1. Instructional Leadership Math Cadre 6 th – 12 th Grade SHIFT 2: COHERENCE THE PROE CENTER

2. Multi-Tiered System of Support (MTSS/RtI) Statewide System of Support Priority Focus Foundational Focus Areas: -Continuous Improvement Process (Rising Star) -Common Core ELA -Common Core Math -Teacher Evaluation -Balanced Assessment

3. Commitments

4. Today’s Outcomes  Use the Standards for Mathematical Practice while problem solving  Define coherence in the Common Core  Identify coherence within the standards  Utilize coherent problem solving structures that build conceptual understanding  Describe instructional strategies for computation  Explore resources that build coherence  Plan for implementation

5. Today’s Agenda  Review Shift 1: Focus  Digging Deeper with the Standards for Mathematical Practice  Shift 2: Coherence  Supporting Diverse Learners: Modes of Representation  Problem Solving Structures  Coherent instructional resources  Common Core Math in the Media

6. Shift 1: FocusREVIEW

7. Focusing on Solving Problems USING THE STANDARDS FOR MATHEMATICAL PRACTICE

8. Solving Problems 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.

9. Debriefing the Activity Discuss: 1) The content 2) The math practices

10. Shift 2: Coherence

11. Coherence

12. Mike McCallum The Importance of Coherence in Math

13. Coherence  Math should make sense  A progression of learning  Coherence supports focus  Use supporting material to teach major content Math should make sense Within a grade level Across many grade levels

14. Coherence WITHIN a grade level The standards within a grade level strategically allow:  Instruction that reinforces major content and utilizes supporting standards Important to remember:  Meaningful introduction to topics so that skills complement one another

15. Coherence WITHIN a grade level Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.2.MD.5 Draw a scaled picture graph and a scaled bar graph to represent a data set with several categories. Solve one- and two- step “how many more” and “how many less” problems using information presented in scaled bar graphs. 3.MD.3 Geometric measurement: understand concepts of area and relate area to multiplication and addition. 3.MD.3 rd cluster Make a line plot to display a data set of measurements in fractions of a unit ( ½, ¼, 1/8). Solve problems involving addition and subtraction of fractions by using information presented in line plots. 4.MD.4

16. Coherence ACROSS grade levels  Students apply skills from previous grade levels to learn new topics in their current grade level  Meaningful math progressions reflect this, building knowledge across the grade levels

17. Coherence ACROSS grade levels One of several staircases to algebra designed in the OA domain.

18. Coherence ACROSS grade levels 4.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. 5.NF.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction. 5.NF.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 6.NS. Apply and extend previous understandings of multiplication and division to divide fractions by fractions. 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 equations to represent the problem. Grade 4 Grade 5 Grade 6 CCSS

19. Coherence Activity

20. Learning Progression: A pathway or stream students travel as they progress toward mastery of the skills. These streams contain carefully sequenced building blocks (content standards) that support students’ progression towards mastery.

21. Benefits of Progressions/Streams  Enables teachers to build instructional sequences  Provide a framework to systematically implement effective formative assessment

22. Common Core Progression Streams 1. Counting and Cardinality (K) 2. Algebraic Thinking (K-HS) 3. Number and Quantity 4. Geometry 5. Functions 6. Statistics and Probability 7. High School Modeling Visual Map

24. Activity: Progression Jigsaw 1) Read the intro for your designated Progression 2) Read your grade level in the designated Progression, highlighting key skills for your grade level 3) Create a flow-chart of skills, visually depicting how the skills in each grade level build on one another in developing conceptual understanding within the Progression (domain) 4) Creatively share your learning progression with the rest of the group

25. Modes of Representation

26. Manipulatives or Tools Pictures/ Graphs Written Symbols Oral/Written Language Real-Life Situations

27. Building Conceptual Understanding Adding Modes of Representation 4545 x 24 x 24

28. Problem Solving Structures WRITE A SIMPLE MULTIPLICATION/ DIVISION WORD PROBLEM.

29. Addition & Subtraction Result UnknownChange UnknownStart Unknown Add to Two bunnies sat on the grass. Three more bunnies hopped there. How many bunnies are on the grass now? 2 + 3 = ? Two bunnies were sitting on the grass. Some more bunnies hopped there. Then there were five bunnies. How many bunnies hopped over to the first two? 2 + ? = 5 Some bunnies were sitting on the grass. Three more bunnies hopped there. Then there were five bunnies. How many bunnies were on the grass before? ? + 3 = 5 Take From Five apples were on the table. I ate two apples. How many apples are on the table now? 5 – 2 = ? Five apples were on the table. I ate some apples. Then there were three apples. How many apples did I eat? 5 – ? = 3 Some apples were on the table. I ate two apples. Then there were three apples. How many apples were on the table before? ? – 2 = 3 Total UnknownAddend UnknownBoth Addends Unknown Put Together/ Take Apart Three red apples and two green apples are on the table. How many apples are on the table? 3 + 2 = ? Five apples are on the table. Three are red and the rest are green. How many apples are green? 3 + ? = 5, 5 – 3 = ? Grandma has five lowers. How many can she put in her red vase and how many in her blue vase? 5 = 0 + 5, 5 = 5 + 0 5 = 1 + 4, 5 = 4 + 1 5 = 2 + 3, 5 = 3 + 2 Difference UnknownBigger UnknownSmaller Unknown Compare (“How many more?” version): Lucy has two apples. Julie has five apples. How many more apples does Julie have than Lucy? (“How many fewer?” version): Lucy has two apples. Julie has five apples. How many fewer apples does Lucy have than Julie? 2 + ? = 5, 5 – 2 = ? (Version with “more”): Julie has three more apples than Lucy. Lucy has two apples. How many apples does Julie have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Lucy has two apples. How many apples does Julie have? 2 + 3 = ?, 3 + 2 = ? (Version with “more”): Julie has three more apples than Lucy. Julie has five apples. How many apples does Lucy have? (Version with “fewer”): Lucy has 3 fewer apples than Julie. Julie has five apples. How many apples does Lucy have? 5 – 3 = ?, ? + 3 = 5

30. Multiplication & Division Unknown ProductGroup Size UnknownNumber of Groups Unknown 3 x 6 = ?3 x ? = 18, and 18 ÷ 3 = ?? x 6 = 18, and 18 ÷ 6 = ? Equal Groups There are 3 bags with 6 plums in each bag. How many plums are there in all? Measurement example. You need 3 lengths of string, each 6 inches long. How much string will you need altogether? If 18 plums are shared equally into 3 bags, then how many plums will be in each bag? Measurement example. You have 18 inches of string, which you will cut into 3 equal pieces. How long will each piece of string be? If 18 plums are to be packed 6 to a bag, then how many bags are needed? Measurement example. You have 18 inches of string, which you will cut into pieces that are 6 inches long. How many pieces of string will you have? Arrays, Area There are 3 rows of apples with 6 apples in each row. How many apples are there? Area example. What is the area of a 3 cm by 6 cm rectangle? If 18 apples are arranged into 3 equal rows, how many apples will be in each row? Area example. A rectangle has area 18 square centimeters. If one side is 3 cm long, how long is a side next to it? If 18 apples are arranged into equal rows of 6 apples, how many rows will there be? Area example. A rectangle has area 18 square centimeters. If one side is 6 cm long, how long is a side next to it? Compare A blue hat costs $6. A red hat costs 3 times as much as the blue hat. How much does the red hat cost? Measurement example. A rubber band is 6 cm long. How long will the rubber band be when it is stretched to be 3 times as long? A red hat costs $18 and that is 3 times as much as a blue hat costs. How much does a blue hat cost? Measurement example. A rubber band is stretched to be 18 cm long and that is 3 times as long as it was at first. How long was the rubber band at first? A red hat costs $18 and a blue hat costs $6. How many times as much does the red hat cost as the blue hat? Measurement example. A rubber band was 6 cm long at first. Now it is stretched to be 18 cm long. How many times as long is the rubber band now as it was at first? General a x b = ?a x ? = p, ad p ÷ a = ?? x b = o, and p ÷ b = ?

31. Problem Solving Structures WHICH PROBLEM SOLVING STRUCTURE DID YOU USE?

32. Instructional Strategies FOR MULTIPLICATION

33. Additive Properties  Repeated Addition  Skip Counting

34. Distributive Property  Area/Area  Partial Product  Box method  Decomposing

35. Advanced Strategies  Double and Halve  (Standard Algorithm)

36. Coherence in Action VIEWS YOU CAN USE

37. Dan Meyer Math Class Needs a Make-over

38. Dan Meyer Resources Dan Meyer creates Coherence for teachers, too. Three-Act Math Tasks on Livebinders

39. Math Common Core in the Media

40. Common Core in the Media

41. Planning for Next Steps TAKING IT BACK TO MY CLASSROOM SHARING WITH MY COLLEAGUES

42. Comments… Questions… Concerns…

43. Cindy Dollman – cdollman@peoriaroe48.netcdollman@peoriaroe48.net Joe Delinski – jdelinski@peoriaroe48.netjdelinski@peoriaroe48.net Kim Glow – kglow@peoriaroe48.netkglow@peoriaroe48.net