1. 04.05.12 Module 2: Content and Curriculum, K-8 Mathematics

2. Agenda Common Core State Standards Overview Standards Progression: A Closer Look –Number Systems: K-5 Base Ten Operations 3-5 Fractions 6-8 Rational Numbers Mathematical Practices in Action Closing 2

3. Math Class Needs a Makeover 3

4. Common Core State Standards “These standards define what students should understand and be able to do in their study of mathematics.” National Governors Association Center for Best Practices and Council of Chief State School Officers (2010) Common Core State Standards for Mathematics 4

5. Two Types of Standards Standards for Mathematical Practice Eight standards common to all grade levels K-12 Mathematical Content Standards Standards that vary by grade level and conceptual category 5

6. Mathematical Habits of Mind “Mathematical habits of mind allow one to bring coherence to many different parts of the curriculum… Building coherence in the development of mathematical ideas across grades is key to improving students’ mathematical learning.” Cuoco, Goldenberg, & Mark (2010) Organizing a Curriculum around Mathematical Habits of Mind 6

7. Pedagogical Shifts with Common Core State Standards Focus Identifies key ideas, understandings, and skills for each grade or course Stresses deep learning, applying concepts and skills within the same grade or course Coherence Articulates a progression of topics across grades Connects ideas with other ideas, enabling students to learn with understanding, develop skill proficiency, and solve problems 7

8. Grade-Level Standards Locate the CCSS for Mathematics handout. Read the introduction for your grade level. Highlight critical focus areas and mark as follows: Expectations similar to CA 1997 Standards !New expectations ?Questions I have regarding these expectations Be ready to discuss. 8

9. Format of the Overview Domains: Overarching ideas that connect topics across the grades Clusters: Illustrate the progression of increasing complexity from grade to grade © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. 9

10. Format of the Standards Domain: Cluster: Standards: Define what students should know and be able to do for each grade level or course. © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. 10

11. Domains and Conceptual Categories Distribution Findell & Foughty (2011) College and Career-Readiness Through the Common Core State Standards for Mathematics 11

12. Example of Domain Flows Elementary to High School Calahan (2011) Delving Deeper into the Common Core Operations & Algebraic Thinking Expressions & Equations Algebra Number: Base Ten Number & Quantity Number: Fractions Geometry Measurement & Data Statistics & Probability Statistics & Probability Functions The Number System K - 56 - 89 - 12 12

13. Base Ten Place Value (K-5) 13

14. Base Ten Place Value (K-5) Chart Ones Tens Hundreds Thousands x 10 14

15. Base Ten Place Value (K-5) Chart Ones Tens Hundreds Thousands x 10 Tenths x 10 Hundredths x 10 Thousandths 15

16. Symmetry with Respect to the Ones Place Ones Tens Hundreds Thousands Tenths Hundredths Thousandths 16

17. Base Ten Place Value (K-5): Ones Tens Hundreds Thousands Tenths Hundredths Thousandths Fractional Notation 17

18. Base Ten Place Value (K-5): Ones Tens Hundreds Thousands Tenths Hundredths Thousandths Decimal Notation 1000 100 10 1 0.1 0.01 0.001 18

19. Base Ten Place Value (K-5): Ones Tens Hundreds Thousands Tenths Hundredths Thousandths 10 3 10 2 10 1 10 0 Exponential Notation 1000 = 10 3 0.001 = 1.0 x 10 -3 10 -1 10 -2 10 -3 19

20. Part 1: Number and Operations Base Ten Progression Grade K: Work with numbers 11-19 to gain foundation for place value. (K.NBT.1) Grade 1: Extend the counting sequence; understand place value. (1.NBT.2 Grade 2: Understand place value and use this understanding and properties of operations to add and subtract. (2.NBT.1) Grade 3 : Use place value understanding and properties of operations to perform multi-digit arithmetic. ( Grade 4: Generalize place value understanding for multi- digit whole numbers; use this understanding to perform multi-digit arithmetic. (4.NBT.1 – 4.NBT.3) Grade 5: Understand the place value system and perform operations with multi-digit whole numbers and with decimals to hundredths. (5.NBT.1 through 5.NBT.4) What part of the activity depends on the expectations of the K-2 domains? 20

21. Connecting to Current Practice Think/Pair/Share How does the place value demonstration with money highlight the progression of expectations in the K-5 Number and Operations in Base Ten Domain? How does the progression of standards within the Base Ten domain correspond to your current practice? What are some things you might do now to help students transition to the CCSS? 21

22. Fractions Domain (3-5) Solve this problem on your own. 22

23. Fraction Misconceptions 23

24. Video Debrief Fractions Misconceptions Think/Pair/Share What were this student’s strengths? What were his misunderstandings regarding the division of fractions? How might you address these misunderstandings? 24

25. “Represent a fraction a/b on a number line diagram by marking off the lengths 1/b from 0…” Represent 2/3 on this number line: Number and Operations: Fractions Progression (3-5) Standard 3.NF.2a 01 25

26. Number and Operations: Fractions Progression (3-5) Standard 4.NF.2 “Compare two fractions… by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2.” 01½ 26

27. Number and Operations: Fractions Progression (3-5) Standard 5.NF.2 “Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers.” 27 Determine whether the number sentence is correct without using common denominators. Explain how you know.

28. Number Systems Progression (6-8) Standard 6.NS.6c “Find and position integers and other rational numbers on a horizontal or vertical number line diagram…” 10 1 0 28

29. Number Systems Progression (6-8) Standards 7.NS.5 and 8.NS.2 “Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram…” 4 Where does this belong? 29

30. Number Systems Progression Area = 1 Area = 4 Where does this belong? 30

31. Number Systems Progression (6-8) Standards 7.NS.5 and 8.NS.2 “Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram…” 4 31

32. Common Core State Standards for Mathematical Practice 21 st Century Skills Communication Collaboration Creativity Critical Thinking Construction of Knowledge 32

33. Part 1: Standards for Mathematical Practice in Action Locate the MARS Tasks and Student Work handout. Complete the task on your own. Once task is completed, discuss the prompts: –What mathematical skills and understanding are needed to successfully complete the task? –Which Standards for Mathematical Practice did you use to solve the task? Use the Standards of Mathematical Practice Look-For Tool to identify practices used. 33

34. Part 2 Mathematical Practices in Action Using the MARS Tasks and Student Work handout, examine the student work provided for the task you completed. Determine which Standards for Mathematical Practice are represented in the student work. Compare your work with the student work. –How are the practices similar? –How are the practices different? More tasks available: –www.insidemathematics.orgwww.insidemathematics.org 34

35. Connecting to Current Practice Think/Pair/Share How does the progression of standards within the Number and Operations in Base Ten, Number and Operations in Fractions, and Number Systems domains correspond to your current practice? What are some things you might do now to help students transition to the CCSS? 35

36. Next Steps Think about what you learned today. Decide on one thing you will do differently to start transitioning to the CCSS. Share your ideas with a partner. 36

37. For Further Investigation California’s Common Core State Standards http://www.cde.ca.gov/ci/cc http://www.cde.ca.gov/ci/cc Common Core State Standards Initiative http://www.corestandards.org http://www.corestandards.org Tools for the Common Core (includes the Progressions) http://www.commoncoretools.wordpress.com http://www.commoncoretools.wordpress.com Inside Mathematics (MARS Tasks & Resources) www.insidemathematics.org www.insidemathematics.org K-5 Math Teaching Resources www.k-5mathteachingresources.com www.k-5mathteachingresources.com 37