UNIVERSITY OF MASSACHUSETTS BOSTON
EDCG 668/416
Fractions on the Number Line
Partitioning number lines into different size intervals can help make sense of equivalent fractions. Work
with a partner on the following questions and ideas.
1. Below are four number lines, each starting at 0 and ending at 1. Partition each number line according
to the following guidelines:
a) Partition lines A, B, C, and D into three equal-length intervals and label each tick mark with the
appropriate fraction.
b) Further partition line B so that it shows a total of six equal-length intervals. Label each tick mark
with the appropriate fraction.
c) Further partition line C so that it shows a total of nine equal-length intervals. Label each tick
mark with the appropriate fraction.
d) Further partition line D so that it shows twelve equal-length intervals. Label each tick mark with
the appropriate fraction.
A
0
1
B
0
1
C
0
1
D
0
1
2. a) When partitioning the number line into n intervals, how many tick or hash marks do you make?
Explain.
b) Using the number lines above, which fractions are equivalent to 32 ? Explain how to use the
number line to find additional fractions equivalent to 32 .
c) Define the term, “equivalent” in a way that is mathematically correct but that would make sense
to a 9 year-old student.
© 2011 Elementary Pre-Service Teachers Mathematics Project • School of Education • Boston University •
Boston, MA 02215 • (617) 353-7105 • [email protected]
Please do not circulate or cite without permission
Fractions on the Number Line
21
3. a) Ziv and Vanessa used different strategies to partition the number line to approximate the location
of 11
30 . Follow their directions and flow chart models to partition the lines below.
Ziv: I first partitioned the unit into fifths. Then I further partitioned each fifth into thirds. Finally, I
partitioned each of the smallest intervals into halves and counted eleven intervals from zero to
.
get to 11
30
1
÷5
÷3
1
5
1
15
÷2
1
30
0
1
Vanessa: I partitioned my number line differently! I first partitioned the 1 into sixths. Then, I
partitioned each sixth into fifths. I also counted eleven intervals from zero.
1
÷6
÷5
1
6
1
30
0
1
b) Why do both methods work to locate 11
?
30
c) Partition the number lines below in two different ways to locate 125 .
0
1
0
1
d) List all of the factors of 30 and of 12.
e) How is partitioning to locate
11
30
and
5
12
related to the factorization of 30 and 12?
© 2011 Elementary Pre-Service Teachers Mathematics Project • School of Education • Boston University •
Boston, MA 02215 • (617) 353-7105 • [email protected]
Please do not circulate or cite without permission
Fractions on the Number Line
22
Group Discussion Questions:
•
•
•
Elementary school students often repeatedly partition intervals on a number line in half. For
example, they divide the interval from 0 to 1 in half, then divide each half in half again, and so
on. Why doesn’t this method work to locate 30ths or 12ths? For what denominators will this
strategy work?
Discuss Question 3e.
Use the number line model to explain how the numbers,
are different.
4
10
and
6
15
, are the same and how they
4. To locate 1¾ on the following number line, follow these directions:
4
5
to find the location of 15 . Next determine the value of the smallest interval shown.
•
Use
•
Use the length of 15 or of the smallest interval to locate 1 and 2 on the number line. Why is
this important?
Use what you know to determine the exact location of 1 43 . Why must you further partition
the number line to locate 1 43 ?
•
a) Mark the location of 1 43 on the number line.
4
5
0
b) Mark the location of
7
15
on the number line below.
0
2
3
c) Mark the location of
9
32
on the number line below.
1 38
0
5.
Determine the number that is represented by the “X” on the number lines below.
a)
0
X
2
5
© 2011 Elementary Pre-Service Teachers Mathematics Project • School of Education • Boston University •
Boston, MA 02215 • (617) 353-7105 • [email protected]
Please do not circulate or cite without permission
Fractions on the Number Line
23
b)
0
X
1
4
6. a) On a number line, mark
thinking.
1
3
and
1
2
, then locate three rational numbers between them. Explain your
0
1
b) A common error that occurs in the elementary grades is for students to state that there are no
numbers between two fractions such as 27 and 37 . Why might they hold this misconception?
Group Discussion Questions:
•
•
•
Explain your methods and reasoning for solving some of the problems from Questions 4 and 5.
Why is it helpful to know the location of 1?
The density property of rational numbers states that there are an infinite number of rational
numbers between any two rational numbers. Why is this an important idea for 5th and 6th graders
to understand? How can teachers use number lines to help students make sense of this idea? Are
the integers dense? Why or why not?
Summarize and Connect:
Write your responses to the following questions.
•
•
What connections exist among factors, multiples, partitioning, and equivalent fractions?
The interpretation of a fraction as a point on the number line that is a set distance from zero is
quite different from the part-whole interpretation. Describe key features of the number line
interpretation using the words: interval, equivalence, distance, unit, location, density, chunk(ing)
and partition(ing).
© 2011 Elementary Pre-Service Teachers Mathematics Project • School of Education • Boston University •
Boston, MA 02215 • (617) 353-7105 • [email protected]
Please do not circulate or cite without permission