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.