MathematicsGrades 6-12
February NTI
February 4, 2013
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Overview of the Day
1. Standards for Mathematical Practice
2. Progressions Documents - Grades 6-8 & 9-12
3. NYSED Assessment Development
Lunch
4a. LearnZillion – Grades 6-8
4b. PARCC Model Content Frameworks -
Grades 9-12
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ObjectivesGain a deeper understanding of the eight Standards for Mathematical Practice, articulate them to others, and implement the MPs in the classroom alongside mathematical content standards.
Identify key areas to focus on in the areas of Ratio and Proportional Relationships ( Grades 6 and 7) and Functions (Grades 8-12).
Become familiar with the PARCC Model Content Frameworks and compare similarities and differences between the PARCC MCF and CCSSM Appendix A.
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“The formulation of the problem is often more essential than its solution, which may be merely a matter of mathematical or experimental skill.”
-Albert Einstein
Standards for Mathematical Practice
GRADES 6-12
Teri Calabrese-Gray, CVES Assistant Superintendent
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Standards for Mathematical Practice
“Chairs in Hall” – Illustrative Mathematics
Three hallways contained 9,876 chairs altogether. One-fifth of the chairs were transferred from the first hall to the second hall. Then, one-third of the chairs were transferred from the second hall to the third hall and the number of chairs in the third hall doubled. In the end, the number of chairs in the three halls became the same. How many chairs were in the second hall to begin with?
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Standards for Mathematical Practice
The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction.
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Standards for Mathematical Practice
The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily.
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Standards for Mathematical Practice
Without a flexible base from which to work, students may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut.
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Standards for Mathematical Practice
In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
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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.
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Standards for Mathematical Practice
ACTIVITY
Using the Standards for Mathematical Practice (MP) handout on your table and the number you selected for your table, please read the corresponding MP. Divide your piece of chart paper in half and on one side label it Student Evidence and on the other side label it Teacher Evidence. Don’t forget to write the number of the MP you selected on your chart paper as well.
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Standards for Mathematical Practice
ACTIVITY (cont’d)
Brainstorm with members at your table what students would be doing in the classroom if this MP was being implemented effectively and list evidence on your chart paper.
Next, brainstorm with members at your table what teachers would be doing in the classroom if this MP was being implemented effectively and list evidence on your chart paper.
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Standards for Mathematical Practice
ACTIVITY (cont’d)
Locate all those tables that worked on the same MP as you and come together as one group and share your work.
Review the evidence and determine if it should stay on the chart or be deleted. Be prepared to report out.
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Standards for Mathematical Practice
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MP 1:
Make sense of problems and persevere in solving them.
Standards for Mathematical Practice
MP 2:
Reason abstractly and quantitatively.
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Standards for Mathematical Practice
MP 3:
Construct viable arguments and critique the reasoning of others.
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Standards for Mathematical Practice
MP 4:
Model with mathematics.
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Standards for Mathematical Practice
MP 5:
Use appropriate tools strategically.
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Standards for Mathematical Practice
MP 6:
Attend to precision.
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Standards for Mathematical Practice
MP 7:
Look for and make use of structure.
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Standards for Mathematical Practice
MP 8:
Look for and express regularity in repeated reasoning.
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Standards for Mathematical Practice
PROBLEM
What is the mean of 5, 8, 9, and 6?
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Standards for Mathematical Practice
Identify the MPs that align with this problem. Discuss at your tables.
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Standards for Mathematical Practice
Pose a different problem that could go deeper but require the same mathematical content knowledge?
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Standards for Mathematical Practice
Explain what students might learn from the second question compared to the first question?
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Standards for Mathematical Practice
Explain what a teacher might learn from how students answer the first question compared to how students answer the second question?
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Integrating the Standards for Mathematical Practice
Inside +=x Mathematics
Watch the video using the Inside Mathematics link above and collect evidence from the lesson that exemplifies the Standards for Mathematical Practice. Focus on both the students and the teacher.
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Integrating the Standards for Mathematical Practice
Inside +=x Mathematics
Compare your evidence with an elbow partner and then engage in a conversation with members at your table to come to consensus as to which MPs you observed.
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Integrating the Standards for Mathematical Practice
Inside +=xMathematics
For a more in-depth study of the Standards for Mathematical Practice, please visit
http://www.insidemathematics.org/index.php/commmon-core-math-intro.
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Standards for Mathematical Practice
MP PLACEMAT ACTIVITY
Everyone will need the MP Placemat and the MP Activity Cards. Read through the MP Activity Cards on your own. Once you have read the activity cards, you need to decide where you would put them on your placemat. You have 16 activity cards and 16 boxes on your placemat.
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Standards for Mathematical Practice
MP PLACEMAT ACTIVITY (cont’d)
First you will have time to work independently and then in small groups at your table. In the end, your table must come to consensus and a representative needs to come up and post their results on the master placemat.
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Common Core State Standards: Progressions
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GRADES 6-12
Kristine S. Cole, SUNY Research Fund Fellow Teri Calabrese-Gray, CVES Assistant Superintendent
Common Core State Standards: Progressions
Let’s look at
Grades 6-7, Ratios and Proportional Relationships
and how it builds to
Grade 8 , High School, Functions
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Common Core State Standards: Progressions
K-W-L
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Common Core State Standards: Progressions The Common Core State Standards in mathematics were built on progressions: narrative documents describing the progression of a topic across a number of grade levels, informed both by research on children's cognitive development and by the logical structure of mathematics. These documents were spliced together and then sliced into grade level standards.
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Common Core State Standards: Progressions This would be useful in teacher preparation and professional development, organizing curriculum, and writing textbooks. Progressions documents also provide a transmission mechanism between mathematics education research and standards. Research about learning progressions produces knowledge which can be transmitted through the progressions document to the standards revision process; questions and demands on standards writing can be transmitted back the other way into research questions.
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Common Core State Standards: Progressions From that point on the work focused on refining and revising the grade level standards. The early drafts of the progressions documents no longer correspond to the current state of the standards.
The progressions can explain why standards are sequenced the way they are, point out cognitive difficulties and pedagogical solutions, and give more detail on particularly knotty areas of the mathematics.
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Common Core State Standards: Progressions
Grades 6-7 – Ratios and Proportional Relationships
Grade 8 and High School – Functions
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Common Core State Standards: Progressions
Grades 6-7
Ratios and Proportional Relationships
• Overview – Pages 2-4• Grade 6 – Pages 5-7• Grade 7 – Pages 8-12
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Common Core State Standards: Progressions
Grade 8 and High School: Functions• Overview and Grade 8 – Pages 2-6
• Battery Charging – Page 5• High School – Interpreting Functions –
Pages 7-10• Interpreting the Graph – Page 7• Cell Phones – Page 8• Warming and Cooling – Page 9
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Common Core State Standards: Progressions
Grade 8 and High School Functions• High School – Building Functions – Pages
11-13 (stop before advanced standards)• Lake Algae – Page 11• Transforming Functions – Page 12
• High School – Linear and Exponential Models – Pages 15-16
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Common Core State Standards: Progressions
ACTIVITY
Each group is responsible for their respective sections of the Progressions documents. Once your table has read their section of the document and engaged in an initial discussion, locate similar tables who read the same section.
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Common Core State Standards: Progressions
ACTIVITY (cont’d)
The goal of the activity is to use the collective knowledge of your group to develop a unique way to present your material to the entire group. Once everyone has completed their work, they will be asked to report out.
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Common Core State Standards: Progressions
Grades 6-7
Ratios and Proportional Relationships
• Grade 6 • Grade 7• Overview
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Common Core State Standards: Progressions
Grade 8 and High School: Functions
• Grade 8 • High School – Interpreting Functions• High School – Building Functions• High School – Linear and Exponential
Models• Overview
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Common Core State Standards: Progressions
Influenza Epidemic
http://www.illustrativemathematics.org/illustrations/637
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Common Core State Standards: Progressions An epidemic of influenza spreads through a city. The figure below is the graph of I=f(w) , where I is the number of individuals (in thousands) infected w weeks after the epidemic begins.
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(Task from Functions Modeling Change: A Preparation for Calculus, Connally et al., Wiley 2010.)
Common Core State Standards: Progressions
1.Estimate f(2) and explain its meaning in terms of the epidemic.
2.Approximately how many people were infected at the height of the epidemic? When did that occur? Write your answer in the form f(a)=b .
(Task from Functions Modeling Change: A Preparation for Calculus, Connally et al., Wiley 2010.)
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Common Core State Standards: Progressions
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Common Core State Standards: Progressions
K-W-L
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PARCC Model Content FrameworksMathematics
GRADES 3–11
Version 3.0
November 2012
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NYS Next Generation Assessments
2012-13 NYS Grades 3-8 Math and ELA assessments are built on the Common Core within the constraints of the NYS testing system
Regents ELA and Math Examinations will be rolled out in 2013-14 and 2014-15
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NYS Next Generation Assessments
All curricular and professional development resources produced by the State Education Department will follow the PARCC MCF, as will State assessments beginning with the 2013-14 school year.
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PARCC Model Content Frameworks
Grade 8
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PARCC Model Content Frameworks
Algebra I
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PARCC Model Content Frameworks
Algebra II
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PARCC Model Content Frameworks
Why were the PARCC Model Content
Frameworks (MCF) developed?
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PARCC Model Content Frameworks
Dual Purpose
Although the primary purpose of the Model Content Frameworks is to provide a frame for the PARCC assessments, they also are voluntary resources to help educators and those developing curricula and instructional materials.
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PARCC Model Content Frameworks
The Model Content Frameworks for Mathematics for each grade is written with the expectation that students develop content knowledge, conceptual understanding and expertise with the Standards for Mathematical Practice. A detailed description of all features of the standards would be significantly lengthier and denser.
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PARCC Model Content Frameworks
The Model Content Frameworks for Mathematics provide guidance for grades 3-8 and high school in the following areas: •Examples of key advances from the previous grade; •Fluency expectations or examples of culminating standards; •Examples of major within-grade dependencies; •Examples of opportunities for connections among standards, clusters or domains;
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PARCC Model Content Frameworks
The Model Content Frameworks for Mathematics provide guidance for grades 3-8 and high school in the following areas: •Examples of opportunities for in-depth focus; •Examples of opportunities for connecting mathematical content and mathematical practices; and •Content emphases by cluster.
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Key Elements of thePARCC Model Content Framework
Illinois State Board of Education
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High SchoolCourse
IndividualEnd of Course
Overviews
Pathway Summary
Tables
Assessment Limits Tables for
Standards Assessed in
More than One Course
Mathematical Practices in Relation to Course Content
Key Advances
Fluency Recommendations
PARCC Model Content Frameworks
Individual End-of-Course Overviews
Each overview shows which standards are assessed on a given end-of-course assessment as well as relative cluster emphases.
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PARCC Model Content Frameworks
Individual End-of-Course Overviews
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PARCC Model Content Frameworks
Key Advances
This category highlights some of the major steps in the progression of increasing knowledge and skill from year to year.
Note that each key advance in mathematical content also corresponds to a widening scope of problems that students can solve.
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PARCC Model Content Frameworks
Math Practices in Relation to Course Content
This category highlights some of the mathematical practices and describes how they play a role in each course. These examples are provided to stress the need to connect content and practices, as required by the standards.
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PARCC Model Content Frameworks
Math Practices in Relation to Course Content
Modeling with mathematics is a theme in all high school courses. Modeling problems in high school center on problems arising in everyday life, society, and the workplace.
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PARCC Model Content Frameworks
Fluency Recommendations
The high school standards do NOT set explicit expectations for fluency nor will the PARCC assessments address fluency, but fluency is important in high school mathematics. For example, fluency in algebra can help students get past the need to manage computational details so that they can observe structure and patterns in problems.
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PARCC Model Content Frameworks
Fluency Recommendations
This section makes recommendations about fluencies that can serve students well as they learn and apply mathematics. These fluencies are highlighted to stress the need for curricula to provide sufficient supports and opportunities for practice to help students gain fluency.
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PARCC Model Content Frameworks
Pathway Summary Tables
Each pathway summary table shows three end-of-course assessments’ standards at a glance. For each non-(+) high school standard, the end-of-course assessment(s) assessing the standard are shown by a dot ( ) symbol. Shading in the pathway summary table indicates high school standards that are appropriate for more than one end-of-course assessment.
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PARCC Model Content Frameworks
Pathway Summary Table
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PARCC Model Content Frameworks
Assessment Limits Table for Standards Assessed on More than one End-of-Course Test
When a high school standard is appropriate for more than one end-of-course test in a given pathway, the need arises to specify just how the assessment of the standard will differ for students in each successive course. This information is provided in the assessment limits tables.
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PARCC Model Content Frameworks
Assessment Limits Table for Standards Assessed on More than one End-of-Course Test
In general, the approach to striking this balance has been to set stricter limits on standards relating to procedural skill and to set less strict limits on standards relating to conceptual understanding and exploration.
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PARCC Model Content Frameworks
Assessment Limits Table for Standards Assessed on More than one End-of-Course Test
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CCSSM Cluster CCSSM Key CCSSM Standard Algebra I Assessment Limits and Clarifications
Algebra II Assessment Limits and Clarifications
Interpret the structure of expressions
A-SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
i) Tasks are limited to numerical expressions and polynomial expressions in one variable.
ii) Examples: Recognize 532 472 as a difference of squares and see an opportunity to rewrite it in the easier-to-evaluate form (5347)(5347). See an opportunity to rewrite a2 9a 14 as (a7)(a2).
i) Tasks are limited to polynomial, rational, or exponential expressions.
ii) Examples: see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). In the equation x2+ 2x + 1 + y2 = 9, see an opportunity to rewrite the first three terms as (x+1)2, thus recognizing the equation of a circle with radius 3 and center (1,0). See (x2 + 4)/(x2 + 3) as ( (x2+3) + 1 )/(x2+3), thus recognizing an opportunity to write it as 1 + 1/(x2 + 3).
PARCC Model Content Frameworks
ACTIVITY
You have been asked to provide an overview of the PARCC Model Content Frameworks document to a community partner/organization (e.g., workforce development board, PTA, Chamber of Commerce, Kiwanis, college faculty)
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PARCC Model Content Frameworks
ACTIVITY (cont’d)
Work with team members to decide how you will deliver your message. This activity is meant to be open-ended, except each group must clearly provide several talking points based on your particular course.
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PARCC Model Content Frameworks
The Model Content Frameworks do NOT contain a suggested scope and sequence by quarter. Rather, they provide examples of key content dependencies (where one concept ought to come before another), key instructional emphases, opportunities for in-depth work on key concepts, and connections to critical practices.
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The PARCC MCF Connection to Assessments
Algebra I focuses on linear, quadratic, and exponential functions with domain in the integers. It also suggests work with the piecewise functions (including step and absolute value), square root and cube root in several standards.
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The PARCC MCF Connection to Assessments
The PARCC MCF does state, “In Algebra I, students will master linear and quadratic functions.” This implies that despite the exposure to some of the more advanced functions, the majority of the time should be spent on linear and quadratic.
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CCSSM Appendix A
The Common Core State Standards Appendix A is posted online at:
http://www.corestandards.org/assets/CCSSI_Mathematics_Appendix_A.pdf
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CCSSM Appendix A
The pathways and courses are models. They provide possible approaches to organizing the content of the CCSSM into coherent and rigorous courses.
States and districts are not expected to adopt these courses as is; rather, they are encouraged to use these pathways and courses as a starting point for developing their own.
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CCSSM Appendix A
Units within each course are intended to suggest a possible grouping of the standards into coherent blocks.
The ordering of the clusters within a unit follows the order of the standards document in most cases, not the order in which they might be taught. Attention to ordering content within a unit will be needed as instructional programs are developed.
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PARCC Model Content Frameworks
ACTIVITY
Using the same groups from the last activity, you are going to compare two documents, the PARCC MCF to the CCSSM Appendix A for each of the three high school courses. Again we are going to focus on the traditional pathway.
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PARCC Model Content Frameworks
ACTIVITY (cont’d
Examine the two documents for similarities and differences for each high school mathematics course. Once your team has completed the analysis, locate others who reviewed the same course and compile your list of similarities and differences and record electronically.
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PARCC Model Content Frameworks
Key Question
“What information needs to be shared with your mathematics department(s) and the larger school community?”
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High School Illustrative Sample Item
http://www.parcconline.org/samples/mathematics/high-school-seeing-structure-quadratic-equation
Seeing Structure in a Quadratic Equation
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High School Illustrative Sample Item
http://www.parcconline.org/samples/mathematics/high-school-seeing-structure-quadratic-equation
High School Sample Illustrative Item: Seeing Structure in a Quadratic Equation
1.Using the Overview of Task Type Slide (slide 91), identify the Type of Task the high school illustrative sample item represents.
2.Identify the most relevant Content Standard(s) this problems aligns to? Explain your reasoning.
3.Identify the most relevant Standards for Mathematical Practice(s) does this align to? Explain your reasoning.
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Additional Resources
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The Mathematics Common Core Toolbox
The Charles A. Dana Center at the University of Texas at Austin and Agile Mind, Inc.
This site is a resource designed to support districts working to meet the challenge and the opportunity of the new standards. Here you will find tools and instructional materials that help you to better understand and to implement the CCSSM.
Additional Resources
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The Mathematics Common Core Toolbox
Key Visualizations
•Algebra I•Geometry•Algebra II
Additional Resources
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The Mathematics Common Core Toolbox
PARCC Prototyping Project
High School Tasks•Cellular growth• Golf balls in water• Isabella’s credit card• Rabbit populations• Transforming graphs of quadratic functions
ASCD Educational Leadership
Teaching Like a Four-Star Chef
by Carol Ann Tomlinson
Don’t confuse the ingredients with the dinner.
You need to design the recipe – you may decide to use the ingredients in a different way than someone else.
Outstanding teachers have “developed the art of making elegant dinners that incorporate, but are not limited to, prescribed ingredients.”
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Thank You!
Don’t forget to complete your +s and s for the day. Tomorrow we will be focusing on Modeling and Dr. Eric Robinson will join us to share his expertise and the work he is doing at the national level.
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