Area: Nonstandard Units
In this video, we’re going to learn how to describe what the area of a shape is and also how to measure area by counting unit squares.
When we talk about area, we’re thinking about shapes. What is the area of a shape? It’s a measure of the space inside a 2D shape. Another way of thinking about it is as the space that a flat shape takes up.
Here’s a 2D shape. Now, if we colored inside the boundary of this shape, we’d say that the part of the shape that has been colored is its area. It’s the space inside the shape. This lady is putting carpet tiles to cover her floor. If we think of the floor of her room as being a flat shape, then we can say that this lady is covering the area of the floor, the space that the floor takes up.
Here’s a 2D shape. It’s actually got six sides, so it’s a hexagon. But they’re not all the same length, so we’d have to call it an irregular hexagon. Now, how would we go about measuring the space inside this shape? How would we find its area? Well, as we’ve seen from the title of this video, we’re going to be thinking about somethings called nonstandard units. Now if we were asked to measure the length of something, depending on what we’re measuring, we might use millimeters, centimeters, and so on. But if you can think back far enough to the very first lessons you had on length, you weren’t taught about millimeters or centimeters. We didn’t worry about standard units of measurement like these. We used nonstandard units.
Perhaps you did things like seeing how many cubes fitted along the length of an object or maybe made a line of counters or even used hand spans. These are all not the usual ways that we measure length in everyday life. But there’re ways that are good when we’re starting off to learn about length. And in the same way, we’re going to be thinking about units of measurement to measure area that aren’t the usual ways we’d measure area in real life, but they’re a good introduction. So what units shall we use to measure the space inside our shape?
There are somethings we need to remember here about units of measurement. Firstly, they all need to be the same size. If we measured the length of something using millimeters, each of those millimeters would be exactly the same size. Or even if we use nonstandard units like a line of counters, it’d be no good if one of those counters was larger than all the others. And even if we measure something using hand spans, we always try to stretch our hand to the same distance and use the same hand every time. So whatever we choose to measure the space inside our shape, it’s going to have to be the same size every time.
The other thing to remember is it’s going to have to fit. Look at this line of cubes we’ve shown for measuring length. We haven’t put them all wonky, have we, or on top of each other. There’s no gaps in between them. They fit exactly. Let’s try measuring the area of our shape using these colored squares of paper. How many can we fit inside the shape? One, two, three. By the way, notice how we’re making sure that there are no gaps in between our squares and none of them overlap at all. They all fit inside the shape. We need to cover all of the area inside the shape. Let’s keep counting, four, five. Five of our squares of blue color paper fit inside the shape. And so we can say that the area of the shape, the space inside it, is five square units or five squares.
Now, here’s an interesting question. What if we take in exactly the same-size shape but this time drew it on squared paper? We wouldn’t need squares of colored paper to put inside the shape. Can you see how we’d find the area? We just need to count the number of squares that we can see inside the shape. And because we’ve drawn it on squared paper, they’re already there for us. There are four in this little area here. And in each row of the rest of the shape, there are four more squares every time. So we can count them quite quickly.
Now, who’s right? Our shape can fit 20 smaller squares inside it. But as we found with the colored squares, it can fit five larger squares. So what’s the area? Five square units or 20 square units? Well, this is the thing about using nonstandard units of measurement. One type of unit isn’t always the same size as another. But as long as we use the same size for the whole measurement, we make sure there’re no gaps, and nothing overlaps, then, really, we can say both these children are correct; they just use different units. Rectangles would fit together and fit inside. So we could say that the area of the shape is 10 rectangles or even if we use this kind of right-angled triangle 10 triangles.
One thing we couldn’t use is circles or plastic counters. I wonder, can you see why? It’s because of these areas of white that we can see, the gaps in between our counters. Although the counters are all the same size, they don’t cover all of the area. And because of the gaps in between, we can’t use circular units of measurement like this. Let’s have a go at measuring the space inside some other 2D shapes then. We’re going to answer some questions where we need to put into practice everything that we’ve learned.
Each flower in Charlotte’s window box needs one square unit of space to grow. How many flowers will fit in the window box?
This question describes Charlotte who has a window box, one of those containers that you sometimes see outside people’s windows where they grow flowers. Now, there are only so many flowers that you can grow in a window box, depends how big it is. And we’re told that each flower in Charlotte’s window box needs one square unit of space to grow. Each flower that Charlotte wants to grow is going to need its own space. We’re given a diagram that shows what this means. We’ve got a square that is one unit tall and one unit wide. It’s one square unit. Now, we’re asked, how many flowers will fit in the window box? In other words, how many flowers will fit in the space inside the window box?
This word space is important. We saw it in our first sentence. It tells us really that this question is all about area. We know that the area of a 2D shape is the space inside it. Now we need to be a little bit careful with this question because a window box in real life isn’t a flat shape. It’s a box, a cuboid without a lid. That’s where all the flowers go, but we can see a picture of our window box that is a 2D shape. We could think of this as showing a bird’s-eye view of our window box, a plan of what it might look like from above. That’s why we can show it as a rectangle. And that’s why we can start thinking about area.
We can see that this rectangle is made up of several squares. They’re square units. And we know from what we’ve read already that each flower that Charlotte wants to grow is going to need one square unit of space. To find out how many flowers are going to fit in the window box, we’re going to need to count the square units. There are four squares across the top row, then another four, and four along the bottom. We could say the area of this plan of our window box is 12 square units. And because each of these 12 square units is just enough space to fit one flower in it, we know the number of flowers that are going to fit in Charlotte’s window box is 12.
Daniel made this shape with square units. Find the area.
To begin with in this question, we’re told that Daniel’s made a shape out of square units. And we’re shown a picture of the shape that he’s made. It’s this six-sided shape or hexagon here. If we turn our heads to look at it, we could even describe it as being a little bit like an L shape. And most importantly, this diagram shows us the square units that Daniel’s made his shape out of. Now, the question asks us to find the area of this shape. We know that the area of a shape is a measure of the space inside it. So this question is really saying to us Daniel has made a shape out of square units. How much space is inside of it?
Now, to measure the area of Daniel’s L-shaped hexagon, we’re going to need to use units that are all the same size. They’re going to need to fit together with no gaps, and they’re going to need to cover all of the space. But wait a moment. Because Daniel has made his shape out of square units, can’t we just count those square units? They’re all the same size, they will fit together, and there are no gaps. They completely cover the space inside the shape, don’t they? Let’s count how many there are. One, two, three, four, five, six, seven, eight, nine, 10, 11, 12, 13, 14, 15, 16, 17. Daniel used 17 square units to make his shape. And so we can measure the space inside his shape or its area by counting these square units. The area of Daniel’s shape is 17 square units.
Natalie thinks that this shape has an area of 15 square units. Is she correct? Yes, because there are 15 tiles. Yes, because the tiles do not overlap. Yes, because the square units cover the whole area. Or no, because the square units do not cover the whole area.
The shape that’s mentioned in this first sentence is this rectangle that we’re shown here. And we’re told that Natalie thinks that this rectangle has an area of 15 square units. Now where do you think she gets this idea from? Well, we know that the area of a flat shape is a measure of the space inside it. And in the space inside Natalie’s rectangle, she’s placed some blue square units. Maybe these are pieces of colored card or something like that. And if we count them, there are 15 of them. This is where Natalie gets the idea of 15 square units from.
So is Natalie correct? Does this shape have an area of 15 square units? This is not just a straightforward yes–no question. We’re given four possible answers, but as well as saying yes or no, each one contains a reason. This question is really asking us, is Natalie correct and why? Let’s go through our four possible answers. Now we know from having counted them that part of our first possible answer is correct. There are actually 15 tiles. This bit’s true. Maybe our first answer is the right one. Let’s have a look at the second. Our second answer also suggests that Natalie might be right, but this time for a different reason. This time, it’s because the tiles do not overlap.
We know that whenever we measure area, it’s important that the units that we use are in their own space. They don’t overlap at all. So it’s definitely important that Natalie hasn’t overlapped any of her square units. But you know, Natalie has made one mistake. To measure the area or the space inside a shape, we need to make sure that all of that space is covered. Our next two possible answers tell us that the square units cover the whole area, but then the square units do not cover the whole area. Which one of these is true? Well, we don’t have to look very hard, do we?
There’s a lot of white space in this rectangle. Natalie’s square units aren’t touching each other at all. They haven’t covered the whole area. So although there are 15 tiles and the tiles that Natalie’s used do not overlap, and these are both very good things, Natalie hasn’t pushed them all up really close to each other so that they cover the whole area. This is the mistake that she’s made. So when Natalie says that the shape has an area of 15 square units, is she correct? No, because the square units do not cover the whole area.
Scarlett, the gardener, charges customers by how many individual square pieces of grass she lays when she creates a new backyard design. Order these yards from least to most expensive.
The first sentence in this question tells us about Scarlett, a gardener. She clearly designs the backyards for different people. And she makes these designs by laying out individual square pieces of grass to make different shapes. This first sentence tells us that Scarlett charges customers according to how many individual square pieces of grass she uses. The more pieces of grass, the more expensive the backyard.
Now, even though we’re told to order the yards from least to most expensive, this question isn’t about money at all. It’s about us understanding which of the shapes we can see contains the most squares. We know that the space inside a shape, and in this question we’re talking about the space inside Scarlett’s designs, is called its area. So really, this question is asking us to put the shapes A, B, C, and D in order of area from smallest to largest area. Now, if you were to look at these four designs, do you think you could spot which one has the smallest area?
Shape A looks like it might be the thinnest, and shape B is definitely the smallest in terms of height. But neither of these two facts help us because what we need to think about is the space inside each shape. And the only way we’re going to find this is by counting the squares that Scarlett’s going to need to use.
Now, we can’t see any squares inside each of these shapes. But because they’ve been drawn on squared paper, we know where the squares belong. In fact, we could draw them on using a pencil. That’s a lot clearer. Now let’s count them. Design A has an area of 10 square units. Let’s label that so we don’t forget it. Design B is made up of 11 square pieces of grass, so we can say the area of this shape is 11 square units. The person who pays Scarlett for design C is going to need to pay her for 12 square pieces of grass. So shape C has an area of 12 square units, and we can see that yard C has an area of nine square units.
Each of the areas is different and so we can put them in order really quickly. The yard that needs the smallest number of square pieces of grass is yard D, with nine square units. Then we have yard A with 10, yard B with 11, and the largest yard, so the one that’s going to cost most money, is yard C. In this question, we were asked to order the yards from least to most expensive. But we realized this question was really asking us to order them from smallest to largest area. In order, the designs are D, A, B, and C.
So what have we learned in this video? We’ve learned that the area of a 2D shape is the space inside it. We’ve also learned how to measure area using nonstandard units and how not to measure it.