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Building a Conceptual Understanding of Fractions in Grades 3-5

Building a Conceptual Understanding of Fractions in Grades 3-5. Nicolae Borota Office of STEM. Agenda. 9:00am -9:30am. 9:30am – 11:30 am. Introduction Goals for today’s presentation My background Involvement of the Office of STEM. Fractions Examine student difficulties

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Building a Conceptual Understanding of Fractions in Grades 3-5

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  1. Building a Conceptual Understanding of Fractions in Grades 3-5 Nicolae Borota Office of STEM

  2. Agenda 9:00am -9:30am 9:30am – 11:30 am Introduction Goals for today’s presentation My background Involvement of the Office of STEM Fractions Examine student difficulties Strategies for Instruction Building students’ conceptual understanding Fractions Examine student misconceptions Assessment examples Build upon the foundational skills Wrap-up Summary Contact information Feedback LunchOn your own…options?? 11:30am – 12:30pm 12:30pm – 2:30pm 2:30pm – 3:00pm

  3. On your mark, get set, DRAW! Ready, set,…

  4. On your mark, get set, DRAW!

  5. On your mark, get set, DRAW! Hold on to your drawing. We will revisit it a little later…

  6. Do this!

  7. Priorities by Grade Level http://www.achievethecore.org/content/upload/Focus%20in%20Math_091013_FINAL.pdf

  8. Priorities in Grades 3-5 http://www.achievethecore.org/content/upload/Focus%20in%20Math_091013_FINAL.pdf

  9. Grade 3 https://docs.google.com/file/d/0B793QJfe3Xc-UnFLWEFhTkVadjA/edit?usp=drive_web&pli=1

  10. Grade 4 https://docs.google.com/file/d/0B793QJfe3Xc-UnFLWEFhTkVadjA/edit?usp=drive_web&pli=1

  11. Grade 5 https://docs.google.com/file/d/0B793QJfe3Xc-UnFLWEFhTkVadjA/edit?usp=drive_web&pli=1

  12. Hear it from the source Fractions in the CCSS

  13. Fraction progression Grade 3: Develop an understanding of fractions as numbers. Grade 4: Understand fraction equivalence and ordering. Build fractions from unit fractions and apply and extend previous understandings of operations on whole numbers. Use decimal notation for fractions and compare fractions.

  14. Fraction progression Grade 5: Use equivalent fractions to add and subtract fractions (including mixed numbers with unlike denominators). Apply and extend previous understandings to multiply fractions and to divide unit fractions.

  15. Area of Concern “National tests show nearly half of eighth-graders aren’t able to put three fractions in order by size.” • (Wall Street Journal article Sept 24, 2013)

  16. What do parents do? “Trouble with fractions is the most common reason parents seek math help for their fourth- and fifth-graders.” • (Larry Martinek, chief instructional officer of Mathnasium Learning Centers)

  17. Why do students have trouble? Discuss with your group and compose a list of the top three mathematical reasons why you think your students have trouble with fractions.

  18. What is so difficult? 3.NF.A.1 3.NF.A.2 Type of Understanding Example of Student thought A Fraction is a Number A fraction represents an amount, not just pieces or a situation. When asked to put the fraction on a number line, a student said “you can’t put it on a number line, because it’s two pieces out of three pieces. It’s not a number.” (Watanabe, 2007)

  19. What is so difficult? 3.NF.A.1 3.NF.A.2a 3.NF.A.3a Type of Understanding Example of Student thought Partitioning Fractions A whole number can be divided into smaller and smaller equal parts. The same fractional quantity can be represented by different fractions. Difficulty seeing how to divide a whole number into equal parts. Difficulty seeing that is equal to , , … and so on. (Watanabe, 2007)

  20. What is so difficult? 3.NF.A.1 3.NF.A.2a Type of Understanding Example of Student thought The Meaning of the Denominator (cont.) The more units a whole is partitioned into, the smaller each one is. fits exactly times into the whole Students may think “ is bigger than because 5 is bigger than 4. Difficulty seeing that fits in the whole 3 times, fits in the whole 4 times. Has trouble seeing that , etc. equal 1. (Watanabe, 2007)

  21. What is so difficult? 3.NF.A.1 3.NF.A.2a 3.NF.A.3a Type of Understanding Example of Student thought Knowing What is the Whole Constructing the whole given a fractional part. Keeping track of the whole. Difficulty making the whole when you give them a fractional part, e.g.: “This paper is , show me the whole.” Sees that the magnitude of a fraction depends on the magnitude of the whole. (e.g., half of a small cookie is not the same as half of a large cookie. (Watanabe, 2007)

  22. What is so difficult? 3.NF.A.2b 3.NF.A.3c Type of Understanding Example of Student thought Confusion about whether the two drawings below represent of a pie or of a pie. Knowing What is the Whole Constructing the whole given a fractional part. Keeping track of the whole. (Watanabe, 2007)

  23. What is so difficult? Type of Understanding Example of Student Thought May think is larger than because 4 is larger than 3 OR is larger than because 9 is larger than 4 OR is the same size as because the difference between the top and the bottom in both fractions is 2. Fraction Size Understands that fraction size is determined by the (multiplicative) relationship between numerator and denominator – not just the numerator, not just by the denominator, and not by the difference between the numerator and the denominator. (Watanabe, 2007)

  24. What is so difficult? Type of Understanding Example of Student thought You cannot have , because there’s only in a whole. Fractions Can Represent Quantities Greater Than One May be difficult for students who have a strong image of a fraction as a piece of something. (Watanabe, 2007)

  25. Sample problem (4th Grade NAEP 2007) 4.NF.A.1 These three fractions are equivalent. Give two more fractions that are equivalent to these. What percent of students got it correct? http://nces.ed.gov/nationsreportcard/itmrlsx/

  26. Sample problem (4th Grade NAEP 2007) 4.NF.A.1 These three fractions are equivalent. Give two more fractions that are equivalent to these. Difficulty: Easy (60.21% Correct) http://nces.ed.gov/nationsreportcard/itmrlsx/

  27. Equivalent fractions 4.NF.A.1 5.NF.A.1 Grade 3 denominators 2,3,4,6,8 Grade 4 denominators 2,3,4,5,6,8,10,12,100 (AMTNJ presentation,Glatzer, 2013)

  28. Sample problem (4th Grade NAEP 2007) 3.NF.A.3 3.NF.A.3d • Mark says of his candy bar is smaller than of the same candy bar. • Is Mark right? Yes No • Draw a picture or use words to explain why you think Mark is right or wrong. What percent of students got this correct? http://nces.ed.gov/nationsreportcard/itmrlsx/

  29. Sample problem (4th Grade NAEP 2007) 3.NF.A.3 3.NF.A.3d • Mark says of his candy bar is smaller than of the same candy bar. • Is Mark right? Yes No • Draw a picture or use words to explain why you think Mark is right or wrong. • Difficulty: Medium (40.85% Correct) http://nces.ed.gov/nationsreportcard/itmrlsx/

  30. Sample problem (4th Grade NAEP 2005) 4.NF.B.3a 3 • Difficulty: Medium (53.03% Correct) What was the most often chosen incorrect response for students in New Jersey? http://nces.ed.gov/nationsreportcard/itmrlsx/

  31. Sample problem (4th Grade NAEP 2005) 4.NF.B.3a 3 6% 51% 27% 16% New Jersey’s Results http://nces.ed.gov/nationsreportcard/itmrlsx/

  32. Identified Lack of Conceptual Understanding • Not viewing fractions as numbers at all, but rather as meaningless symbols that need to be manipulated in arbitrary ways to produce answers that satisfy a teacher. (Institute of Education Science– Practice Guide – Fractions, 2010)

  33. Identified Lack of Conceptual Understanding • Not viewing fractions as numbers at all, but rather as meaningless symbols that need to be manipulated in arbitrary ways to produce answers that satisfy a teacher. • Focusing on numerators and denominators as separate numbers rather than thinking of the fraction as a single number. (Institute of Education Science– Practice Guide – Fractions, 2010)

  34. Identified Lack of Conceptual Understanding 3.NF.A.2 • Not viewing fractions as numbers at all, but rather as meaningless symbols that need to be manipulated in arbitrary ways to produce answers that satisfy a teacher. • Focusing on numerators and denominators as separate numbers rather than thinking of the fraction as a single number. • Confusing properties of fractions with those of whole numbers. (Institute of Education Science– Practice Guide – Fractions, 2010)

  35. Recommendations • Build on students’ informal understanding of sharing and proportionality to develop initial fraction concepts. (Institute of Education Science– Practice Guide – Fractions, 2010)

  36. 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. (Institute of Education Science– Practice Guide – Fractions, 2010)

  37. 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 makes sense. (Institute of Education Science– Practice Guide – Fractions, 2010)

  38. Pattern Block Man What fraction is blue?

  39. Pattern Block Man 3.NF.A.1 Let’s discuss your answers.

  40. Conceptual Understanding is paramount Wait to introduce problem-solving until students understand what the numerators and denominators actually represent. Unit fractions are so critical at this stage!!

  41. Japanese view of fractions “Fractions should be introduced to represent one portion of an equally divided object, or to represent a fractional part of some quantity.” (Japanese Course of Study Teaching Guide. Takahashi, Watanabe, & Yoshida, 2004)

  42. According to the CCSS 3.NF.A.1 Develop understanding of fractions as numbers. 1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. (Common Core State Standards in Mathematics, Grade 3)

  43. According to the CCSS Develop understanding of fractions as numbers. 1. Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b. No coincidence that the concept of unit fractions is first under that heading. (Common Core State Standards in Mathematics, Grade 3)

  44. According to the CCSS 3.NF.A.2 Develop understanding of fractions as numbers. 2. Understand a fraction as a number on the number line; represent fractions on a number line diagram. (Common Core State Standards in Mathematics, Grade 3)

  45. Unit fractions as units 3.NF.A.1 After this kind of introduction, the idea that represents a collection of two units should be taught, as if is thought of as a unit. (Thompson and Saldanha, 2003)

  46. Write it to see it 3.NF.A.1 Students may be able to grasp the notion of unit fractions as being units if you write it as you would any other unit. vs. 2 thirds

  47. Write it to see it 3.NF.A.1 Students may be able to grasp the notion of unit fractions as being units if you write it as you would any other unit. 2 thirds 2 miles 2 feet 2 birds

  48. Write it to see it 3.NF.A.1 4.NF.A.3a 2 thirds + 5 thirds = 7 thirds 2 miles + 5 miles = 7 miles 2 birds + 5 birds = 7 birds

  49. Revisiting your drawing As promised, we will now review your drawing from earlier. Let’s see what people came up with to represent 2/3. Let us discuss.

  50. Pizzas are not all they are cut up to be How many of you have needed to cut a pizza up into fifths, sixths, sevenths, eighths, or ninths? Much more common is having something of length and breaking it into fractional pieces. Fraction bars….number lines…

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