Dodge City Public Schools Grades 7 - 12 August 17, 2011 Elaine Watson, Ed.D .

Dodge City Public SchoolsGrades 7 - 12

August 17, 2011 Elaine Watson, Ed.D.

International Center for Leadership in Education

Common Core Standards for Mathematical

Practice

IntroductionsIntroduce yourself:

NameInstructional LevelOn a scale of 1 – 5, with

1 representing very little knowledge5 representing expert knowledgewhere 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 CoreThe new standards support improved curriculum and instruction due to increased:FOCUS, via critical areas at each grade levelCOHERENCE, through carefully developed

connections within and across gradesCLARITY, with precisely worded standards that

cannot be treated as a checklistRIGOR, 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 K 1 2 3 4 5Counting and CardinalityOperations and Algebraic ThinkingNumbers and Operations in Base TenNumbers and Operations FractionsMeasurement and DataGeometry

Structure of Common Core Content Standards 6 - 8

Domain 6 7 8Ratio and Proportional RelationshipsThe Number SystemExpressions and EquationsFunctionsGeometryStatistics and Probability

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


Number and Quantity Overview

• The Real Number System• Quantities• The Complex Number System• Vector and Matrix Quantities


Algebra Overview

• Seeing Structures in Expressions• Arithmetic with Polynomials and

Rational Expressions• Creating Equations• Reasoning with Equations and

Inequalities


Functions Overview

• Interpreting Functions• Building Functions• Linear, Quadratic, and

Exponential Models• Trigonometric Functions


Geometry Overview

• Congruence• Similarity, Right Triangles, and

Trigonometry• Circles• Expressing Geometric Properties

with Equations• Geometric Measurement and

Dimension• Modeling with Geometry


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

Problem SolvingReasoning and

ProofCommunicationRepresentation

Connections

Natl. Resource Council

Adding it Up

Adaptive ReasoningStrategic Competence

Conceptual Understanding

Procedural FluencyProductive 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!


SummativeMulti-stateAssessment Resources forTeachers and Educational Researchers

SMARTER Balanced Assessment

Consortium

(SBAC)


SBACSummative

Assessments

Computer Adaptive Testing (CAT)

Performance Events


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.


Performance Events

In-depth performance taskWill require students to think critically in order to solve a

non-traditional problem

Interpret a situation Develop a

plan Communicate the solution


*No grade level was provided for these samples. Practice Tests will be available in the 2013-2014 school year

• Look over three SBAC Sample Items*

• Discuss reactions in a small group

• Report out

The International Center for Leadership in Education

Rigor/Relevance Framework

Thinking Continuum

Acquisition of Knowledge

Assimilation of Knowledge

Knowledge Taxonomy1.Awareness2.Comprehension3.Analysis4.Synthesis5.Evaluation

Action Continuum

Acquisition of

Knowledge

Application of

Knowledge

Application Model1. Knowledge in one discipline2. Application within discipline3. Application across disciplines4. Application to real-world predictable

situations5. Application to real-world

unpredictable situations

1 2 3 4 5

Application

Knowledge

1

2

3

45

6

1

2

3

45

6

1 2 3 4 5

A

1

2

3

45

6

1 2 4 5

A B3

1

2

3

45

6

1 2 4 5

A B

C

3

1

2

3

45

6

1 2 4 5

A B

C

3

D

1

2

3

45

6

1 2 4 5

A B

C

3

D

A B

C DKNOWLEDGE

A P P L I C A T I O N

A B

C DKNOWLEDGE


• Express probabilities as fractions, percents, or decimals.

• Classify triangles according to angle size and/or length of sides.

• Calculate volume of simple three- dimensional shapes.

• Given the coordinates of a quadrilateral, plot the quadrilateral on a grid.

• 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.

• 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.

• 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.

A B

C DKNOWLEDGE


• Analyze the graphs of the perimeters and areas of squares haing different-length sides.
















A B

C DKNOWLEDGE


















A BC D

KNOWLEDGE


















A B

C DKNOWLEDGE


















Standards for Mathematical PracticeStudents 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 solutionAnalyze givens, constraints, relationships and goalsMake conjectures about the form and meaning of the

solutionPlan a solution pathway rather than simply jump into

a solution attemptConsider analogous problemsTry special cases and simpler forms of original problem


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



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?SimilaritiesDifferences

2. Reason Abstractly and Quantitatively


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 contextualizeTo pause as needed during the symbolic manipulation

in order to look back at the referent values in the problem

2. Reason Abstractly and Quantitatively


Reason Quantitatively, which entails habits of:Creating a coherent representation of the problem

at handConsidering the units involvedAttending to the meaning of quantities, not just

how to compute themKnowing and flexibly using different properties of

operations and objects

3.Construct viable arguments and critique the reasoning of others


Understand and use… stated assumptions, definitions, and previously established results… when constructing arguments



Make conjectures and build a logical progression of statements to explore the truth of their conjectures.

Able to analyze situations by breaking them into casesby recognizing and using counterexamples

Justify their conclusions, communicate to others, and respond to the arguments of others



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



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 MathematicsMathematically 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 MathematicsModeling 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 MathematicsExamples 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 MathematicsExamples 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 carAnalyzing the growth of a savings account

balance or of a bacterial colony

4.Model with MathematicsModels 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 MathematicsModels we devise are also constrained by:Limitations of our mathematical, statistical,

and technical skillsLimitations of our ability to recognize

significant variables and relationships among them

4.Model with MathematicsPowerful tools for modeling:Diagrams of various kindsSpreadsheetsGraphing technologyAlgebra

4.Model with MathematicsBasic Modeling Cycle

Problem Formulate

Compute Interpret

Validate Report


Problem• Identify variables in the

situation• Select those that

represent essential features


Formulate• Select or create a geometrical, tabular,

algebraic, or statistical representation that describes the relationships between the

variables


Compute• Analyze and perform

operations on these relationships to draw

conclusions


Interpret• Interpret the result of the mathematics in terms of the

original situation


Validate• Validate the

conclusions by comparing them with the

situation…


EITHER OR

Validate

Re - FormulateReport on conclusions

and reasoning behind them

5.Use appropriate tools strategically

• Pencil and paper

• Concrete models

• Ruler, compass, protractor

• Calculator

• Spreadsheet• Computer Algebra

System• Statistical

Package• Dynamic Geometry

Software

Mathematically proficient students use:


Mathematically proficient students are:Sufficiently familiar enough with the tools

for their grade level to Know how to use themKnow what is to gain by using themKnow their limitations


Mathematically proficient students can:Analyze graphs and solutions from

graphing calculatorsCan detect possible errors through

estimation and other mathematical knowledge


Mathematically proficient students can:Analyze graphs and solutions from graphing

calculatorsCan explore different assumptions and

consequencesCan detect possible errors through

estimation and other mathematical knowledge


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 precisionMathematically proficient students; Try to communicate precisely to others

Use clear definitionsState the meaning of symbols they useUse the equal sign consistently and appropriatelySpecify units of measureLabel axes

6.Attend to precisionMathematically proficient students; Try to communicate precisely to others

Calculate accurately and efficientlyExpress numerical answers with a degree of

precision appropriate for the problem contextGive carefully formulated explanations to each

otherCan 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 repeatedLook for both general methods and for

shortcutsMaintain oversight of the process while

attending to the details.

Contact Information International Center for Leadership in Education 1587 Route 146 Rexford, NY 12148 (518) 399-2776http://www.LeaderEd.comElaine Watson, Ed.D. Email: [email protected]

http://www.LeaderEd.com/

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Dodge City Public Schools Grades 7 - 12 August 17, 2011 Elaine Watson, Ed.D .

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