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Understanding and Teaching Fractions
Sybilla Beckmann
Department of MathematicsUniversity of Georgia
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Goals for the presentation today on fractions
An opportunity to think together about:
The Common Core definition of fraction and fractions on number
lines;
The reasoning underlying equivalent fractions;
The transition from whole number to fraction multiplication;
Connecting fractions with division;
Attending closely to the wording of problems.
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The Importance of a Focus on Fractions (Rational
Numbers)
NCTM Focal Points
National Mathematics Advisory Panel
IES Practice Guide: Assisting Students Struggling with Mathematics:
Response to Intervention (RtI) for Elementary and Middle Schools
IES Practice Guide: Developing Effective Fractions Instruction for
Kindergarten Through 8th Grade
Common Core State Standards for Mathematics
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IES Practice GuidesRtI Math and Fractions Guides
RtI Math, Recommendation 2:
Instructional materials for students receiving interventions should focus
intensely on in-depth treatment of . . . rational numbers in grades 4 8.
. . .
Fractions, Recommendation 2:
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 fractionconcepts from the early grades onward.
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Common Core State Standards for Mathematics
Two domains for fractions and rational numbers:
Number and OperationsFractions, Grades 3 5
The Number System, Grades 6 8.
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3.NF Number and OperationsFractions, Grade 3
3.NF Develop understanding of fractions as numbers.
3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part whena whole is partitioned into b equal parts; understand a fraction a/b as
the quantity formed by a parts of size 1/b.
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Definition of fraction
Why does the CC define fractions AB as A parts, each of size1B?
Why are fractions notdefined as A out of B?
The latter definition does not extend to improper fractions, such as 54
.
What would 5 out of 4 mean?
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Definition of fraction
Why does the CC define fractions AB as A parts, each of size1B?
Why are fractions notdefined as A out of B?
The latter definition does not extend to improper fractions, such as 54
.
What would 5 out of 4 mean?
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Definition of fractionFirst define fractions with numerator 1 (unit fractions)
0
0 1
1 whole
14 4
141
41
4
1
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Definition of fraction
1 whole
1
4 4
1
4
1
4
1
4
3
0
0 1
41
42
4
3
44
45
3 parts,
each 1/4
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Watch out for errors with fractions on number lines
0 1
A common misconception:
students count tick marks instead of
attending to length.
The student put 4 tick marks inside theinterval instead of dividing the interval
into 4 equal parts.
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Definition of fraction
Your turn: Explain the definition of fraction to yourself or to a neighbor.
Show and describe: 15 then35 then
65 or
Show and describe: 16 then 56 then 76
CC Fraction Definition:
The fraction 1B is the quantity formed by 1 part when a whole is
partitioned into B equal parts.
The fraction AB is the quantity formed by A parts of size 1B.
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Visual representations to pave the way for the abstract
version
RtI Math, Recommendation 5 is on visual representations
Use visual representations such as number paths, number lines,
arrays, strip diagrams, other simple drawings or pictorial
representations to scaffold learning and pave the way for
understanding the abstract version of the representation.
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Equivalent fractions
4.NF.1
Explain why a fraction a/b is equivalent to a fraction (na)/(nb) by
using visual fraction models, with attention to how the number and sizeof the parts differ even though the two fractions themselves are the
same size. Use this principle to recognize and generate equivalent
fractions.
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E i l f i
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Equivalent fractions
4 times as many shaded parts
4 times as many parts in all
more parts, but they are smaller
the same amountis shaded
1 whole
2
3
2
3
24
34
8
12
= =
split each part into 4 parts
Explain the 24
Explain the 34Why , arent we dividing?
Explain both = signs
Describe the next step
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E i l t f ti
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Equivalent fractions
4 times as many shaded parts
4 times as many parts in all
more parts, but they are smaller
the same amountis shaded
1 whole
2
3
2
3
24
34
8
12
= =
split each part into 4 parts
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E i l t f ti i G d 5
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Equivalent fractions in Grade 5
5.NF.5b
. . . [relate] the principle of fraction equivalence a/b = (na)/(nb) to
the effect of multiplying a/b by 1.
For example:
2
3=
2
3 1 =
2
3
4
4=
2 4
3 4=
8
12
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Si l d bl i i t ti
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Simple word problems give meaning to operations
RtI Math, Recommendation 4 is on solving word problems based on
common underlying structures.
Simple word problems give meaning to mathematical operations suchas subtraction or multiplication. When students are taught the
underlying structure of a word problem, they not only have greater
success in problem solving but can also gain insight into the deeper
mathematical ideas in word problems.
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Fraction multiplication
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Fraction multiplication
5.NF Apply and extend previous understandings of multiplication and
division to multiply and divide fractions.
3.OA.1:Interpret products of whole numbers, e.g., interpret 57 as the total
number of objects in 5 groups of 7 objects each. For example, describe
a context in which a total number of objects can be expressed as 57.
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Extending multiplication to fractions
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Extending multiplication to fractions
A B is the amount in A groups of B each, A of B.
3
liters
3
liters
3
liters
3
liters
4 of 3
4 3
Amount in 4 bottles, if a bottle is 3 liters 4 of 3 4 3 liters
Amount in 4 bottles, if a bottle is 13 liter 4 of13 4
13 liters
Amount in 14 bottle, if a bottle is13 liter
14 of
13
14
13 liters
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Fraction multiplication
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Fraction multiplication
1 whole
1/3
1/4 of 1/31/4 of 1/3 is 1/12
1/4 1/3 = 1/12
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Connecting division and fractions
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Connecting division and fractions
5.NF.3
Interpret a fraction as division of the numerator by the denominator
(a/b = ab).
For example:
3 5 =3
5
How can we see this relationship?
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Connecting division and fractions
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Connecting division and fractions
1 whole submarine sandwich
3 subs divided equally among 5 people
1
5
1
5
1
5
3
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1
5
1 persons share is 3/5 of a sub
1 persons share is 3 5+
1
5+ =
3
535 =
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Attending closely to the wording of problems
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Attending closely to the wording of problems
Is this a word problem for 23 12 ?
There was23 of a pizza left over. Ben ate
12 of the pizza that was left.
Then how much pizza was left?
First I showed 2/3. Then when you
take away half of that you have 1/3 left.
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Attending closely to the wording of problems
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Attending closely to the wording of problems
Compare the wording, meaning, and way of solving these word
problems:
1 Anna had 12 cup of juice in her glass. She drank13 of it. How much
juice is left?
2 Anna had 12 cup of juice in her glass. She drank13 of a cup of
juice. How much juice is left?
3 Anna had 13 of a cup of juice in her glass. After she got some more
juice, she had 12
of a cup. How much more juice did she get?
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Thank you!
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Thank you!
Questions? Comments?
[Problem 1 sounds like its solved by 12
13
but it isnt. Anna drank 13
of
12 of a cup, which is 16 of a cup. She has 12 16 = 26 = 13 of a cup left.Problem 2 is solved by 12
13 . She has
16 of a cup left.
Problem 3 sounds like it might involve addition, but is a 13 + J=12
problem, which is solved by 12 13 . Notice that the action in this
problem reverses the action in the previous problem, so must have thesame solution.]
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