Which fraction in each pair is greater?
Give one or more reasons. Try not to use drawings or models.
Do not use common denominators or cross-multiplication.
Rely on concepts.
A. 4 or 4
5
9
G. 7 or 5
12
12
B. 4 or 5
7
7
H. 3 or 3
5
7
C. 3 or 4
8
10
I. 5 or 6
8
10
D. 5 or 5
3
8
J. 9 or 4
8
3
E. 3 or 9
4
10
K. 4 or 7
6
12
F. 3 or 4
8
7
L. 8 or 7
9
8
Presented at 2011 KSDE Summer CCSS Academy
3-5 Math Sessions
Conceptual Thought Patterns for Comparison
The first two comparison schemes listed here rely on the
meanings of the top and bottom numbers in fractions and on
the relative sizes of unit fractional parts. The third and fourth
1
ideas use the additional ideas of 0, 2 , and 1 as convenient
anchors or benchmarks for thinking about the size of fractions.
1. More of the same-size parts. To compare 3 and ,5 it is
8
8
easy to think about having 3 of something and also 5 of
5
the same thing. It is common for children to choose 8 as
larger simply because 5 is more than 3 and the other
numbers are the same. Right choice, wrong reason.
Comparing 38 and 58 should be like comparing 3 apples and
5 apples.
2. Same number
of parts
but parts of different sizes. Consider
3
3
the case of 4 and 7 . If a whole is divided into 7 parts, the
parts will certainly be smaller than if divided into only 4
parts. Many children will select as 37 larger because 7 is
more than 4 and the top numbers are the same. That
approach yields correct choices when the parts are the
same size, but it causes problems in this case. This is like
comparing 3 apples with 3 melons. You have the same
number of things, but melons are larger.
Presented at 2011 KSDE Summer CCSS Academy
3-5 Math Sessions
3. More 3and less than
one-half
or one
whole. The fraction
5
5
7
pairs 7 versus 8 and 4 versus 8 do not lend themselves
to either of the previous thought processes. In the first
pair, 37 is less than half of the number of sevenths needed
to make a whole, and so is less than a half. Similarly, is
more than a half. Therefore,
is the larger fraction. The5
3
second pair is determined7by noting that one fraction is8
5
less than 1 and the other is greater
than 1.
8
9
10
4. Distance from one-half or one whole. Why is
greater
3
than 4 ? Not because the 9 and 10 are big numbers,
although you will find that to be a common student
response. Each is one fractional part away from one
whole, and tenths are smaller than fourths. Similarly,
5
4
notice that 8 is smaller than 6 because it is only one-eighth
4
more than a half, while 6 is a sixth more than a half. Can
3
5
you use this basic idea to compare 5 and 9 ? (Hint: each is
1
5
7
half of a fractional part more than 2 .) Also try 7 and 9 .
How did your reasons for choosing fractions compare to
these ideas? It is important that you are comfortable with these
informal comparison strategies as a major component of your
own number sense as well as for helping children develop
theirs.
Tasks you design for your students should assist them in
developing these and possible other methods of comparing two
fractions. It is important that the ideas come from your
students and their discussions. To teach “the four ways to
compare fractions” would be adding four more mysterious
rules and would be defeating for many students.
Presented at 2011 KSDE Summer CCSS Academy
3-5 Math Sessions