Video Transcript
Comparing Decimals: Tenths and
Hundredths
In this video, we’re going to learn
how to use models and place value tables to compare numbers with up to two decimal
places.
These two children are taking part
in a standing long jump competition. We can see that the boy has already
jumped 1.15 meters or one and fifteen hundredths of a meter. And we’ve just watched the girl
jump 1.2 meters or one and two-tenths of a meter. Now, both of these measurements are
represented by decimal numbers. They have both a whole part and a
fraction part. And these two parts are separated
by a decimal point. So both children have jumped one
and a bit meters. But in order for us to see who’s
jumped the furthest, we’re going to need the ability to be able to compare these two
decimals together. By the end of this video, we’ll
have learned enough skills to be able to do this and to be able to say who’s jumped
the furthest altogether.
One way we could compare decimals
together is to use models to help us. For example, if we want to find out
which number is larger, 0.6 or 0.9, we could try sketching a model that shows six
tenths and nine tenths. And then we could compare these
parts visually. We could start by sketching a
square like this to represent one whole. And then if we divided it into 10
equal parts, each part would be worth one-tenth or 0.1. So to model 0.6, we need to shade
in six-tenths, and to model 0.9, we’d need to shade in nine tenths. Now we can see just by looking at
the shaded parts which is the largest. Nine-tenths is greater than
six-tenths. Now there are lots of different
models that we can use to represent decimal numbers. Let’s have a go at comparing two
decimals that have been modeled using an abacus.
Complete the following using the
symbol for is greater than, is less than, or is equal to.
An abacus is a way of modeling a
number. And in this question, we’ve got two
to look at. We need to compare both numbers
together and then choose the correct symbol to show whether the first number is
greater than the second, whether it’s less than the second number, or whether the
two numbers are exactly the same. Our first number shows two ones and
three-tenths. It’s a decimal number. We know this because it’s made up
of a whole part — two ones or a whole number — but also a fractional part. That’s a part that’s worth less
than one. And three-tenths of this fractional
part, three-tenths are less than one.
And if we were to write this number
using digits, we need a two in the ones place, then a decimal point to show that
we’re moving into the fractional part, and then a three in the tenth place. Two ones and three-tenths is the
same as 2.3. Now let’s look at our second
number. This is made up of one one and
seven-tenths. So we have less ones, but a few
more tenths in this number. And again, it’s a decimal. This is because it’s made up of
that whole part and a part that’s worth less than one. This time, we could represent this
model using digits with a one in the ones place. Again, we’re going to need a
decimal point to separate it from the fraction part and then a seven in the tenths
place to show our seven-tenths. One and seven-tenths is the same as
1.7.
So how are we going to compare
these two decimals that have been modeled on each abacus? Should we compare the number of
beads in the tenths place first? Or should we think about the
ones? Well, we need to apply the same
rule that we use when we compare whole numbers. And that’s to compare the digits
that have the largest value first and work our way from left to right. So we shouldn’t start by comparing
the tenths. A tenth is less than one. We need to start by looking at the
number of ones that we have.
Our first abacus shows two in the
ones place, but, in our second abacus, there’s only one bead in the ones place. It doesn’t matter how many tenths
each number has, because we’ve compared the ones. And we can see that the first
number is larger than the second. 2.3 or two ones and three-tenths is
greater than 1.7 or one one and seven-tenths. The correct symbol to use in
between these two models is the one that represents is greater than.
Another way that we could compare
decimals is by using a number line. For example, if we want to find out
whether sixty-nine hundredths is greater than ninety-six hundredths, we could sketch
a number line from zero to one and then label where each number belongs. 0.69 or sixty-nine hundredths is
just before 0.70 or seventy hundredths. But 0.96 or ninety-six hundredths
is really close to one whole. So we’ve used our number line to
show that sixty-nine hundredths is not greater than ninety-six hundredths. 0.69 is less than 0.96.
Let’s move on to a third way that
we can compare decimals together, and you’ll be familiar with this from comparing
whole numbers. This is to use a place value table
to compare the digits in our numbers. Let’s go through an example
question where we need to do this.
Complete the following using the
symbol for is equal to, is less than, or is greater than.
In this question, we’re given two
numbers, and they’ve been written for us in place value tables. Let’s read them. Our first number contains one one,
four-tenths, and eight hundredths, 1.48. Our second number also has a single
one, it also has four-tenths, but this time it has six hundredths, 1.46. These two numbers are decimals. They have a whole part. That’s the number of ones. But then they also have a part
that’s worth less than one. And to separate the whole part from
the fractional part, we use a decimal point.
Now, in between our two decimals,
there’s a box. And we need to complete the
statement using one of the comparison symbols we’re given equal to, less than, or
greater than. In other words, we need to compare
both these decimals together. Now both of these decimals have
been written in place value tables to help us. So we can compare them digit by
digit. Now shall we start with the
hundredths and work from right to left? Or shall we start with the ones and
work from left to right? Well, if we take a moment to think
about what each one of these columns is worth, we know that our ones column
represents one whole.
Now, as we move to the right, each
new column is worth 10 times less than the one before. So a tenth is 10 times less than a
whole. It’s what we get if we take a whole
and split it up into 10 equal parts. And then a hundredth is 10 times
less again. It’s what we get if we take one of
our tenths and split that up into 10 equal parts. So knowing that, which part of our
number do you think is most important? It’s the part that’s worth the
most. We need to start by comparing our
ones column and then work from left to right. Our first decimal number contains a
one in the ones place. But can you see we’ve also got this
in our second number too? So we can’t separate our numbers
just by looking at the ones. We’re going to need to move on and
look at the tenths.
Both numbers have a four in the
tenths place. We still can’t separate them. The beginning of both our decimals
is the same, isn’t it, one and four-tenths. We’re going to need to compare the
hundredth digits. Below these are the digits that
have the least value in our numbers. They’re actually going to turn out
to be the ones that make the difference. This tells us that the difference
or a gap between our two numbers is very small. It’s just going to be a matter of
hundredths. Our first number has an eight in
the hundredths place, but our second number only has a six. And eight hundredths are greater
than six hundredths.
The difference between our two
numbers is only two hundredths, which is a tiny difference. But it’s enough for us to be able
to say which number is larger. We’ve compared these two decimal
numbers digit by digit using place value tables. One, four tenths, and eight
hundredths is greater than one, four tenths, and six hundredths. Or to say it a different way, 1.48
is greater than 1.46. The correct symbol to use in
between these two decimals is the one that represents is greater than.
So we’ve looked at three different
ways we can compare decimals using a model, comparing them on a number line, and
comparing them digit by digit using a place value table. But what if our decimals are
represented in different ways? Let’s try a question where they
are.
Complete using the symbol for is
less than, is equal to, or is greater than. Five-tenths what seventy
hundredths.
In this question, we’ve got two
values that we need to compare together. And there are a combination of both
digits and words. We’ve got five-tenths and then
there’s seventy hundredths. Now a very easy mistake to make
here would be just to look at the numerals and say to ourselves, “Well, I know that
five is a lot less than 70, so five-tenths must be less than seventy
hundredths.” Now, in a moment we’re going to
compare these values, and it might be the case that the first one is less than the
second, but it’s not because five is less than 70. And that’s because these values are
talking about two different place value units. We’ve got five-tenths and seventy
hundredths.
Let’s sketch a couple of models to
help us see what these values actually mean. We could represent five-tenths by
taking a whole square splitting it into 10 equal parts and then shading five of
them. There are no whole squares, but
there are five-tenths of a square. So we’d write this value as 0.5 if
we were writing it in digits. Our second value is a number of
hundredths. Now, just like all the other
columns, the hundredths place can only fit one digit in. But we’re told that this value is
worth seventy hundredths. If we were to sketch this as a
model again, we’d need our whole square and, this time, to split it into 100 equal
parts. Then we’d need to shade in 70 of
them.
So once again, we’ve shaded in zero
whole squares and we’d write seventy hundredths with a seven in the tenths place and
a zero in the hundredths place. If we drew another model, can you
see that seventy hundredths is actually the same as seven-tenths? That’s why that digit seven goes in
the tenths place. Now we’ve sketched these two
models; we can compare the values quite easily. We can see that five-tenths is less
than seventy hundredths. As we’ve just seen, seventy
hundredths is the same as seven-tenths. And we know that five-tenths must
be less than seven-tenths.
Another way we could find the same
answer is by looking at our place value table and comparing the digits. Both numbers have zero ones, but
our first number has five-tenths and our second number has seven-tenths. So it turns out that five-tenths is
less than seventy hundredths. But it’s not because five is less
than 70. To find the answer, we needed to
work out what five-tenths and also seventy-hundredths were worth. And we use both models and also a
place value table to help compare them. Five-tenths is less than seventy
hundredths. And so the symbol that we need to
use in between these two values to compare them is the one that represents less
than.
If you remember right at the very
start of this video, we had a standing long jump competition with two decimal
measurements that we wanted to compare. Decimals crop up quite a lot in
sports: the length that somebody jumps, the time that it takes them to run a certain
distance, or, as we’ll see in the next question, the number of yards that they can
throw a football.
Peyton Manning averaged 7.89
passing yards per attempt in 2006 and Carson Palmer averaged 7.76. Who had the higher average?
This question is a really good
example of how decimals are sometimes used in sport. Now, perhaps you’ve already heard
of the two sportsmen that are mentioned in this question. But if not, Peyton Manning and
Carson Palmer were both quarterbacks in American football. And one of the key things that a
quarterback has to do is to make forward passes by throwing the football. And that’s where the two decimal
measurements in this question come from. We’re told that Peyton Manning
averaged 7.89 passing yards every time he attempted a throw and Carson Palmer
averaged 7.76 yards. And our question asks us who had
the higher average. In other words, we need to compare
these two decimal measurements together. Let’s do so by drawing a place
value table.
We’re going to need to include
columns for ones, tenths, and hundredths because both of these values are decimals
with two decimal places. In other words, they show both
tenths and hundredths. Now, back in 2006, Peyton Manning’s
average was 7.89 yards. We know this is a decimal number
because there’s a decimal point in it, but also because it contains a whole part,
seven ones, which represents seven whole yards. And then there’s a fraction part,
which is a part worth less than one, eight-tenths and nine hundredths. We could read this as 7.89 but also
seven and eighty-nine hundredths.
Now let’s complete Carson Palmer’s
average, which we can see was 7.76. So that’s seven whole yards, a
seven in the ones place, and then, in the fraction part, we’ve got seven-tenths and
six hundredths, seven and seventy-six hundredths. So let’s find out which one of
these numbers is larger. The column with the greatest value
is going to be the ones column, so we need to compare these digits first. And we can see that both players
threw seven whole yards.
So far, our numbers are exactly the
same, and this is often the case with sports measurements. When you’ve got two players that
are very close together in ability, the first few digits might be the same. There’s often only a very small
difference between them. So let’s move on to the tenths
digits. 7.89 contains eight-tenths, and
7.76 contains seven-tenths. And eight is greater than
seven. And we can stop right there. We don’t need to move to the
hundredths column. We can separate these two numbers
by looking at their tenths. Because eight-tenths is greater
than seven-tenths, we can say that 7.89 is greater than 7.76. Well, to say the numbers a
different way, seven and eighty-nine hundredths is greater than seven and
seventy-six hundredths. And so, the player that had the
higher average was Peyton Manning.
So what have we learned in this
video? We’ve learned how to use models and
place value tables to compare numbers with up to two decimal places.
