Download math cccs - Chester School District

Common Core State
Standards in Mathematics
Overview Grades 6-8
Seth Korman
Middle School Supervisor of
Curriculum and Instruction
Mathematics and Science
Wayne Township Public Schools
[email protected]
Foxtrot
“Math Class needs a Makeover”
with Dan Meyer
http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html
Common Core State Standards
 Fewer, clearer, and higher, to best drive effective policy
and practice;
 Aligned with college and work expectations, so that all
students are prepared for success upon graduating from high
school;
 Inclusive of rigorous content and application of knowledge
through high-order skills, so that all students are prepared for
the 21st century;
 Internationally benchmarked, so that all students are
prepared for succeeding in our global economy and society; and
 Research and evidence-based.
Common Core State Standards for
Mathematics
Two types of Standards:
 Mathematical Practice
o Recurring throughout the grades
o Describes habits of mind for the mathematics student
o Integrated into instruction
 Mathematical Content: different at each grade level
o K-8 standards presented by grade level
o Organized into domains that progress over several grade levels
o Grade introduction give 2-4 focal points per grade level
Standards for
Mathematical Practice
 Make sense of problems and persevere in solving them
 Reason abstractly and quantitatively
 Construct viable arguments and critique the reasoning of
others
 Model with mathematics
 Use appropriate tools strategically
 Attend to precision
 Look for and make use of structure
 Look for and express regularity in repeated reasoning
http://www.corestandards.org/Math/Practice
1. Make sense of problems and
persevere in solving them
6. Attend to precision
Problem solving with precision
Grouping the practice standards
2. Reason abstractly and
quantitatively
3. Construct viable
arguments and critique
the reasoning of others
Reasoning and Explaining
4. Model with mathematics
5. Use appropriate tools
strategically
Modeling and using tools
7. Look for and make use
of structure
8. Look for and express
regularity in repeated
reasoning
Seeing structure and
generalizing
Math Standards Key Shifts
Previous State Math Standards
Many topics in each grade
Common Core State Standards
Focus on 2-3 concepts per grade
Topics somewhat arbitrarily placed in different Coherent progression of topics from one grade
grades
to the next
Emphasis on basic facts
Fluency with basic facts coupled with deeper
understanding of why things happen
Focus on one way to get the right answer
Examining multiple ways to represent
mathematical concepts and solutions
Isolated learning
Emphasis on real-world application
Overview of K-8 Standards
The K- 8 standards:
The K-5 standards provide students with a solid
foundation in whole numbers, addition, subtraction,
multiplication, division, fractions and decimals
The 6-8 standards describe robust learning in
geometry, algebra, and probability and statistics
Modeled after the focus of standards from highperforming nations, the standards for grades 7 and
8 include significant algebra and geometry content
Students who have completed 7th grade and
mastered the content and skills will be prepared for
algebra, in 8th grade or after
http://www.achieve.org/files/CCSSJune22010FINAL.ppt
Overview of CCSS in NJ
Grade 1
Grade2
Grade3
Grade4
Grade5
Grade6
Grade7
Grade8
C
19
17
22
20
11
28
22
18
N
1
5
0
2
2
1
2
0
+
1
4
5
5
11
14
10
4
-
0
0
1
0
1
0
1
2
C: Currently required/very similar to required content in 2008 NJCCCS
N: New content, not previously required
+: New requirement, previously at a higher grade level
-: New requirement, previously at a lower grade level
www.nj.gov/education/aps/cccs/math/CCSSguidance.pdf
Design and Organization of CCSS
Design and Organization of CCSS
Focal points at each grade level
www.corestandards.org/Math/Content/6/introduction
CCSS Domains Across Grades 6-8
DOMAIN/GRADE
6
7
Ratios and
Proportional
Relationships
X
X
The Number
System
X
X
X
Expressions and
Equations
X
X
X
Geometry
X
X
X
Statistics and
Probability
X
X
X
Functions
8
X
Domain
 Larger groups of related standards
 Overarching “big ideas” that connect topics across
the grades
 Descriptions of the mathematical content to be
learned, elaborated through clusters and standards
 The only domain that appears from K-8 is Geometry
Clusters
o May appear in multiple grade levels with increasing
developmental standards as the grade levels
progress
o Indicate WHAT students should know and be able
to do at each grade level
o Reflect both mathematical understandings and
skills, which are equally important
o Groups of related Standards
Standards
Content statements
Progressions of increasing complexity from grade
to grade
What students understand and should be able to
do. Related standards will increase in complexity
over the years
Major Shifts within the Mathematics
CCSS
1. Focus: Focus strongly where the standards focus
2. Coherence: Think across the grade levels, and link to
major topics
3. Rigor: In major topics, pursue conceptual
understanding, procedural skill fluency, and application
Focus Strongly where the Standards
Focus
 Significantly narrow the scope of content and deepen
how time and energy is spent in the math classroom
 Focus deeply on what is emphasized in the standards,
so that students gain strong foundations
Focus
 Move away from “mile wide, inch deep” curricula
 Teach less, learn more
“Less topic coverage can be associated with higher scores on those topics covered because
students have more time to master the content that is taught.”
-Ginsburg et al., 2005
http://engageny.org/resource/common-core-in-mathematics-shift-1-focus
Priorities or Key Areas to Focus in
Mathematics
Grade
Priorities in Support of Rich Instruction and
Expectations of Fluency and Conceptual
Understanding
K–2
Addition and subtraction, measurement
using whole number quantities
3–5
Multiplication and division of whole
numbers and fractions
6
7
8
Ratios and proportional reasoning; early
expressions and equations
Ratios and proportional reasoning;
arithmetic of rational numbers
Linear algebra
Coherence: Think Across Grades, and
Link to Major Topics Within Grades
 Carefully connect the learning within and across
grades so that students can build new understanding
on foundations built in previous years
 Begin to count on solid conceptual understanding of
core content and build on it. Each standard is not a
new event, but an extension of previous learning.
CCSS Progression
CCSS Progression
progression
Rigor: In Major Topics, Purse Conceptual
Understanding, Procedural Skill and
Fluency, and Application
 The CCSS requires a balance of the following:
o Solid conceptual understanding
o Procedural skill and fluency
o Application of skills in problem solving situations
 Pursuit of all of the above requires equal intensity in
time, activities, and resources
Conceptual Understanding
Teach more than “how to get the answer” and
instead support students’ ability to access concepts
from a number of perspectives
Students are able to see math as more than a set of
mnemonics or discrete procedures
Conceptual understanding supports the other aspects
of rigor (fluency and application)
Fluency
The standards require speed and accuracy in
calculation
Teachers structure class time and/or homework time
for students to practice core functions such as singledigit multiplication so that they are more able to
understand and manipulate more complex concepts
Required Fluencies in K-8
Grade Level
Required Fluency
K
Add/subtract within 5
1
Add/subtract within 10
2
Add/subtract within 20 (know single-digit sums from memory)
Add/subtract within 100
3
Multiply/divide within 100 (know single-digit products from memory)
Add/subtract within 1000
4
Add/subtract within 1,000,000
5
Multi-digit multiplication
6
Multi-digit division
Multi-digit decimal operations
7
Solve px + q = r, p(x + q) = r
8
Solve simple 2 x 2 systems by inspection
Intentional Design Limitations of
CCSS
What the standards do NOT define:
• How teachers should teach
• All that can or should be taught
• The interventions needed for students below grade
level
• The full range of support for ELL and students with
special needs
• Everything needed to be college and career ready
Application
 Students can use appropriate concepts and procedures for
application even when not prompted to do so
 Teachers provide opportunities at all grade levels for
students to apply math concepts in “real world” situations
 Work with grade level colleagues in other disciplines such
as science, ensure that students are using grade-levelappropriate math to make meaning of and access science
content
Grade 6 Content & Pacing
Grade 6 Pacing Guide
Unit Title
1. Using Expressions and Equations
2. Operating with Positive Rational
Numbers
3. Understanding Positive & Negative
Numbers
4. Applications of Geometry
Pacing
4 weeks
4 weeks
2 weeks
3 weeks
5. Ratios and Rates
3 weeks
6. Algebraic Reasoning
5 weeks
7. Statistics and Distribution
5 weeks
Standards
6.EE.1
6.EE.2
6.EE.3
6.EE.4
6.NS.1
6.NS.2
6.NS.3
6.NS.4
6.G.2
6.NS.5
6.NS.6
6.NS.7
6.NS.8
6.G.1
6.G.3
6.G.4
6.RP.1
6.RP.2
6.RP.3
6.EE.6
6.EE.7
6.EE.9
6.SP.1
6.SP.2
6.SP.3
6.SP.4
6.SP.5
6.E E.5
6.EE.6
6.EE.7
6.E E.8
Grade 6: EE
Grade 6: EE
Grade 7 Content & Pacing
Unit Title
1. Operating with Rational Numbers (add/sub)
2. Operating with Rational Numbers (mult/div)
Pacing
Standards
4 weeks
7.NS.1
7.NS.3
3 weeks
7.NS.2
7.NS.3
7.EE.2
7.EE.3
7.G.2
7.G.3
7.G.4
7.G.5
7.G.6
7.RP.1
7.RP.2
7.RP.3
7.G.1
3. Two and Three Dimensional Geometry
4 weeks
4. Proportional Relationships
5 weeks
5. Algebraic Reasoning II
4 weeks
7.EE.1
7.EE.2
7.EE.4
3 weeks
7.SP.1
7.SP.2
7.SP.3
7.SP.4
3 weeks
7.SP.5
7.SP.6
7.SP.7
7.SP.8
6. Inferences about Populations
7. Probability
Grade 7: EE
Grade 7: Number
Grade 7: Geometry
Grade 8 Content & Pacing
Unit Title
Pacing
Standards
8.NS.1
8.NS.2
8.EE.1, 2, 3, 4
8.G.6 8.G.8
8.G.7 8.EE.2
8.G.1 8.G.5
8.G.2 8.G.3
8.G.4
8.EE.5 8.F.2
8.EE.6 8.F.3
8.EE.7 8.F.4
8.F.1
8.F.5
1. Real Numbers
3 weeks
2. Pythagorean Theorem
4 weeks
3. Congruence and Similarity
4 weeks
4. Linear Relationships
5 weeks
5. Systems of Linear Relationships
3 weeks
8.EE.7 8.F.2
8.EE.8 8.F.4
6. Volume
2 weeks
8.G.9
3 weeks
8.SP.1 8.SP.3
8.SP.2 8.SP.4
7. Patterns in Data
Grade 8: Geo
Grade 8: Geo
PARCC
Beyond creating assessments, PARCC wants to create:
Content frameworks to guide educators
Model instructional units anchored around the PARCC
Assessment Component
Sample Assessment Tasks that mirror tasks on PARCC
Professional Development modules
PARCC, “The Partnership for Assessment of Readiness for College and Careers”, April
2011, 15.
Content Emphasis by Cluster Grade 6
Major
Ratios and Proportional Relationships
 Understand ratio concepts and use
ratio reasoning to solve problems
The Number System
 Apply and extend previous
understanding of numbers to the
system of rational numbers
 Apply and extend previous
understandings of multiplication and
division to divide fractions by
fractions
Expressions and Equations
 Apply and extend previous
understandings of arithmetic to
algebraic expressions
 Reason about and solve one-variable
equations and inequalities
 Represent and analyze quantitative
relationships between dependent
and independent variables
Supporting
Geometry
 Solve real-world and
mathematical problems
involving area, surface
area, and volume
Additional
Statistics and Probability
 Develop understanding of
statistical variability
 Summarize and describe
distributions
The Number System
 Compute fluently with multidigit numbers and find
common factors and multiples
Content Emphasis by Cluster Grade 7
Major
Supporting
Ratios and Proportional Relationships
 Analyze proportional relationships
and use them to solve real-world and
mathematical problems
Statistics and Probability
 Use random sampling
to draw inferences
about a population
 Investigate chance
processes and develop,
use, and evaluate
probability models
The Number System
 Apply and extend previous
understandings of operations with
fractions to add, subtract, multiply,
and divide rational numbers
Expressions and Equations
 Use properties of operations to
generate equivalent expressions
 Solve real-life and mathematical
problems using numerical and
algebraic expressions and equations
Additional
Statistics and Probability
 Draw informal comparative
inferences about two
populations
Geometry
 Solve real-life and
mathematical problems
involving angle measure, area,
surface area, and volume
 Draw, construct and describe
geometrical figures and
describe the relationships
between them
Content Emphasis by Cluster Grade 8
Major
Expressions and Equations
 Work with radicals and integer
exponents
 Understand the connections
between proportional relationships,
lines, and linear equations
 Analyze and solve linear equations
and pairs of simultaneous linear
equations
Functions
 Define, evaluate, and compare
functions
Geometry
 Understand and apply the
Pythagorean Theorem
 Understand congruence and
similarity using physical models,
transparencies, or geometry
software
Supporting
The Number System
 Know that there are
numbers that are not
rational, and
approximate them by
rational numbers
Additional
Geometry
 Solve real-life and
mathematical problems
involving volume of cylinders,
cones, and spheres
Functions
 Use functions to model
relationships between
quantities
Statistics and Probability
 Investigate patterns of
association in bivariate
data
http://www.achievethecore.org/d
ownloads/Math%20Shifts%20and%
20Major%20Work%20of%20Grade.
pdf
PARCC Assessments
PARCC is developing an assessment system comprised of four components. Each
component will be computer-delivered and will leverage technology to incorporate
innovations.
 Two summative, required assessment components designed to
o Make “college- and career-readiness” and “on-track” determinations
o Measure the full range of standards and full performance continuum
o Provide data for accountability uses, including measures of growth
 Two non-summative, optional assessment components designed to
o Generate timely information for informing instruction, interventions, and
professional development during the school year
o An additional third non-summative component will assess students’ speaking
and listening skills
Assessments
Beginning of
School Year
End of
School Year
Flexible administration
Diagnostic
Assessment
Mid-Year
Assessment
PerformanceBased
Assessment
End-of-Year
Assessment
Speaking and
Listening
Assessment
Key:
Optional
Required
Optional Assessments
During the Year
Beginning of
School Year
End of
School Year
Flexible administration
Diagnostic
Assessment
Mid-Year
Assessment
 Flexible indicator of  Performance-based
items and tasks
student knowledge
and skills
 Emphasis on hard-tomeasure standards
 Allows instruction,
supports and
 Individual states may
professional
consider including as a
development to be
summative
tailored to improve
component
student learning
Two Required Assessments Yield Overall
Score
Beginning of
School Year
End of
School Year
PerformanceBased
Assessment
 After 75 percent of the
school year
 Extended tasks,
applications of concepts
and skills
 Solving multistep
problems requiring
abstract reasoning,
precision, perseverance
and strategic use of tools
End-of-Year
Assessment
 After 90 percent of the
school year
 Innovative, short-answer
items
 Short items that address
both concepts and skills
PARCC Overview of Assessment
Design
Administered in two components:
o A performance-based assessment (PBA) component,
administered after approximately 75% of the school
year
o An end of year assessment (EOY) component,
administered after approximately 90% of the school
year.
PARCC Summative Assessment
 PBA will include both short- and extended- response
questions that focus on conceptual knowledge and
skills, and the mathematical practices of reasoning
and modeling
 EOY will be comprised of primarily short-answer
questions focused on conceptual knowledge, skills,
and understandings.
PARCC Number and Length of
Testing Sessions
PBA component will have two sessions
approximately 50 minutes each
EOY component will have two sessions
approximately 55 minutes each
www.parcconline.org
www.parcconline.org/samples/item-task-prototypes
Overview of Mathematics Task Types
Task Type
Description of Task Type
I. Tasks assessing
concepts, skills and
procedures
•
•
•
•
Balance of conceptual understanding, fluency, and application
Can involve any or all mathematical practice standards
Machine scorable including innovative, computer-based formats
Will appear on the End of Year and Performance Based Assessment
components
II. Tasks assessing
expressing
mathematical
reasoning
•
Each task calls for written arguments / justifications, critique of
reasoning, or precision in mathematical statements (MP.3, 6).
Can involve other mathematical practice standards
May include a mix of machine scored and hand scored responses
Included on the Performance Based Assessment component
III. Tasks assessing
modeling /
applications
•
•
•
•
•
•
•
Each task calls for modeling/application in a real-world context or
scenario (MP.4)
Can involve other mathematical practice standards.
May include a mix of machine scored and hand scored responses
Included on the Performance Based Assessment component
Overview of Formative Components
Early Assessments:
- Diagnostic Assessment designed to be an indicator of student knowledge
and skills so that instruction, supports and professional development can
be tailored to meet student needs.
- Designed to be administered close to the beginning of the year.
- Will provide an early snapshot of achievement, knowledge and skills so
that educators can tailor instruction, supports for students, and
professional development to meet students’ needs.
- Mid-Year Assessment comprised of performance-based items and tasks,
with an emphasis/focus on hard-to-measure standards. After study,
individual states may consider including as a summative component
- Designed to be administered near the middle of the school year.
- Teachers could score this assessment to get quick feedback on student
learning relative to the CCSS.
Formative Components
 are formative assessments developed by PARCC with its grant
funds
 are available to all PARCC states and their local districts
 are intended to be administered early and midway through the
school year, however, allow for flexible administration-- they
can be administered at locally determined times, including at
the discretion of the classroom teacher
 can be scored quickly -- some can be computer administered
and scored, others can be scored by the classroom teacher – so
that teachers can have timely information that can inform
instruction for their students
Conclusion
The standards and the assessments
are not intended to be new names of
old ways of doing things. They were
created to take the next step in
education.
Resources
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www.corestandards.org/
www.parcconline.org
http://www.ccsstoolbox.com/
http://www.illustrativemathematics.org/
http://commoncore.americaachieves.org/
http://map.mathshell.org/materials/index.php
http://insidemathematics.org/
http://www.thinkfinity.org/welcome
http://schools.nyc.gov/Academics/CommonCoreLibrary/TasksUnitsSt
udentWork/default.htm
 http://www.smarterbalanced.org/
 http://katm.org/wp/common-core/