Transcript Fractions: Conceptual Understanding for Grades 3-5

Presented by:
Jenny Ray, Mathematics Specialist
Kentucky Department of Education
Northern KY Cooperative for Educational Services
Jenny C. Ray Math Specialist, NKY Region
Kentucky Department of Education
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Children’s Ideas about
Fractions:

Show me where ½ could be on the
number line below:
Why do students
sometimes choose
this part of the
number line?
0
1
2
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2
Children’s Ideas about Whole
Numbers:
3>2
1=1


ALWAYS.
ALWAYS.
So…how can it be that 1/3 > ½ ?
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When students can’t ‘remember’ a
procedure, they resort to performing any
operation they know they can do…
•
Estimate the answer:
• A) 1
• B) 2
• C) 19
• D) 21
• E) I don’t know.
12/
13
+ 7/8
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National Assessment of Educational Progress (NAEP)
results show an apparent lack of understanding of
fractions by 9, 13, and 17 yr olds.
Estimate the answer:
12/
13
+ 7/8
Only 24% of the 13-yr-olds responding chose
the correct answer, “2”.
 55% selected 19 or 21

 These students seem to be operation on the
fractions without any mental referents to aid their
reasoning.
Results from the 2nd Mathematical Assessment of the
National Assessment of Educational Progress
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Perhaps you’ve seen this reasoning…
1/
2
+ 1/3 = 2/5
 If students have an understanding of the
value of the fractions on a number line, or
as parts of a whole, then they can argue the
unreasonableness of this answer.
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How can students learn to think
quantitatively about fractions?
•
•
•
•
•
•
Based on research…
“…students should know something about the relative
size of fractions.
They should be able to order fractions with the same
denominators or same numerators as well as
to judge if a fraction is greater than or less than 1/2.
They should know the equivalents of 1/2 and other
familiar fractions.
The acquisition of a quantitative understanding of
fractions is based on students' experiences with
physical models and on instruction that emphasizes
meaning rather than procedures.” (Bezuk & Cramer,
1989)
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FRACTION MANIPULATIVES
Hands on experiences help students develop a conceptual
understanding of fractions’ numerical values.
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Learning Activity:
Fraction Circles
The white circle is 1. What is the value
of each of these pieces?
1 yellow
3 reds
1 purple Now…change the unit:
3 greens The yellow piece is 1.
What is the value of
those pieces?
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Learning Activity:
Using Counters
Eight counters equal 1, or 1 whole.
What is the value of each set of counters?

1 counter
2 counters
4 counters
6 counters

12 counters



Now, change the unit:
Four counters equal 1.
What is the value of each set of
counters?
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Learning Activity:
Cuisinaire Rods
The green Cuisenaire rod equals 1.
What is the value of each of these rods?
red
Change the unit:
black
The dark green rod is 1.
white
Now what is the value of
dark green
those rods?
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Learning Activity:
Number Lines
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A “new” way of thinking/teaching…

“Many pairs of fractions can be
compared without using a formal
algorithm, such as finding a common
denominator or changing each fraction
to a decimal.”
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Comparing without an algorithm
•
Pairs of fractions with like denominators:
1/4 and 3/4
3/5 and 4/5
• Pairs of fractions with like numerators:
1/3 and 1/2
2/5 and 2/3
• Pairs of fractions that are on opposite
sides of 1/2 or 1:
3/7 and 5/9
3/11 and 11/3
• Pairs of fractions that have the same
number of pieces less than one whole:
2/3 and 3/4
3/5 and 6/8
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Comparing 3/7 and 5/9…a
student’s response:
•
•
•
The fractions in the third category are on
"opposite sides" of a comparison point.
One fourth-grade student compared 3/7
and 5/9 in the following manner (Roberts
1985):
"Three-sevenths is less. It doesn't cover
half the unit. Five-ninths covers over
half."
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Comparing 6/8 and 3/5: A student’s
response…
•
•
•
•
A fourth-grade student compared 6/8 and 3/5 in this
way (Roberts 1985):
"Six-eighths is greater. When you look at it, then you
have six of them, and there'd be only two pieces left.
And then if they're smaller pieces like, it wouldn't have
very much space left in it, and it would cover up a lot
more.
Now here [3/5] the pieces are bigger, and if you have
three of them you would still have two big ones left. So
it would be less."
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Conceptual Understanding

Notice that each child's reasoning from the
previous two examples is based on an
internal image constructed for fractions.

Hands-on experiences with fractional parts,
both smaller than and greater than one, helps
to create this conceptual knowledge, so that
procedures that they develop make sense.
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Exploring fractions with the same
denominators
•
Use circular pieces. The whole circle is
the unit.
– A. Show 1/4
– B. Show 3/4
Are the pieces the same size?
How many pieces did you use to show 1/4?
How many pieces did you use to show 3/4?
Which fraction is larger? How do you know?
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Comparing fractions to ½ or 1
•
•
Use circular pieces. The whole circle is the
unit.
A. Show 2/3
B. Show 1/4
•
•
•
•
Which fraction covers more than one-half of the
circle?
Which fraction covers less than one-half of the
circle?
Which fraction is larger? How do you know?
Compare these fraction pairs in the same way.
– 2/8 and 3/5
– 1/3 and 5/6
– 3/4 and 2/3
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Grade 3
Also MP6: Attending to Precision
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Area Representations…
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“The goal is for students to see unit fractions as the basic
building blocks of fractions, in the same sense that the number 1
is the basic building block of the whole numbers.” Progressions for
the Common Core; commoncoretools.org (2011)
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3.NF.2
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3.NF.3abc
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3.NF.3d
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Grade 4

“Grade 4 students learn a fundamental
property of equivalent fractions: multiplying
the numerator and denominator of a
fraction by the same non-zero whole
number results in a fraction that represents
the same number as the original fraction.”
Progressions for the Common Core; commoncoretools.org (2011)
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4.NF.1
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4.NF.2

Compare 5/8 and 7/12
 Equivalent fractions..60/96 and 56/96

Compare 7/8 and 13/12
 Use the benchmark of 1.
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4.NF.3
Students in grade 4 are able to compose and decompose fractions using
unit fractions and can write them as equations.
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4.NF.3c
Adding mixed numbers with like
denominators by reasoning about the
size of the numbers, not by using a
“GPS” system…
7½+3½=
2¾ +5¾=

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4.NF.4

5 x 1/3 =

5 x 2/3 =
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4.NF.5

Express a fraction with denominator 10
as an equivalent fraction with
denominator 100, and use this
technique to add 2 fractions with
respective denominators 10 and 100.

2/10 + 35/100 =
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4.NF.6

Use decimal notation for fractions with
denominators 10 or 100.

27/10
27/100

2.7
0.27
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4.NF.7
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5.NF.1
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5.NF.2
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5.NF.4b
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5.NF.7

Apply and extend previous
understandings of division to divide unit
fractions by whole numbers and whole
numbers by unit fractions.

5 divided by ½…
½ divided by 3…
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Resources for Activities
Illustrativemathematics.org
Illuminations (NCTM)
Rational Number Project
nzmaths
Mars/Shell Centre
Teaching Channel
www.jennyray.net
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