Finding the missing pieces: Middle Grades Fractions

1. Finding the missing pieces:Middle Grades Fractions Solve using a visual fraction model and give a context:

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3. Finding the missing pieces Middle Grades Fractions Jeanne Simpson NCTM 2014

4. Focus Strategies and resources to embed fractional reasoning while maintaining the rigor and relevance required in the middle grades Common Core Standards

5. topics • Essential fraction concepts • Connections to 6-8 content • Strategies to fit the two together fractionprogression.wikispaces.com

6. The trouble with fractions…

7. notes practice KCF songs basic facts practice review rhymes practice quizzes reteaching review tutoring clear steps posters practice review mixed practice

8. Why can’t students remember? “Students who are asked to practice the algorithm over and over…stop thinking. They sacrifice the relationships in order to treat the numbers simply as digits.” Imm, Fosnot, Uittenbogaard (2012)

9. Why are fractions important? Difficulty with learning fractions is pervasive and is an obstacle to further progress in mathematics and other domains dependent on mathematics, including algebra. It has also been linked to difficulties in adulthood, such as failure to understand medication regimens. National Mathematics Panel Report, 2008

10. TheAlabama Department of Education’sinitiative to improve math and science teaching (K-12) Alabama Math, Science, and Technology Initiative

11. Goals for AMSTI Math Students should: • Use multiple and alternative strategies to solve problems. • Communicate mathematical understanding of content and process standards, both orally and in writing. • Become mathematical problem solvers. • Learn to reason mathematically • Be able to make real world connections through the application of mathematical knowledge.

12. practice discourse inquiry songs basic facts practice review rhymes manipulatives quizzes reteaching review tutoring student writing practice in context review mixed practice projects

13. Where are the 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.

14. The structure is the standardsDaro, McCallum, Zimba The natural distribution of prior knowledge in classrooms should not prompt abandoning instruction in grade level content, but should prompt explicit attention to connecting grade level content to content from prior learning. To do this, instruction should reflect the progressions on which the CCSSM are built. For example, the development of fluency with division using the standard algorithm in grade 6 is the occasion to surface and deal with unfinished learning with respect to place value. Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking. http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422

15. The structure is the standardsDaro, McCallum, Zimba This is a basic condition of teaching and should not be ignored in the name of standards. Nearly every student has more to learn about the mathematics referenced by standards from earlier grades. Indeed, it is the nature of mathematics that much new learning is about extending knowledge from prior learning to new situations. For this reason, teachers need to understand the progressions in the standards so they can see where individual students and groups of students are coming from, and where they are heading. But progressions disappear when standards are torn out of context and taught as isolated events. http://commoncoretools.me/2012/02/16/the-structure-is-the-standards/#more-422

16. Progressions – The missing piece?

17. Progression for 3-5 Number and operations - Fractions • The meaning of fractions • Partitioning • Unit fractions • Models • Number lines • Equivalent fractions • Comparing fractions • Operations

18. Ongoing Assessment project (Ogap) • Understand research on how students learn specific math concepts. • Strengthen content knowledge in these areas. • Understand how research is reflected in math programs and CCSS. • Increase ability to analyze evidence in student work to inform instruction. Developed as a part of the Vermont Mathematics Partnership Ongoing Assessment Project (OGAP) funded by NSF EHR-0227057 and the US DOE (S366A20002).

19. continuous assessment • Preassess before each topic and continuously engineer discussions, activities, and tasks all along that are purposefully designed to elicit specific student understandings. • Use data to differentiate Tier 1 instruction and to ensure that 80% of students are thriving in Tier 1. • Provide interventions via small groups.

20. Sample preassessment

21. RTI and More…

22. Fraction resources

23. The meaning of fractions

24. Developing effective fractions instruction in kindergarten through 8th grade Recommendations • Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts. • Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward. • Help students understand why procedures for computations with fractions make sense. • Develop students’ conceptual understanding of strategies for solving ratio, rate, and proportion problems before exposing them to cross-multiplication as a procedure to use to solve such problems. • Professional development programs should place a high priority on improving teachers’ understanding of fractions and of how to teach them.

25. The Meaning of fractions What does this number mean?

26. Circleof the triangles

27. 8th Grade students

28. The Meaning of fractions • The top number counts. • The bottom number tells what is counted. • Together they represent one number.

29. Models for fractions Petit, Laird, Marsden (2010)

30. Partitioning The Act of Dividing More sharers means smaller shares. Measures must be equal, but shares don’t have to be identical.

31. Which of the following show fourths? Why? (Small, 2014, p. 14)

32. Circle the fraction that is closest to

33. Pattern block fractionsCynthia Lanius

34. 6th Grade Partitioning • 6.G.1 – Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes… • 6.G.4 – Represent 3D figures using nets…find the surface area. • 6.RP.1 – Understand a ratio.. • Begin the school year with a study of area. • Assess partitioning skills and understandings. • Focus on area model for fractions. • Differentiate instruction based on assessments. • Use Number Talks for whole class discussions on items that <80% of class understands. • Work with small groups on other concepts.

35. 8th Grade Partitioning Pythagorean Theorem – Circle Sandwich • A square is inscribed in a circle which is inscribed in a square as shown. Note that the vertices of the inner square meet the midpoints of the outer square’s sides. • Consider the area of the region left by removing the interior of the small square from the interior of the big square. Is the area of the blue region more or less than of that?

36. Number lines Help students recognize that fractions are numbers and that they expand the number system beyond whole numbers. Use number lines as a central representational tool in teaching this and other fraction concepts from the early grades onward.

37. Place these fractions on the number line below in their correct position.

38. Place these fractions on the number line below in their correct position.

39. 6th Grade number lines • 6.NS.5 - Understand that positive and negative numbers are used together to describe quantities having opposite directions or values… • 6.NS.6 – Understand a rational number as a point on the number line… • 6.NS.7 – Understanding ordering and absolute value of rational numbers. • 6.NS.8 – Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane… • 6.G.3 - Draw polygons in the coordinate plane given coordinates for vertices; use coordinates to find the length of a side joining points with the same first or second coordinate.

40. 7th and 8th Grade number lines 7th Grade 8th Grade Irrational numbers Graphing equations and functions Transformations Statistical graphs • Integers • Graphing proportional relationships

41. Equivalent fractions reducing

42. The experts say…. “Do not tell students that their answer is incorrect if not in the simplest or lowest terms. This also misinforms students about the equivalence of fractions. If you want the answer in simplified form, provide feedback to the student that the answer is correct but must be simplified.” Van de Walle, Bay-Williams, Lovin, Karp (2014, p. 119)

43. Equivalent areas

44. Equivalent lengths

45. Equivalence in 6th - 8th Grade • Expressions • Equations • Exponents • Irrational numbers • Slope 8.EE.6 Illustrative Mathematics

46. Comparing fractions

47. Which fraction is greater? • or More of the same-size parts • or Same number of parts, different sizes • or More or less than one whole • or More or less than one half • or Distance from one whole

48. Operations

49. Do your students remember procedures?

50. Maybe a different procedure would help?