Mathematics
Grades 6-12
February NTI
February 4, 2013
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Overview of the Day
1. Standards for Mathematical Practice
2. Progressions Documents - Grades 6-8 & 9-12
3. NYSED Assessment Development
Lunch
4a. LearnZillion – Grades 6-8
4b. PARCC Model Content Frameworks Grades 9-12
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Objectives
Gain a deeper understanding of the eight Standards for
Mathematical Practice, articulate them to others, and
implement the MPs in the classroom alongside
mathematical content standards.
Identify key areas to focus on in the areas of Ratio and
Proportional Relationships ( Grades 6 and 7) and
Functions (Grades 8-12).
Become familiar with the PARCC Model Content
Frameworks and compare similarities and differences
between the PARCC MCF and CCSSM Appendix A.
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“The formulation of the
problem is often more
essential than its
solution, which may be
merely a matter of
mathematical or
experimental skill.”
-Albert Einstein
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Standards for Mathematical Practice
GRADES 6-12
Teri Calabrese-Gray, CVES Assistant Superintendent
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Standards for Mathematical Practice
“Chairs in Hall” – Illustrative Mathematics
Three hallways contained 9,876 chairs
altogether. One-fifth of the chairs were
transferred from the first hall to the second
hall. Then, one-third of the chairs were
transferred from the second hall to the third
hall and the number of chairs in the third
hall doubled. In the end, the number of
chairs in the three halls became the same.
How many chairs were in the second hall to
begin with?
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Standards for Mathematical Practice
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. Designers of curricula, assessments, and
professional development should all attend to the
need to connect the mathematical practices to
mathematical content in mathematics instruction.
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Standards for Mathematical Practice
The Standards for Mathematical Content are a
balanced combination of procedure and
understanding. Expectations that begin with the
word “understand” are often especially good
opportunities to connect the practices to the
content. Students who lack understanding of a
topic may rely on procedures too heavily.
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Standards for Mathematical Practice
Without a flexible base from which to work,
students may be less likely to consider
analogous problems, represent problems
coherently, justify conclusions, apply the
mathematics to practical situations, use
technology mindfully to work with the
mathematics, explain the mathematics accurately
to other students, step back for an overview, or
deviate from a known procedure to find a
shortcut.
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Standards for Mathematical Practice
In this respect, those content standards which set
an expectation of understanding are potential
“points of intersection” between the Standards
for Mathematical Content and the Standards for
Mathematical Practice. These points of
intersection are intended to be weighted toward
central and generative concepts in the school
mathematics curriculum that most merit the time,
resources, innovative energies, and focus
necessary to qualitatively improve the curriculum,
instruction, assessment, professional development,
and student achievement in mathematics.
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Standards for Mathematical Practice
1. Make sense of problems and persevere in solving
them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning
of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
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Standards for Mathematical Practice
ACTIVITY
Using the Standards for Mathematical Practice
(MP) handout on your table and the number you
selected for your table, please read the
corresponding MP. Divide your piece of chart
paper in half and on one side label it Student
Evidence and on the other side label it Teacher
Evidence. Don’t forget to write the number of the
MP you selected on your chart paper as well.
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Standards for Mathematical Practice
ACTIVITY (cont’d)
Brainstorm with members at your table what
students would be doing in the classroom if this
MP was being implemented effectively and list
evidence on your chart paper.
Next, brainstorm with members at your table what
teachers would be doing in the classroom if this
MP was being implemented effectively and list
evidence on your chart paper.
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Standards for Mathematical Practice
ACTIVITY (cont’d)
Locate all those tables that worked on the same
MP as you and come together as one group and
share your work.
Review the evidence and determine if it should
stay on the chart or be deleted. Be prepared to
report out.
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Standards for Mathematical Practice
MP 1:
Make sense of problems
and persevere in solving
them.
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Standards for Mathematical Practice
MP 2:
Reason abstractly and
quantitatively.
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Standards for Mathematical Practice
MP 3:
Construct viable
arguments and critique the
reasoning of others.
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Standards for Mathematical Practice
MP 4:
Model with mathematics.
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Standards for Mathematical Practice
MP 5:
Use appropriate tools
strategically.
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Standards for Mathematical Practice
MP 6:
Attend to precision.
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Standards for Mathematical Practice
MP 7:
Look for and make use of
structure.
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Standards for Mathematical Practice
MP 8:
Look for and express
regularity in repeated
reasoning.
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Standards for Mathematical Practice
PROBLEM
What is the mean of 5, 8, 9, and
6?
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Standards for Mathematical Practice
Identify the MPs that align with this
problem. Discuss at your tables.
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Standards for Mathematical Practice
Pose a different problem that could go
deeper but require the same
mathematical content knowledge?
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Standards for Mathematical Practice
Explain what students might learn from
the second question compared to the
first question?
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Standards for Mathematical Practice
Explain what a teacher might learn
from how students answer the first
question compared to how students
answer the second question?
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Integrating the Standards for
Mathematical Practice
Inside +=x
Mathematics
Watch the video using the Inside
Mathematics link above and collect
evidence from the lesson that exemplifies
the Standards for Mathematical Practice.
Focus on both the students and the teacher.
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Integrating the Standards for
Mathematical Practice
Inside +=x
Mathematics
Compare your evidence with an elbow
partner and then engage in a conversation
with members at your table to come to
consensus as to which MPs you observed.
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Integrating the Standards for
Mathematical Practice
Inside +=x
Mathematics
For a more in-depth study of the Standards
for Mathematical Practice, please visit
http://www.insidemathematics.org/index.
php/commmon-core-math-intro.
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Standards for Mathematical Practice
MP PLACEMAT ACTIVITY
Everyone will need the MP Placemat and the MP
Activity Cards. Read through the MP Activity
Cards on your own. Once you have read the
activity cards, you need to decide where you would
put them on your placemat. You have 16 activity
cards and 16 boxes on your placemat.
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Standards for Mathematical Practice
MP PLACEMAT ACTIVITY (cont’d)
First you will have time to work
independently and then in small groups at
your table. In the end, your table must come
to consensus and a representative needs to
come up and post their results on the master
placemat.
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Common Core State
Standards: Progressions
GRADES 6-12
Kristine S. Cole, SUNY Research Fund Fellow
Teri Calabrese-Gray, CVES Assistant Superintendent
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Common Core State
Standards: Progressions
Let’s look at
Grades 6-7, Ratios and Proportional
Relationships
and how it builds to
Grade 8 , High School, Functions
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Common Core State
Standards: Progressions
K-W-L
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Common Core State
Standards: Progressions
The Common Core State Standards in
mathematics were built on progressions:
narrative documents describing the
progression of a topic across a number of
grade levels, informed both by research on
children's cognitive development and by the
logical structure of mathematics. These
documents were spliced together and then
sliced into grade level standards.
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Common Core State
Standards: Progressions
This would be useful in teacher preparation and
professional development, organizing
curriculum, and writing textbooks. Progressions
documents also provide a transmission
mechanism between mathematics education
research and standards. Research about learning
progressions produces knowledge which can be
transmitted through the progressions document to
the standards revision process; questions and
demands on standards writing can be transmitted
back the other way into research questions.
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Common Core State
Standards: Progressions
From that point on the work focused on refining
and revising the grade level standards. The early
drafts of the progressions documents no longer
correspond to the current state of the standards.
The progressions can explain why standards are
sequenced the way they are, point out
cognitive difficulties and pedagogical
solutions, and give more detail on particularly
knotty areas of the mathematics.
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Common Core State
Standards: Progressions
Grades 6-7 – Ratios and Proportional
Relationships
Grade 8 and High School – Functions
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Common Core State
Standards: Progressions
Grades 6-7
Ratios and Proportional
Relationships
•
•
•
Overview – Pages 2-4
Grade 6 – Pages 5-7
Grade 7 – Pages 8-12
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Common Core State
Standards: Progressions
Grade 8 and High School: Functions
•
•
Overview and Grade 8 – Pages 2-6
• Battery Charging – Page 5
High School – Interpreting Functions –
Pages 7-10
• Interpreting the Graph – Page 7
• Cell Phones – Page 8
• Warming and Cooling – Page 9
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Common Core State
Standards: Progressions
Grade 8 and High School Functions
• High School – Building Functions – Pages
11-13 (stop before advanced standards)
• Lake Algae – Page 11
• Transforming Functions – Page 12
• High School – Linear and Exponential
Models – Pages 15-16
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Common Core State
Standards: Progressions
ACTIVITY
Each group is responsible for their
respective sections of the Progressions
documents. Once your table has read
their section of the document and
engaged in an initial discussion, locate
similar tables who read the same
section.
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Common Core State
Standards: Progressions
ACTIVITY (cont’d)
The goal of the activity is to use the
collective knowledge of your group to
develop a unique way to present your
material to the entire group. Once
everyone has completed their work,
they will be asked to report out.
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Common Core State
Standards: Progressions
Grades 6-7
Ratios and Proportional
Relationships
•
•
•
Grade 6
Grade 7
Overview
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Common Core State
Standards: Progressions
Grade 8 and High School: Functions
•
•
•
•
•
Grade 8
High School – Interpreting Functions
High School – Building Functions
High School – Linear and Exponential
Models
Overview
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Common Core State
Standards: Progressions
Influenza Epidemic
http://www.illustrativemathematics.org/illust
rations/637
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Common Core State
Standards: Progressions
An epidemic of influenza spreads through a city.
The figure below is the graph of I=f(w) , where I is
the number of individuals (in thousands) infected
w weeks after the epidemic begins.
(Task from Functions Modeling
Change: A Preparation for Calculus,
Connally et al., Wiley 2010.)
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Common Core State
Standards: Progressions
1. Estimate f(2) and explain its meaning in
terms of the epidemic.
2. Approximately how many people were
infected at the height of the epidemic? When
did that occur? Write your answer in the form
f(a)=b .
(Task from Functions Modeling Change: A Preparation for Calculus, Connally et al., Wiley 2010.)
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Common Core State
Standards: Progressions
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Common Core State
Standards: Progressions
K-W-L
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PARCC Model Content
Frameworks
Mathematics
GRADES 3–11
Version 3.0
November 2012
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NYS Next Generation Assessments
2012-13 NYS Grades 3-8 Math and
ELA assessments are built on the
Common Core within the constraints of
the NYS testing system
Regents ELA and Math Examinations
will be rolled out in 2013-14 and 201415
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NYS Next Generation Assessments
All curricular and professional
development resources produced by
the State Education Department will
follow the PARCC MCF, as will State
assessments beginning with the 201314 school year.
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PARCC Model Content Frameworks
Grade 8
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PARCC Model Content Frameworks
Algebra I
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PARCC Model Content Frameworks
Algebra II
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PARCC Model Content Frameworks
Why were the PARCC Model
Content
Frameworks (MCF) developed?
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PARCC Model Content Frameworks
Dual Purpose
Although the primary purpose of the Model
Content Frameworks is to provide a frame
for the PARCC assessments, they also are
voluntary resources to help educators and
those developing curricula and
instructional materials.
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PARCC Model Content Frameworks
The Model Content Frameworks for
Mathematics for each grade is written with
the expectation that students develop
content knowledge, conceptual
understanding and expertise with the
Standards for Mathematical Practice. A
detailed description of all features of the
standards would be significantly lengthier
and denser.
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PARCC Model Content Frameworks
The Model Content Frameworks for Mathematics
provide guidance for grades 3-8 and high school in
the following areas:
• Examples of key advances from the previous
grade;
• Fluency expectations or examples of culminating
standards;
• Examples of major within-grade dependencies;
• Examples of opportunities for connections among
standards, clusters or domains;
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PARCC Model Content Frameworks
The Model Content Frameworks for Mathematics
provide guidance for grades 3-8 and high school in
the following areas:
• Examples of opportunities for in-depth focus;
• Examples of opportunities for connecting
mathematical content and mathematical practices;
and
• Content emphases by cluster.
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Key Elements of the
PARCC Model Content Framework
Key
Advances
Fluency
Recommendations
High
School
Course
Individual
End of Course
Overviews
Pathway
Summary
Tables
Assessment
Limits Tables for
Standards
Assessed in
More than One
Course
Mathematical Practices in Relation to Course
Content
Illinois State Board of Education
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PARCC Model Content Frameworks
Individual End-of-Course Overviews
Each overview shows which standards are
assessed on a given end-of-course
assessment as well as relative cluster
emphases.
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PARCC Model Content Frameworks
Individual End-of-Course Overviews
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PARCC Model Content Frameworks
Key Advances
This category highlights some of the major
steps in the progression of increasing
knowledge and skill from year to year.
Note that each key advance in mathematical
content also corresponds to a widening scope
of problems that students can solve.
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PARCC Model Content Frameworks
Math Practices in Relation to Course
Content
This category highlights some of the
mathematical practices and describes how
they play a role in each course. These
examples are provided to stress the need to
connect content and practices, as required by
the standards.
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PARCC Model Content Frameworks
Math Practices in Relation to Course
Content
Modeling with mathematics is a theme in all
high school courses. Modeling problems in
high school center on problems arising in
everyday life, society, and the workplace.
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PARCC Model Content Frameworks
Fluency Recommendations
The high school standards do NOT set
explicit expectations for fluency nor will the
PARCC assessments address fluency, but
fluency is important in high school
mathematics. For example, fluency in algebra
can help students get past the need to
manage computational details so that they
can observe structure and patterns in
problems.
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PARCC Model Content Frameworks
Fluency Recommendations
This section makes recommendations about
fluencies that can serve students well as they
learn and apply mathematics. These fluencies
are highlighted to stress the need for curricula
to provide sufficient supports and
opportunities for practice to help students
gain fluency.
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PARCC Model Content Frameworks
Pathway Summary Tables
Each pathway summary table shows three
end-of-course assessments’ standards at a
glance. For each non-(+) high school
standard, the end-of-course assessment(s)
assessing the standard are shown by a dot
( ) symbol. Shading in the pathway summary
table indicates high school standards that are
appropriate for more than one end-of-course
assessment.
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PARCC Model Content Frameworks
Pathway Summary Table
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PARCC Model Content Frameworks
Assessment Limits Table for Standards
Assessed on More than one End-of-Course Test
When a high school standard is appropriate
for more than one end-of-course test in a
given pathway, the need arises to specify just
how the assessment of the standard will differ
for students in each successive course. This
information is provided in the assessment
limits tables.
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PARCC Model Content Frameworks
Assessment Limits Table for Standards
Assessed on More than one End-of-Course Test
In general, the approach to striking this
balance has been to set stricter limits on
standards relating to procedural skill and to
set less strict limits on standards relating to
conceptual understanding and exploration.
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PARCC Model Content Frameworks
Assessment Limits Table for Standards
Assessed on More than one End-of-Course Test
CCSSM Cluster CCSSM Key CCSSM Standard
Interpret the
structure of
expressions
A-SSE.2
Use the structure of an
expression to identify ways to
rewrite it. For example, see x4 –
y4 as (x2)2 – (y2)2, thus
recognizing it as a difference of
squares that can be factored as
(x2 – y2)(x2 + y2).
Algebra I Assessment Limits and
Clarifications
Algebra II Assessment Limits and
Clarifications
i) Tasks are limited to numerical
expressions and polynomial
expressions in one variable.
i) Tasks are limited to polynomial,
rational, or exponential expressions.
ii) Examples: see x4 – y4 as (x2)2 –
ii) Examples: Recognize 532 - 472 as (y2)2, thus recognizing it as a
a difference of squares and see an difference of squares that can be
factored as (x2 – y2)(x2 + y2). In the
opportunity to rewrite it in the
2
2
easier-to-evaluate form (53+47)(53- equation x + 2x + 1 + y = 9, see an
opportunity to rewrite the first
47). See an opportunity to rewrite
three terms as (x+1)2, thus
2
a + 9a + 14 as (a+7)(a+2).
recognizing the equation of a circle
with radius 3 and center (-1,0). See
(x2 + 4)/(x2 + 3) as ( (x2+3) + 1
)/(x2+3), thus recognizing an
opportunity to write it as 1 + 1/(x2 +
3).
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PARCC Model Content Frameworks
ACTIVITY
You have been asked to provide an overview
of the PARCC Model Content Frameworks
document to a community partner /
organization (e.g., workforce development
board, PTA, Chamber of Commerce, Kiwanis,
college faculty)
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PARCC Model Content Frameworks
ACTIVITY (cont’d)
Work with team members to decide how you
will deliver your message. This activity is
meant to be open-ended, except each group
must clearly provide several talking points
based on your particular course.
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PARCC Model Content Frameworks
The Model Content Frameworks do NOT
contain a suggested scope and sequence by
quarter. Rather, they provide examples of
key content dependencies (where one
concept ought to come before another), key
instructional emphases, opportunities for
in-depth work on key concepts, and
connections to critical practices.
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The PARCC MCF
Connection to Assessments
Algebra I focuses on linear, quadratic,
and exponential functions with domain in
the integers. It also suggests work with
the piecewise functions (including step
and absolute value), square root and
cube root in several standards.
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The PARCC MCF
Connection to Assessments
The PARCC MCF does state, “In Algebra
I, students will master linear and
quadratic functions.” This implies that
despite the exposure to some of the more
advanced functions, the majority of the
time should be spent on linear and
quadratic.
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CCSSM Appendix A
The Common Core State Standards
Appendix A is posted online at:
http://www.corestandards.org/assets/CC
SSI_Mathematics_Appendix_A.pdf
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CCSSM Appendix A
The pathways and courses are models. They
provide possible approaches to organizing the
content of the CCSSM into coherent and rigorous
courses.
States and districts are not expected to adopt
these courses as is; rather, they are encouraged to
use these pathways and courses as a starting
point for developing their own.
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CCSSM Appendix A
Units within each course are intended to suggest a
possible grouping of the standards into coherent
blocks.
The ordering of the clusters within a unit follows
the order of the standards document in most
cases, not the order in which they might be taught.
Attention to ordering content within a unit will be
needed as instructional programs are developed.
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PARCC Model Content Frameworks
ACTIVITY
Using the same groups from the last activity,
you are going to compare two documents, the
PARCC MCF to the CCSSM Appendix A for
each of the three high school courses. Again
we are going to focus on the traditional
pathway.
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PARCC Model Content Frameworks
ACTIVITY (cont’d
Examine the two documents for similarities
and differences for each high school
mathematics course. Once your team has
completed the analysis, locate others who
reviewed the same course and compile your
list of similarities and differences and record
electronically.
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PARCC Model Content Frameworks
Key Question
“What information needs to be
shared with your mathematics
department(s) and the larger school
community?”
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High School Illustrative Sample Item
http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation
Seeing Structure in a Quadratic Equation
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High School Illustrative Sample Item
http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation
High School Sample Illustrative Item: Seeing Structure
in a Quadratic Equation
1. Using the Overview of Task Type Slide, identify the
Type of Task the high school illustrative sample item
represents.
2. Identify the most relevant Content Standard(s) this
problems aligns to? Explain your reasoning.
3. Identify the most relevant Standards for Mathematical
Practice(s) does this align to? Explain your reasoning.
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High School Illustrative Sample Item
http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation
High School Sample Illustrative Item: Seeing Structure
in a Quadratic Equation
Task Type I: Tasks assessing concepts, skills and
procedures
Alignment: Most Relevant Content Standard(s)
• A-REI.4. Solve quadratic equations in one variable.
a) Use the method of completing the square to
transform any quadratic equation in x into an
equation of the form (x – p)2 = q that has the same
solutions. Derive the quadratic formula from this form.
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High School Illustrative Sample Item
http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation
High School Sample Illustrative Item: Seeing Structure
in a Quadratic Equation
Alignment: Most Relevant Content Standard(s)
• A-REI.4. Solve quadratic equations in one variable.
b) Solve quadratic equations by inspection (e.g., for x2 =
49), taking square roots, completing the square, the
quadratic formula, and factoring, as appropriate to
the initial form of the equation. Recognize when the
quadratic formula gives complex solutions and write
them as a  bi for real numbers a and b.
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High School Illustrative Sample Item
http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation
High School Sample Illustrative Item: Seeing Structure
in a Quadratic Equation
Alignment: Most Relevant Mathematical Practice(s)
• Students taking a brute-force approach to this task will
need considerable symbolic fluency to obtain the
solutions. In this sense, the task rewards looking for and
making use of structure (MP.7).
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Additional Resources
The Mathematics Common Core Toolbox
The Charles A. Dana Center at the University of
Texas at Austin and Agile Mind, Inc.
This site is a resource designed to support
districts working to meet the challenge and the
opportunity of the new standards. Here you will
find tools and instructional materials that help you
to better understand and to implement the
CCSSM.
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Additional Resources
The Mathematics Common Core Toolbox
Key Visualizations
• Algebra I
• Geometry
• Algebra II
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Additional Resources
The Mathematics Common Core Toolbox
PARCC Prototyping Project
High School Tasks
• Cellular growth
• Golf balls in water
• Isabella’s credit card
• Rabbit populations
• Transforming graphs of quadratic functions
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ASCD Educational Leadership
Teaching Like a Four-Star Chef
by Carol Ann Tomlinson
Don’t confuse the ingredients with the dinner.
You need to design the recipe – you may decide to
use the ingredients in a different way than
someone else.
Outstanding teachers have “developed the art of
making elegant dinners that incorporate, but are
not limited to, prescribed ingredients.”
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Thank You!
Don’t forget to complete your +s and s
for the day. Tomorrow we will be
focusing on Modeling and Dr. Eric
Robinson will join us to share his
expertise and the work he is doing at
the national level.
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