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

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Dodge City Public Schools

Grades 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 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 K 1 2 3 4 5

Counting and Cardinality

Operations and Algebraic Thinking

Numbers and Operations in Base Ten

Numbers and Operations Fractions

Measurement and Data

Geometry

Structure of Common Core Content Standards 6 - 8

Domain 6 7 8

Ratio and Proportional Relationships

The Number System

Expressions and Equations

Functions

Geometry

Statistics 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!

Summative

Multi-state

Assessment

Resources for

Teachers 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

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.Awareness

2.Comprehension

3.Analysis

4.Synthesis

5.Evaluation

Action Continuum

Acquisition of

Knowledge

Application of

Knowledge

Application Model1. 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

1 2 3 4 5

Application

Knowledge

1 2 3 4 5

1 2 4 5

C DKNOWLEDGE

A P P L I C A T I O N

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.

C DKNOWLEDGE

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

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

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 car

Analyzing 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 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 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 - Formulate

Report 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 calculators

Can detect possible errors through estimation and other mathematical knowledge

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

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

Try to communicate precisely to othersUse clear definitionsState the meaning of symbols they useUse the equal sign consistently and appropriatelySpecify units of measureLabel axes

6.Attend to precision

Try to communicate precisely to othersCalculate 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

Look closely to discern a pattern or structureIn x2 + 9x + 14, can see the 14 as 2 x 7 and

the 9 as 2 + 7Can 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.

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: elaine.watson0729@gmail.com

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

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