Repeating Decimals Repeated

In the previous post, I discussed the connection decimals have with fractions, and specifically went over when fractions have repeating decimals. I explained how we can tell when a fraction is a repeating decimal, and that concept is so important and fundamental to mathematics, I decided to bring up repeating decimals again to talk about it more deeply. And that concept is finding patterns. We know when we have a repeating decimal, and what those repeating decimals are, by finding repetition in the remainder during long division.

One of the best fractions to illustrate this point of finding and using patterns in math are the ones with 7 as the denominator, x/7. For a homework assignment, I explored repeating decimals and fractions. So I started with fractions with 3 in the denominator, then 6, then 7, and then 11. The fractions with 3 or 6 in the denominator had a simple pattern with one or two repeating decimals. With 7 as the denominator, the pattern was more complex with 6 repeating decimals. But an even more interesting pattern came to light. Usually when the numerator changes, so do the numbers in the fraction's decimal form. That wasn't the case when the numerator changed with 7 as the denominator. Instead, the numbers were the same, they were just in a different order that could be determined by the sequence or pattern of the remainders, as shown in the figure below.

To show how the fraction x/7 is a special case to students, I would start with 1 in the numerator, or x=1. In the previous post about repeating decimals, I showed the long division process using both decimals and whole numbers. But in the this post, I only used whole number because I wanted to keep the number of figures down. But during a class, I would do both. Now, the number 7 can't go into the number 1. So we add a zero, shown in red, to get 10. 7 goes into 10 once, with a remainder of 3. 7 can't go into 3, so I drop down a zero, shown in purple, to get 30. 7 goes into 30 four times, with a remainder of 2. I need to drop down a zero again, shown in turquoise, to get 20. Then 7 goes into 20 twice, with a remainder of 6.

At this point, a student might say that this fraction doesn't have a repeating decimal. This is another good reason to explore this example with students. It shows that a repeating decimal can have any number of repeating digits in it's answer, not just 1 or 2. So, I show the students we need to continue by dropping down another 0, shown in green, to get 60. 7 goes into 60 eight times, with a remainder of 4. Drop down another zero, shown in orange, to get 40. 7 goes into 40 five times, with a remainder of 5. Drop down another 0, shown in pink, to get 50. 7 goes into 50 seven times, with a remainder of 1, which is the number we started out with in the numerator. And thus, we finally have the repeating reminder that indicates which numbers are repeating in the decimal form of this fraction. This should help illustrate that fractions will either have a finite number of digits or an infinite number of repeating digits, whether it's 1 or 12, in their decimal form.

Next, I would ask the students to change the numerator and divide by 7 to see if they notice any patterns when they do so. As shown in the examples below, I have compared 1/7 to the other fractions that have 7 as the denominator. I've color coordinated the drop-down-zeros to show how the order of the remainders, and consequently the order of the decimal numbers, changes when the numerator changes. I have also labeled the remainder and decimal order based on 1/7 to help illustrate this pattern even more. 

In 1/7, there are 7 numbers, 0-1-4-2-8-5-7, before the remainder 1 repeats itself again in the long division process. This indicates that there are 6 repeating digits in the decimal form: 1-4-2-8-5-7, as indicated by the long bar over these 6 numbers.  As we can see, whenever the numerator changes, that numerator corresponds to one of the remainders when we divided 1 by 7. For example, the numerator 2 corresponds with remainder #3 from 1/7, and the numerator 3 corresponds with remainder #2. Consequently, this shifts the order of the remainders, which shifts the order of the digits in the decimal. So, for 2/7, the remainder and decimal order is 3, 4, 5, 6, 1, 2. For 3/7, the remainder and decimal order is 2, 3, 4, 5, 6, 1 as shown in the figure below. 

 For 4/7, the remainder and decimal order is 5, 6, 1, 2, 3, 4. And for 5/7, the remainder and the decimal order is 6, 1, 2, 3, 4, 5, as shown in the figure below.

Lastly, we have 6/7. And the remainder and decimal order for this fraction is 4, 5, 6, 1, 2, 3, as shown in the figure below.

Hopefully, the students will notice a pattern at the very least, and the class can share their reasons for why the pattern happens the way it does. The figures above should help them along in their reasoning and understanding. By the end of this example, I hope they achieve a better understanding of repetition patterns with remainders to determine a repeating decimal. Also, I hope they gain a better ability to notice patterns in math in general, since patterns greatly help in understanding math concepts and solving problems. 

Figures were created using GeoGebra.