.
Participants engage in professional reading as given in the handout.
Next, participants talk first in small groups and then share with the larger
group.
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
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Grades 6-9, The Number System: Crafting Teaching Sequences for Extended Interventions PPT
Grades 6-9, The Number System: Crafting Teaching Sequences for Extended Interventions Facilitators Guide
Grade 8 Module 7 Mid-Module Assessment
Additional Suggested Resources
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A Story of Ratios Year Long Curriculum Overview
A Story of Ratios CCLS Checklist
A Story of Functions Year Long Curriculum Overview
A Story of Functions CCLS Checklist