Teaching Notes 2.7: Expressing Repeating Decimals
as Fractions or Mixed Numbers
The process of expressing a repeating decimal as a fraction requires students to write an equation,
rewrite the decimal, and solve an equation. Confusion with any of these steps can lead to
mistakes.- Explain that there are two types of nonterminating decimals: those whose digits repeat in a
specific pattern and those whose digits do not repeat in a specific pattern. - Explain that if the digits repeat in a specific pattern, a bar can be placed over the repeat-
ing digits and the decimal can be expressed as a fraction. For example, 0. 777 ...= 0 .7and
0. 313131 ...= 0 .31. Unlike terminating decimals such as 0.7, which equals
7
10
and 0.31,which equals31
100
, the repeating decimals cannot be written over a power of 10. Also, empha-size that ifn= 0 .7, then 10n= 7 .7. Similarly, ifn= 0 .31, then 100n= 31 .31. An example
of a nonterminating decimal whose digits do not repeat in a specific pattern isπ.- Review the information and example on the worksheet with your students. Explain that they
should find two equations so that when one equation is subtracted from the other, a ter-
minating decimal is in the equation that results. Note that multiplying an equation by any
nonzero number produces an equation that has the same solution as the original equation.
Also note that multiplying by 10 ‘‘moves’’ the decimal point one place to the right; multiply-
ing by 100 ‘‘moves’’ the decimal two places to the right, and so on.
EXTRA HELP:
Check your answer by dividing the numerator by the denominator to see if the quotient is the
same as the repeating decimal.ANSWER KEY:
(1)1
9
(2)
23
99
(3)
8
11
(4)
5
9
(5) 2
9
11
(6)
1
33
(7) 6
512
999
(8) 7
6
37
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(Challenge)Manny is correct. Roseanne’s example of 0. 3 =1
3
is equivalent to3
9
.
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60 THE ALGEBRA TEACHER’S GUIDE