Rethinking Fractions:Implications for Teaching and      Learning Algebra            Gail Burrill      Michigan State Unive...
Fractions1. On the portion of the number line below, a dot shows   where 1/2 is. Use another dot to show where 3/4 is.  (N...
An OverviewWhy a different approach?What does this approach look like?The role of interactive dynamic technologyWhat might...
Procedural & Conceptual         KnowledgeUnderstanding requires both procedural andconceptual knowledgeConceptual supports...
Conceptual Knowledge:–   Makes connections more visible,–   enables reasoning about the mathematics,–   less susceptible t...
Research on learning algebra:Making links to the classroom 1988 NCTM Yearbook on Algebra:Common Mistakes in Algebra (Marqu...
A fractionIs typically thought of as three things:   Quotient,   Part to Whole, or   RatioRethinking Fractions:      Based...
Grade 3: UnderstandA fraction 1/b is the quantity formed by 1 part when a whole is  partitioned into b equal parts; a frac...
Grade 3: UnderstandA fraction 1/b is the quantity formed by 1 part when a whole is  partitioned into b equal parts; a frac...
Grade 3: Understand3. equivalence of fractions in special cases, and compare fractions   by reasoning about their size.   ...
Grade 3: Understand3. equivalence of fractions in special cases, and compare fractions   by reasoning about their size.   ...
Grade 3: GeometryPartition shapes into parts with equal areas.Express the area of each part as a unit fraction ofthe whole
Grade 4: FractionsExtend understanding of fraction equivalence and ordering.  1. Explain why a fraction a/b is equivalent ...
Grade 4: FractionsExtend understanding of fraction equivalence and ordering.  1. Explain why a fraction a/b is equivalent ...
Grade 4: Fractions3. A fraction a/b with a > 1 is a sum of fractions 1/b.   a. addition and subtraction of fractions is jo...
Grade 4: Fractions3. A fraction a/b with a > 1 is a sum of fractions 1/b.   a. addition and subtraction of fractions is jo...
Fraction as a point on a number line                           QuickTimeª and a                             decompressor  ...
QuickTimeª and a                           decompressor                 are needed to see this picture.Describe in words w...
EqualityThe equality a/b = k/l means that the twopoints a/b and k/l are precisely the samepoint on the number line.Locate ...
A unit other than 1If the weight of an object X is our unit, then5/7 would mean 5 parts of X after it hasbeen partitioned ...
Unit squaresThe area of the unit square is by definition1.If two regions are congruent, then their areasare equal. (If two...
Area of unit square as unit 1Prove 15/6 = 5/2
Area of unit square as unit 1Prove 15/6 = 5/25/2 is 2 and one half so it is2 and a half unit squares                      ...
Find the fraction if        QuickTimeª and a                                       The shaded area is one unit          de...
Find the fraction if                             The shaded area is one unit     QuickTimeª and a      decompressorare nee...
Exercises1. Indicate the approximate position of each of the followingon the number line, and also write it as a mixed num...
ExercisesIndicate the approximate position of each of the following onthe number line, and also write it as a mixed number...
Equivalence        QuickTimeª and a                              QuickTimeª and a          decompressor                   ...
Addition    QuickTimeª and a m copies of 1/L plus k copies      decompressor picture.are needed to see this of 1/L is (m +...
Multiplication                                     QuickTimeª and a                                      decompressor     ...
Peopl e learn by   Engaging in a concrete experience   Observing reflectively   Developing an abstract   conceptualization...
Research on learning algebra:Making links to the classroom 1988 NCTM Yearbook on Algebra:Common Mistakes in Algebra (Marqu...
References Mathematics. (2010). Council of Chief StateCommon Core State StandardsSchool Officers & National Governors Asso...
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Fractions to Algebra

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Gail Burrill's presentation at Green Lake 2012 of H. Wu's work with understanding fractions

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  • Gail slide 3 of 19
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Students learn if they are actively involved in choosing and evaluating strategies, considering assumptions, and receiving feedback. they encounter contrasting cases- notice new features and identify important ones. Struggle with a concept before they are given a lecture Develop both conceptual understandings and procedural skills NCTM High School Focal Points suggest reasoning and sense making should be the central focus of high school math
  • Gail slide 3 of 19
  • Fractions to Algebra

    1. 1. Rethinking Fractions:Implications for Teaching and Learning Algebra Gail Burrill Michigan State University Burrill@msu.edu
    2. 2. Fractions1. On the portion of the number line below, a dot shows where 1/2 is. Use another dot to show where 3/4 is. (NAEP 2003) QuickTimeª and a decompressor are needed to see this picture. 37% correct, grade 4 64% correct grade 82. Jose ate 1/2 of a pizza.Ella ate 1/2 of another pizza.Jose said that he ate more pizza than Ella, but Ellasaid they both ate the same amount. Use words andpictures to show that Jose could be right. (NAEP, 1992) Grade 4, 57% omitted; 8%satisfactory
    3. 3. An OverviewWhy a different approach?What does this approach look like?The role of interactive dynamic technologyWhat might we gain for algebra?
    4. 4. Procedural & Conceptual KnowledgeUnderstanding requires both procedural andconceptual knowledgeConceptual supports the development ofprocedural but not the other way around NRC, 2001
    5. 5. Conceptual Knowledge:– Makes connections more visible,– enables reasoning about the mathematics,– less susceptible to common errors,– less prone to forgetting.Procedural Knowledge:– strengthens and develops understanding– allows students to concentrate on relationships rather than working out results NRC, 1999; 2001
    6. 6. Research on learning algebra:Making links to the classroom 1988 NCTM Yearbook on Algebra:Common Mistakes in Algebra (Marquis, 1988) a2. b5 = (ab)7 (x+4)2 = x2+ 16 x+y y x r x+r = + = x+z z y s y+s 3a-1= 110 of 22 were related to 3afractions
    7. 7. A fractionIs typically thought of as three things: Quotient, Part to Whole, or RatioRethinking Fractions: Based on Chapter 2, Fractions by H. Wu Department of Mathematics #3840 University of California, Berkeley Berkeley, CA 94720-3840 http://www.math.berkeley.edu/_wu/
    8. 8. Grade 3: UnderstandA fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts; a fraction a/b is the quantity formed by a parts of size 1/b. a fraction is a number on the number line; represent fractions on a number line diagram. 2. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
    9. 9. Grade 3: UnderstandA fraction 1/b is the quantity formed by 1 part when a whole is partitioned into b equal parts; a fraction a/b is the quantity formed by a parts of size 1/b. a fraction is a number on the number line; represent fractions on a number line diagram. 2. a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts. Recognize that each part has size 1/b and that the endpoint of the part based at 0 locates the number 1/b on the number line. b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.
    10. 10. Grade 3: Understand3. equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. two fractions as equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent, c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. d. Compare two fractions with the same numerator or denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole.
    11. 11. Grade 3: Understand3. equivalence of fractions in special cases, and compare fractions by reasoning about their size. a. two fractions are equivalent (equal) if they are the same size, or the same point on a number line. b. Recognize and generate simple equivalent fractions. Explain why the fractions are equivalent, c. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. Express whole numbers as fractions, and recognize fractions that are equivalent to whole numbers. d. Compare two fractions with the same numerator or denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole.
    12. 12. Grade 3: GeometryPartition shapes into parts with equal areas.Express the area of each part as a unit fraction ofthe whole
    13. 13. Grade 4: FractionsExtend understanding of fraction equivalence and ordering. 1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models. Use this to recognize and generate equivalent fractions. 2. Compare two fractions with different numerators and denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole.Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
    14. 14. Grade 4: FractionsExtend understanding of fraction equivalence and ordering. 1. Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models. Use this to recognize and generate equivalent fractions. 2. Compare two fractions with different numerators and denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole.Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
    15. 15. Grade 4: Fractions3. A fraction a/b with a > 1 is a sum of fractions 1/b. a. addition and subtraction of fractions is joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. c. Add and subtract mixed numbers with like denominators,4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. A fraction a/b is a multiple of 1/b. 5/4 = 5 × (1/4). b. A multiple of a/b is a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
    16. 16. Grade 4: Fractions3. A fraction a/b with a > 1 is a sum of fractions 1/b. a. addition and subtraction of fractions is joining and separating parts referring to the same whole. b. Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. c. Add and subtract mixed numbers with like denominators,4. Apply and extend previous understandings of multiplication to multiply a fraction by a whole number. a. A fraction a/b is a multiple of 1/b. 5/4 = 5 × (1/4). b. A multiple of a/b is a multiple of 1/b, and use this understanding to multiply a fraction by a whole number.
    17. 17. Fraction as a point on a number line QuickTimeª and a decompressor are needed to see this picture.N/3 is a multiple of1/3 k/p is the length of the concatenation of k segmentsA whole number is each of which has length 1 p .also a fractionNo difference k/p is k copies of 1/p .between proper andimproper fractions
    18. 18. QuickTimeª and a decompressor are needed to see this picture.Describe in words where each fraction is on the numberline and sketch a picture to show its location.(a) 7/9 (b) 6/11 (c) 9/4 (d) 13/5(e) 10/ 3(f) k/5 , where k is a whole number between 11 and14.(g) k/6 , where k is a whole number between 25 and 29.
    19. 19. EqualityThe equality a/b = k/l means that the twopoints a/b and k/l are precisely the samepoint on the number line.Locate 16/3; 84/17. Which one is larger andwhy?
    20. 20. A unit other than 1If the weight of an object X is our unit, then5/7 would mean 5 parts of X after it hasbeen partitioned into7 parts of equal weight.Unit is the weight of a piece of ham thatweighs three pounds. 1/3 of the weightwould be one pound.Unit is a pack of 15 pencils; 2/3 of a unitwould be to partition into three equal partsof 5 pencils or 10 pencils.
    21. 21. Unit squaresThe area of the unit square is by definition1.If two regions are congruent, then their areasare equal. (If two shapes are congruent, thenone figure can be rotated, translated orreflecting to fit on exactly on top of theother.)Find four representations of 1/4.Find two representations of 5/6, 7/4, 9/4.
    22. 22. Area of unit square as unit 1Prove 15/6 = 5/2
    23. 23. Area of unit square as unit 1Prove 15/6 = 5/25/2 is 2 and one half so it is2 and a half unit squares QuickTimeª and a QuickTimeª and a decompressor decompressor are needed to see this picture. are needed to see this picture.
    24. 24. Find the fraction if QuickTimeª and a The shaded area is one unit decompressorare needed to see this picture. QuickTimeª and a decompressor are needed to see this picture. a b c
    25. 25. Find the fraction if The shaded area is one unit QuickTimeª and a decompressorare needed to see this picture. QuickTimeª and a decompressor are needed to see this picture. a b c
    26. 26. Exercises1. Indicate the approximate position of each of the followingon the number line, and also write it as a mixed number. (a)67/4 . (b) 205/11 (c) 459/23 (d) 1502/24 .2. (a) After driving 352 miles, we have done only two-thirds ofthe driving for the day. How many miles did we plan to drivefor the day?(b) After reading 168 pages, I am exactly four-fifths of the waythrough. How many pages are in the book?(c) Helena was three quarters of the way to school after havingwalked 22 5 miles from home. How far is her home fromschool?3. (a) I have a friend who earns two dollars for every threetimes she walks her parents’ dog. She knows that this week sheshe will walk the dog twelve times. How much will she earn?
    27. 27. ExercisesIndicate the approximate position of each of the following onthe number line, and also write it as a mixed number. (a) 67/4 .(b) 205/11 (c) 459/23 (d) 1502/24 .2. (a) After driving 352 miles, we have done only two-thirds ofthe driving for the day. How many miles did we plan to drivefor the day?(b) After reading 168 pages, I am exactly four-fifths of the waythrough. How many pages are in the book?(c) Helena was three quarters of the way to school after havingwalked 22 5 miles from home. How far is her home fromschool?3. (a) I have a friend who earns two dollars for every threetimes she walks her parents’ dog. She knows that this week sheshe will walk the dog twelve times. How much will she earn?
    28. 28. Equivalence QuickTimeª and a QuickTimeª and a decompressor decompressor are needed to see this picture. are needed to see this picture. QuickTimeª and a QuickTimeª and a decompressor decompressor are needed to see this picture. are needed to see this picture. 5/2=15/3 km/kl = m/l and m/l = km/klUse the representation of 1 as the area of a unit squareto give a proof of why 6/14 = 3/7 and 30/12 = 5/2 .
    29. 29. Addition QuickTimeª and a m copies of 1/L plus k copies decompressor picture.are needed to see this of 1/L is (m + k) copies of 1/L QuickTimeª and a decompressor are needed to see this picture. QuickTimeª and a decompressor are needed to see this picture.
    30. 30. Multiplication QuickTimeª and a decompressor QuickTimeª and a decompressor are needed to see thiare needed to see this picture.
    31. 31. Peopl e learn by Engaging in a concrete experience Observing reflectively Developing an abstract conceptualization based upon the reflection Actively experimenting/testing based upon the abstraction Zull, 2002
    32. 32. Research on learning algebra:Making links to the classroom 1988 NCTM Yearbook on Algebra:Common Mistakes in Algebra (Marquis, 1988) a2. b5 = (ab)7 (x+4)2 = x2+ 16 x+y y x r x+r = + = x+z z y s y+s 3a-1= 110 of 22 were related to 3afractions
    33. 33. References Mathematics. (2010). Council of Chief StateCommon Core State StandardsSchool Officers & National Governors Association.www.corestandards.org/Marquis, J. Common mistakes in algebra.The Ideas of Algebra K-12. 1988Yearbook. Coxford, A. (Ed). Reston, VA. National Council of Teachers ofMathematics.National Assessment for Educational Progress (2003, 1992). ReleasedItems. National Center for Educational Statistics.National Research Council (2001). Adding It Up. Kilpatrick, J., Swafford,J., & Findell, B. (Eds.) Washington DC: National Academy Press. Alsoavailable on the web at www.nap.edu.National Research Council. (1999). How People Learn: Brain, mind,experience,and school. Bransford, J. D., Brown, A. L., & Cocking, R. R.(Eds.). Washington, DC: National Academy Press.Wu, H. (2006). Fractions, Chapter 2. In Understanding Numbers inElementary School Mathematics, Amer. Math. Soc., 2011.]//www.math.berkeley.edu/_wu/James Zull, ( 2002). The Art of Changing the Brain: Enriching the Practiceof Teaching by Exploring the Biology of Learning. Association forSupervision and Curriculum Development, Alexandria, Virginia.

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