Topic 2: Comparing and Ordering Repeating Decimals
for use after
Accentuate the Negative
Investigation 4
Remember that decimals can be
terminating or repeating
.For example,
0.75 the decimal equivalent to
is a terminating decimal.
If you divide 2 by 11, you will get the repeating decimal 0.181818. . .
You can show that the digits repeat forever by putting a bar over the digits
that repeat.
Exercises
Find the decimal equivalent to each fraction. Tell if the decimal is
terminating or repeating.
1.
2.
3.
4.
5. a.
Which decimal is greater,
or
? Explain.
b. Which decimal is greater,
or
? Explain.
c. Describe a process for deciding which of two positive repeating
decimals is greater.
Copy each pair of numbers in Exercises 6
–
9. Insert
R
or
S
to make a
true statement.
6.
j
7. 0.75 j
8. 8.333 j
9.
j
Order each set of decimals from least to greatest.
10.
11.
0.75,
12. 2.33333, 2.033333, 2
13.
5.003,
5.003, 5.03
0.705,
0.75
0.05, 0.105,
0.005
8.03
0.85
0.805
0.01
0.001
0.75
0.16
0.17
0.05
0.06
1
6
3
8
2
3
5
11
0.181818 c 5 0.18
3
Q
4
R
14. a. Which decimal is greater, -
or -
? Explain.
b. Which decimal is greater, -
or -
? Explain.
c. Describe a process for deciding which of two negative repeating
decimals is greater.
Copy each pair of numbers in Exercises 15–18. Insert
R
or
S
to make a
true statement.
15. -
j -
16. -0.75 j -
17. -4.555 j -
18. -
j -
Order each set of decimals from least to greatest.
19. -
-
-
20. -0.99, ---
0.909,
0.900
,
1
0.06, 0.106,
0.006
4.05
0.0008
0.008
0.01
0.001
0.75
0.16
0.17
0.05
0.06
Comparing and Ordering Repeating
Decimals
Guided Instruction
Mathematical Goals
•
Compare and order repeating decimals.
Vocabulary
•
terminating decimal
•
repeating decimal
At a Glance
Students have had some experiences writing fractions as repeating decimals,
so Exercises 1–4 help students to recall that information. The remainder of
the exercises focus on describing a procedure for comparing two positive
repeating decimals, or two negative repeating decimals.
You will have to be careful that students do not incorrectly generalize a
procedure for ordering repeating decimals. When you order terminating
decimals, like 0.2458 and 0.267, you look for the first decimal place where
the digits are not equal and identify the decimal with the greater digit in
that place as the greater value. The same procedure works for repeating
decimals, provided that the repeating digits are not all 9’s. For example,
0.05 ?
, but 0.05 =
.
Here is the reason for this:
Let
x
=
or 0.4999. . .
Then 10
x
=
or 4.9999. . .
So 9
x
= 10
x
-
x
9
x
= 4.9999. . . - 0.4999. . .
9
x
= 4.5
x
=
or
Students at this point in CMP2 are not expected to know that
= 1.
The examples in these exercises were chosen to avoid that exception.
When students order any numbers, it is often helpful for them to think
about which value is farther to the right on a standard number line. The
greater of two values is always the value farther to the right on a standard
number line. This is especially helpful when ordering negative numbers. For
example,
,
, but
-
. -
.
Before Exercise 1, ask:
•
If two fractions have the same decimal representation, are the fractions
equivalent?
•
How many digits are there in a repeating decimal?
•
How can you tell by looking at the denominator of a proper fraction
whether that fraction has a terminating or repeating decimal
representation?
Before Exercise 5, ask:
•
How do you know which of two positive terminating decimals is
greater?
•
How do you know which of two positive repeating decimals is greater?
0.16
0.17
0.16
0.17
0.9
1
2
4.5
9
4.9
0.49
0.04
0.049
PACING
1 day
continued on next page
Assignment Guide for Topic 2
Core
1–13
Other
14–20
Note:
In Exercises 15-19, you may want to ask
students to write out six to eight of the decimal
digits to be sure that they are repeating the
correct string of digits in each case.
Answers to Topic 2
Exercises
1.
, repeating
2.
, repeating
3.
0.375, terminating
4.
, repeating
5. a.
; the hundredths place is greater.
b.
; the hundredths place is greater.
c.
Compare corresponding place values of
two positive repeating decimals until the
digit in one is greater than the other. The
decimal with the greater digit is the greater
decimal.
Note:
Mathematically, this has to
be qualified by saying “so long as the digits
after this value are not all 9’s.”
6.
.
7.
0.75 ,
8.
8.333 .
9.
.
10.
,
11.
, 0.75,
12.
2, 2.033333, 2.33333
13.
5.003,
,
14. a.
-
; when you graph them on a number
line,
-
is closer to zero than -
.
b.
-
; when you graph -
on a number
line, it is closer to zero than
-
.
c.
When both decimals are negative, the
decimal with the greater absolute value has
the lesser value; that is, it is farther to the
left of zero.
15.
-
, -
16.
-0.75 . -
17.
-
4.555 , -
18.
-
. -
19.
-
, - , -
20.
-
1, -0.99, -
0.909
, -0.900
0.106
0.06
0.006
4.05
0.0008
0.008
0.01
0.001
0.75
0.17
0.16
0.16
0.05
0.06
0.05
5.003 5.03
0.705
0.75
0.005 0.05 0.105
8.03
0.85
0.805
0.01
0.001
0.75
0.17
0.06
0.16
0.45
0.6
Before Exercise 14, ask:
•
How do you know which of two negative terminating decimals is
greater?
•
How do you know which of two negative repeating decimals is greater?
•
How are the procedures for comparing terminating and repeating
decimals the same?
•
How are the procedures for comparing terminating and repeating
decimals different?
Guided Instruction
continued