Common Core State Standards support video for Grade 3 in math: this is standard 3.MD.7d. This third grade standard says to relate the area to the operations of multiplication and addition; recognize area as additive; find areas of rectilinear figures by decomposing them into non-overlapping rectangles, and adding the areas of the non-overlapping parts, applying this technique to solve real-world problems.
Now, that’s a lot to the standard, so let’s start off just looking at this from an additive perspective. You’d want your students to have something, something solid to work with, so maybe some little algebra tiles or whatever you have on hand. The students are switching over from non-standard units to standard units, so it’s important to associate these tiles with that. So, you might want to relate this to a square unit, which could be something like square inches, square centimeters. But again, the idea would be that, to keep it very simple, is that it’s a 1 by 1 unit of some type, but it’s a square unit that links over back to area.
Next, what we want to do is take all of those tiles that were all over the place, and configure them in such a way where you make a rectangle to start getting to this idea of rectilinear figures. So, in this case, we have made it to where it’s a 1 by 5 rectangle. So, again, they can just count these up for now since we are still pretty much in an additive mode. Then what we can do is take, let’s say, for example, that array that we have, that 1 by 5, and let’s have three of them. So, now, what the students can do is, starting from an additive perspective, go ahead and take these and just count them up to where again, you’d have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 13, 14, 15.
Now, at this stage in third grade, students are switching over to multiplication. So, get them used to the idea that we can look at this as a 3 by 5 rectangle that has 15 square units for the area, where again, we’re looking at, in this case, the horizontal rows. We’d have three sets of five, or we can also look at this from another perspective, where it’s five groups of three, where the kids would be looking at this in terms of five rows of three.
Now, we need to go over to the idea of rectilinear figures, which are figures that are made up of straight lines, but not necessarily just nice, simple rectangles, something like this figure here, where you have actually a combination of rectangles. So, what we need to do is to give them experience with this type of figure where again, it’s more than just one simple rectangle. So, using this one as an example, some students might decide to do this, where they split it up into these three rectangles. And again, depending on where they are, you might have them draw all of the smaller rectangles that are possible. So, in this case, there’s six total.
And then over here; now, here’s where the students have got to do a little bit of figuring. They can tell that, well, this also has to be a 2 over here. So, now this is a 2 by 2, and again, they can either just go by computation or actually have them draw it in. Now we have 4 square units there, and then they can do the same thing with the bigger rectangle. But, again, the key here is that they’re going to have to figure out, well, what is this length here? So, if they look over here, they could tell that there was a 2 there, which means that this is going to have to be a 4. So, now, basically we have a 4 by 8 rectangle, and they can either do it with simple multiplication, that there’s 32 more. Or again, you can just have them draw it all out and have the 32 little squares. So, now we have a total of 4 and 6 and 32, which would give us 42.
Using the same figure, you know that kids will think differently, so some kids might approach it by doing something like this instead. And so, we basically go through the same process where kids have to figure out what the different sides are. Well, this rectangle here, you’ve got your sides already; it’s 3 by 6. And again, depending on where they are, they can do it through multiplication or doing the counting. So, we have a total of 18 there (3 by 6). Here we know the sides, 2 by 6. So, then, here we have 12 more to tack on.
And then, this is where they have to do a little bit of work here, because they don’t know what this is. If we look across, the whole distance is 8, and we have a 3 here, and a 2 there. That’s a total of five. Let’s see, 5 from 8, so this has to be a 3. And then, we have to do some work here too. We had six total on the right- or the left-hand side. So, how far is it to here? Well, let’s see, it was a distance of two there, a total of six. Six minus 2 is 4. So, then, this is a 4, so we have a 4 by 3 rectangle, which would give us another 12. And so, here we get the same total of 42 square units.
You might have a few kids that will surprise you. They might look at this more from a subtraction perspective rather than additive. What they might do is assume that you have the big rectangle that’s 6 by 8, which would be a total of 48. And then, they’ll take this remaining rectangle and actually subtract whatever area that has. So again, it’s a matter of figuring out, well let’s see, what’s the dimensions? We already have the 2. We need to figure out what this is, and just like before, we have 3 and 2, that’s 5. We need to make up the difference with 8, so this has to be a 3. So, that’s a 3 by 2 rectangle, which is a total of 6 for the area. And guess what? These kids have come up with the same solution but using a pretty different approach.
The standard says that we need to apply this to real-world problems. The ones that we did before, those could have represented, say, a kitchen floor where you have, all your little tiles would be the tiles that you put in the kitchen. And that’s a good idea to do this in that manner, to have something in real life that kids can relate to. They can relate those square tiles to square units, be it square feet or whatever. So, in this kind of situation, you might have something similar, a real-life situation to where this would be a kitchen. And let’s say you’re going to have an island in the middle, and you give the kids the task of figuring out how many tiles that you’re actually going to need to find the area where you want to put your tiles.
There’s different ways to approach this, just like we did before with the other example. And, just very quickly, the most sensible thing for kids to do, they would probably figure out the total area, which is for the 8 by 6 rectangle, which would give you 48. And then, for the shaded area in the middle, well, that’s 4 by 2, which gives us 8. And then, if we take the difference between the two, that’s 40 square units.