Transcript Slide 1

CCSSI FOR MATHEMATICS
“STANDARDS OF PRACTICE”
Collegial Conversations
GRADES 6 – 8
Today’s Goal
 To explore the Standards for Content and
Practice for Mathematics
 Begin to consider how these new CCSS
Standards are likely to impact your classroom
practices
What are the Common Core State
Standards?
 Aligned with college and work expectations
 Focused and coherent
 Included rigorous content and application of
knowledge through high-order skills
 Build upon strengths and lessons of current state
standards
 Internationally benchmarked so that all students are
prepared to succeed in our global economy and society
 Research and evidence based
 State led- coordinated by NGA Center and CCSSO
Focus
• Key ideas, understandings, and skills are
identified
• Deep learning of concepts is emphasized
– That is, time is spent on a topic and on
learning it well. This counters the “mile wide,
inch deep” criticism leveled at most current
U.S. standards.
Benefits for States and Districts
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Allows collaborative professional development based
on best practices
Allows development of common assessments and other
tools
Enables comparison of policies and achievement
across states and districts
Creates potential for collaborative groups to get more
economical mileage for:
– Curriculum development, assessment, and
professional development
Common Core Development
• Initially 48 states and three territories
signed on
• As of November 29, 2010, 42 states have
officially adopted
• Final Standards released June 2, 2010, at
www.corestandards.org
• Adoption required for Race to the Top
funds
Michigan’s Implementation Timeline
• Held October and November of 2010 rollouts
• District curricula and assessments that provide a
K-12 progression for meeting the MMC
requirements will require minimal adjustments to
meet CCSS
• Curriculum and assessment alignment in SY10-11
• Implementation SY11-12
• New assessment 2014-15 (Smarter Balanced
Assessment Consortium or SBAC – replaces MEAP
and MME)
Background
Responsibilities of States in the Consortium
Each State that is a member of the Consortium in 2014–
2015 also agrees to do the following:
 Adopt common achievement standards no later than the 2014–2015 school
year,
 Fully implement the Consortium summative assessment in grades 3–8 and
high school for both mathematics and English language arts no later than
the 2014–2015 school year,
 Adhere to the governance requirements,
 Agree to support the decisions of the Consortium,
 Agree to follow agreed-upon timelines,
 Be willing to participate in the decision-making process and, if a Governing
State, final decisions, and
 Identify and implement a plan to address barriers in State law, statute,
regulation, or policy to implementing the proposed assessment system and
address any such barriers prior to full implementation of the summative
assessment components of the system.
Technology Approach
SBAC Item Bank
• Partitioned into a secure item bank for
summative assessments and a non-secure
bank for the interim/benchmark assessments:
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Traditional selected-response items
Constructed-response items
Curriculum-embedded performance events
Technology-enhanced items (modeled after
assessments in use by the U.S. military, the
architecture licensure exam, and NAEP)
HOW TO READ THE GRADE LEVEL
STANDARDS
Domains are large groups of related
standards. Standards from different
domains may sometimes be closely
related. Look for the name with the code
number on it for a Domain.
Common Core Format
Clusters are groups of related standards.
Standards from different clusters may
sometimes be closely related, because
mathematics is a connected subject.
• Clusters appear inside domains.
Standards
define what
students
should be
Common
Core
Format
able to understand and be able to do –
part of a cluster.
They are content statements. An example content statement is:
“Interpret and compute quotients of fractions, and solve word problems
involving division of fractions by fractions, e.g., by using visual fraction
models and equations to represent the problem. For example, create a
story context for (2/3) ÷ (3/4) and use a visual fraction model to show
the quotient; use the relationship between multiplication and division to
explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b)
÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people
share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3
of a cup of yogurt? How wide is a rectangular strip of land with length
3/4 mi and area 1/2 square mi?,” 6. NS.1. The “NS” stands for “Number
System”. Please refer to page three in your grade level appropriate Common
Core document.
•Progressions of increasing complexity from grade to grade
Common Core - Clusters
• May appear in multiple grade levels in the K-8
Common Core. There is increasing development
as the grade levels progress
• What students should know and be able to do
at each grade level
• Reflect both mathematical understandings and
skills, which are equally important
Common Core Format
K-8
High School
Grade
Conceptual Category
Domain
Domain
Cluster
Cluster
Standards
(There are no preK Common Core Standards)
Standards
Format of K-8 Standards
Grade Level
Domain
Format of K-8 Standards
Standard
Cluster
Standard
Cluster
Mathematics » Grade 6 » Introduction
In Grade 6, instructional time should focus on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of
ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which
includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking.
1.
Students use reasoning about multiplication and division to solve ratio and rate problems about quantities. By viewing equivalent ratios and rates as deriving
from, and extending, pairs of rows (or columns) in the multiplication table, and by analyzing simple drawings that indicate the relative size of quantities,
students connect their understanding of multiplication and division with ratios and rates. Thus students expand the scope of problems for which they can use
multiplication and division to solve problems, and they connect ratios and fractions. Students solve a wide variety of problems involving ratios and rates.
2.
Students use the meaning of fractions, the meanings of multiplication and division, and the relationship between multiplication and division to understand and
explain why the procedures for dividing fractions make sense. Students use these operations to solve problems. Students extend their previous understandings
of number and the ordering of numbers to the full system of rational numbers, which includes negative rational numbers, and in particular negative integers.
They reason about the order and absolute value of rational numbers and about the location of points in all four quadrants of the coordinate plane.
3.
Students understand the use of variables in mathematical expressions. They write expressions and equations that correspond to given situations, evaluate
expressions, and use expressions and formulas to solve problems. Students understand that expressions in different forms can be equivalent, and they use the
properties of operations to rewrite expressions in equivalent forms. Students know that the solutions of an equation are the values of the variables that make
the equation true. Students use properties of operations and the idea of maintaining the equality of both sides of an equation to solve simple one-step
equations. Students construct and analyze tables, such as tables of quantities that are in equivalent ratios, and they use equations (such as 3x = y) to describe
relationships between quantities.
4.
Building on and reinforcing their understanding of number, students begin to develop their ability to think statistically. Students recognize that a data
distribution may not have a definite center and that different ways to measure center yield different values. The median measures center in the sense that it is
roughly the middle value. The mean measures center in the sense that it is the value that each data point would take on if the total of the data values were
redistributed equally, and also in the sense that it is a balance point. Students recognize that a measure of variability (interquartile range or mean absolute
deviation) can also be useful for summarizing data because two very different sets of data can have the same mean and median yet be distinguished by their
variability.
Students learn to describe and summarize numerical data sets, identifying clusters, peaks, gaps, and symmetry, considering the context in which the data were
collected. Students in Grade 6 also build on their work with area in elementary school by reasoning about relationships among shapes to determine area,
surface area, and volume. They find areas of right triangles, other triangles, and special quadrilaterals by decomposing these shapes, rearranging or removing
pieces, and relating the shapes to rectangles. Using these methods, students discuss, develop, and justify formulas for areas of triangles and parallelograms.
Students find areas of polygons and surface areas of prisms and pyramids by decomposing them into pieces whose area they can determine. They reason about
right rectangular prisms with fractional side lengths to extend formulas for the volume of a right rectangular prism to fractional side lengths. They prepare for
work on scale drawings and constructions in Grade 7 by drawing polygons in the coordinate plane.
Mathematics » Grade 7 » Introduction
In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2)
developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving
scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface
area, and volume; and (4) drawing inferences about populations based on samples.
1. Students extend their understanding of ratios and develop understanding of proportionality to solve single- and multi-step problems. Students
use their understanding of ratios and proportionality to solve a wide variety of percent problems, including those involving discounts, interest,
taxes, tips, and percent increase or decrease. Students solve problems about scale drawings by relating corresponding lengths between the
objects or by using the fact that relationships of lengths within an object are preserved in similar objects. Students graph proportional
relationships and understand the unit rate informally as a measure of the steepness of the related line, called the slope. They distinguish
proportional relationships from other relationships.
2. Students develop a unified understanding of number, recognizing fractions, decimals (that have a finite or a repeating decimal representation),
and percents as different representations of rational numbers. Students extend addition, subtraction, multiplication, and division to all rational
numbers, maintaining the properties of operations and the relationships between addition and subtraction, and multiplication and division. By
applying these properties, and by viewing negative numbers in terms of everyday contexts (e.g., amounts owed or temperatures below zero),
students explain and interpret the rules for adding, subtracting, multiplying, and dividing with negative numbers. They use the arithmetic of
rational numbers as they formulate expressions and equations in one variable and use these equations to solve problems.
3. Students continue their work with area from Grade 6, solving problems involving the area and circumference of a circle and surface area of
three-dimensional objects. In preparation for work on congruence and similarity in Grade 8 they reason about relationships among twodimensional figures using scale drawings and informal geometric constructions, and they gain familiarity with the relationships between angles
formed by intersecting lines. Students work with three-dimensional figures, relating them to two-dimensional figures by examining crosssections. They solve real-world and mathematical problems involving area, surface area, and volume of two- and three-dimensional objects
composed of triangles, quadrilaterals, polygons, cubes and right prisms.
4. Students build on their previous work with single data distributions to compare two data distributions and address questions about differences
between populations. They begin informal work with random sampling to generate data sets and learn about the importance of representative
samples for drawing inferences.
Mathematics » Grade 8 » Introduction
In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling
an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a
function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle,
similarity, and congruence, and understanding and applying the Pythagorean Theorem.
1. Students use linear equations and systems of linear equations to represent, analyze, and solve a variety of problems. Students recognize
equations for proportions (y/x = m or y = mx) as special linear equations (y = mx + b), understanding that the constant of proportionality (m) is
the slope, and the graphs are lines through the origin. They understand that the slope (m) of a line is a constant rate of change, so that if the
input or x-coordinate changes by an amount A, the output or y-coordinate changes by the amount m·A. Students also use a linear equation to
describe the association between two quantities in bivariate data (such as arm span vs. height for students in a classroom). At this grade, fitting
the model, and assessing its fit to the data are done informally. Interpreting the model in the context of the data requires students to express a
relationship between the two quantities in question and to interpret components of the relationship (such as slope and y-intercept) in terms of
the situation.
Students strategically choose and efficiently implement procedures to solve linear equations in one variable, understanding that when they use
the properties of equality and the concept of logical equivalence, they maintain the solutions of the original equation. Students solve systems of
two linear equations in two variables and relate the systems to pairs of lines in the plane; these intersect, are parallel, or are the same line.
Students use linear equations, systems of linear equations, linear functions, and their understanding of slope of a line to analyze situations and
solve problems.
2. Students grasp the concept of a function as a rule that assigns to each input exactly one output. They understand that functions describe
situations where one quantity determines another. They can translate among representations and partial representations of functions (noting
that tabular and graphical representations may be partial representations), and they describe how aspects of the function are reflected in the
different representations.
3. Students use ideas about distance and angles, how they behave under translations, rotations, reflections, and dilations, and ideas about
congruence and similarity to describe and analyze two-dimensional figures and to solve problems. Students show that the sum of the angles in a
triangle is the angle formed by a straight line, and that various configurations of lines give rise to similar triangles because of the angles created
when a transversal cuts parallel lines. Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the
Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find
distances between points on the coordinate plane, to find lengths, and to analyze polygons. Students complete their work on volume by solving
problems involving cones, cylinders, and spheres.
K – 5 DOMAINS
Domains
Grade Levels
Counting and Cardinality
K only
Operations and Algebraic
Thinking
1-5
Number and Operations in
Base Ten
1-5
Number and Operations Fractions
3-5
Measurement and Data
1-5
Geometry
1-5
MIDDLE GRADES DOMAINS
Domains
Grade Levels
Ratio and Proportional
Relationships
6-7
The Number System
6-8
Expressions and Equations
6-8
Functions
8
Geometry
6-8
Statistics and Probability
6-8
Michigan GLCE vs. CCSS
Grade
Topic
Whole Number: Meaning
Whole Number: Operations
Measurement Units
Common Fractions
Equations & Formulas
Data Representation & Analysis
2-D Geometry: Basics
2-D Geometry: Polygons & Circles
Measurement: Perimeter, Area & Volume
Rounding & Significant Figures
Estimating Computations
Whole Numbers: Properties of Operations
Estimating Quantity & Size
Decimal Fractions
Relation of Common & Decimal Fractions
Properties of Common & Decimal Fractions
Percentages
Proportionality Concepts
Proportionality Problems
2-D Geometry: Coordinate Geometry
Geometry: Transformations
Negative Numbers, Integers, & Their Properties
Number Theory
Exponents, Roots & Radicals
Exponents and Orders of Magnitude
Measurement: Estimation & Errors
Constructions Using Straightedge & Compass
3-D Geometry
Geometry: Congruence & Similarity
Rational Numbers & Their Properties
Patterns, Relations & Functions
Proportionality: Slope & Trigonometry
Uncertainty & Probability
Real Numbers: Their Subsets & Properties
Topic intended in Michigan GLCE
Topic intended in CCSS
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MAJOR SHIFTS K - 5
Numeration and operation intensified, and introduced
earlier
•Early place value foundations in Kindergarten
•Regrouping as composing/decomposing in Grade 2
•Decimals to hundredths in Grade 4
All three types of measurement simultaneously
•Non-standard, English and metric
Emphasis on fractions as numbers
Emphasis on number line as visualization/structure
HOW IS THERE LESS?
•Backed off of algebraic patterns K – 5
•Backed off of statistics and probability in
K–5
•Delayed content like percent and ratios
and proportions
MAJOR SHIFTS 6 - 8
Ratio and proportion focused on in Grade 6
•Ratio, unit rates, converting measurement, tables of values,
graphing and missing value problems
Percents introduced Grade 6
Statistics introduced Grade 6
•Statistical variability (measures of central tendency, distributions,
interquartile range, mean and absolute deviation and data shape)
Rational numbers in Grade 7
One-third of algebra for all students in Grade 8
HOW IS THERE LESS?
•The Common Core Standards are not
less in the middle grades and will only be
fewer if what happens in elementary
leads to more students knowing the
content and avoiding the repetition.
Fractions, Grades 3–6
 3. Develop an understanding of fractions as numbers.
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4. Extend understanding of fraction equivalence and ordering.
4. Build fractions from unit fractions by applying and extending
previous understandings of operations on whole numbers.
4. Understand decimal notation for fractions, and compare
decimal fractions.
5. Use equivalent fractions as a strategy to add and subtract
fractions.
5. Apply and extend previous understandings of multiplication
and division to multiply and divide fractions.
6. Apply and extend previous understandings of multiplication
and division to divide fractions by fractions.
Statistics and Probability, Grade 6
Develop understanding of statistical variability
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Recognize a statistical question as one that anticipates variability in the
data related to the question and accounts for it in the answers. For
example, “How old am I?” is not a statistical question, but “How old are
the students in my school?” is a statistical question because one
anticipates variability in students’ ages.
Understand that a set of data collected to answer a statistical question
has a distribution which can be described by its center, spread, and
overall shape.
Recognize that a measure of center for a numerical data set summarizes
all of its values with a single number, while a measure of variation
describes how its values vary with a single number.
Algebra, Grade 8
Graded ramp up to Algebra in Grade 8
• Properties of operations, similarity, ratio and proportional
relationships, rational number system.
Focus on linear equations and functions in Grade 8
•
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.
– Use functions to model relationships between quantities.
High School
Conceptual themes in high school
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Number and Quantity
Algebra
Functions
Modeling
Geometry
Statistics and Probability
College and career readiness threshold
•
(+) standards indicate material beyond the threshold; can
be in courses required for all students.
THE REASON
WHY WE ARE HERE
TODAY!
CCSSM
Mathematical
Practices
The Common Core proposes a set of
Mathematical Practices that all teachers should
develop in their students. These practices are
similar to NCTM’s Mathematical Processes from
the Principles and Standards for School
Mathematics.
Design and Organization
Mathematical Practice – expertise students
should acquire: (Processes & proficiencies)
• NCTM five process standards:
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Problem solving
Reasoning and Proof
Communication
Connections
Representations
NCTM Process Standards and the
CCSS Mathematical Practice Standards
NCTM Process Standards
CCSS Mathematical Practices
Problem Solving
Make sense of problems and persevere
in solving them.
Use appropriate tools strategically
Reasoning and Proof
Reason abstractly and quantitatively.
Critique the reasoning of others.
Look for and express regularity in
repeated reasoning
Communication
Construct viable arguments
Connections
Attend to precision.
Look for and make use of structure
Representations
Model with mathematics.
Design and Organization
• Mathematical proficiency (National Research
Council’s report Adding It Up)
– Adaptive reasoning
– Strategic competence
– Conceptual understanding (comprehension of
mathematical concepts, operations, relations)
– Procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently, and appropriately)
– Productive disposition (ability to see mathematics as
sensible, useful, and worthwhile
Mathematics/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
Mathematics/Standards for Mathematical Practice
“The Standards for Mathematical Practice describe varieties of expertise that
mathematics educators at all levels should seek to develop in their students.
These practices rest on important “processes and proficiencies” with
longstanding importance in mathematics education.” CCSS, 2010
Standards for Mathematical Practice
• Carry across all grade levels
• Describe habits of a mathematically expert student
Standards for Mathematical Content
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•
K-8 presented by grade level
Organized into domains that progress over several grades
Grade introductions give 2-4 focal points at each grade level
High school standards presented by conceptual theme (Number &
Quantity, Algebra, Functions, Modeling, Geometry, Statistics & Probability
Standards of Mathematical Practice
1. Choose a partner at your table and “Pair Share” the
Standards of Practice between you and your partner.
2. When you and your partner feel you understand
generally each of the standards, discuss the following
question:
What implications might the standards
of practice have on your classroom?
Transition from Current State Standards & Assessments
to New Common Core Standards
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Develop Awareness
Needs Assessment/Gap Analysis
Planning
Capacity Building
Job-embedded Professional Development
Transition Planning
Next Steps:
• Alignment of CCSS with curriculum
• Gap analysis (content and skills that vary from
the MEAP and MME)
• What instructional practices will facilitate the
transition?
• What new assessment strategies will be
needed?
• Professional development needs?
Transition Planning
• Gather in teams from your schools and discuss
– What are your immediate needs as a classroom teacher
being asked to implement the CCSS?
– What professional development is needed?
– What initial gaps come to mind and how do you address
these gaps?
– As a school team, look at the eight Standards for
Mathematical Practice. What do they look like? Sound
like? What will students need in order to implement them?
What will teachers need? What are the implications for
assessment and grading?
Select a recorder, time keeper and someone to report out for
your group.
Questions?
Please contact:
PUT YOUR
INFORMATION HERE!
Have a great day!