Video Transcript
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.
