Common Core State Standards
Session 2
K-5 Mathematics
Bell Work Activity
• Handout #1
– The Common Core State
Standards for
Mathematics from A to Z
• List words that begin with
each letter of the alphabet
that identify aspects of the
Common Core State
Standards for Mathematics.
2
WELCOME
Introductions
• Dr. Karol Yeatts
• BCI
• Bethany Mathers
• BEESS
• Liz Conn
• BEESS
• Martha Gioielli
• DA
• Mark Drennan
• SALA
Agenda
• The Common Core State
Standards for Mathematics
from A to Z
• Exploring the Standards for
Mathematical Practice
• Exploring the Common
Core State Standards for
Mathematics Critical Areas
• Exploring the Mathematics
Progressions
3
“We live in a time of vast changes that
include accelerating globalization,
mounting quantities of information,
the dominating influence of science
and technology, and the clash of
civilizations. Those changes call for
new ways of learning and thinking in
school, business, and the professions.”
-Howard Gardner
Five Minds for the Future (2007)
4
Expected Outcomes
•
Enhance knowledge base of the Common Core
Standards for Mathematics.
•
Become familiar with the structure of the
Common Core State Standards for Mathematics.
Enhance knowledge of the Common Core
Standards for Mathematical Practice.
Understand how the critical areas bring focus to
key mathematical concepts for students to learn
at each grade level.
Consider how the learning progressions can be
used to inform curriculum and guide instruction.
5
Digital Checkpoint
Session 2
K-5 Mathematics
Digital Checkpoint
• Poll Everywhere
• http://www.pollev
erywhere.com/
7
How To Vote via Texting
TIPS
1. Standard texting rates only (worst case US $0.20)
2. We have no access to your phone number
3. Capitalization doesn’t matter, but spaces and spelling do
How to Vote via Web Response
9
Rate your level of understanding
of the Common Core State Standards
for Mathematics?
Send to: 37607
Text Code # to respond
•
•
•
•
•
No understanding
Limited understanding
Fair understanding
Good understanding
Excellent understanding
Submit response at
http://PollEV.com
63586
63623
63667
63668
63669
10
Virtual Parking Lot
Send to: 37607
Text: 220274 plus your
message
Submit response at
http://PollEV.com/
11
Digital Resources
for the Common Core
• Apple
• http://itunes.apple.com/us/app/commoncore-standards/id439424555
• Android
• https://play.google.com/store/apps/details?i
d=com.edutater.corestandards
• Common Core State Standards for
Mathematics
• http://www.corestandards.org/
12
Aligned
Building
Foundation
CCSSO
Coherence
Clarity
Application Balanced
Critical Areas
Clusters
Guided
by
Principles
McCallum
Mathematical
Practice
Habits
of
Mind
International
Benchmarked
Illustrative
Mathematics
Daro
Developmental
Level
Domains
Joint
effort
Organized Progressions
National
Procedural
Focus Opportunities fluency
NGA
Proficiency
PARCC
Sensemaking
Standards
Teachers
Timeline
Vision
Understanding
Ensuring
Education
Focus
Evidencedbased
Fluency
Knowledge
Quality
Learnerfocused
Life-long
skills
Robust,
Relevant,
Real-world
Researchbased
Rigor
Whole
Child
Approach
X
YOU
Zimba
High School Conceptual Categories
• Number and Quantity (N)
• Algebra (A)
• Functions (F)
• Modeling (*)
• Geometry (G)
• Statistics and Probability (S)
A-Z
http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf
15
Domains for K-12
K
1
2
3
4
5
6
7
8
HS
Counting
and
Cardinality
(CC)
Number and Operations in Base Ten (NBT)
Number and
Operations-Fractions
(NF)
The Number System
Ratios and
Proportional
Relationships
(RP)
Number
and
Quantity
Functions (F)
Operations and Algebraic Thinking (OA)
Expressions and
Equations (EE)
Algebra
Geometry (G)
Measurement and Data (MD)
A-Z
Statistics and Probability (SP)
Domain
Cluster
Headings
Domain
Cluster
Headings
Domain
Cluster
Headings
Domain
Cluster
Headings
A-Z
17
Domain
Cluster
Headings
Standards
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Florida’s Numbering of
the Common Core State Standards
MACC.K.CC.2.5
Subject Grade Domain Cluster Standard
Subject
Mathematics Common Core
Grade
K
Domain
Counting and Cardinality
Cluster
Count to tell the number of objects
Standard
Count to answer “how many?” questions about as many as 20
things arranged in a line, a rectangular array, or a circle, or as
many as 10 things in a scattered configuration; given a number
from 1–20, count out that many objects.
A-Z 19
Standards for Mathematical Practice
“The Standards for Mathematical Practice are
unique in that they describe how teachers need to
teach to ensure their students become
mathematically proficient. We were purposeful in
calling them standards because then they won’t
be ignored.”
- Bill McCallum
20
Standards for Mathematical Practice
Make sense of problems
and persevere in solving
them
Use appropriate tools
strategically
Reason abstractly and
quantitatively
Attend to precision
Construct viable
arguments and critique
the reasoning of others
Look for and make
sense of structure
Model with mathematics
Look for and express
regularity in repeated
reasoning
A-Z
21
Florida’s Common Core State Standards
Implementation Timeline
Year/Grade Level
K
1
2
3-8
9-12
2011-2012
FL
L
L
L
L
2012-2013
FL
FL
L
L
L
2013-2014
CCSS fully implemented
FL
FL
FL
BL
BL
2014-2015
CCSS fully implemented and
assessed
FL
FL
FL
FL
FL
F - full implementation of CCSS for all content areas
L – begin full implementation of content area literacy standards including: (1) use of
informational text, text complexity, quality and range in all grades (K-12), and (2) CCSS
Literacy Standards in History/Social Studies, Science, and Technical Subjects (6-12)
B - blended instruction of CCSS with Next Generation Sunshine State Standards (NGSSS);
last year of NGSSS assessed on FCAT 2.0
A-Z
http://www.fldoe.org/bii/pdf/CCSS-ImplementationTimeline.pdf
22
Standards for Mathematical Practice
Session 2
K-5 Mathematics
Poll Question
How many Common Core Standards for
Mathematical Practice are there?
Send to: 37607
Text: 63844 plus your
message
Submit response at
http://PollEV.com
24
Standards for Mathematical Practice
Overarching Habits of Mind of a
Productive Mathematical Thinker
1. Make sense of problems and persevere in solving them
6. Attend to precision
Reasoning
and
Explaining
Modeling
and
Using Tools
2. Reason abstractly
and quantitatively
3. Construct viable
arguments and
critique the reasoning
of others
4. Model with
mathematics
5. Use appropriate
tools strategically
Seeing
Structure and
Generalizing
7. Look for and make
use of structure
8. Look for and
express regularity in
repeated reasoning
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25
Standards for Mathematical Practice
• Develops dispositions and habits of mind
– “Characteristic of an educated person”
•
•
•
•
Precision in thought
Precision in the use of language and terms
Precision of argument
Sense making happens through conversations
http://www.youtube.com/watch?v=9pKcO9E4Flw&feature=relmfu
A-Z
http://youtu.be/9pKcO9E4Flw
26
The Standards for
Mathematical Practice
• Please locate the Common Core State
Standards for Mathematics.
Page 6
• Take a moment to examine the first three
words of the narrative description for each of
the 8 mathematical practices.
• What do you notice?
Mathematically Proficient Students…
27
Digital Task
Your Digital Task is to:
• Read your assigned Mathematical Practice.
• Identify the words (verbs) that illustrate the
student actions for this practice.
• Text the words on one continuous line with
spaces between each word.
Example: #..... create
analysis model describe
demonstrate….
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Digital Task Text Numbers
Send to: 37607
Text a CODE to respond
followed by the verbs
Practice #1
Practice #2
Practice #3
Practice #4
Practice #5
Practice #6
Practice #7
Practice #8
Submit responses at
http://PollEV.com
CODE
63946
31998
36509
50239
66363
66469
66559
66572
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Mathematical Practice 1 - Make sense of
problems and persevere in solving them.
Text 63946 to 37607
Submit responses at
http://PollEV.com
Mathematical Practice 2 - Reason
Abstractly and Quantitatively
Text 31998 to 37607
Submit responses at
http://PollEV.com
Mathematical Practice 3 - Construct
viable arguments and critique the
reasoning of others.
Text 36509 to 37607
Submit responses at
http://PollEV.com
Mathematical Practice 4 - Model with
mathematics.
Text 50239 to 37607
Submit responses at
http://PollEV.com
Mathematical Practice 5 - Use
appropriate tools strategically.
Text 66363 to 37607
Submit responses at
http://PollEV.com
Mathematical Practice 6 - Attend to
precision.
Text 66469 to 37607
Submit responses at
http://PollEV.com
Mathematical Practice 7 - Look for and
make sense of structure.
Text 66559 to 37607
Submit responses at
http://PollEV.com
Mathematical Practice 8 - Look for and
express regularity in repeated reasoning.
Text 66572 to 37607
Submit responses at
http://PollEV.com
Expectations
1
Students planning solution pathways, monitoring and
evaluating their progress and asking “Does this make
sense?”
2
Students knowing and using different properties of
operations and objects and creating a coherent
representation of the problem at hand.
3
Students understanding and using definitions. Students
justifying and explaining their thinking and listening to
arguments of others and deciding if they make sense..
4
Students applying and using mathematics to solve
problems connected to real-life situations. Students
using models to represent, analyze and interpret
results.
38
Expectations
5
Students being familiar with tools appropriate for their
grade or course and using technology tools to explore
and deepen their understanding of concepts. Students
being able to make sound decisions about when each
of these tools might be helpful.
6
Students communicating precisely to others. Students
calculating accurately and efficiently, expressing
numerical answers with a degree of precision
appropriate for the problem context.
7
Students being able to look closely to discern a pattern or
structure. Students being able to shift perspectives.
8
Students evaluating the reasonableness of their results.
Student Maintaining oversight of the process, while
attending to the details.
39
http://www.wordle.net/show/wrdl/5360414/Mathematical_Practice_Actions
40
Consider the Learners
• Over 240,000 ELLs in Florida
• Almost every district has ELLs
• 300 languages are spoken among ELLs
• 79% of ELLs are in Mainstream/Inclusion
model classrooms
• ELLs are learning in the same classrooms
as non-ELLs
41
Making the Content Comprehensible
• Use the standards vocabulary as a teaching tool.
“Generalize, develop, describe, analyze, apply,
measure,” etc. are all words ELLs will hear in the
classroom and need to understand.
• ELLs may know how to perform the skill using
their language, they just may not yet have the
English vocabulary.
• Use pictures, graphs, and charts whenever
possible.
• Make use of root words and cognates.
42
Classroom Strategies
• Group ELLs with non-ELLs to work together.
• Allow more wait time for ELLs to respond.
– Silence does not necessarily mean the
student does not know the answer, the ELL
may be translating the answer and need more
time.
• Remember that ELLs from different countries
may display mathematical functions in different
ways.
43
Critical Areas and Mathematics Progressions
Session 2
K-5 Mathematics
K-5 Domains and Critical Areas
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Kindergarten Domains
Counting and Cardinality
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
1st Grade Domains
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
2nd Grade Domains
Operations and Algebraic Thinking
Number and Operations in Base Ten
Measurement and Data
Geometry
3rd Grade Domains
Operations and Algebraic Thinking
Number and Operation in Base Ten
Number and Operation: Fractions
Measurement and Data
Geometry
4th Grade Domains
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
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
Operations and Algebraic Thinking
Number and Operations in Base Ten
Number and Operations: Fractions
Measurement and Data
Geometry


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

5th Grade Domains
Operations and Algebraic Thinking
Number and Operations in Base Ten
Number and Operations: Fractions
Measurement and Data
Geometry
Kindergarten Critical Areas
 Representing and comparing whole numbers, initially with sets of objects.
 Describing shapes and space.
 More learning time in Kindergarten should be devoted to number than to other topics
1st Grade Critical Areas
 Developing understanding of addition, subtraction, and strategies for addition and subtraction within 20.
 Developing understanding of whole number relationships and place value, including grouping in tens and
ones.
 Developing understanding of linear measurement and measuring lengths as iterating length units.
 Reasoning about attributes of, and composing and decomposing geometric shapes.
2nd Grade Critical Areas
 Extending understand of base-ten notation.
 Building fluency with addition and subtraction.
 Using standard units of measure.
 Describing and analyzing shapes.
3rd Grade Critical Areas
 Developing understanding of multiplication and division strategies for multiplication within 100.
 Developing understanding of fractions, especially unit fractions (fractions with numerator 1).
 Developing understanding of the structure of rectangular arrays and of area.
 Describing and analyzing two-dimensional shapes.
4th Grade Critical Areas
 Developing understanding and fluency with multi-digit multiplication, and developing understanding of
dividing to find quotients involving multi-digit dividends.
 Developing understanding of fractions equivalence, addition and subtraction of fractions with like
denominators, multiplication of fractions by whole numbers.
 Understanding that geometric figures can be analyzed and classified based on their properties, such as
having parallel sides, perpendicular sides, particular angle measures, and symmetry.
5th Grade Critical Areas
 Developing fluency with addition and subtraction of fractions, developing understanding of the
multiplication of fractions and of division of fractions in limited case (unit fractions divided by whole
numbers and whole numbers divided by unit fractions).
 Extending division to 2-digit divisors, integrating decimal fractions into the place value system and
developing understanding of operations with decimals to hundredths, and developing fluency with whole
number and decimal operations.
 Developing understanding volume.
45
A-Z
Two critical areas in Kindergarten
In Kindergarten, instructional time should focus on
two critical areas:
1.
• Representing, relating, and operating on
whole numbers, initially with sets of objects
2.
• Describing shapes and space
More learning time in Kindergarten should
be devoted to number than to other topics.
Page 9
http://www.corestandards.org/thestandards/mathematics/kindergart
en/introduction/
46
Identify the Kindergarten Critical Area
Cluster Heading
Know number names and the count sequence.
Critical Area
#1 Numbers
Count to tell the number of objects.
#1 Numbers
Describe and compare measurable attributes.
#2 Shapes
#1 Numbers
#2 Shapes
Compare numbers.
Identify and describe shapes.
Understand addition as putting together and
adding to, and understand subtraction as taking
apart and taking from.
Work with numbers 11-19 to gain foundations for
place value.
Classify objects and count the number of objects
in each category
Analyze, compare, create, and compose shapes.
#1 Numbers
#1 Numbers
#2 Shapes
#2 Shapes
47
Four critical areas in 1st Grade
In Grade 1, instructional time should focus on four
critical areas:
1.
developing understanding of addition, subtraction, and
strategies for addition and subtraction within 20
2.
developing understanding of whole number relationships
and place value, including grouping in tens and ones
3.
developing understanding of linear measurement and
measuring lengths as iterating length units
4.
reasoning about attributes of, and composing and
decomposing geometric shapes
Page 14
http://www.corestandards.org/the
-standards/mathematics/grade1/introduction/
48
Identify the 1st Grade Critical Areas
Cluster Heading
Represent and solve problems involving
addition and subtraction.
Use place value understanding and properties
of operations to add and subtract.
Represent and interpret data.
Add and subtract within 20.
Tell and write time.
Work with addition and subtraction equations.
Reason with shapes and their attributes.
Understand place value.
Measure lengths indirectly and by iterating
length units.
Extend the counting sequence.
Understand and apply properties of operations
and the relationship between addition and
subtraction.
Critical Area
#1 Operations
#2 Base Ten
#3 Measurement
#1 Operations
#3 Measurement
#1 Operations
#4 Geometry
#2 Base Ten
#3 Measurement
#2 Base Ten
#1 Operations
49
Four critical areas in 2nd Grade
In Grade 2, instructional time should focus on four
critical areas:
1.
extending understanding of base-ten notation
2.
building fluency with addition and subtraction
3.
using standard units of measure
4.
describing and analyzing shapes
Page 17
http; ://www.corestandards.org/thestandards/mathematics/grade2/introduction/
50
Four critical areas in 3rd Grade
In Grade 3, instructional time should focus on four
critical areas:
1.
developing understanding of multiplication and division
and strategies for multiplication and division within 100
2.
developing understanding of fractions, especially unit
fractions (fractions with numerator 1)
3.
developing understanding of the structure of rectangular
arrays and of area
4.
describing and analyzing two-dimensional shapes
Page 21
http; ://www.corestandards.org/thestandards/mathematics/grade2/introduction/
51
Mathematics Progressions
Whole Numbers to Fractions in Grades 3-6
http://youtu.be/w7h64xjN-PM
http://www.youtube.com/watch?v=w7h64xjNPM&list=PLD7F4C7DE7CB3D2E6&index=7&featu
re=plpp_video
http://commoncoretools.files.wor
dpress.com/2012/02/ccss_progr
ession_nf_35_2011_08_12.pdf
52
Three critical areas in 4th Grade
In Grade 4, instructional time should focus on three critical
areas:
1.
developing understanding and fluency with multi-digit
multiplication, and developing understanding of dividing to find
quotients involving multi-digit dividends
2.
developing an understanding of fraction equivalence, addition
and subtraction of fractions with like denominators, and
multiplication of fractions by whole numbers
3.
understanding that geometric figures can be analyzed and
classified based on their properties, such as having parallel sides,
perpendicular sides, particular angle measures, and symmetry.
Page 27
http://www.corestandards.org/thestandards/mathematics/grade53
4/introduction
Building Fluency
What is meant by fluency?
Mathematics Fluency: A Balanced Approach (1:56)
http://www.youtube.com/watch?v=ZFUAV00bTwA&list=PLD7F4C7DE7
CB3D2E6&index=13&feature=plpp_video
http://youtu.be/ZFUAV00bTwA
54
Key Fluencies
Grade
Standard
Key Fluency
K
MACC.K.OA.1.5 Add/subtract within 5
1
MACC.1.OA.3.6 Add/subtract within 10
2
MACC.2.OA.2.2 Add/subtract within 20
MACC.2.NBT.2.5 Add/subtract within 100 (pencil and paper)
3
MACC.3.NBT.1.2 Add/subtract within 1,000
MACC.3.OA.3.7 Multiply/divide within 100
4
MACC.4.2.4
Critical Area #1
5
MACC.5.NBT.2.5
Critical Area #1
6
7
8
MACC.6.NS.2.2
MACC.6.NS.2.3
MACC.7.EE.2.4a
MACC.8.EE.3.8b
Add/subtract within 1,000,000
Develop fluency with efficient procedures for multiplying
whole numbers
Multi-digit multiplication
Developing fluency with addition and subtraction of
fractions
Multi-digit division
Multi-digit decimal operations
Solve px + q = r, p(x + q) = r
Solve simple 22 systems by inspection
A-Z
55
Three critical areas in 5th Grade
In Grade 5, instructional time should focus on three critical
areas:
1.
developing fluency with addition and subtraction of fractions, and
developing understanding of the multiplication of fractions and of
division of fractions in limited cases (unit fractions divided by whole
numbers and whole numbers divided by unit fractions)
2.
extending division to 2-digit divisors, integrating decimal fractions into
the place value system and developing understanding of operations
with decimals to hundredths, and developing fluency with whole
number and decimal operations
3.
developing understanding of volume
Page 33
http://www.corestandards.org/the
-standards/mathematics/grade5/introduction
56
Mathematics Progressions
The Importance of
Mathematics Progressions
Operations on Whole Number Progressions
http://youtu.be/a-P9KQdhE0U
http://www.youtube.com/watch?v=aP9KQdhE0U&feature=BFa&list=PLD7F4C7DE7CB3D2E6
57
Mathematics Progressions Project
•Kindergarten Counting and Cardinality
•Number and Operations in Base Ten
•Number and Operations—Fractions
•K–5 Operations and Algebraic Thinking
•Measurement and Data
•Geometry Progression (coming soon!)
•Expressions and Equations
•Ratios and Proportional Relationships
•Statistics and Probability, Grades 6–8
http://commoncoretools.me/category/progressions/
58
Give me a lever long enough
and a fulcrum to place it on,
and I can move the world.
‐‐Archimedes
59
Reflective Thoughts
How will you use the Common Core Standards for
Mathematical Practices to inform your curriculum
and guide your instruction?
How will the Critical Areas and the Cluster headings
help to inform your curriculum and guide your
instruction?
How will you use the Learning Progressions to
inform your curriculum and guide your instruction?
60