Terminating and Repeating Decimals
Student Probe
5
Write as a decimal.
8
Answer: 0.625
Lesson Description
In this lesson students use long division to convert
fractions into repeating or terminating decimals.
Calculator use in encouraged.
Rationale
All rational numbers, including mixed numbers, can be
converted into repeating or terminating decimals by
dividing the numerator by the denominator. This division
process can be done with or without the use of a
calculator. It is important for students to realize that
there situations where a fractional representation is
more efficient and there are times when a decimal
representation is more efficient. Students need to be
flexible in their thinking to decide which is more
appropriate in a given situation.
Additionally, some fractions and their decimal
equivalents are used so often that it is efficient for
students to know them without having to convert from
one form to another. Some examples of these include:
1
1
3
1
1
2
0.5,
0.25,
0.75,
0.125
0.33,
0.66.
2
4
4
8
3
3
With repeated use students will become familiar with
these conversions, so it is not necessary to require
memorization.
At a Glance
What: Converting fractions into repeating
or terminating decimals
Common Core Standard: CC.7.NS.2d.
Apply and extend previous
understandings of multiplication and
division and of fractions to multiply and
divide rational numbers. (d) Convert a
rational number to a decimal using long
division; know that the decimal form of a
rational number terminates in 0s or
eventually repeats.
Mathematical Practices:
Use appropriate tools strategically.
Attend to precision.
Look for and express regularity in
repeated reasoning.
Who: Students who cannot convert a
fraction into a repeating or terminating
decimal
Grade Level: 7
Prerequisite Vocabulary: numerator,
denominator, rational number, equivalent
fractions
Prerequisite Skills: division of whole
numbers
Delivery Format: Individual, small group
Lesson Length: 30 minutes
Materials, Resources, Technology:
calculator
Student Worksheets: none
Preparation
Allow students to have access to calculators. Simple, four-function calculators are appropriate
for this lesson.
Lesson
The teacher says or does…
1. What are some fractions that
1
are equivalent to ?
2
2. How can we write
5
as a
10
Expect students to say or do… If students do not, then the
teacher says or does…
5
Answers will vary, but listen
What about
? Is it
5
10
for
.
1
10
equivalent to ?
2
0.5
Prompt students, if
necessary.
decimal?
3. So we are saying
1 5
0.5 ,
2 10
1
0.5 .
2
4. What does the “fraction bar”
mean?
1
That means is the same as
2
1 2.
Put 1 2 in your calculator
and tell me what answer it
returns.
That is another way to figure
1
out
0.5 .
2
3
5. What does mean?
8
Use your calculator to find
3
the decimal equivalent of .
8
or
6. Find the decimal equivalent
1
of .
3
Division.
What operation does the
fraction bar indicate?
0.5
Monitor students.
3 8
What operation is the
fraction bar telling us to
do?
3 8 0.375 .
1 3 0.333333333 …
Monitor students.
Monitor students. Prompt
if necessary.
The teacher says or does…
7. Notice that the 3’s keep
going! This is called a
repeating decimal. Can we
write all of the 3’s?
Mathematicians have a way
to show that the 3’s repeat
forever.
It is written like this: 0.33 .
The bar is placed over the
part of the decimal that
repeats to show that it
repeats those digits forever.
8. There are two kinds of
rational numbers: the ones
1
that repeat such as , and
3
the ones that end, or
1
3
terminate, like and .
2
8
9. Find the decimal equivalent
5
of .
6
Is it a repeating decimal or a
terminating decimal?
How do you know?
10. Find the decimal equivalent
5
of
.
16
Is it a repeating decimal or a
terminating decimal?
How do you know?
11. Let’s find the decimal
3
equivalent of 2 . We know
5
we have 2 wholes, so we only
3
need to consider the .
5
Is it terminating or repeating?
Expect students to say or do… If students do not, then the
teacher says or does…
No.
5
0.83
6
5
means 5 6 .
6
Repeating, because the 3
repeats forever.
5
means 5 16 .
16
5
0.3125
16
Terminating, because it ends.
2
3
2
5
3 5
Terminating
2 0.6 2.6
3
3 5.
5
The teacher says or does…
Expect students to say or do… If students do not, then the
teacher says or does…
12. Repeat the steps above with a
variety of fractions and mixed
numbers. Include both
terminating and repeating
decimals.
Teacher Notes
k
k n . This can
n
seem unusual to students who have been thinking of fractions as parts of wholes.
2. Make sure that students understand the notation for repeating decimals and use it
correctly.
3. When converting mixed numbers to decimals, students should understand that only the
fractional part of the number is converted. The whole number remains unchanged.
4. Notice that any fraction with a denominator of 7 will have a repeating block of 6 digits.
Students will sometimes fail to notice all of the digits.
1. This lesson relies on the interpretation of fractions as indicted division,
Variations
Students may be interested in the types of rational numbers that have repeating or terminating
1 1 1
1
decimal forms. Let students create a table of the first 20 unit fractions, , , ,..., , and their
2 3 4
20
decimal equivalents (using a calculator). Students should then investigate the patterns that
emerge. (Teacher Note: denominators of fractions which terminate have prime factors of only
2 and/or 5.)
Formative Assessment
Write
5
as a decimal.
12
Answer: 0.416
References
Mathematics Preparation for Algebra. (n.d.). Retrieved August 10, 2010, from Doing What
Works:
Van de Walle, J. A., & Lovin, L. H. (2006). Teaching Student-Centered Mathematics Grades 5-8
Volume 3. Boston, MA: Pearson Education, Inc.