what type of decimal expansion does
have????
Answers
Answer:
Sarah learned that in order to change a fraction to a decimal, she can use the standard division algorithm and divide the numerator by the denominator. She noticed that for some fractions, like 14 and 1100 the algorithm terminates at the hundredths place. For other fractions, like 18, she needed to go to the thousandths place before the remainder disappears. For other fractions, like 13 and 16, the decimal does not terminate. Sarah wonders which fractions have terminating decimals and how she can tell how many decimal places they have.
Convert each of the following fractions to decimals to help Sarah look for patterns with her decimal conversions:
12,13,14,15,16,110,111,112,115.
Which fractions on the list have terminating decimals (decimals that eventually end in 0's)? What do the denominators have in common?
Which fractions on the list have repeating decimals? What do the denominators have in common?
Which fractions pq (in reduced form) do you think have terminating decimal representations? Which do you think have repeating decimal representations?
IM Commentary
The goal of this task is to convert some fractions to decimals and then make conjectures about which fractions have terminating decimal expansions (as well as the length of those decimals). The teacher may wish to focus more on the conversion process and less on identifying which fractions have terminating or repeating decimal representations. In this case, only part (a) is needed and the key is that all remainders which appear in the long division algorithm are less than the divisor: this means that the decimal must either terminate (when there is a remainder of 0) or repeat (when we see the same non-zero remainder twice). The other parts of this task are looking forward to an additional level of structure, namely identifying which fractions have terminating decimals and which have repeating decimals.
The Standards for Mathematical Practice focus on the nature of the learning experiences by attending to the thinking processes and habits of mind that students need to develop in order to attain a deep and flexible understanding of mathematics. Certain tasks lend themselves to the demonstration of specific practices by students. The practices that are observable during exploration of a task depend on how instruction unfolds in the classroom. While it is possible that tasks may be connected to several practices, the commentary will spotlight one practice connection in depth. Possible secondary practice connections may be discussed but not in the same degree of detail.
Mathematical Practice Standard 8, Look for and express regularity in repeated reasoning, illuminates the work of students as they compute, calculate, or manipulate numbers and/or symbols. In this task, seventh graders are looking for patterns, considering generalities and limitations, and making sense of their observations. As students convert the given fractions to decimals, they will be looking for patterns so that they can ascertain which fractions have terminating or repeating decimal expansions. They will notice that a decimal terminates when there is a remainder of zero and it repeats when they find themselves repeating the same calculations over and over again. Students may dig a little deeper to distinguish the similarities and differences of the denominators of terminating and repeating decimal expansions. As students make conjectures about what they are observing, they should be encouraged to test more examples. After they study many examples and search for regularity, they may conclude that the prime factors of the denominators of fractions with terminating decimals are 2 and/or 5. Whereas, when the denominator has a prime factor other than 2 or 5, the decimal repeats.
Solution
We show the long division process on the most difficult of these fractions, namely 112:
Step-by-step explanation:
Notice that the remainder after subtracting 8×12 (hundredths) is the same as the remainder after subtracting 3×12 (thousandths), namely 4. This means that the 3 in the decimal repeats: we continue to take away 3 groups of 12 (in the ten thousandths place, hundred thousandths place, and so on) and the remainder is always 4. The decimal expansions of all of the fractions are listed below (those which repeat can be found in the same way as 112 above and those which terminate are found when the long division process ends):
Step-by-step explanation:
13 / ( 2² × 5)
As the denominator is of the form 2^m × 5^n where m and n are whole numbers.
Hence, it is terminating decimal expansion.
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