Teaching Fractions with Learning Progressions

1. Kristin Frang, Mathematics Consultant MAISD Regional Math & Science Center Laura Wilson, Teacher Leader Vicksburg Community Schools Teaching Fractions with Learning Progressions

2. SeriesGoals • Explore a framework that balances conceptual understanding, procedural fluency and application. • Identify critical components to understanding fractions and instructional strategies that support their development. • Understand students’ development of core concepts and ways of reasoning about fractions

3. YourQuestions • How do I get in so much higher order thinking? • MAISD CCSSM wiki • Overwhelming, but every journey begins with a step.

4. Today’sAgenda • Review last session & homework • Major Emphasis in CCSS • SBAC Performance Task • Organizing Instruction • Levels 0-2 Student Reasoning LUNCH • Levels 3-4 Student Reasoning • Matching Instruction to Student Reasoning

5. MindStreaming Find a partner at your table: • Partner A talks for 1 min about the topic • Partner B listens and encourages Partner A The role reverses: • Partner B talks for 1 min about the topic • Partner A listens and encourages Partner B

7. Equal Partitioning

8. Iterating

9. Unit Fractions

10. Fraction Comparison

11. Big Ideas of Fraction Concepts

12. Chapter 1 Thoughts

13. It is difficult for students to move from working with whole numbers to working with fractions because a fractional quantity is described with two numbers, not one, and understanding fraction requires one to explicitly comprehend a relationship between two quantities – the whole and its parts. Battista By focusing on relationships, children come to see fractions as connected to other things they know about number and operationEmpson & Levi

14. CCSSCriticalArea 3rd grade – page 21 4th grade – page 27 5th grade- page 33

15. ClusterHeadings Cluster headings function like topic sentences in a paragraph in that they state the point of, and lend additional meaning to, the individual content standards that follow…Materials do not simply treat the Standards as a sum of individual content standards and individual practice standards. K-8 Publishers’ Criteria for the CCSS for Mathematics

16. Connections at a single grade level can be used to improve focus, by tightly linking secondary topics in the major work of the grade. K-8 Publishers’ Criteria for the CCSS for Mathematics

17. Focus on Major Work: In any single grade, students and teachers using the materials as designed spend the large majority of their time, approximately 3/4 , on the major work of each grade level… In addition, major work should especially predominate in the first half of the year K-8 Publishers’ Criteria for the CCSS for Mathematics

19. Numberwork(10 – 15 min)

20. NumberWorkExample Differences of (Purpose: differences) • Ask students to write pairs of numbers with a given difference (e.g. difference of 7) for a predetermined number of minutes • Share responses and repeat several times. • (e.g. 32/39, 54/61, 1009/1016) • Variation: Use decimals and fractional differences

21. NumberWorkExamples

22. Number work is … commonly done whole class beginning of math block can also be tied into ‘calendar’ routinesfor younger students work was done independently, but chosen specifically for individual students.

23. Balanced Mathematics Problem Solving Conceptual Understanding Formative Assessment Skills, Facts, Procedures Flexible Set of Thinking & Reasoning Kristin Frang, MAISD Regional Mathematics & Science Center Adapted from Cathy Seely, Balance is Basic and Larry Ainsworth, Five Easy Steps to a Balanced Math Program

26. ProblemSolving • All content domains • Large & small group settings • May serve multiple purposes, such as: • Introducing new concepts • Challenging students to develop and apply effective strategies • Providing a context for improving skills

27. ProblemSolving Utilizes problems that: • Emerge from the students’ environment • Can be found in the school’s curricular materials • Allow for multiple solution strategies • Lead to justification and generalization

28. InspectingEquations Focus is on learning about how the equal sign expresses equality relationships. Students:

29. InspectingEquationsExample Can students accurately solve equations with repeated variables? g + g + 4 = 16 h + h – 3 = 11 a + a + a = 15 Do students take advantage of familiar number relationships? 25 + 47 + 75 = y 98 + 69 + 2 = y

30. FactFluency Illuminates a child’s sense of number relationships and the thinking strategies he/she uses to calculate single-digit computation • Defines numbers sizes to use in other blocks • Identify size of numbers a student can compute mentally • Defines numbers sizes to work on for individual fluency & maintenance

31. 3-2-1Reflection • 3 things you want to remember from our time this morning • Highlight 2 things you want to focus on implementing first • Choose 1 and create a plan to help you implement in your current unit

32. Problem solving should be the central focus of the mathematics curriculum. Problem solving is not a distinct topic, but a process that should permeate the entire program and provide the context in which concepts and skills can be learned

33. AccessibleCognitiveJumps

34. LevelsofSophisticationinStudentReasoning

35. Level 0(Sublevel 0.1 & 0.2) Tell what fraction is shaded. Response: 1

36. Level0(Sublevel 0.1 & 0.2) Three people want to share the pizza below equally. Show how much each person gets. Level 0.1 Level 0.2

37. Level1 Studentrecognizesonlyfamiliarpicturesoffractions

38. Level1 Shape A looks like a half. The rest are not halves

39. Level1 I think A,B,C and D look like ¾ . E and F don’t look like ¾.

40. Level 2 • Count all parts, count shaded parts • Don’t explicitly focus on whole • Difficult time finding fractions of sets • Don’t use iteration/partitioning into equal parts • Can make comparisons by drawing pictures

41. Level 2 A, Band F are halves because one part is shaded and one part is not.

42. Level 2 • Count all parts, count shaded parts • Don’t explicitly focus on whole • Difficult time finding fractions of sets • Don’t use iteration/partitioning into equal parts • Can make comparisons by drawing pictures

43. Level 2 A, B, C and D are ¾ because they all have 4 parts and 3 are shaded. E and F are not ¾ because they have to many pieces.

44. Level1vs.Level2

45. Level3 • Partition into equal parts only when whole is explicitly specified • Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain • Reasoning is restricted to shape, do not reason about quantities • Use splitting and iterating

46. Level3 • A is one-half because there are 2 equal pieces and 1 is shaded. Same thing on B. • C, move square in bottom row to empty spot so the gray & white parts are the same (see red arrow) • D, move the bottom shaded squares to the left to make one half. • E, move all shaded squares to the left. • F is not one half because the gray and white parts aren’t equal.

47. Level3 • Partition into equal parts only when whole is explicitly specified • Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain • Reasoning is restricted to shape, do not reason about quantities • Use splitting and iterating

48. Level3 • B and D because they both have 4 equal parts and 3 shaded. • The parts aren’t equal in A and C. • E is ¾ because you can divide it so that 3 out of 4 equal parts are shaded. • F is not because there are 6 squares shaded.

50. Level3 • Partition into equal parts only when whole is explicitly specified • Difficulty find fractions of sets, improper fractions and performing arithmetic operations because whole is more difficult to maintain • Reasoning is restricted to shape, do not reason about quantities • Use splitting and iterating