Challenges and Opportunites
of Common Core: Fraction
Teaching and Learning Tad Watanabe
Kennesaw State University
science.kennesaw.edu/~twatanab
Stages of Fraction Teaching &
Learning in CCSS •  Laying foundations: Grades 1 ~ 2
•  Understanding fractions as
numbers: Grades 3 ~ 4
•  Fraction arithmetic: Grades 4 ~ 6 Laying Foundations •  1.G.3
Partition circles and rectangles into
two and four equal shares, describe
the shares using the words halves,
fourths, and quarters, and use the
phrases half of, fourth of, and quarter
of. Describe the whole as two of, or
four of the shares. Understand for
these examples that decomposing
into more equal shares creates smaller
shares. Laying Foundations •  2.G.3
Partition circles and rectangles into
two, three, or four equal shares,
describe the shares using the words
halves, thirds, half of, a third of,
etc., and describe the whole as two
halves, three thirds, four fourths.
Recognize that equal shares of
identical wholes need not have
the same shape. Some Key Ideas •  Equal partitioning
•  Describing shares
•  Exploring the relationship between
the size of shares and the number
of shares Equal Partitioning Challenge
–  How do students know that shares are
equal?
Equal Partitioning Challenge
–  How do students know that shares are
equal?
Equal Partitioning Challenge
–  It’s about quantities, not shapes.
Reasoning Empirical
More abstract
Both
and
are half of the same
sized share. Therefore
they are equal. Describing Shares Singapore Textbook, Standard Edition, 2B, p. 62 Describing Shares •  CCSS does not specify the use of
fraction notations in Grades 1 & 2.
•  Gunderson & Gunderson (1957)
–  At the beginning of fraction instruction,
we should write out the fraction words
rather than use standard notation: 3fourths, instead of ¾. Number and Size of Shares Singapore Textbook, Standard Edition, 2B, p. 62 Number and Size of Shares Challenge
– Fractions as
relationships and
fractions as
quantities
– Both pizzas are
divided into halves
and quarters.
– Which half is
bigger?
– Which is bigger,
half or qurater?
Fractions as Numbers •  Grade 3 Overview
Students develop an understanding of fractions,
beginning with unit fractions. Students view fractions
in general as being built out of unit fractions, and
they use fractions along with visual fraction models
to represent parts of a whole. Students understand
that the size of a fractional part is relative to the size
of the whole. For example, 1/2 of the paint in a
small bucket could be less paint than 1/3 of the
paint in a larger bucket, but 1/3 of a ribbon is longer
than 1/5 of the same ribbon because when the
ribbon is divided into 3 equal parts, the parts are
longer than when the ribbon is divided into 5 equal
parts. Students are able to use fractions to represent
numbers equal to, less than, and greater than one.
They solve problems that involve comparing
fractions by using visual fraction models and
strategies based on noticing equal numerators or
denominators. Fractions as Numbers Opportunity: Unitary view of fractions
•  3.NF.1
Understand a fraction 1/b as the quantity
formed by 1 part when a whole is
partitioned into b equal parts;
understand a fraction a/b as the quantity
formed by a parts of size 1/b. Unitary View of Fractions Tokyo Shoseki (2006), 3B, p. 57 Unitary View of Fractions Tokyo Shoseki (2006), 3B, p. 60 Fractions on the number line •  3.NF.2
Understand a fraction as a number on
the number line; represent fractions on a
number line diagram. Fractions on the number line Challenge
–  1 as a whole
–  1 as a point on the number line vs. 1
as the distance from 0, or the length of
a unit interval
Bridging to number line Tokyo Shoseki (2006), 3B, p. 60 Fractions as Numbers •  3.NF.3
Explain equivalence of fractions in special
cases, and compare fractions by reasoning
about their size.
•  4.NF.1
Explain why a fraction a/b is equivalent to a
fraction (n x a)/(n x b) by using visual fraction
models, with attention to how the number
and size of the parts differ even though the
two fractions themseselves are the same size.
Use this principle to recognize and generate
equivalent fractions.
Equivalent Fractions Tokyo Shoseki (2006), 4B, p. 44 Equivalent Fractions Challenge (Grade 4)
If we divide each thrid into 2 equal
parts…
Therefore,
.
But, why “x” when we “divide”? Comparing Fractions •  3.NF.3.d
Compare two fractions with the same
numerator or the same denominator by
reasoning about their size. Recognize
that comparisons are valid only when the
two fractions refer to the same whole.
Record the results of comparisons with
the symbols >, =, or <, and justify the
conclusions, e.g., by using a visual
fraction model. Comparing Fractions as
Numbers •  Whole = 1
•  Key understandings
–  The numerator tells us how many units
there are.
–  The denominator tell us the size of the
unit in terms of the number of equal
parts 1 must be partitioned.
–  The size of fractions are determined by
considering BOTH. Important Misconception •  Given two proper fractions, the one
with the smaller difference between
the numerator and the denominator
is larger.
Example
2/3 > 2/5 because 3-2 < 5-2
Important Misconception •  Given two proper fractions, the one
with the smaller difference between
the numerator and the denominator
is larger.
Example
2/3 > 2/5 because 3-2 < 5-2
Complement Singapore Textbook, Standard Edition, 3B, p. 85 Comparing Fractions (Gr. 4) •  4.NF.2
… or by comparing to a benchmark
fraction such as ½. … Singapore Textbook, Standard Edition, 3B, p. 88 Whole Numbers and Fractions •  3.NF.3.c
Express whole numbers as fractions, and
recognize fractions that are equivalent to
whole numbers. Examples: Express 3 in
the form of 3 = 3/1; recognize that 6/1
= 6; located 4/4 and 1 at the same point
of a number line diagram.
Grade 3 footnote
Grade 3 expectations in this domaing
(NF) are limited to fractions with
denominators 2, 3, 4, 6, and 8. 1 as denominator •  Perhaps it may be better to
introduce it in the context of
quotient meaning of fraction in
Grade 5, i.e., 5 = 5 ÷ 1 = 5/1. Tokyo Shoseki (2006), 5B, pp. 51 - 52 Grade 3 Overview … For example, 1/2 of the paint in a small
bucket could be less paint than 1/3 of the
paint in a larger bucket, but 1/3 of a ribbon
is longer than 1/5 of the same ribbon
because when the ribbon is divided into 3
equal parts, the parts are longer than when
the ribbon is divided into 5 equal parts. Improper fractions, mixed
numbers, whole numbers Tokyo Shoseki (2006), 4B, pp. 40 & 43 Addition/Subtraction Opportunity
–  Unitary View of Fractions Addition/Subtraction Tokyo Shoseki (2006), 3B, p. 62 Addition/Subtraction Tokyo Shoseki (2006), 3B, pp. 62 - 63 Addition/Subtraction Tokyo Shoseki (2006), 5A, p. 44 Addition/Subtraction •  You can only add/subtract numbers
when they are referring to the same
unit.
–  Grade 1: addition/subtraction of
multiples of 10
Examples: 30 + 40; 80 – 60
–  Grade 2 – 5: solving problems
involving measurement
–  Grade 5: decimal number addition/
subtraction
Example: 0.3 + 0.4; 0.8 – 0.6 Multiplication/Division Opportunity
–  Clear distinction between multiplying
fractions by whole numbers (Grade 4)
and multiplying by fractions (Grade 5) Meaning of Multiplication •  3.OA.1
Interpret products of whole numbers,
e.g., interpret 5 x 7 as the total
number of objects in 5 roups of 7
objects each.
(# of groups) x (group size)
= product Multiplication •  Grade 4: multiply a fraction by a
whole number
5 x ¾, but not ¾ x 5
•  Grade 5: multiplying by a fraction
¾ x 5 as well as ¾ x ½
Meaning of Multiplication •  5 x ¾ means 5 groups of ¾ in each
group.
•  Does not require an adjustment in
the meaning of multiplication.
•  5 groups of 3 ¼-units, or 5x3 ¼units.
5x¾
= 5 x (3 x ¼)
= (5 x 3) x ¼
Meaning of Multiplication •  ¾ x (###) means ¾ times as
much of (###) – 5.NF.5
•  4.OA.1
Interpret a multiplication equation
as a comparison, e.g., interpret 35
= 5 x 7 as a statement that 35 is 5
times as many as 7 and 7 times as
many as 5.
Meaning of Multiplication Challenge
•  4.OA.1
Interpret a multiplication equation as
a comparison, e.g., interpret 35 = 5 x
7 as a statement that 35 is 5 times as
many as 7 and 7 times as many as 5.
So, can both 5 x ¾ and ¾ x 5 be
interpreted as 5 times as much of ¾?
Multiplying by Fractions •  5.NF.4.a
Interpret product (a/b) x q as a
parts of partition of q into b equal
parts; equivalently, as the result of
a sequence of operations a x q ÷ b. Multiplying by Fractions Challenge
(2/3) x 4
–  2/3 of 4
–  partition 4 into 3 equal parts
–  take 2 of the equal parts
(2/3) x 4 = 2 x (4 ÷ 3)
Multiplying by Fractions Challenge
(2/3) x (4/5)
–  2/3 of 4/5
–  partition 4/5 into 3 equal parts
–  take 2 of the equal parts
(2/3) x (4/5) = 2 x {(4/5) ÷ 3}
Multiplying by Fractions Challenge
(2/3) x (4/5)
–  2/3 of 4/5
–  partition 4/5 into 3 equal parts
–  take 2 of the equal parts
(2/3) x (4/5) = 2 x {(4/5) ÷ 3}
5.NF.7: Apply and extend previous
understandings of division to divide unit
fractions by whole numbers and whole
numbers by unit fractions.
Division of Fractions •  Dividing a whole number by a unit
fraction
–  How many unit fractions are in so
many wholes? (measurement division)
•  Dividing a fraction by a whole
number
–  How much is in each group if a
fraction is partitioned into so many
parts? (fair sharing division) Division of Fractions •  4/5 ÷ 2 = ?
–  If 4 1/5-units are partitioned into 2
groups, how much is in each group?
–  4 ÷ 2 = 2 1/5-units in each group.
–  4/5 ÷ 2 = (4 ÷ 2)/5 Division of Fractions •  3/5 ÷ 2 = ?
–  If 3 1/5-units are partitioned into 2
groups, how much is in each group?
–  3 ÷ 2 = not a whole number!
•  3/5 = 6/10
–  If 6 1/10-units are partitioned into 2
grous, how much is in each group?
–  6 ÷ 2 = 3 1/10-units in each group
–  3/5 ÷ 2 = 6/10 ÷ 2 = (6 ÷ 2)/10 = 3/10 Division of Fractions •  In general,
a/b ÷ n = a/(b × n) Multiplication and Division Opportunity: double number line
(2/3) x (4/5)
–  2/3 of 4/5
–  partition 4/5 into 3 equal parts
–  take 2 of the equal parts
Division of Fractions 2/5 ÷ 3/4 = ? Division of Fractions Division of Fractions •  Mathematical practice: repeated
reasoning
2/5 ÷ 3/4 = ?
–  How much is for 1/4?
–  Take 4 of that amount to find how
much is for 1. Division of Fractions •  Mathematical practice: repeated
reasoning
2/5 ÷ 3/4 = ?
Multiplicatin & Division Opportunity: Developing a unified
understanding of multiplication,
division, and proportion. •  4/5 x 2/3
•  2/5 ÷ 3/4 Unit Approach •  With 6 gallons of gasoline, Mike’s car
can travel 250 miles. How far can he
travel with 4 gallons of gasoline?
Finde per-one quantity – partitive division