This is Common Core State Standards Support Video in Mathematics. The standard is 4.NF.2. This standard reads: Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.
Let’s look at the last of the statements that deals with recording the results and using the symbols and using visual models. This will be addressed throughout the video because we will be using the symbolism, and we will be using visuals. Let’s look at the second statement that deals with recognizing that comparisons are valid only when the two fractions refer to the same whole. Now the reality is that a lot of times we’re going to be dealing with more than one figure, so we’re actually dealing with not just the same whole but a lot of times with a congruent whole. So students should think of it this way that yes, comparisons are valid only when the two fractions refer either to the same whole or to congruent wholes.
For example, let’s say we’re comparing 3/5 and 4/6. Knowing that these two wholes are exactly the same size justifies being able to make the determination here simply by comparing the two physically. So if we compare them, we can tell just by looking that 3/5 is smaller than 4/6. So then we can reach the conclusion that 3/5 is in fact smaller than 4/6. In looking at the standards, back in third grade, there is another standard that’s very similar to this. In fact, if you look at the second and the third statements in each of these standards, they’re exactly the same one: the statements dealing with comparisons being valid only when the two fractions refer to the same whole and the other statement that refers to recording the results and using visual models. The only difference here is the initial statement.
Back in third grade, they compared fractions with the same numerators or denominators as opposed to here in fourth grade. Now we’re dealing with comparing fractions with different numerators and different denominators. And also back in third grade, they reasoned about the size as far as how they thought about the comparison. As opposed to now in the fourth grade, they’re going to compare fractions by creating common denominators or common numerators or by comparing to a benchmark fraction such as 1/2.
So let’s focus on the first statement that deals with comparing two fractions with different numerators and different denominators by creating common denominators or numerators or by comparing to a benchmark fraction. So let’s start with two fractions, and we have different numerators and different denominators. Let’s compare to a benchmark fraction. Let’s look at that strategy. In this case, let’s use 1/2. So we start off with two congruent wholes, and then we have the actual representation of 2/5 and 4/6. We find where 1/2 is, which is easier to do with the 4/6 because there are an even number of parts. Then of course, by visual inspection, we can tell that 2/5 is less than 1/2 while 4/6 is bigger than 1/2. So, subsequently 2/5 has to be smaller than 4/6.
Now let’s compare fractions with different numerators and different denominators and again, by comparing to a benchmark fraction such as 1/2. Again, we start off with our 2 congruent wholes, get our representations. But we have a problem here. Both fractions are bigger than 1/2, so that’s not going to do us any good. Now another benchmark is 1. So which of these two fractions will be closer to 1? Now if we use our physical models and look at this from a linear perspective, we know that since 1/5 is a bigger part than 1/6, that 1/5 is going to be a longer segment here than 1/6. So using the same kind of logic, then it follows that 2/5 would be bigger than 2/6. So now the reasoning would be that, well 2/5 is going to be further away from 1, which means that it has to be closer to 0, which then of course means that our 3/5 has to be smaller than the 4/6.
Notice that we actually used another strategy here. We created common numerators, not with the original fractions themselves, but with the ones that we created when we were comparing to 1. In fact, that’s really the only reason that this worked is because we had common numerators. That way we could validly compare these distances from 1.
Now let’s look at the strategy that deals with creating common denominators or numerators. But first we need to look at some of the prerequisite knowledge. First of all, hidden in the background is the important role of the multiplicative identity, you know, the idea that any number times 1 is going to result in the same number. The multiplicative identity is the foundation for both the simplification of fractions and the creation of equivalent fractions, so a very critical idea. An additional item of knowledge that must be ingrained is the idea that the number 1 can be represented as a fraction in unlimited ways. So for example, we could represent 1 as 1/1, 2/2, 3/3, 4/4, 5/5, and the list goes on and on. So it’s important that students understand that any number over itself would be 1.
There are a couple of other items that are really critical. First, students must understand the representation. The majority of situations will be part-whole contexts, and here the numerator involves the number of chosen parts while the denominator represents the total number of parts contained in the whole. So it’s very important that students know the distinctions between the two. Second, students must understand the inverse relationship in the denominator. What we mean here is that the larger the number of parts, the smaller each part will be. And it’s a switch in thinking for students because they’re used to comparing whole numbers.
So for example, they know that 6 is bigger than 4, but that’s going to throw them off here. Again, the inverse relationship between the number of parts and the size of the parts is critical. So for example, here we have this circle cut up into 4 parts whereas this one over here is cut up into 6 parts; fewer number of parts over here with the 1/4, so these parts are larger than these over here where there’s more parts. Understanding that, now kids will understand why it is that 1/6 is smaller than 1/4. Again, a very important idea that needs to be ingrained with these kids—again, this inverse relationship that’s involved with the denominators. The more parts that you have, the smaller each part is going to have to be and vice versa.
Let’s look at the strategy of comparing common numerators. Here the numerators are already the same. We’re comparing 3/9 and 3/6. But again, we have that additional reasoning problem here that students have to realize that over here, with the 3/6 we’ve got 6 pieces as opposed to over here, we have 9 pieces. So these pieces have got to be smaller. So I’ve got 3 pieces in each case, but the ones here with the 6 are bigger. So obviously the 3/6 is bigger than the 3/9, but it takes a lot of thinking for kids to straighten that out. In fact, here using the strategy of comparing to a benchmark would probably be easier because 3/6 is 1/2 and 3/9 is a fraction smaller than 1/2.
Let’s go ahead and do an example where we create common numerators. Let’s say we’re comparing 5/6 to 7/9. Real important idea, the idea of the multiplicative identity: students have to understand that what I am doing here is multiplying by 1. So we’re not changing the value of either one, because again multiplying by 1 results in the same number. Now the easiest way to get a common numerator would be to simply multiply the two numerators; 5 times 7 to be 35. So 35 will be our common numerator.
Now the thought process is what do I have to multiply 5 by to get 35? Well, a 7. So I need to change that 1 to 7/7. And then over here, in order to get 35, I have to multiply 7 times 5. So we have to change that 1 to 5/5. So we do our computations, and so now we’re actually comparing 35/42 to 35/45. And we notice that 42 compared to 45, there are fewer parts. So over here, these parts are larger. So now we have 35 of these larger parts compared to 35 of these smaller parts. So now the reasoning is that 35/42 is bigger than 35/45. So 5/6 would be larger than 7/9. Care must be taken here with the reasoning. Remember the inverse relationship involved in the denominator, that again, the more parts you have the smaller those parts are going to have to be.
Now let’s look at the strategy that you’re probably a lot more familiar with and that is creating common denominators. So let’s say we’re comparing 5/6 and 7/9. Again, we are multiplying by 1 so we’re not changing the value of either one. As far as a common denominator, the easiest way is to just multiply the two denominators. Six times 9 is 54, so we are going to have a common denominator of 54. So we have to multiply 6 times 9 to get our 54, so we have to multiply the 5/6 by 9/9. Over here we need to multiply 9 times 6 to get 54. So we need to multiply 7/9 by 6/6, which of course was 1.
Now we do our computation. We get 45/54 for our 5/6 and for 7/9, we get 42/54. Now the nice thing is that we’re comparing apples to apples. We have 45 of these 54ths compared to 42 of those same 54ths. Obviously 45 is bigger than 42, so the reasoning is that 45/54 will be larger than 42/54. Note that no attempt was made to use the lowest common denominator. Again, all we did was just multiply the original denominators to get our common denominator, and that’s all that’s expected at this level. It’s not until Grade 5 that students will add fractions with unlike denominators. But they will not be using lowest common denominators, according to standard 5.NF.1. Also, the concept of least common multiple, which in fractions would be the lowest common denominator, is not addressed until Grade 6 in standard 6.NS.4.
So again, at this grade level, as far as getting common numerators and common denominators, all they have to do is multiply—numerator times numerator to get a common numerator or denominator times denominator to get your common denominator. In summary, this is a very important standard. It’s critical. It’s very important because of number sense. Students have to get a sense of the relative size of fractions when you’re comparing and especially as it also relates to whole numbers. There is a difference; they have to make a transition in making their comparisons: again, critical standard, especially from the perspective of number sense.