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TEACHING DIVISION OF FRACTIONS Teruni Lamberg, Ph.D. University of Nevada, Reno 1

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Discussion What are some challenges you have experienced teaching division of fractions and multiplication of fractions? 2

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Common Core Division of Fractions CCSS.Math.Content.5.NF.B.6 Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. CCSS.Math.Content.5.NF.B.6 CCSS.Math.Content.5.NF.B.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. 1 CCSS.Math.Content.5.NF.B.7 3

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How divison of fractions is typically learned The invert and multiply method (Sharp &Adams, 2002; Ball, 2007). Even though this method is widely taught and used, individuals still struggle with understanding why this method works Ball (1990;1997). “Invert and multiply” method is taught as rote procedures in school (Borko, Eisenhart, Brown, Underhill, Jones & Agard, 1992) 4

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The challenges of teaching division of fractions for conceptual understanding Teaching division of fractions is difficult because visualizing division of fractions and reconciling the algorithm is not easy (Perlwitze, 2005; Ball 1990). 5

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Misconceptions of division of fractions “When you divide whole numbers, the resulting answer gets smaller” 4÷2=2 “When you divide fractions the resulting answer gets bigger.” 1/2÷1/4 =2 6

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Unitizing and its role in solving fractions Unitizing involves identifying the referent unit and re-conceptualizing the unit during the process of problem solving (Behr, Khoury, Harel Post and Lesh, 1997, 1993; Lamon, 1999). 7

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Whole Number Division What do you do when you divide whole number? 9 ÷3= Partitive Division Measurement Division 8

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Dividing Fractions 1÷ 3 = ? Think of a context problem? How would you solve it? 9

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Try this! Think of a context and how would you solve it? Try this! Think of a context and how would you solve it? 3÷2 =? 10

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Levels of Analysis and Sense making Phase 1: Making Thinking Explicit (Explaining Reasoning) Phase 2: Analyzing Each Other's Solution (Analyzing Low Level to More Sophisticated Reasoning ) Phase 3: Developing New Mathematical Insights (Abstract Mathematical Concepts) 11 Source: PDToolkit Lamberg

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Continuum: Levels of Understanding and Student Strategies Inefficient strategies Efficient Strategies 11111 11111 11111 11111 5+5+5+5 5x4=20 20 ÷5=4 Simpler Representations (Concrete) Abstract Representations ** + ** 2 + 2 2 apples and 2 apples two groups of two apples two plus two 12.

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Whole Number Divided by Fractions 13 CCSS.Math.Content.5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.CCSS.Math.Content.5.NF.B.7b http://www.youtube.com/watch?v=wzbZeALhnrs

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Number lines Why invert and multiply? 14

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Fraction Divided by a Whole Number CCSS.Math.Content.5.NF.B.7aCCSS.Math.Content.5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3. Try this! ¼ ÷ 4 http://www.youtube.com/watch?v=Xda7b7 wq2-w 15

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Try this! Create a model, connect model to algorithm ½ ÷ 2 = 19

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Fraction bar and Numberline Tool http://www.mathsisfun.com/numbers/fract ion-number-line.html http://www.mathsisfun.com/numbers/fract ion-number-line.html http://www.mathplayground.com/Thinking Blocks/thinking_blocks_modeling%20_to ol.html http://www.mathplayground.com/Thinking Blocks/thinking_blocks_modeling%20_to ol.html 20

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Try This! 21 2 ½ ÷ ¼ = ?

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2 ½ ÷ ¼ 22

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2/3÷2/6= 23

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Area Model ¾ ÷2/3= 24

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¾ ÷ 2/3=? Identify a/b in relation to whole unit “What does ¾ look like?” 3/4 Identify the “unit of measure” :What does 2/3 of the same unit used in step 1 look like? 2/3 25

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¾ (a/b) become new referent unit Transition to multiplicative thinking Partition (a/b) into c/d “How many 2/3 size units are there in ¾?” ¾ x3/2 How many 1/8 in 2/3? How many 1/8 in ¾? How many 2/3’rd are there in ¾ Figuring out common denominator 26

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Answer in relation to c/d unit as a reference unit 1/1/8 27

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Try the following problems! Draw Area Model and connect to algorithm. Try also using a number line? 4/6÷1/3= 1 ½ ÷ 1/8= 29

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AREA MODEL FOR MUTIPLYING FRACTIONS

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Multiplication of Fractions http://nlvm.usu.edu/en/nav/frames_asid_1 94_g_3_t_1.html?from=search.html http://nlvm.usu.edu/en/nav/frames_asid_1 94_g_3_t_1.html?from=search.html 32

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Thank you Teruni Lamberg, Ph.D University of Nevada, Reno terunil@unr.edu Blog: http://www.mathdiscussions.wordpress.com http://www.mathdiscussions.wordpress.com 33

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We need a common denominator to add these fractions.

We need a common denominator to add these fractions.

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