Mathematics

6.G.2, Part 1, Transcript

This is Common Core State Standards Support Video in mathematics. The standard is 6.G.2. This standard reads: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas volume is equal to length times width times height (V=lwh) and volume is equal to the area of the base times height (V=bh) to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

So let’s start by looking at this first part of the standard that deals with packing a right rectangular prism with unit cubes. So of course it’s important that kids understand your basic unit cube is going to be 1 by 1 by 1 as far as its dimensions. So if we’re going to pack let’s say, this 4-by-5-by-6 rectangular prism okay, we’ll pack it. Notice that we have one row of six of these cubic units, but then of course, we have a dimension of 5 here. Our depth is 5, so we actually have 30 cubic units along this first layer, if you will. But we have four of those, so we have 30 cubic units in each one of these. So we have a total of 120 cubic units. So if we actually look at the count compared to what we get with the formula, it does come out to be the same both ways. Also, we can make a connection to the other form of the volume formula—area of the base times the height. The area of the base is 30, and then when we multiply that times our 4 on the height, we do get 120 also.

Now students need to also get used the idea of what happens when you partition a rectangular prism. So if we take this rectangular prism that has a width of 6, a depth of 5, and a height of 4, and let’s partition this to where the height is going to be 3 instead of 4. So we’re going to knock out that bottom part, and so we’re going to lose 30 cubic units. So our new prism will be 3 by 5 by 6. If we examine the smaller prism that’s 3 by 5 by 6, our volume is 90 cubic units compared to the 120 cubic units of the original rectangular prism. So we removed 30 cubic units.

Let’s do another partitioning. Let’s say we partition it this way to where we’re going to partition the width. So notice our depth is 5, so we’re going to have five of these cubic units along the bottom here, but we have a stack of four of them, because the height is 4. So that’s a total of 20 cubic units. So our new prism would be 4 by 5 by 5. If we apply our formula, that’s 100 cubic units for that volume, compared to 120 cubic units of our original. So we did remove 20 cubic units. So again, we did it both ways by comparing the results from the formulas and also by looking at what happens with our actual little cubic units.

Let’s try another partitioning. This time, let’s partition it this way where we’re cutting off some of the depth. So here, looking at in terms of let’s say, stacks. So we have a stack of four of these here, and then another one here and so forth. The dimension this way along the width is 6, and we have six of these stacks of four for 24 cubic units. And if we apply our formula to our new rectangular prism compared to the original, we get 96 cubic units for the 6-by-4-by-4 prism compared to the original 120. So it does check out that we lost 24 cubic units either way, either by looking at the difference of the volumes of the formula or if we look at the actual cubic units that were lost.So we took a rectangular prism, and we partitioned it along the height. We partitioned it along the width, and we partitioned the depth also. And so the result was a smaller prism that’s 3 by 4 by 5, and so what happens here, we have a 120 cubic units compared to 60 cubic units. So we lost a total of 60, since 120 minus 60 is 60. But wait a minute, the three partitions that we did earlier resulted in subtracting 30 units, cubic units, then 20 cubic units, and then 24 cubic units. Now that’s a total of 74. If you subtract 74 from 120, that’s 46, not 60. So what happened?

Well let’s go back and look. Here’s our first partitioning that we did. We did in fact lose 30 cubic units; 120 minus 90, so that checks out. Okay, here’s the difference. Here’s where you really have to be careful. Notice what we are partitioning now is the resulting prism from that first partitioning. This one is 3 by 5 by 6, not the original 4 by 5 by 6. So that’s where the difference is right here, okay. So then we did our second partitioning and we would have a 3-by-5-by-5 right rectangular prism, and that volume based on the formula is 75. And if we take 75 from 90, we’ve lost 15 cubic units.

Now we take that resulting prism, which is 3 by 5 by 5 and let’s partition that along the depth so that we have a 3-4-5 right rectangular prism as a result. Using the formula, that volume is 60 compared to the 75 over here. So we lost 15 cubic units. So, overall we started off with a 120, then we lost 30, then we lost 15, then we lost another 15. So we lost a total of 60 from the 120 that resulted in 60 cubic units. So again the difference in the two processes was that the first time we did three different partitions but back with the original 4-by-5-by-6 right rectangular prism as opposed to this second series of partitions where every time we did a partition, it was with the resulting prism, not with the original.

6.G.2, Part 2, Transcript

Now the partitioning we’ve done up to this point has all been with whole numbers, but we have to adhere to the standard. And it says that we have to find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths and show that the volume is the same as would be found by multiplying the edge lengths of the prism. So we have to do two different things. We have to deal with edge lengths that have fractions in them. More than likely, we’ll just use this original 4-by-5-by-6 right rectangular prism, but we need to deal with some mixed numbers. But we’re also going to have to show that the volume, by looking at all the little cubic units, is going to be the same as if we used the formula.

So let’s partition it again along the base, the height. Let’s cut it to 3 1/2 and 1/2. Now look what happens. We’ve created some new units here along the bottom that are 1 by 1 by 1/2, because the height is only 1/2, not 1 like it was before. Now each one of these is going to be 1 times 1 times 1/2—1/2 cubic unit. So let’s see what happens with the volume. Well, let’s see, if we just use the formula and just multiply 6 times 5 times 1/2, that’s 15. So we have 15 cubic units in this layer, if you will. And then, the other three stacks or layers all of those cubic units are going to have a height of 1, so there they’re going to be 5 by 6 by 1 on the height.

So we’ve got 30 cubic units, but we have three of those (layers). So now, if we just look at the units, look what happens. We have 30 plus 30 plus 30 plus another 15. That is 105 cubic units. Now notice that we have 90 of these 1-by-1-by-1 cubic units, and we had 30 of the 1 by 1 by 1/2. Note that 90 plus 30 is 120, so we still have 120 of these small prisms in this new right rectangular prism, just like we did with the original 4 by 5 by 6. Notice along the height that we still have a stack of four. The only difference is that instead of all of them having a height of 1, three of them have a height of 1, well, three of the layers and one of the layers, all those little cubic units have a height of 1/2 instead. What we can also do is go back and connect to the optional formula for the volume, the area of the base times height. So here we took 30 times 3 1/2; we’ll also come out with the 105 cubic units.

Now it’s hard to tell what the expectation is based on the standard. At this grade level, it might be a reasonable expectation that they would be dealing with right rectangular prisms where two of the dimensions are whole numbers and only one of them has a fraction involved. But we don’t know that, so let’s keep going. So let’s take this resulting right rectangular prism, and let’s partition it again. So let’s partition the width from 6 to where it’s 5 1/3, and we’re losing 2/3 over here. So here’s what happens. All of the units along here that were 1 by 1 by 1 are now going to be 1 by 1 by 1/3. And the ones that have a height of 1, we’re going to have three stacks or three layers of those.

And at the very bottom, we’re going to have the ones that have a height of 1/2. So their dimensions are going to be 1/2 by 1 by 1/3. The depth is 5. So we’re going to have five of these 1-by-1-by-1/3 units along here. We’ve got three stacks of those. And then these smaller ones, the 1/2 by 1 by 1/3, we’re only going to have one layer of those along here. So we have that result. And then as far as the size of each of these, well, these would have a height of 1, 1 by 1 by 1/3. Each one of those is 1/3 cubic unit. The ones that have a height of 1/2 are going to have a volume of 1/6 cubic unit.

So now let’s work on determining the total volume. Okay, on this lower layer, where the height is 1/2 for all of these small units, if we were to take our formula to just figure out this volume, we multiply it all out and we get 13 1/3. Now if we look at this in terms of just counting up what we have as far as the units, notice that here that this is 5 by 5. So we have 25 of these cubic units that are 1 by 1 with a height of 1/2. And then along here, we will have five of these that are 1/2 by 1 by 1/3. And when we figure out that volume, that’s 5/6 of a cubic unit. So then if we combine, if we add 12 1/2 with 5/6 we do get 13 1/3 just like we did with the formula.

So that takes care of the first layer along the bottom. What about the others? These are going to have a height of 1. So if we use the formula for the volume of a prism, this one is going to have dimensions 5 1/3 by 5 by 1. So we multiply that out, that’s 26 2/3. Now notice a relationship between the two volumes here. Notice that the 26 2/3 is actually twice as much as 13 1/3. And that makes sense, because what we had here was a situation where the height here is 1, the height here is 1/2, and that’s a 2 to 1 relationship. So this makes sense.

Now if we look at this in terms of the units again as opposed to using the formula, again, we use right here where it’s 5 by 5. So we’ve got 25 of these smaller units that are 1 by 1 by 1, and along here, we’ve got five of these that are 1 by 1 by 1/3. Those would be all along in here. And if we add 25 with 5/3, we get 26 and 2/3, just like we did when we used the formula. But we have 3 of these, so we need to multiply that volume times 3. It turns out it’s a nice even 80. So now those three stacks or layers came out to be 80, but we have to go back and combine it with that bottom layer where the volume was 13 1/3. And so, if we combine that together, we get 93 1/3. Then if we compare that to the result from just using the formula, it does come out to be the same. So it checks out.

So we have shown that taking all of our little cubic units and combining them all together, we do get the same result as we did from the formula. Now let’s stop and see what has happened so far with the units. On the first partitioning, we created a second type of unit that was 1 by 1 by 1/2 because of the difference in the height. Then when we did our second partitioning on the resulting prism, now we created two more different sized units that were 1 by 1 by 1/3. We had three stacks of those, and then we have another different sized unit that’s 1/2 by 1 by 1/3 and we had one layer of those. So we’ve got four different units now; units of four different sizes.

Now if you look at the overall picture, the 1 by 1 by 1s, we have 5 by 5. So we have 25 of these per layer for a total of 75. The 1 by 1 by 1/3 along here, we had five per layer, and we had three of those layers for 15. Then on the very bottom stack where the height is 1/2 for all the units, we had 5 by 5. That’s 25. So we had 25 of these that are 1 by 1 by 1/2. And along this row right here, we had five of these that were 1/2 by 1 by 1/3. If we look at what happens if we combine the total numbers, as far as all the different little sized cubic units, notice that it adds up to being 120. That compares way back to the original where we had a 4 by 5 by 6 that had 120 cubic units. But all of those were 1 by 1 by 1. Now with this one that we’ve done two different partitions on, we still have 120 little prisms. But the difference is they’re different sizes. They’re not all 1 by 1 by 1. We have four different sizes here.

Again, we don’t know the expectation at this level, sixth grade. Three dimensions might be a bit much, but let’s go a third dimension. Okay, let’s take our resulting prism that’s 3 1/2 by 5 by 5 1/3, and let’s do one more partitioning. Let’s partition the depth from 5 to 4 3/4 and we’re going to knock out 1/4. What happens here? Well we’re going to partition it along here to where it’s just this row in the front that’s impacted. These would be the ones that are a height of 1. Same thing is going to happen for our very bottom layer where all those cubic units have a height of 1/2. So let’s look at the ones with a height of 1 first.

Okay, when we partition along here, the 1 by 1 by 1 little cubic units are going to get cut in such a way where now they are 1 by 3/4 by 1. We’re going to have five of these per layer, because again, we’re only dealing with this front row right here. And then don’t forget that over here we had a unit that was 1 by 1 by 1/3, and that’s going to get partitioned to where it’s 1 by 3/4 by 1/3, and we only have one of those per layer. Something very similar is going to happen to the very bottom stack or layer, the ones that have a height of 1/2. Again the ones that are impacted are just along this front row. We have our original units, were 1 by 1 by 1/2, and they’re going to get partitioned in such a way where the dimensions are 1/2 by 3/4 by 1. And we have 5 of these per layer along this row. And then let’s not forget that over here we had a unit that was 1/2 by 1 by 1/3. And again, that was going to get partitioned to where that one got cut all three ways, so it’s now 1/2 by 3/4 by 1/3. I only have one of those in that layer.

Okay here comes the fun part. Now the volume, if we use the formula, is 5 1/3 times 4 3/4 times 3 1/2. That results in 88 2/3 cubic units. But now, the tough task is going to be to take all of those little fractional prisms, all those smaller cubic units, and see if all of that adds up to be 88 2/3. So again, we have to go by what the standard says. We have to pack it with unit cubes of the appropriate edge lengths and show that that volume is the same as we would get from the formula. Okay, so here’s what we’re going to do. We are going to create a smaller right rectangular prism and place it inside of that one. And we’re going to place it in such a way where it’s the top that’s flush with the bigger prism, and this back face is flush, and this face along here is flush to where you have a space down here at the bottom. You have a space along here at the front, and you have a space along here on the left-hand side.

Now the dimensions of this smaller right rectangular prism on the inside are the nice whole numbers. So that one is going to be 5 by 4 by 3. And it’s real important that we realize this, because this smaller right rectangular prism on the inside with the whole number dimensions, that’s going to be our benchmark and our reference for all of this. So that’s a real important consideration. Okay one step at a time. Let’s go with the volume of the smaller rectangular prism on the inside with the whole number dimensions. Now we just need to first figure out how many of these do we have. Well it’s 5 by 4, so we have 20 of them, and we have three of them as far as the layers or the stacks. So we have a total of 60 of these inside of this rectangular prism. Each one of those was 1 cubic unit. So we have a total of 60 cubic units for the volume of our smaller right rectangular prism on the inside with the whole number dimensions.

Next let’s look at the cubic units that are here directly underneath this smaller right rectangular prism. So it’s all of these under here. Now all of these are 1 by 1 by 1/2. Okay, this is 5 by 4, so there are 20 of those, and I’ve only got one stack, one layer, because again we’re talking about this space under here. So we’ve got 20 of those. Each one of those is 1/2 cubic unit each. So we multiply, and we get 10 cubic units for that volume. Okay, now let’s take this corridor, if you will, along here. Okay 1 by 1 by 1/3; these units, there are four of these per row here, because again the dimension here is 4. So we have four of these, but there are three layers of them, so we’re going to have 12 of these. And then along the very bottom down here, we’ve got these units that are 1/2 by 1 by 1/3. Again there are four of them this way, only 1 layer. So we have 12 of these and four of those.

Now let’s figure out the volume. Each one of these is 1/3 cubic unit. Do our multiplication. We get 4. Each one of these is 1/6 cubic unit; multiply by 4 and we get 2/3 in simplest form. Okay now let’s look at this front corridor if you will, right in here. Okay, we have along here on the top, we have five of these, and there are three layers. These are the ones that are 1 by 3/4 by 1. So again, we are going to end up with a total of 15 in here. And then this smaller unit that’s 1 by 3/4 by 1/2, that will be here along the base. And there are five of those, and there’s just one layer. Each one of these, 1 by 1 by 3/4, has a volume of 3/4. So we multiply. We get 11 1/4. And then these here, we only have five, and the dimensions are 1 by 3/4 by 1/2. The volume for each one is 3/8; do our multiplication, we get 1 7/8.

Now it sort of appears like we’re done. Remember that we were using this smaller right rectangular prism, this smaller box inside of the bigger one, as our framework and our reference. We figured out what was inside that box. We figured out what was underneath it. We figured out the volume along this corridor over here, and we figured out the volume along this corridor in here. But we’re not done. We also have a little corner stack here that we didn’t account for yet. Okay so in here, we’re going to have these units that are 1 on the height and then 3/4 this way along the depth and then 1/3 along the width. We’ve got three layers of those, so we have three of those, and then here at the very bottom, we have one of these that got cut three times, all three partitions, so that it’s 1/2 by 3/4 by 1/3. And again, we just have one of those.

Okay, figuring out the volume, these units that are 1 by 3/4 by 1/3, if you multiply that out, that’s 1/4 cubic unit each times three of them. And then here, this one and only, this unit here that’s 1/2 by 3/4 by 1/3, multiply that out. That’s 1/8 of a cubic unit each, and there’s only one of those. Now we can finalize this. If we take all of these volumes that resulted from having so many of these different sized little prisms, we add it up. It is 88 2/3. So it checks out with what we get from the formula.

Now if we look at the quantity, how many of these different sized units do we have? If you add all of this up, that’s 120. Interesting! And it’s just like we had way back at the beginning. If we were to round all of these up to where we had that original 4-by-5-by-6 right rectangular prism, that’s 120 of those cubic units, but they were all 1 by 1 by 1. Here we also have 120 of these smaller right rectangular prisms, but they are different sizes. So we really looked in depth at this first part of the standard, and the tough part was again, taking all of those unit cubes of different sizes and showing that the volume was the same as it would be by just using the formula for the volume.

We can’t forget this last part, and this is pretty much up to the teachers. It’s just coming up with some real-life situations that would use the formula for the volume in some type of real-world context. So let’s take, let’s say this was an example: We have a shipping box that’s 24 inches tall, 36 inches wide, and 30 inches deep. However, because sheets of shipping foam are used to line the interior faces, the actual interior dimensions are 22 1/2 inches tall, 34 2/3 inches wide, and 28 2/3 inches deep. So what’s the actual available volume for shipping purposes? So of course, your students would then have to figure out the volume to solve the problem.

And one more example—let’s say you have a solid rectangular prism, and it’s made out of styrofoam and it’s cut into two pieces as shown. Now if each cubic inch of styrofoam can support 1/6 of a pound in water, how much weight can each of these styrofoam blocks support? So again the students would have to find the volume of each one. For this one here, they’d have to multiply 32 times 20 times 21 4/5 to figure out that volume. Then of course, they would multiply by 1/6 to see how much weight this one could support. And then they would do the same thing with this smaller block here that’s 20 by 32 by 9 1/5.

So there’s a lot to this standard. It’s pretty complex because of all the partitioning and so forth that might be involved. And we’re not sure without sample test items if students would be expected to deal with a right rectangular prism with only one dimension that has a fractional edge length or two, or maybe even three. It’s hard to tell, so we went ahead and we took it all the way out to having three dimensions that all had a mixed number involved.

6.G.2, Part 3, Transcript

We need to go back and look at this standard and pay attention to this whole idea of unit cubes. The reason is because that phrase unit cubes as interpreted makes a huge difference in the interpretation of this standard and what’s possible in terms of instruction and assessment at the sixth grade level. Our concern is this idea of finding the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths. So we’re not going to worry about the second part of this standard that deals with applying the formulas, volume is equal to length times width times height (V=lwh).

Now there is a fifth grade standard that’s very similar to this one and notice what the difference is. The one in the fifth grade, you also have to find the volume of a right rectangular prism by packing with unit cubes, but the sides are whole numbers. So there’s no problem there, because your unit cubes are all understood to be 1 by 1 by 1.

Now let’s examine this idea again, this idea of unit cubes. In this situation, we have a right rectangular prism that’s 4 by 5 by 6. We’re going to fill it, we’re going to pack it with unit cubes that are 1 by 1 by 1. So we’re going to start off and start filling it. We start filling in the first row. So now we’ve done that. And so, what’s going to happen is we’re going to have on the bottom layer 30 of those unit cubes. Now the height is 4, so we’re going to have a total of four of those layers stacked on top of each other. We’re looking at this in terms of the unit cubes, 1 by 1 by 1, and we have a total of 120 of these unit cubes. Based on the formula, it checks out. We would have 120 cubic units. And again, these cubic units are understood to be unit cubes that are 1 by 1 by 1.

Now this idea of unit cubes, the way that we addressed it previously, we had a right rectangular prism, and we even looked at an example where all three dimensions had fractional edge lengths. Now what happened was that the 1-by-1-by-1 unit cubes, because of all this partitioning, got cut up into all these different sized units. That’s how we approached it to keep it simpler. When you look at applying the formula, we multiply 5 1/3 times 4 3/4 times 3 1/2, and we got 88 2/3 cubic units. Now, the understanding here is that 88 of those would be 1-by-1-by-1 unit cubes, plus we’d also have 2/3 of another 1-by-1-by-1 unit cube.

Now the standard says to find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes. Okay, well let’s see what we could do here. If we had one where let’s say the height was 4 but it’s going to become a height of 3 1/2. And again, we’re talking about 1-by-1-by-1 unit cubes. Well we have a problem, because that bottom layer, we have units that are 1 by 1 by 1/2. So the height messes things up here. On the bottom layer, we’re going to have 6 by 5 by 1/2, and if we just apply the formula, that comes out to 15 cubic units. Now what can we do? Well let’s see. Let’s look at the bottom layer. What we can do is split those up like so, and then what we can also do is this if we take all of these horizontal arrays and split them up this way. Now what we can do is take all these on the right and stack them on top of the ones on left, and look at what we’ve done. We’ve actually created unit cubes, because here we had a height of 1/2 and 1/2 which is 1. So we have created unit cubes that are 1 by 1 by 1, except now, the dimensions of this bottom layer are going to be 3 by 5 by 1.

If we go back now and apply this, here’s what we have. Now it depends on how we’re going to interpret this. If we’re going to literally stack these inside this right rectangular prism, it’s really not going to be possible, because these have a height of 1. We can’t squeeze those in here, because the height that we have to work with is 1/2. So we did create unit cubes but not in a way where we could actually stack them inside this right rectangular prism. If we’re going to look at this in terms of physically being able to stack the unit cubes in this right rectangular prism, then we’re going to have to do something different with this 1-by-1-by-1/2 unit.

So let’s take a look and see what we can do, and again with the idea of unit cubes. We have a height of 1/2, so it makes sense that our unit cubes that are going to be of equal lengths on each dimension are going to have to be 1/2 by 1/2 by 1/2. So if we take this a step further, I’m going to have to take our dimensions and split them up to where we have chunks of 1/2. Then we can create 1/2-by-1/2-by-1/2 unit cubes, and as you can tell, we’re going to have four of these 1/2-by-1/2-by-1/2 unit cubes for each of those 1-by-1-by-1/2 units of volume, and each of these 1/2-by-1/2-by-1/2 unit cubes has a volume of 1/8 of a 1-by-1-by-1 unit cube.

Let’s take a look at what happens now. We have our bottom layer where we have 6 by 5 by 1/2, and we have the 1-by-1-by-1/2 units. Each one of those has a volume of 1/2, and we have 30 of them. So if we multiply that, we have 15 cubic units for our volume for that layer. Now what we did to adhere to this idea of unit cubes that have equal dimensions, we were able to take each one of those, and we could put four of those 1/2-by-1/2 -by-1/2 unit cubes in it. Since we have 30 of these 1 by 1 by 1/2, and we could put four of those in each one, we’ve got 120 of them. And each one of those 1/2-by-1/2-by-1/2 unit cubes has a volume of 1/8. If we multiply 120 by 1/8, we have 15 cubic units. Now those cubic units are understood to be 1-by-1-by-1 unit cubes, and that’s what happens mathematically. But the reality here physically is that we have to do all this creative stuff to get unit cubes that were all the same dimension, in this case 1/2 by 1/2 by 1/2.

Now let’s take a more complicated scenario. What if we had a situation where we might have two of the dimensions that have fractional edge lengths? So instead of a length of 6 here, let’s say it was 5 1/3 instead. This is connecting back to the previous video where, look what happened; we have these partitions created that are going to be 1/3 by 1 by 1, and we’ve got a stack of three of those. But then on the bottom layer, the height was 1/2 originally for those units. So now we’re going to create yet another size of partition here. So again, we had one type of unit that was created, 1/3 by 1 by 1. And we have all of these stacked in there. For the bottom layer, we’re going to create some new units that are 1/3 by 1 by 1/2, and we have one stack of those.

So here’s our dilemma. We have 15 of these units that are 1/3 by 1 by 1, and we have five of these even smaller units that are 1/3 by 1 by 1/2. So we have these two to deal with. So let’s take this one first, the 1/3 by 1 by 1. It makes sense that our unit cube is to be 1/3 by 1/3 by 1/3. Again we have the dimensions all the same. Now I could only get one across here, but then this way, we can have three that way, and then the height is 1. So I can stack three total. So we have a total of nine of these 1/3-by-1/3-by-1/3 unit cubes that I could stack in each one of these 1/3-by-1-by-1 units. If I do the mathematics, each one of these 1/3-by-1/3-by-1/3 unit cubes is going to have a volume of 1/27, and of course, that’s of a 1-by-1-by-1 unit cube. If we do a little bit of manipulation here, it would take 27 of these 1/3-by-1/3-by-1/3 unit cubes to fill a 1-by-1-by-1 unit cube. So we go back to the original situation. We have 15 of these 1/3-by-1-by-1 units. I could partition this into small unit cubes that are 1/3 by 1/3 by 1/3, and I could put nine of those into each of these 1/3-by-1-by-1 units. Since I have 15 of these, and there are 9 in each one, that’s 135 total that I would need of these 1/3-by-1/3-by-1/3 unit cubes.

Now let’s look at the other situation here. We have the other units that are 1/3 by 1 by 1/2. The common denominator is 6, so I’m going to need unit cubes that are 1/6 by 1/6 by 1/6. This is a little bit more complicated if we’re dealing with six. This dimension was 1/3 along here, so I could put 2 there. We have 1 here for the width and I could put six across there. Then the height is 1/2, that’s 3/6, so I can stack three that way. So we’re going to end up with a total of 36 of these. So I would need 36 of these 1/6-by-1/6-by-1/6 unit cubes to fill each one of these larger 1/3-by-1-by-1/2 units. It would take 216 of these 1/6-by-1/6-by-1/6 unit cubes to fill a 1-by-1-by-1 unit cube.

Let’s look at the big picture. Here we have the 1/3-by-1-by-1 units. Now I could fill each one of those with 1/3-by-1/3-by-1/3 unit cubes, and I had 135 total of those here. Then, the second situation, I had these other sizes, 1/3 by 1 by 1/2; created unit cubes that are 1/6 by 1/6 by 1/6 there. There are 36 in each one of those. There are five, so there are 180 of those 1/6-by-1/6-by-1/6 unit cubes. Let’s look at what would happen now if we look at all the different sizes of these partitions. If we have a rectangular prism that’s 3 1/2 by 5 by 5 1/3, and on this bottom layer, we had units that were partitioned to be 1 by 1 by 1/2, we have to do this with unit cubes that are 1/2 by 1/2 by 1/2. We were able to put four of these in each one of those. In each one of these 1-by-1-by-1/2 partitions, I have four of the 1/2-by-1/2-by-1/2 unit cubes. There are 25 of them. So we have 100 of these 1/2-by-1/2-by-1/2 unit cubes in this bottom layer.

Then we had these other units created that filled in these three stacks here. These are 1/3-by-1/3-by-1/3 unit cubes. We have nine in each one. We have 15 of these total, so we have 135 of these 1/3-by-1/3-by-1/3 unit cubes. Then last but not least, we had the 1/3-by-1-by-1/2 partitions. We filled these with 1/6-by-1/6-by-1/6 unit cubes. There are 36 in each one. There are five of those, so we have 180 of those 1/6-by-1/6-by-1/6 unit cubes.

Now notice that this isn’t the end of the story. All we’ve done is figured out how many of those small unit cubes that we have. Now we still have the additional step of actually finding the volume based on all of these different sized unit cubes. So like here, we have 100 of these 1/2-by-1/2-by-1/2 unit cubes. But we have to take 100 and multiply it by 1/8, because it takes eight of these 1/2-by-1/2-by-1/2 unit cubes to make 1-by-1-by-1 unit cubes. So we have to take this and multiply 100 by 1/8. Over here, each one of these is 1/27 of a 1-by-1-by-1 unit cube. So we would have to take 135 times 1/27. And over here with the 1/6-by-1/6-by-1/6 unit cubes, each one of these is 1/216 of a unit cube. So we would have to take the 180 and multiply it by 1/216.

So you can see that this would be quite a mess. That was with right rectangular prisms with two of the three sides having fractional edge lengths. What we looked at so far was the possibility of having two sides of the right rectangular prism having fractional edge lengths. But what if you had one that had all three dimensions that had fractional edge lengths? You could have the possibility of something like this created where you have a partition that’s 1/3 by 3/4 by 1/2. Look what would be involved here. The lowest common denominator here is 12, so we would need unit cubes that are 1/12 by 1/12 by 1/12. If we do the math, it would take 216 of these little bitty 1/12-by-1/12-by-1/12 unit cubes to pack this unit here, and we would need 1728 of these little 1/12-by-1/12-by-1/12 unit cubes to pack a 1-by-1-by-1 unit cube.

So if we had an example like we did in the previous video, something like this, a right rectangular prism that’s 3 1/2 by 4 3/4 by 5 1/3, this is what happened. We had all these different sizes of partitions that were created from a 1-by-1-by-1 unit cube. So we’re dealing with a whole bunch of different sizes. If we’re going to follow this to the letter where we’re going to find the volume by packing it with unit cubes, and by unit cubes again, we mean that all three dimensions have to be the same, this is what we’re dealing with. We’d have to have unit cubes that are all these different dimensions. For example, this one here would have to be filled with unit cubes that are 1/4 by 1/4 by 1/4.

As you can see, this would be quite complicated, especially at the sixth grade level. In an assessment situation, it’s hard to imagine that a sixth grader would be given the task of doing this with three dimensions. In all likelihood, you would think that they might get a rectangular prism that might have one of the dimensions that has fractional edge lengths to make it more doable. So again, we had to go back and revisit this to again concentrate on this idea of unit cubes being cubes that all have to have the same dimension. Without having assessment samples, it’s hard to determine exactly how this might be tested. It just depends on how you interpret this idea of unit cubes. But if we’re going to go with the formal more precise definition, this is what we would have to be dealing with.