Connecticut Core Standards for Mathematics

1. Connecticut Core Standards for Mathematics Systems of Professional Learning Module 2 Grades 6-12: Focus on Content Standards

2. Focus on Standards for Mathematical Content Outcomes • By the end of this session you will have: • Strengthened your working relationship with peer Core Standards Coaches. • Deepened your understanding of the practice standards specified in the CCS-Math. • Examined the implications of the language of the content standards for teaching and learning. • Identified and modified CCS-aligned instructional tasks that combine both the content and practice standards. • Analyzed the progression of topics in the content standards, both within and across grade levels.

3. Focus on Standards for Mathematical Content Outcomes (cont'd) • By the end of this session you will have: • Deepened your understanding of the potential of the CCS-Math to change mathematics teaching and learning. • Gained an understanding of some of the challenges involved in implementing the CCS-Math. • Explored strategies for supporting teachers as they make changes to their classroom practices. • Made plans for next steps in your CCS-Math implementation.

4. Today’s Agenda Morning Session • Welcome and Introductions • Sharing Implementation Experiences • The Language of the Content Standards • The Progression of the Content Standards Afternoon Session • Meeting the Expectations of the Content Standards through Cognitively Rigorous Tasks • Supporting Change • Next Steps Post-Assessment, Session Evaluation, & Wrap Up

5. Introductory Activity:Pre-Assessment – CCS-Math • Please complete the Pre-Assessment Page 4

6. Sharing Implementation Experiences • Section 1 Page 6

7. Coherence Instructional Shifts for Mathematics Rigor Focus • The Standards for Mathematical Content • The Standards for Mathematical Practice Two Areas

8. Focus on the major work for each grade Focus

9. Coherence The Standards are designed around coherent progressions and conceptual connections Analyze proportional relationships and use them to solve real-world and mathematical problems. Understand the connections between proportional relationships, lines, and linear equations. Create equations that describe numbers or relationships.

10. Coherence The CCS-Math are designed around coherent progressions and conceptual connections All Roads Lead to Algebra…… Math Concept Progression K-12

11. The major topics at each grade level focus on: Rigor • CONCEPTUAL UNDERSTANDING • More than getting answers • Not just procedures • Accessing concepts to solve problems • PROCEDURAL SKILL • AND FLUENCY • Speed and accuracy • Used in solving more complex problems • Supported by conceptual understanding • APPLICATION OF MATHEMATICS • Using math in real-world scenarios • Choosing concepts without prompting

12. Developing Mathematical Expertise The Standards for Mathematical Practice • Make sense of problems and persevere in solving them • Reason abstractly and quantitatively • Construct viable arguments and critique the reasoning of others • Model with mathematics • Use appropriate tools strategically • Attend to precision • Look for and make use of structure • Look for and express regularity in repeated reasoning

13. Sharing Experiences Implementing CCS-Math Practice Standards Positive Highlights Challenges Lessons Learned Page 6

14. The Language of the Content Standards • Section 2 Page 9

15. What Do These Students Understand? Part 1 Pages 9-12

16. From the CCSS-M Authors Watch Video

17. The major topics at each grade level focus on: Rigor • CONCEPTUAL UNDERSTANDING • More than getting answers • Not just procedures • Accessing concepts to solve problems • PROCEDURAL SKILL • AND FLUENCY • Speed and accuracy • Used in solving more complex problems • Supported by conceptual understanding • APPLICATION OF MATHEMATICS • Using math in real-world scenarios • Choosing concepts without prompting

18. Conceptual Understanding “Conceptual understanding refers to an integrated and functional grasp of mathematical ideas.” (Adding it Up: Helping Children Learn Mathematics, 2001)

19. Conceptual Understanding Example CCSS.Math.Content.6.NS.A.1Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. Question: What is (3/4) ÷ (1/8)?

20. Conceptual Understanding Question: What is (3/4) ÷ (1/8)? Student Response: I got the answer 6 by flipping the 2nd fraction over and then multiplying across the top and across the bottom.

21. Conceptual Understanding Example Standard 7.RP.2a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. Question: Josh is 10 years old and Reina is 7. Explain whether or not you can use a proportion to find Reina’s age when Josh is 18. Student Response: In 8 years, Reina will be 15. You can’t use a proportion because the ratio of their ages isn’t constant.

22. Procedural Skill and Fluency “Procedural skill and fluency is demonstrated when students can perform calculations with speed and accuracy.” (Achieve the Core) “Fluency promotes automaticity, a critical capacity that allows students to reserve their cognitive resources for higher-level thinking.” (Engage NY)

23. Procedural Skill and Fluency Which steps can be used to solve for the value of y? A. divide both sides by , then subtract 57 from both sides B. subtract 57 from both sides, then divide both sides by C. multiply both sides by , then subtract 57 from both sides D. subtractfrom both sides, then subtract 57 from both sides http://www.engageny.org/sites/default/files/resource/attachments/grade_7_math_released_questions.pdf

24. Procedural Skill and Fluency Example Standard 7.EE.4a: Solve word problems leading to equations of the form px + q = r and p(x+q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. Question: A rectangle has a perimeter of 54 cm. Its length is 6 cm. What is its width? Student Response: I know that the length and width add up to 27. The width has to be 19 because 27 – 6 = 19.

25. Application of Mathematics • The Standards call for students to use math flexibly for applications. • Teachers provide opportunities for students to apply math in authentic contexts. • Teachers in content areas outside of math, particularly science, ensure that students are using math to make meaning of and access content. Frieda & Parker, 2012 Achieve the Core, 2012

26. Application of Mathematics Example Sophia’s dad paid $43.25 for 12.5 gallons of gas. What is the cost of one gallon of gas? Retrieved from Illustrative Mathematics http://www.illustrativemathematics.org/

27. What Do These Students Understand? Part 2 Pages 9–12

28. Think About It… • How does the approach of the CCS-Math content differ from previous approaches to mathematics teaching and learning? • How might you help teachers to understand these differences?

29. Let’s Take A Break… …Be back in 10 minutes

30. The Progression of the Content Standards • Section 3 Page 16

31. The Organization of the Standards

32. Domain Distribution http://www.definingthecore.com

33. Domain Progression For More Information: http://commoncoretools.me/category/progressions/

34. Exploring the Content Standards Ratios and Proportional Relationships and Functions The Number System Expressions and Equations Geometry Statistics and Probability

35. Exploring the Content Standards 8.EE.3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than th eother. 8.EE.2 Apply Use square root and cube root symbols to represent solutions to equations of the form x2=p wheand x3=p where p is a positive rational number. 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. Page 16

36. Explore the Content Standards 8.NS.1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeat eventually… 7.NS.1 Apply and extend previous understandings of addition and subtractionto add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 6.NS.1 Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g. by using visual fraction models and dquations to represent the problem. Page 17

37. Exploring the Content Standards 7.EE.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 7.NS.2 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 7.RP.2 Recognize and represent proportional relationships between quantities. Page 18

38. From the Authors Watch Video

39. Reflect • How might you help teachers at your school to fully understand the progressions of the content standards? • What questions do you anticipate teachers having about the content standards? Page 19

40. Bon Appétit

41. Meeting the Expectations of the Content Standards by Teaching with Cognitively Rigorous Tasks • Section 4 Page 21

42. Math Class Needs a MakeoverDan Meyer Watch Video

43. Kites Activity A store sells kits to make kites. All the kites are quadrilaterals. Some are what we call “kite-shaped.” Others are rectangles, squares, rhombi, and four sided shapes with no particular characteristics. A kit has string, paper, and two sticks to form the skeleton of the kite. The store owner needs to know what sticks to put in the kits for each shape, and how to tell the purchaser how to put the sticks together for each shape. Your job is to give the store owner information about making squares, rectangles, trapezoids, and typical kite shapes. For each shape, list the sticks needed and how they should be put together. Use the paper strips as your sticks and connect them using the brads to make your kite shapes.

44. Take a Look… Sam uses one-inch frames for pictures for which the length is 2 inches longer than the width, as shown. Write an algebraic expression for the area of the picture alone. Write an algebraic expression for the area of the picture and frame together. c. If the area of a frame is 24 square inches, what are the dimensions of the picture?

45. Take a Look… Sam uses one-inch frames for pictures for which the length is 2 inches longer than the width as shown. If the area of a frame is 24 square inches, what are the dimensions of the picture?

46. Take a Look… Sam uses one-inch frames for pictures for which the length is 2 inches longer than the width. If the area of a frame is 24 square inches, what are the dimensions of the picture?

47. Take a Look… Sam uses one-inch frames for pictures for which the length is 2 inches longer than the width. If the picture’s dimensions are both whole numbers, show that the area of the frame has to be a multiple of 4.

48. The BIG Question How can I help teachers incorporate cognitively rigorous mathematics tasks that will benefit ALL students?

49. Strategies for Differentiating Cognitively Rigorous Tasks Scaffolding Open Questions Parallel Tasks C-R-A Page 23

50. Scaffolding A circle has its center at (6, 7) and goes through the point (1, 4). A second circle is tangent to the first circle at the point (1, 4) and has the same area. What are the coordinates for the center of the second circle? Show your work or explain how you found your answer. • What can be added to the problem? • What can happen during the implementation?