Mathematics Grades 6-12 February NTI February 4, 2013 EngageNY.org 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 EngageNY.org 2 Objectives Gain 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. EngageNY.org 3 “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 EngageNY.org 4 Standards for Mathematical Practice GRADES 6-12 Teri Calabrese-Gray, CVES Assistant Superintendent EngageNY.org 5 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? EngageNY.org 6 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. EngageNY.org 7 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. EngageNY.org 8 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. EngageNY.org 9 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. EngageNY.org 10 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. EngageNY.org 11 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. EngageNY.org 12 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. EngageNY.org 13 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. EngageNY.org 14 Standards for Mathematical Practice MP 1: Make sense of problems and persevere in solving them. EngageNY.org 15 Standards for Mathematical Practice MP 2: Reason abstractly and quantitatively. EngageNY.org 16 Standards for Mathematical Practice MP 3: Construct viable arguments and critique the reasoning of others. EngageNY.org 17 Standards for Mathematical Practice MP 4: Model with mathematics. EngageNY.org 18 Standards for Mathematical Practice MP 5: Use appropriate tools strategically. EngageNY.org 19 Standards for Mathematical Practice MP 6: Attend to precision. EngageNY.org 20 Standards for Mathematical Practice MP 7: Look for and make use of structure. EngageNY.org 21 Standards for Mathematical Practice MP 8: Look for and express regularity in repeated reasoning. EngageNY.org 22 Standards for Mathematical Practice PROBLEM What is the mean of 5, 8, 9, and 6? EngageNY.org 23 Standards for Mathematical Practice Identify the MPs that align with this problem. Discuss at your tables. EngageNY.org 24 Standards for Mathematical Practice Pose a different problem that could go deeper but require the same mathematical content knowledge? EngageNY.org 25 Standards for Mathematical Practice Explain what students might learn from the second question compared to the first question? EngageNY.org 26 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? EngageNY.org 27 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. EngageNY.org 28 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. EngageNY.org 29 Integrating the Standards for Mathematical Practice Inside +=x Mathematics For a more in-depth study of the Standards for Mathematical Practice, please visit http://www.insidemathematics.org/index. php/commmon-core-math-intro. EngageNY.org 30 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. EngageNY.org 32 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. EngageNY.org 33 Common Core State Standards: Progressions GRADES 6-12 Kristine S. Cole, SUNY Research Fund Fellow Teri Calabrese-Gray, CVES Assistant Superintendent EngageNY.org 34 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 EngageNY.org 35 Common Core State Standards: Progressions K-W-L EngageNY.org 36 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. EngageNY.org 37 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. EngageNY.org 38 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. EngageNY.org 39 Common Core State Standards: Progressions Grades 6-7 – Ratios and Proportional Relationships Grade 8 and High School – Functions EngageNY.org 40 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 EngageNY.org 41 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 EngageNY.org 42 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 EngageNY.org 43 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. EngageNY.org 44 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. EngageNY.org 45 Common Core State Standards: Progressions Grades 6-7 Ratios and Proportional Relationships • • • Grade 6 Grade 7 Overview EngageNY.org 46 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 EngageNY.org 47 Common Core State Standards: Progressions Influenza Epidemic http://www.illustrativemathematics.org/illust rations/637 EngageNY.org 48 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. (Task from Functions Modeling Change: A Preparation for Calculus, Connally et al., Wiley 2010.) EngageNY.org 49 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.) EngageNY.org 50 Common Core State Standards: Progressions EngageNY.org 51 Common Core State Standards: Progressions K-W-L EngageNY.org 52 PARCC Model Content Frameworks Mathematics GRADES 3–11 Version 3.0 November 2012 EngageNY.org 53 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 201415 EngageNY.org 54 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 201314 school year. EngageNY.org 55 PARCC Model Content Frameworks Grade 8 EngageNY.org 56 PARCC Model Content Frameworks Algebra I EngageNY.org 57 PARCC Model Content Frameworks Algebra II EngageNY.org 58 PARCC Model Content Frameworks Why were the PARCC Model Content Frameworks (MCF) developed? EngageNY.org 59 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. EngageNY.org 60 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. EngageNY.org 61 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; EngageNY.org 62 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. EngageNY.org 63 Key Elements of the PARCC Model Content Framework Key Advances Fluency Recommendations High School Course Individual End of Course Overviews Pathway Summary Tables Assessment Limits Tables for Standards Assessed in More than One Course Mathematical Practices in Relation to Course Content Illinois State Board of Education EngageNY.org 64 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. EngageNY.org 65 PARCC Model Content Frameworks Individual End-of-Course Overviews EngageNY.org 66 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. EngageNY.org 67 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. EngageNY.org 68 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. EngageNY.org 69 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. EngageNY.org 70 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. EngageNY.org 71 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. EngageNY.org 72 PARCC Model Content Frameworks Pathway Summary Table EngageNY.org 73 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. EngageNY.org 74 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. EngageNY.org 75 PARCC Model Content Frameworks Assessment Limits Table for Standards Assessed on More than one End-of-Course Test CCSSM Cluster CCSSM Key CCSSM Standard 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). Algebra I Assessment Limits and Clarifications Algebra II Assessment Limits and Clarifications i) Tasks are limited to numerical expressions and polynomial expressions in one variable. i) Tasks are limited to polynomial, rational, or exponential expressions. ii) Examples: see x4 – y4 as (x2)2 – ii) Examples: Recognize 532 - 472 as (y2)2, thus recognizing it as a a difference of squares and see an difference of squares that can be factored as (x2 – y2)(x2 + y2). In the opportunity to rewrite it in the 2 2 easier-to-evaluate form (53+47)(53- equation x + 2x + 1 + y = 9, see an opportunity to rewrite the first 47). See an opportunity to rewrite three terms as (x+1)2, thus 2 a + 9a + 14 as (a+7)(a+2). 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). EngageNY.org 76 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) EngageNY.org 77 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. EngageNY.org 78 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. EngageNY.org 79 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. EngageNY.org 80 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. EngageNY.org 81 CCSSM Appendix A The Common Core State Standards Appendix A is posted online at: http://www.corestandards.org/assets/CC SSI_Mathematics_Appendix_A.pdf EngageNY.org 82 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. EngageNY.org 83 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. EngageNY.org 84 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. EngageNY.org 85 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. EngageNY.org 86 PARCC Model Content Frameworks Key Question “What information needs to be shared with your mathematics department(s) and the larger school community?” EngageNY.org 87 High School Illustrative Sample Item http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation Seeing Structure in a Quadratic Equation EngageNY.org 88 High School Illustrative Sample Item http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation High School Sample Illustrative Item: Seeing Structure in a Quadratic Equation 1. Using the Overview of Task Type Slide, 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. EngageNY.org 89 High School Illustrative Sample Item http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation High School Sample Illustrative Item: Seeing Structure in a Quadratic Equation Task Type I: Tasks assessing concepts, skills and procedures Alignment: Most Relevant Content Standard(s) • A-REI.4. Solve quadratic equations in one variable. a) Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. EngageNY.org 90 High School Illustrative Sample Item http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation High School Sample Illustrative Item: Seeing Structure in a Quadratic Equation Alignment: Most Relevant Content Standard(s) • A-REI.4. Solve quadratic equations in one variable. b) Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula, and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi for real numbers a and b. EngageNY.org 91 High School Illustrative Sample Item http://www.parcconline.org/samples/mathematics/high-school-seeingstructure-quadratic-equation High School Sample Illustrative Item: Seeing Structure in a Quadratic Equation Alignment: Most Relevant Mathematical Practice(s) • Students taking a brute-force approach to this task will need considerable symbolic fluency to obtain the solutions. In this sense, the task rewards looking for and making use of structure (MP.7). EngageNY.org 92 Additional Resources 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. EngageNY.org 93 Additional Resources The Mathematics Common Core Toolbox Key Visualizations • Algebra I • Geometry • Algebra II EngageNY.org 94 Additional Resources 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 EngageNY.org 95 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.” EngageNY.org 96 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. EngageNY.org 97 EngageNY.org 98