Transcript 7-8-hs-ekw-dodge-city

Common Core
Standards for Mathematical
Practice
Dodge City Public Schools
Grades 7 - 12
August 17, 2011
Elaine Watson, Ed.D.
International Center for Leadership in Education
Introductions
Introduce yourself:
 Name
 Instructional Level
 On a scale of 1 – 5, with
 1 representing very little knowledge
 5 representing expert knowledge
 where do you lie with respect to an
understanding of the eight Standards for
Mathematical Practice?
Desired Outcomes
After this three hour presentation, participants will have an
introductory understanding of:
 The difference and connection between the
Standards for Mathematical Practice
and the
Standards for Mathematical Content
 How the Content Standards will be assessed beginning in the
2014-2015 school year
 Be familiar with the format and terminology of the Standards for
Mathematical Practice
 Understand how the ICLE Rigor Relevance Framework can be
used as a tool to plan instruction that will reinforce students’
acquisition of the Standards for Mathematic Practice
Common Core
The new standards support improved curriculum and
instruction due to increased:
 FOCUS, via critical areas at each grade level
 COHERENCE, through carefully developed connections
within and across grades
 CLARITY, with precisely worded standards that cannot be
treated as a checklist
 RIGOR, including a focus on College and Career Readiness
and Standards for Mathematical Practice throughout Pre K –
12.
Common Core
Standards for
Mathematical
Practice
Standards for
Mathematical
Content
Same for All Grade
Levels
Specific to Grade Level
Grade 7 Overview
Grade 8 Overview
High School Overview
Structure of Common Core
Content Standards K - 5
Domain
Counting and Cardinality
Operations and Algebraic Thinking
Numbers and Operations in Base Ten
Numbers and Operations Fractions
Measurement and Data
Geometry
K
1
2
3
4
5
Structure of Common Core
Content Standards 6 - 8
Domain
Ratio and Proportional Relationships
The Number System
Expressions and Equations
Functions
Geometry
Statistics and Probability
6
7
8
Structure of Common Core
Content Standards HS
High School Content Standards are listed in
conceptual categories
Number and Quantity
Algebra
Functions
Modeling
Geometry
Statistics and Probability
Structure of Common Core
Content Standards HS
Number and Quantity
Overview
•
•
•
•
The Real Number System
Quantities
The Complex Number System
Vector and Matrix Quantities
Structure of Common Core
Content Standards HS
Algebra Overview
• Seeing Structures in Expressions
• Arithmetic with Polynomials and
Rational Expressions
• Creating Equations
• Reasoning with Equations and
Inequalities
Structure of Common Core
Content Standards HS
Functions Overview
• Interpreting Functions
• Building Functions
• Linear, Quadratic, and Exponential
Models
• Trigonometric Functions
Structure of Common Core
Content Standards HS
Geometry Overview
• Congruence
• Similarity, Right Triangles, and
Trigonometry
• Circles
• Expressing Geometric Properties
with Equations
• Geometric Measurement and
Dimension
• Modeling with Geometry
Structure of Common Core
Content Standards HS
Statistics and Probability
Overview
• Interpreting Categorical and
Quantitative Data
• Making Inferences and Justifying
Conclusions
• Conditional Probability and the Rules
of Probability
• Using Probability to Make Decisions
Eight Standards for
Mathematical Practice
Describe practices that mathematics educators should
seek to develop in their students
NCTM Process
Standards
Natl. Resource
Council
Adding it Up
Problem Solving
Reasoning and Proof
Communication
Representation
Connections
Adaptive Reasoning
Strategic Competence
Conceptual Understanding
Procedural Fluency
Productive Disposition
Eight Standards for
Mathematical Practice
 Describe ways in which student practitioners of the
discipline of mathematics increasingly ought to engage
with the subject matter as they grow in mathematical
maturity
 Provide a balanced combination of procedure and
understanding
 Shift the focus to ensure mathematical understanding
over computation skills
Quick Common Core
Assessment Overview
 Adopted by all but 6 States
 New assessments are being developed by two consortia (SBAC
and PARCC) who are affiliated with member states
 Kansas is affiliated with Smarter Balanced Assessment
Consortium (SBAC)
 New assessments will be administered starting in 2014-15 each
year for Grades 3 – 8 and at least once in High School.
 Changes in how we instruct students needs to begin NOW!
Quick Common Core
Assessment Overview
Summative
Multi-state
Assessment
Resources for
Teachers and
Educational
Researchers
SMARTER
Balanced Assessment
Consortium
(SBAC)
Quick Common Core
Assessment Overview
SBAC
Summative
Assessments
Computer Adaptive
Testing (CAT)
Performance Events
Quick Common Core
Assessment Overview
Computer Adaptive
Testing (CAT)
1. Students are given a short series of
moderately difficult grade level test
items.
2. Depending upon students initial
performance, delivers items that are
either more or less difficult.
3. Process continues until the student’s
exact level of proficiency is
determined.
Quick Common Core
Assessment Overview
Performance
Events
In-depth performance task
Will require students to think
critically in order to solve a nontraditional problem
Interpret a
situation
Develop a
plan
Communicate
the solution
Quick Common Core
Assessment Overview
• Look over three SBAC Sample Items*
• Discuss reactions in a small group
• Report out
*No grade level was provided for these samples. Practice Tests will
be available in the 2013-2014 school year
The International Center for
Leadership in Education
Rigor/Relevance
Framework
Thinking Continuum
Assimilation of Knowledge
Acquisition of Knowledge
Knowledge Taxonomy
1. Awareness
2. Comprehension
3. Analysis
4. Synthesis
5. Evaluation
Action Continuum
Acquisition
of
Knowledge
Application
of
Knowledge
Application Model
1. Knowledge in one discipline
2. Application within discipline
3. Application across disciplines
4. Application to real-world predictable
situations
5. Application to real-world unpredictable
situations
6
Knowledge
5
4
3
2
Application
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C
D
A
B
APPLI CATI O N
K
N
O
W
L
E
D
G
E
•
•
•
•
Analyze the graphs of the
perimeters and areas of squares
having different-length sides.
Determine the largest rectangular
area for a fixed perimeter.
Identify coordinates for ordered
pairs that satisfy an algebraic
relation or function.
Determine and justify the
similarity or congruence for two
geometric shapes.
C
• Express probabilities as fractions,
percents, or decimals.
• Classify triangles according to
angle size and/or length of sides.
• Calculate volume of simple threedimensional shapes.
• Given the coordinates of a
quadrilateral, plot the quadrilateral
on a grid.
A
•
•
•
•
Obtain historical data about local
weather to predict the chance of snow,
rain, or sun during year.
Test consumer products and illustrate
the data graphically.
Plan a large school event and
calculate resources (food,
decorations, etc.) you need to
organize and hold this event.
Make a scale drawing of the
classroom on grid paper, each group
using a different scale.
D
• Calculate percentages of advertising in
a newspaper.
• Tour the school building and identify
examples of parallel and perpendicular
lines, planes, and angles.
• Determine the median and mode of real
data displayed in a histogram
• Organize and display collected data,
using appropriate tables, charts, or
graphs.
B
APPLI CATI O N
K
N
O
W
L
E
D
G
E
• Express probabilities as fractions, percents, or
decimals.
• Obtain historical data about local
• Analyze the graphs of the
weather
predict thesize
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• Classify
triangles
according
totoangle
perimeters and
areas of squares
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haing different-length sides.
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• Test consumer products and illustrate
• Determine the largest rectangular
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• Calculate
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D
using a different scale.
A
• Calculate percentages of advertising in
a newspaper.
• Tour the school building and identify
examples of parallel and perpendicular
lines, planes, and angles.
• Determine the median and mode of real
data displayed in a histogram
• Organize and display collected data,
using appropriate tables, charts, or
graphs.
B
APPLI CATI O N
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N
O
W
L
E
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• Calculate percentages of advertising
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• Organize and display collected data,
• Express probabilities as fractions,
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• Classify triangles according to
graphs.
angle size and/or length of sides.
C
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• Calculate volume of simple threedimensional shapes.
• Given the coordinates of a
quadrilateral, plot the quadrilateral
on a grid.
APPLI CATI O N
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• Classify triangles according to
angle size and/or length of sides.
• Calculate volume of simple threedimensional shapes.
• Given the coordinates of a
quadrilateral, plot the quadrilateral
on a grid.
A
a newspaper.
• Tour the school building and identify
examples of parallel and perpendicular
lines, planes, and angles.
• Determine the median and mode of real
data displayed in a histogram
• Organize and display collected data,
using appropriate tables, charts, or
graphs.
B
APPLI CATI O N
• Obtain historical data about local weather to
• Analyze the graphs of the
predict
the chance of snow, rain, or sun
K
perimeters and areas of squares
having different-length sides.
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• Identify coordinates for ordered
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C
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• Classify triangles according to
angle size and/or length of sides.
• Calculate volume of simple threedimensional shapes.
• Given the coordinates of a
quadrilateral, plot the quadrilateral
on a grid.
A
D
• Tour the school building and identify
examples of parallel and perpendicular
lines, planes, and angles.
• Determine the median and mode of real
data displayed in a histogram
• Organize and display collected data,
using appropriate tables, charts, or
graphs.
B
APPLI CATI O N
Standards for Mathematical Practice
Students will be able to:
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.
1. Make Sense of Problems
and Persevere in Solving
Mathematically proficient students:
 Explain to self the meaning of a problem and look for
entry points to a solution
 Analyze givens, constraints, relationships and goals
 Make conjectures about the form and meaning of the
solution
 Plan a solution pathway rather than simply jump into a
solution attempt
 Consider analogous problems
 Try special cases and simpler forms of original problem
1. Make Sense of Problems
and Persevere in Solving
Mathematically proficient students:
 Monitor and evaluate their progress and change course if
necessary…“Does this approach make sense?”
 Persevere in Solving
 Transform algebraic expressions
 Change the viewing window on graphing calculator
 Move between multiple representations:
 Equations, verbal descriptions, tables, graphs, diagrams
1. Make Sense of Problems
and Persevere in Solving
Mathematically proficient students:
 Check their answers
 “Does this answer make sense?”
 Does it include correct labels?
 Are the magnitudes of the numbers in the solution in the
general ballpark to make sense in the real world?
 Verify solution using a different method
 Compare approach with others:
 How does their approach compare with mine?
 Similarities
 Differences
2. Reason Abstractly and
Quantitatively
Mathematically proficient students:
 Make sense of quantities and their relationships in a problem
situation
 Bring two complementary abilities to bear on problems involving
quantitative relationships:
 The ability to decontextualize
 to abstract a given situation, represent it symbolically,
manipulate the symbols as if they have a life of their own
 The ability to contextualize
 To pause as needed during the symbolic manipulation in
order to look back at the referent values in the problem
2. Reason Abstractly and
Quantitatively
Mathematically proficient students:
 Reason Quantitatively, which entails habits of:
 Creating a coherent representation of the problem at
hand
 Considering the units involved
 Attending to the meaning of quantities, not just how to
compute them
 Knowing and flexibly using different properties of
operations and objects
3.Construct viable arguments
and critique the reasoning of
others
Mathematically proficient students:
 Understand and use…
stated assumptions,
definitions,
and previously established results…
when constructing arguments
3.Construct viable arguments
and critique the reasoning of
others
Mathematically proficient students:
 Make conjectures and build a logical progression of
statements to explore the truth of their conjectures.
 Able to analyze situations
 by breaking them into cases
 by recognizing and using counterexamples
 Justify their conclusions, communicate to others, and
respond to the arguments of others
3.Construct viable arguments
and critique the reasoning of
others
Mathematically proficient students:
 Reason inductively about data, making plausible
arguments that take into account the context from
which the data arose
 Compare the effectiveness of two plausible arguments,
distinguish correct logic or reasoning from that which is
flawed
3.Construct viable arguments
and critique the reasoning of
others
Mathematically proficient students:
 Can listen or read the arguments of others,
decide whether they make sense,
and ask useful questions
to clarify or improve the arguments
4.Model with Mathematics
Mathematically proficient students:
Model with mathematics.
Modeling is the process of choosing and using
appropriate mathematics and statistics…
to analyze empirical situations
to understand them better,
and to improve decisions.
4.Model with Mathematics
Modeling a situation is a creative process that
involves making choices.
Real world situations are not organized and labeled
for analysis…they do not come with a manual or an
answer in the back of the book!
When making mathematical models, technology is
valuable for varying assumptions, exploring
consequences, and comparing predictions with data.
4.Model with Mathematics
Examples of problem situations that need to be
modeled mathematically in order to solve:
 Estimating how much water and food is needed for
emergency relief in a devastated city of 3 million
people, and how it might be distributed
 Planning a table tennis tournament for 7 players at
a club with 4 tables, where each player plays
against each other player
4.Model with Mathematics
Examples of problem situations that need to be
modeled mathematically in order to solve:
 Designing the layout of the stalls in a school fair so
as to raise as much money as possible
 Analyzing the stopping distance for a car
 Analyzing the growth of a savings account balance
or of a bacterial colony
4.Model with Mathematics
Models devised depend upon a number of factors:
 How precise do we need to be?
 What aspects do we most need to undertand,
control, or optimize?
 What resources of time and tools do we have?
4.Model with Mathematics
Models we devise are also constrained by:
 Limitations of our mathematical, statistical, and
technical skills
 Limitations of our ability to recognize significant
variables and relationships among them
4.Model with Mathematics
Powerful tools for modeling:
 Diagrams of various kinds
 Spreadsheets
 Graphing technology
 Algebra
4.Model with Mathematics
Basic Modeling Cycle
Problem
Formulate
Compute
Validate
Interpret
Report
4.Model with Mathematics
Basic Modeling Cycle
Problem
• Identify variables in the
situation
• Select those that represent
essential features
4.Model with Mathematics
Basic Modeling Cycle
Formulate
• Select or create a geometrical,
tabular, algebraic, or statistical
representation that describes the
relationships between the
variables
4.Model with Mathematics
Basic Modeling Cycle
Compute
• Analyze and perform operations
on these relationships to draw
conclusions
4.Model with Mathematics
Basic Modeling Cycle
Interpret
• Interpret the result of the
mathematics in terms of the
original situation
4.Model with Mathematics
Basic Modeling Cycle
Validate
• Validate the
conclusions by
comparing
them with the
situation…
4.Model with Mathematics
Basic Modeling Cycle
EITHER
Re - Formulate
Validate
OR
Report on
conclusions and
reasoning
behind them
5.Use appropriate tools
strategically
Mathematically proficient students use:
• Pencil and paper
• Concrete models
• Ruler, compass,
protractor
• Calculator
• Spreadsheet
• Computer Algebra
System
• Statistical Package
• Dynamic Geometry
Software
5.Use appropriate tools
strategically
Mathematically proficient students are:
 Sufficiently familiar enough with the tools for
their grade level to
 Know how to use them
 Know what is to gain by using them
 Know their limitations
5.Use appropriate tools
strategically
Mathematically proficient students can:
 Analyze graphs and solutions from graphing
calculators
 Can detect possible errors through estimation
and other mathematical knowledge
5.Use appropriate tools
strategically
Mathematically proficient students can:
 Analyze graphs and solutions from graphing
calculators
 Can explore different assumptions and
consequences
 Can detect possible errors through estimation
and other mathematical knowledge
5.Use appropriate tools
strategically
Mathematically proficient students;
 Can identify relevant external resources, such
as digital content on websites and use them to
pose or solve problems
 Are able to use technological tools in order to
explore and deepen their understanding of
concepts
6.Attend to precision
Mathematically proficient students;
 Try to communicate precisely to others





Use clear definitions
State the meaning of symbols they use
Use the equal sign consistently and appropriately
Specify units of measure
Label axes
6.Attend to precision
Mathematically proficient students;
 Try to communicate precisely to others
 Calculate accurately and efficiently
 Express numerical answers with a degree of precision
appropriate for the problem context
 Give carefully formulated explanations to each other
 Can examine claims and make explicit use of definitions
7. Look for and make use of
structure
Mathematically proficient students;
 Look closely to discern a pattern or structure
 In x2 + 9x + 14, can see the 14 as 2 x 7 and the 9 as
2+7
 Can see complicated algebraic expressions as being
composed of several objects:
5 – 3 (x – y)2 is seen as 5 minus a positive number times a
square, so its value can’t be more than 5 for any real
numbers x and y
8. Look for and express regularity
in repeated reasoning.
Mathematically proficient students;
 Notice if calculations are repeated
 Look for both general methods and for
shortcuts
 Maintain oversight of the process while
attending to the details.
Contact Information
 International Center for Leadership in Education
1587 Route 146
Rexford, NY 12148
(518) 399-2776
http://www.LeaderEd.com
 Elaine Watson, Ed.D.
Email: [email protected]