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|
|Figure 2: Dividing 5/6|
|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|
|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.