Transcript Unitizing approach to division of fractions

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 Division
What do you do when you divide whole
number?
 9 ÷3=
Partitive Division

Measurement Division
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Dividing Fractions

1÷ 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 3: Developing New
Mathematical Insights
(Abstract Mathematical
Concepts)
Source:
PDToolkit
Lamberg
Phase 2: Analyzing Each
Other's Solution
(Analyzing Low Level to More
Sophisticated Reasoning)
Phase 1: Making Thinking Explicit
(Explaining Reasoning)
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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
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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
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Number lines

Why invert and multiply?
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Fraction Divided by a Whole
Number
CCSS.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
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Try this! Create a model, connect
model to algorithm
½
÷2=
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Fraction bar and Numberline Tool

http://www.mathsisfun.com/numbers/fract
ion-number-line.html

http://www.mathplayground.com/Thinking
Blocks/thinking_blocks_modeling%20_to
ol.html
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Try This!
2½÷¼=?
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2½÷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
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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

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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 Fractions

http://nlvm.usu.edu/en/nav/frames_asid_1
94_g_3_t_1.html?from=search.html
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Thank you

Teruni Lamberg, Ph.D
University of Nevada, Reno
[email protected]
Blog: http://www.mathdiscussions.wordpress.com
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