© 2011 Elementary Pre-Service Teachers Mathematics Project • School of Education • Boston University •

Boston, MA 02215 • (617) 353-7105 • schapin@bu.edu

Please do not circulate or cite without permission

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.

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 23? Explain how to use the

number line to find additional fractions equivalent to 23 .

c) Define the term, “equivalent” in a way that is mathematically correct but that would make sense

to a 9 year-old student.

0 1

A

B

C

D

0 1

0 1

0 1

Fractions on the Number Line 21

© 2011 Elementary Pre-Service Teachers Mathematics Project • School of Education • Boston University •

Boston, MA 02215 • (617) 353-7105 • schapin@bu.edu

Please do not circulate or cite without permission

3. a) Ziv and Vanessa used different strategies to partition the number line to approximate the location

of 1130 . 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 1130

.

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.

b) Why do both methods work to locate 1130

?

c) Partition the number lines below in two different ways to locate 512

.

d) List all of the factors of 30 and of 12.

e) How is partitioning to locate 1130 and

512 related to the factorization of 30 and 12?

0 1

0 1

0 1

0 1

1 15

1

15

130

÷ 5 ÷ 3 ÷ 2

1 130

÷ 6 ÷ 5 16

Fractions on the Number Line 22

© 2011 Elementary Pre-Service Teachers Mathematics Project • School of Education • Boston University •

Boston, MA 02215 • (617) 353-7105 • schapin@bu.edu

Please do not circulate or cite without permission

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, 410 and 6

15 , are the same and how they

are different.

4. To locate 1¾ on the following number line, follow these directions:

• Use 45 to find the location of 1

5 . Next determine the value of the smallest interval shown.

• 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 34

. Why must you further partition

the number line to locate 1 34 ?

a) Mark the location of 1 34

on the number line.

45

b) Mark the location of 715 on the number line below.

23

c) Mark the location of 932 on the number line below.

1 38

5. Determine the number that is represented by the “X” on the number lines below.

a)

25

0

0

0

0 X

Fractions on the Number Line 23

© 2011 Elementary Pre-Service Teachers Mathematics Project • School of Education • Boston University •

Boston, MA 02215 • (617) 353-7105 • schapin@bu.edu

Please do not circulate or cite without permission

b)

14

6. a) On a number line, mark 13

and 12, then locate three rational numbers between them. Explain your

thinking.

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 3

7 . 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 6

th 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).

0 X

0 1