1. 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

2. Introductions Introduce yourself: Name Instructional Level On a scale of 1 – 5, with 1 representing very little knowledge 5 representing expert knowledge where do you lie with respect to an understanding of the eight Standards for Mathematical Practice?

3. 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

4. Common Core The 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.

5. Common Core Standards for Mathematical Practice Standards for Mathematical Content Same for All Grade Levels Specific to Grade Level

6. Grade 7 Overview

7. Grade 8 Overview

8. High School Overview

9. Structure of Common Core Content Standards K - 5 DomainK12345 Counting and Cardinality Operations and Algebraic Thinking Numbers and Operations in Base Ten Numbers and Operations Fractions Measurement and Data Geometry

10. Structure of Common Core Content Standards 6 - 8 Domain678 Ratio and Proportional Relationships The Number System Expressions and Equations Functions Geometry Statistics and Probability

11. 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

12. Structure of Common Core Content Standards HS Number and Quantity Overview The Real Number System Quantities The Complex Number System Vector and Matrix Quantities

13. Structure of Common Core Content Standards HS Algebra Overview Seeing Structures in Expressions Arithmetic with Polynomials and Rational Expressions Creating Equations Reasoning with Equations and Inequalities

14. Structure of Common Core Content Standards HS Functions Overview Interpreting Functions Building Functions Linear, Quadratic, and Exponential Models Trigonometric Functions

15. Structure of Common Core Content Standards HS Geometry Overview Congruence Similarity, Right Triangles, and Trigonometry Circles Expressing Geometric Properties with Equations Geometric Measurement and Dimension Modeling with Geometry

16. Structure of Common Core Content Standards HS 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

17. Eight Standards for Mathematical Practice Describe practices that mathematics educators should seek to develop in their students NCTM Process Standards Problem Solving Reasoning and Proof Communication Representation Connections Natl. Resource Council Adding it Up Adaptive Reasoning Strategic Competence Conceptual Understanding Procedural Fluency Productive Disposition

18. 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

19. 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!

20. Quick Common Core Assessment Overview Summative Multi-state Assessment Resources for Teachers and Educational Researchers SMARTER Balanced Assessment Consortium (SBAC)

21. Quick Common Core Assessment Overview SBAC Summative Assessments Computer Adaptive Testing (CAT) Performance Events

22. Quick Common Core Assessment Overview 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.

23. Quick Common Core Assessment Overview Performance Events In-depth performance task Will require students to think critically in order to solve a non- traditional problem Interpret a situation Develop a plan Communicate the solution

24. Quick Common Core Assessment Overview * 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

25. The International Center for Leadership in Education Rigor/Relevance Framework

26. Thinking Continuum Acquisition of Knowledge Assimilation of Knowledge

27. Knowledge Taxonomy  Awareness  Comprehension  Analysis  Synthesis  Evaluation

28. Action Continuum Acquisition of Knowledge Application of Knowledge

29. Application Model  Knowledge in one discipline  Application within discipline  Application across disciplines  Application to real-world predictable situations  Application to real-world unpredictable situations

30. 12345 Application Knowledge 1 2 3 4 5 6

31. 1 2 3 4 5 6 12345 A

32. 1 2 3 4 5 6 1245 A B 3

33. 1 2 3 4 5 6 1245 A B C 3

34. 1 2 3 4 5 6 1245 A B C 3 D

35. 1 2 3 4 5 6 1245 A B C 3 D

36. A B C D KNOWLEDGEKNOWLEDGE A P P L I C A T I O N

37. A B C D KNOWLEDGEKNOWLEDGE 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.

38. A B C D KNOWLEDGEKNOWLEDGE A P P L I C A T I O N Analyze the graphs of the perimeters and areas of squares haing 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. 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.

39. A B C D KNOWLEDGEKNOWLEDGE A P P L I C A T I O N 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.

40. A B C D KNOWLEDGEKNOWLEDGE A P P L I C A T I O N 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. 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. 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.

41. A B C D KNOWLEDGEKNOWLEDGE A P P L I C A T I O N 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. 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. 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.

42. Standards for Mathematical Practice Students 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.

43. 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 solution Analyze givens, constraints, relationships and goals Make conjectures about the form and meaning of the solution Plan a solution pathway rather than simply jump into a solution attempt Consider analogous problems Try special cases and simpler forms of original problem

44. 1. Make Sense of Problems and Persevere in Solving 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

45. 1. Make Sense of Problems and Persevere in Solving Mathematically proficient students: 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? Similarities Differences

46. 2. Reason Abstractly and Quantitatively Mathematically proficient students: 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 contextualize To pause as needed during the symbolic manipulation in order to look back at the referent values in the problem

47. 2. Reason Abstractly and Quantitatively Mathematically proficient students: Reason Quantitatively, which entails habits of: Creating a coherent representation of the problem at hand Considering the units involved Attending to the meaning of quantities, not just how to compute them Knowing and flexibly using different properties of operations and objects

48. 3.Construct viable arguments and critique the reasoning of others Mathematically proficient students: Understand and use… stated assumptions, definitions, and previously established results… when constructing arguments

49. 3.Construct viable arguments and critique the reasoning of others Mathematically proficient students: Make conjectures and build a logical progression of statements to explore the truth of their conjectures. Able to analyze situations by breaking them into cases by recognizing and using counterexamples Justify their conclusions, communicate to others, and respond to the arguments of others

50. 3.Construct viable arguments and critique the reasoning of others Mathematically proficient students: 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

51. 3.Construct viable arguments and critique the reasoning of others Mathematically proficient students: Can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments

52. 4.Model with Mathematics Mathematically 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.

53. 4.Model with Mathematics Modeling 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.

54. 4.Model with Mathematics Examples 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

55. 4.Model with Mathematics Examples 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

56. 4.Model with Mathematics Models 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?

57. 4.Model with Mathematics Models 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

58. 4.Model with Mathematics Powerful tools for modeling: Diagrams of various kinds Spreadsheets Graphing technology Algebra

59. 4.Model with Mathematics Basic Modeling Cycle Problem Formulate Compute Interpret Validate Report

60. 4.Model with Mathematics Basic Modeling Cycle Problem Identify variables in the situation Select those that represent essential features

61. 4.Model with Mathematics Basic Modeling Cycle Formulate Select or create a geometrical, tabular, algebraic, or statistical representation that describes the relationships between the variables

62. 4.Model with Mathematics Basic Modeling Cycle Compute Analyze and perform operations on these relationships to draw conclusions

63. 4.Model with Mathematics Basic Modeling Cycle Interpret Interpret the result of the mathematics in terms of the original situation

64. 4.Model with Mathematics Basic Modeling Cycle Validate Validate the conclusions by comparing them with the situation…

65. 4.Model with Mathematics Basic Modeling Cycle EITHER OR Validate Re - Formulate Report on conclusions and reasoning behind them

66. 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:

67. 5.Use appropriate tools strategically Mathematically proficient students are: Sufficiently familiar enough with the tools for their grade level to Know how to use them Know what is to gain by using them Know their limitations

68. 5.Use appropriate tools strategically Mathematically proficient students can: Analyze graphs and solutions from graphing calculators Can detect possible errors through estimation and other mathematical knowledge

69. 5.Use appropriate tools strategically 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

70. 5.Use appropriate tools strategically 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

71. 6.Attend to precision Mathematically proficient students; Try to communicate precisely to others Use clear definitions State the meaning of symbols they use Use the equal sign consistently and appropriately Specify units of measure Label axes

72. 6.Attend to precision Mathematically proficient students; Try to communicate precisely to others Calculate accurately and efficiently Express numerical answers with a degree of precision appropriate for the problem context Give carefully formulated explanations to each other Can examine claims and make explicit use of definitions

73. 7. Look for and make use of structure Mathematically proficient students; Look closely to discern a pattern or structure In x 2 + 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

74. 8. Look for and express regularity in repeated reasoning. Mathematically proficient students; Notice if calculations are repeated Look for both general methods and for shortcuts Maintain oversight of the process while attending to the details.

75. 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