teaching division of fractions teruni lamberg, ph.d. university of nevada, reno 1
TRANSCRIPT
TEACHING DIVISION OF FRACTIONS
Teruni Lamberg, Ph.D.University of Nevada, Reno
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Discussion
What are some challenges you have experienced teaching division of fractions and multiplication of fractions?
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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.7 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1
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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)
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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).
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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
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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).
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Whole Number DivisionWhat do you do when you divide
whole number? 9 ÷3=Partitive Division
Measurement Division
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Dividing Fractions1÷ 3 = ?
Think of a context problem?
How would you solve it?
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Try this! Think of a context and how would you solve it?
3÷2 =?
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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)
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Source: PDToolkit Lamberg
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
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.
Whole Number Divided by Fractions
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• 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.
http://www.youtube.com/watch?v=wzbZeALhnrs
Number linesWhy invert and multiply?
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Fraction Divided by a Whole NumberCCSS.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=Xda7b7wq2-w 15
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Try this! Create a model, connect model to algorithm
½ ÷ 2 =
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Fraction bar and Numberline Toolhttp://
www.mathsisfun.com/numbers/fraction-number-line.html
http://www.mathplayground.com/ThinkingBlocks/thinking_blocks_modeling%20_tool.html
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Try This!
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2 ½ ÷ ¼ = ?
2 ½ ÷ ¼
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2/3÷2/6=
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Area Model
¾ ÷2/3=
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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
¾ (a/b) become new referent unitTransition to multiplicative thinkingPartition (a/b) into c/d“How many 2/3 size unitsare there in ¾?”¾ x3/2How many 1/8 in 2/3?How many 1/8 in ¾?How many 2/3’rd are there in ¾Figuring out common denominator
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Answer in relation to c/d unit as a reference unit 1/1/8
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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=
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AREA MODEL FOR
MUTIPLYING FRACTIONS
Multiplication of Fractionshttp://
nlvm.usu.edu/en/nav/frames_asid_194_g_3_t_1.html?from=search.html
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Thank youTeruni Lamberg, Ph.D University of Nevada, Reno [email protected]
Blog: http://www.mathdiscussions.wordpress.com
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