This is Common Core State Standards support video in mathematics. The standard is 5.NF.B.3. This standard states: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers (for example, by using visual fraction models or equations to represent the problem). For example, interpret 3/4 as the result of dividing three by four, noting that 3/4 multiplied by four equals three, and that when three wholes are shared equally among four people, each person has a share of size 3/4. If nine people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?
Let’s look at this idea of interpreting a fraction as division. Note that this is the third different interpretation of what the fraction symbol a/b represents. Earlier in the standards, fractions as part of a whole is addressed in standard 3.NF.A.1, that talks about understanding a fraction 1/b as the quantity formed by one part when a whole is partitioned into b equal parts. The standards also address a fraction as a rational number, and this happens in standard 3.NF.A.2. It states: Understand a fraction as a number on the number line; represent fractions on a number line diagram. Fractions as expressions of relationships rather than quantities, in other words, ratios, are not introduced until Grade 6. This happens in standard 6.RP.A.1 that states: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
Now, let’s look at this statement in this standard: solve word problems involving division of whole numbers. Students have had experience with this. It was addressed in the standards back in 3.OA.A.3. This standard talks about using multiplication and division within 100 to solve word problems. If we look at the section of this standard that talks about the answers, it states, “leading to answers in the form of fractions or mixed numbers.” Back in fourth grade, the standards did address this in standard 4.NBT.B.6. This standard talks about finding whole-number quotients and remainders. In the same grade level, Grade 5, there’s standard 5.NBT.B.6, and it’s very similar to standard 4.NBT.B.6. Really, the only difference is that the fourth-grade standard deals with one-digit divisors and the fifth-grade standard deals with two-digit divisors. Also, notice that in the fourth-grade standard, we’re dealing with whole-number quotients and remainders whereas in the fifth-grade standard, 5.NBT.B.6, it deals strictly with whole-number quotients.
So, the question would be, well, when do you address division with two-digit divisors that results in mixed-number quotients? There is a standard in the next grade, in sixth grade, 6.NS.B.2, that doesn’t specifically mention it, but it’s inferred. The standard states: fluently divide multidigit numbers using the standard algorithm. So, you would think that, getting different kinds of quotients, whole number quotients and whole-number quotients with remainders would be part of that. It stands to reason though, that there’s no barrier here in fifth grade to prevent us from using two-digit divisors and getting something other than a whole-number quotient. If we compare these two standards (they’re both in the fifth grade), notice that the fifth-grade standard, 5.NBT.B.6, gets a little bit more detailed and complex because it talks about division of unit fractions by non-zero whole numbers, and it also talks about division of whole numbers by unit fractions.
So, in addition to standard 5.NF.B.3, we’re going to have the expectation that students will be dividing and involve fractions also. Now, let’s go back and look at the standard, and let’s look at this example dealing with 3/4. In looking at it, it kind of leaves you scratching your head a little bit. Interpreting 3/4 as the result of dividing 3 by 4, okay no problem there. Note that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among four people, each person has a share of size 3/4. Let’s look at 3/4 itself. The typical context would be where, okay, we start with a whole. We divide it into four equal parts, and this particular fraction talks about three out of those four equal parts. In this case, we shaded three out of the four. Notice that in this context, the whole is always going to be one.
We look at what this standard is all about, and it’s about interpreting a fraction as division. So, the question is well, how is this different? How is 3 ÷ 4 different than just 3/4? Let’s change this problem to where it fits the context, and let’s make it four people wanting to share three sacks of rice equally. So, this is what would happen. We start off with three wholes, and we know we have to divide this among four people. So, we know that we need to go ahead and divide it up into four equal parts each. Let’s shade these different colors to differentiate one whole from the other. We have to split it up four ways, so let’s draw four circles to represent the four people that are going to be taking these shares. Of course, let’s split them up into four equal parts each because we know that we’re going to be talking about fourths.
Now, let’s take this first circle, and let’s take that one and divide that among the four people. So, here we go. Each one of those gets 1/4 of the grey-shaded area. Now let’s do the same thing with this second one. There we go. Now let’s take the third sack and share that equally among the four people. Note that the question asks how many sacks of rice should each person get? So, when we answer the question, we’re talking about the share that one person gets, and now we can make the connection back to the original fraction of 3/4, and again, note that the whole for the given response is one because we’re talking about the share that one person gets, and it turns out that each person will not get a whole sack. They’ll only get 3/4 of one.
So, in reviewing this again very quickly, we can also put this into decimal form where 3 ÷ 4 would give us a quotient of .75. But, if we take it a step further, if we put that over 1, that’s allowed. We can divide anything by one and not change the result. Notice that the .75 refers to the quantity, and then the 1 in the denominator refers to each person. Every one person gets .75, and note our diagram, that it’s a little bit different though than what the original 3/4 would have been. When we look at this example, it seems a little bit redundant and confusing, especially when the division begins as a proper fraction. However, the key here is to understand how the division changes the context and to comprehend the interpretation of the quotient, especially when the division involves a proper fraction.
Going over this one more time, again here’s the huge difference. If we just have the fraction 3/4, that means we have one whole. We split it up into four equal parts and we’re talking about three of those four equal parts. Now that’s different from 3 ÷ 4 because there, we start off with three wholes. We start off with 3 ones, and we’re going to split that up into four equal parts; and when we do that and we share those, notice that each of those fourths for each individual circle is a different color. So, again, that is a huge difference. But, when you come back around to looking at what your final response would be, you’re talking about a whole of one. But again, notice that the context was different. The result isn’t quite exactly the same even though numerically they do appear to be.
Now let’s focus on the part of the standard that talks about solving word problems. The example that we used, we made it to fit the second example that’s involved with this standard that deals with the 3/4. However, the second example deals with nine people wanting to share a 50-pound sack. These types of problems will typically involve an improper fraction. In other words, the numerator is larger than the denominator, and the quotient will typically be greater than one, and again, that’s what the primary example here is. Again, you would have nine people sharing a 50-pound sack, and you would end up when you solve the problem having 50/9 as your improper fraction.
Notice that the standard calls for answers possibly being in the form of mixed numbers, and that would be the situation with that last example where you would have nine people sharing a 50-pound sack. The very last statement in this standard brings up an interesting point. It’s pretty evident that this standard can also be utilized to strengthen the estimation skills and number sense of students, especially when it comes to students being able to figure out if an answer is reasonable. Also, there can be some contexts that would involve rounding a fraction to a whole number that’s appropriate to the situation.
To complete this, let’s look at some word problem examples that involve division of whole numbers, and we get answers in the form of fractions or mixed numbers. Let’s take this example. One of the stages of the Tour de France bicycle race is a distance of 138 miles. If a racer averages 24 miles per hour, how much time in hours will it take to complete that stage? Just to get a handle on what type of solution this might have, students know that 24 x 10 is 240, so we know that the solution should be less than 10 because we’re dealing with 138 miles, which is, of course, less than 240. As far as the computation, we need to take 138 and divide it by 24 because we’re doing 24 miles per hour. So we do the computation. We’re not going to get a whole number solution here. We have a remainder of 18. Now, if we make that into a fraction, that would be 18/24, which can be simplified. We have a possibility of dividing by six. So, we do that. We divide the numerator and the denominator by six, and that simplifies to 3/4. So, it should take 5 3/4 hours to complete that stage.
Notice that the standard talks about using usual visual fraction models. So, let’s add that to this. What students can do since multiplication by 24 isn’t one of those automatic recall facts, so let’s take some of those multiples: 24 x 2 is 48, 24 x 3 is 72, then 96, then 120. Ah, 24 x 6 is 144. We’re dealing with 138 miles, so we can automatically tell that 138 is going to fall between 120 and 144, which means that our answer should be between 5 and 6 hours. So, now we have a real good idea.
Now we need to use a visual model to represent what’s going on. So, let’s do this in chunks of 24 miles, and the time involved would be 1 hour. So, let’s do that again, one more time, again, again. So, we’re at 5 hours, and we’ve covered 120 miles. If we go 1 more hour, we’ll be at 144 miles, and the 138 falls in between there somewhere. Now the question is well, what part of an hour are we dealing with? Let’s take that section of the visual model and let’s fill in our numbers. Start with 120. End at 144. If we take the difference between those two, of course, that is 24, and now let’s substitute our value, 138.
What is the difference between 138 and 120? We take the difference, which is 18. So, now we know that in terms of miles that would be 18 miles out of the 24, and if we put that in fractional form and we simplify that like we did earlier, we’ll get 3/4. So, we got 3/4 of the distance. That would be how far that would be, the 18 miles out of the 24. But, now we have to relate that to time. This is actually a little bit simpler because we’re dealing with just 1 hour rather than 24 miles, so that’s just going to be 3/4 of 1 hour. So, we have 5 hours plus 3/4 of another one for our final solution of 5 3/4 hours.
Let’s take a second context. A school bus has a maximum capacity of 48 passengers. There are 208 students and teachers that will be going on a field trip. How many buses will be needed? Well, we have to take the 208 and divide that by 48 since each bus can carry 48 passengers. Let’s see, we get 4. That’s 192. Okay, we don’t have a whole number solution again. So, what we have here is 4 with 16 left over, and as a mixed number that’s 4 16/48. We can simplify that. We can divide both of them by 8. So, we get 4 2/6, but that can be simplified a little bit further to 1/3. So, our solution is 4 1/3, mathematically anyway.
Logic dictates that we cannot use 1/3 of a bus. Mathematically, the rules say that we should round down to four buses, but that would not be enough. So, based on the context, even though mathematically, we’re supposed to round down. But, in this context, we actually have to round it up to 5 buses to be able to carry those extra 16 people. Now, let’s address this part about a visual model. Okay, so we’ll take our buses. Okay, that first bus can carry 48. The second bus carries 48, which puts us at 96. A third bus, 3 forty eights puts us at 144 passengers. One more bus puts us at 192. Still not enough—we need one more bus.
Now, at full capacity that would be 240 passengers, but we only have 208. So, we’re going to have four full buses and that’s 192 folks that that can carry. The difference between 208 and 192 is the 16. So, that last bus, that fifth bus will only need to carry 16 people, and again, as a fraction that would be 16/48, which simplifies to 1/3. But, it’s not possible to have just 1/3 of a bus. So, again the context here necessitates that we do some rounding, and it has to be up, not down. So, we have our four buses, and we need a fifth bus because four wasn’t enough. But, rather than 48 people, the last bus only needs to carry 16. So, our final solution would be that we need a total of five buses with four of them at full capacity of 48 passengers each, and one bus will only be at 1/3 capacity, in this case 16 passengers.
What’s interesting with a problem like this is that we can take this standard and extend it. We can enrich this problem. What you could possibly do is ask students, well, we don’t want four full buses and then one of them not even half full. So, what could we do to split the passengers up more evenly? So, now that would make it into a context where if you divided 208 by 5, you would get 41 with some left over. Again, a situation like this would be different thinking where you want the students to divide the passengers up as evenly as they can among the five buses, and that, again, would be an interesting extension to this problem.
If we look at the standards for mathematical practice, in looking at the first four, by doing the activities to address this standard, we would be making sense of problems and persevere in solving them. Students would reason abstractly and quantitatively. They’ll also construct viable arguments and critique the reasoning of others if you have them grouped together in teams, and they would be modeling with the math. In looking at the other four standards for mathematical practice, students would attend to precision and they would look for and express regularity in repeated reasoning.