Repeating Decimals

There is a direct relationship between fractions and decimals. All fractions can be rewritten as decimals, and those decimals (but not all) can be rewritten as fractions. However, the relationship between fractions and decimals may not be clear to students. The first exposure students have with using decimal notation is with money, since American money is written like this: $1.43. With the dollar sign written in front of it, people know 1.43 is representing money and people say that value as 1 dollar and 43 cents. However, money doesn't give students the sense that decimals can represent fractions. But without the dollar sign, the money context is removed, and $1.43 becomes 1.43, which is communicated as one and forty-three hundredths. In this context, the relationship between decimals and fractions becomes clearer, since 1 43/100 is communicated in a similar way: one and forty-three one-hundredths.

To illustrate this point further for my students, I would show them how fractions are hidden division problems. For example, 3/4 can be said as three-fourths or 3 divided by 4. The answer to 3 divided by 4, which is 0.75, is pretty straightforward and the answer doesn't have a remainder. The main focus of this blog post is introducing students to the concept of the repeating decimal, which is when a fraction in decimal form doesn't end neatly, but instead the answer has repeating digits that goes on indefinitely.

The first example I would use to illustrate repeating decimals to my students is 5\6, as shown in Figure 1. Now, I would tell my students that dividing by a number that is bigger than the number being divided, has a similar process as having a smaller number dividing a bigger number. So, I would use the same language that the students are familiar with when doing long division. So, 6 goes into 5 zero times. So, I would put a 0 as the first digit. Since 5 is in the ones place, then I  must use the tenth place (I've designed this lesson assuming I've already discussed place values with decimals with my students). So, I need to place a decimal point at the time when I use the tenth place to show where the whole digits stop and the decimal digits begin. And that spot is right after the 0, as shown in Figure 1. Now, when I was being taught decimals using long division, the teacher had used whole numbers when subtracting. But my professor, John Golden, showed my class a way using decimals. So I will illustrate both ways in my blog post and to my students. That way, students who prefer one method over the other can choose for themselves. In this blog post, the left side of the figures will use whole numbers and the right side will use decimals.

Figure 1: Dividing 5/6

To continue, we need to add a zero, as shown in Figure 2. This concept might be easier to grasp when using decimals, since 5 and 5.0 has the same value, while 5 and 50 does not. But either way, the value for tenth place in our answer will be the same. 6 goes into 50, 8 times, since 6 x 8 = 48. Then I subtract and get 2. Now, I will use decimals. 6 goes into 5.0, 0.8 times, since 6 x 0.8 = 4.8. Then I subtract and get 0.2. This also serves as a good example to show students the trick of shifting the decimal point over one place when multiplying with tenths.

Figure 2: Dividing 5/6

Again, I need to add another 0, so 50 becomes 500, and 5.0 becomes 5.00. Then, I can drop down the 0 to get 20 or 0.20, as shown in Figure 3. Either way the number in the hundredth place of the answer will be the same. 6 goes into 20, 3 times, since 6 x 3 =18. Then I subtract to get to 2. For the decimal side, 6 goes into 0.20, 0.03 times, since 6 x 0.03 = 0.018. Then I subtract to get 0.02. As we can see, especially with the whole number example, we got the number 2 again (the remainder) after we subtracted the two previous numbers. So, I would ask the students, "do you think we will get the same number for the thousandths place in the answer?" To check, I would go through the steps of long division one last time.

Figure 3: Dividing 5/6

To begin one more round of long division, I need to add another 0 again. So, 500 become 5,000 and 5.00 becomes 5.000. I drop the zero down, to get 20 and 0.020. Again, 6 goes into 20, 3 times, since 6 x 3 = 18. And 6 goes into 0.020, 0.003 times, since 6 x 0.003 = 0.018. So, 3 goes into the thousandth place, and when I subtract, we will get the number 2 as the remainder again, as shown in Figure 4. This will show the students that the answer will have a repeating decimal that will continue indefinitely. It will also show the students how to recognize a repeating decimal when it happens, and that's by seeing remainders repeating during long division. I would then show them the proper notation for a repeating decimal, and that is placing a bar over the number or numbers that are repeating, as shown in Figure 4.

Figure 4: Dividing 5/6

In my next example, I will show the students how a series if numbers can repeat. So, I will use the fraction 1/11 to show this. I'm dividing 1 by 11, so 11 goes into 1, zero times. So, 0 goes in the ones place. Again, since I need to use the tenth place to continue, I need to place the decimal point next to the 0 to indicate where the decimal values start, as shown in Figure 5. Again, I will be using whole numbers for long division on the left side of the figure and decimals on the right.

Figure 5: Dividing 1/11

Now, I need to add a 0. So, 1 becomes 10 on the left, and 1 becomes 1.0 on the right, as shown in Figure 6. 11 goes into 10 and 1.0, zero times again. So, 0 goes in the tenth place. Then I subtract, and get 10 on the left side, and 1.0 on the right side, as shown in Figure 6.

Figure 6: Dividing 1/11

Again, I need to add another 0, so 10 becomes 100 and 1.0 becomes 1.00. I drop the 0 down, and get 100 and 1.00 at the bottom. Now, 11 goes into 100, 9 times, since 11 x 9 = 99. I subtract the previous two numbers and get 1 as the remainder. For the right side, 11 goes into 1.00, 0.09 times, since 11 x 0.09 = 0.99 times. I subtract the previous numbers, and I get 0.01 as the remainder. Since I have the number 1 in the remainder again, I know I will have to repeat 0 and 9 again indefinitely. So, 09 is my repeating decimal and I show that by placing the bar above those two numbers in the answer, as shown in Figure 7.

Figure 7: Dividing 1/11

Hopefully by the end of both of these examples my students will have a better understanding of the relationship between fractions and decimals. Furthermore, my students should be able to recognize when they will have a repeating decimal by noticing when they have a repeating remainder. 

The images were created using GeoGebra.