Sunday, March 22, 2015

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.

Sunday, March 8, 2015

Instrumental and Relational Understanding with Fractions

For my teaching class, I read an article "Relational Understanding and Instrumental Understanding," by Richard R. Skemp from Mathematics Teaching in Middle School, September 2006, Vol. 12 No. 2, pages 88 - 95. It defines instrumental understanding as the ability to follow rules and know when to use them, while relational understanding is defined as the ability to know and explain the concept. When I learned multiplying and dividing fractions, it was just a bunch of rules. For multiplication, just multiply the numerator and denominator across to get the answer, as shown in Figure 1. For division, the rule is flip the numerator with the denominator of the second fraction, and then multiply across, as shown in Figure 1. But learning math shouldn't be memorizing a bunch of rules without knowing how they work. It should be about true understanding; knowing the how behind the rule. So, I've designed a lesson showing how the rules work with visual explanations, which will incorporate student instrumental understanding with relational understanding.
Figure 1: Multiplication and Division of Fractions
 First, I would start with dividing 1 by 2 using one circle as the unit. The unit defines what the whole is. In Figure 2, we have 1 circle divided by 2. Students usually know what a half of something is, since division is associated with splitting an amount into groups. So, instinctively they should say that the answer of 1 divided by 2 is 1/2 because the circle is being split by a green line into two groups of 1/2, as shown in Figure 2. Then I would ask the students: does the division rule of fractions fit in with this example and how? Some might see the connection to the flip rule. But if not, then I would show them how dividing by 2 is the same as multiplying by 1/2, which gives us the division flip rule.
Figure 2: Dividing 1 by 2
 Next, I would take the answer from the last problem and have 1/2 divided by 2, which is 1/4. According to Figure 2 above, the circle is already split into halves. Then, I would split the two halves of the circle by 2 as demonstrated with the red line in Figure 3. This will give us 1/4. To show that the division flip rule still works, I will flip the 2 to make it 1/2 and then multiply 1/2 with 1/2 to get 1/4.
Figure 3: Dividing 1/2 by 2
In the past 2 problems, I've only been dividing one circle repeatedly. Now, I'm going to show how the division flip rule still works when dealing with numbers greater than 1. In Figure 4, there are 11 circles, which are being divided by 2. So, the circles will be split into two equal groups, where 5 circles and a half of one circle go into each group which is 5 1/2, or 11/2. Again, dividing 11 by 2 is the same as 11 being multiplied by the fraction, 1/2.
Figure 4: Dividing 11 by 2
Then, I would take the answer from the previous problem, and have 5 1/2 divided by 2 to get 2 3/4, or 11/4. 4 circles get divided equally, so both groups get 2 circles. 1 circle gets split into two halves, so each half goes into a group. And the one-half gets split into two one-fourths, so each fourth goes into a group. Add one group up (2+1/2+1/4), and we get 2 3/4 circles, as shown in Figure 5. And the division flip rule still applies, since 11 x 1 = 11, and 2 x 2 = 4, which gives us 11/4, or 2 3/4.
Figure 5: Dividing 5 1/2 by 2
I have shown how the division flip rule works for whole numbers. Now, I'm going to show that the division flip rule is the same when dividing by fractions. So, again we will take the answer from the previous problems and have 2 3/4 divided by 1/2, which will get 11/2 or 5 1/2 as shown in Figure 6. This is the same as multiplying 2 3/4 by 2, which is the opposite of multiplying by 1/2. So, instead of the answer getting smaller, like in the previous problems, the answer will get bigger. Specifically, it will be the number that got multiplied by 1/2 in the previous problem, which is 5 1/2.
Figure 6: Dividing 2 3/4 by 1/2
So far, I've been multiplying and dividing by unit-fractions. Next, I will show the students how the rules for multiplying and dividing fractions hold for any fraction. So, I will give them the problem 2/3 x 3/4 = 1/2. So, we take the circle and divide it into thirds, and then divide each third into fourths, which gives us a circle split into twelfths, as shown in the Figure 7. In the numerator, 2 x 3 = 6, so we count 6 pieces, which will give us 6/12 or 1/2, which is the purple line in Figure 7. The rule for multiplication holds, since 2 x 3 = 6 and 3 x 4 = 12, giving us 6/12 or 1/2. The red lines indicate the thirds, the blue line indicate the fourths, and the purple line indicate the half of the second circle in Figure 7.
Figure 7: Multiplying 2/3 by 3/4
Since division is the opposite of multiplication, I would again show how dividing the previous answer will get the number that was divided before. So, the problem is 6/12 divided by 3/4 = 2/3. So, we have a circle that is split into twelfths. Then, we split each twelfth into thirds, and then we use the division flip rule. This makes the problem 6/12 x 3/4. And since 6 x 4 = 24, we count 24 pieces in the second circle which gives us 2/3, as shown in Figure 8. The red lines in the second circle indicate thirds. Again, the division flip rule still holds, since 6 x 4 = 24 and 12 x 3 = 36, which gets 24/36 or 2/3.
Figure 8: Dividing 6/12 by 3/4
Hopefully, by the end of this lesson, my students will have the visual explanation that will help them understand how these fraction rules work. That way, the students will have the relational understanding to go with the instrumental understanding.

Figures were created using GeoGebra