WEBVTT
00:00:02.990 --> 00:00:08.750
In this video we’re gonna be comparing fractions that either have the same numerator or the same denominator.
00:00:09.150 --> 00:00:13.670
We’ll be using less than, greater than, and equal signs to express the differences.
00:00:13.970 --> 00:00:20.540
And we’ll also be reminding ourselves that we can only sensibly compare fractions when they’re referring to the same whole.
00:00:20.960 --> 00:00:25.210
For example, half a mouse is not equal to half of an elephant.
00:00:27.520 --> 00:00:31.510
First though let’s quickly remind ourselves what numerators and denominators are.
00:00:32.910 --> 00:00:40.960
A numerator is what we call “the upstairs number” on a fraction and a denominator is what we call “the downstairs number” on a fraction.
00:00:43.630 --> 00:00:46.500
Now let’s get a scrumptious looking chocolate cake.
00:00:46.920 --> 00:00:48.390
That’s a whole cake.
00:00:48.690 --> 00:00:56.050
And if we split it into one piece and eat one of those pieces, then we’ve eaten the whole cake on our own.
00:00:56.830 --> 00:00:58.740
Yum, but ouch!
00:01:00.380 --> 00:01:02.830
So one represents the whole cake.
00:01:03.070 --> 00:01:10.010
And we’ve eaten one-oneth or one over one — one out of one — of that cake.
00:01:11.730 --> 00:01:14.550
Now let’s take another cake exactly like the last cake.
00:01:14.820 --> 00:01:20.040
But this time we’re gonna split it into two equal pieces so that we can share the cake between two people.
00:01:21.200 --> 00:01:26.210
Each person would get a half, one over two, a half of the cake.
00:01:28.080 --> 00:01:32.800
We take another similar cake and split it into four pieces to share between four people.
00:01:34.270 --> 00:01:39.390
So each person gets one out of the four pieces; that’s one-quarter of the cake.
00:01:40.810 --> 00:01:44.320
Now we got a third cake and we split that into six equal pieces.
00:01:45.270 --> 00:01:51.730
And each person getting a piece of that cake would have one out of the six pieces; that’s a sixth of a cake.
00:01:53.020 --> 00:02:01.040
So if we share a cake between two people, each person gets a bigger share of the cake than they would do if we shared it between four people.
00:02:01.720 --> 00:02:08.090
And if we share a cake between four people, each person gets a bigger share of the cake than they would if they would have shared it between six people.
00:02:09.580 --> 00:02:14.750
So we can use this symbol here to represent greater than or larger than.
00:02:14.750 --> 00:02:17.400
So a half is greater than a quarter.
00:02:18.700 --> 00:02:21.610
And a quarter is greater than a sixth.
00:02:23.000 --> 00:02:25.240
Now that sign might look a bit confusing to start off with.
00:02:25.240 --> 00:02:29.520
But what you gotta remember is it’s got a great big end and it’s got a small end.
00:02:30.550 --> 00:02:36.020
And the big end goes against the big number and the small end goes against the small number.
00:02:36.690 --> 00:02:41.580
Likewise over here a quarter is bigger, so the big end of the sign goes to the quarter.
00:02:41.800 --> 00:02:45.530
A sixth is smaller, so the small end of the sign goes against the sixth.
00:02:47.330 --> 00:02:50.580
Now there’s another version of that sign as well called the less than sign.
00:02:51.000 --> 00:02:57.610
So we know a quarter is less than a half, so the small end of the sign goes against the smaller number.
00:02:57.890 --> 00:03:00.410
The big end of the sign goes against the bigger number.
00:03:01.130 --> 00:03:09.750
A sixth is less than a quarter, so the small end of the sign goes against the smaller number and the larger end goes against the larger number.
00:03:11.280 --> 00:03:14.360
Now some of you might be thinking whoa, hold on a second!
00:03:14.880 --> 00:03:20.590
You’re saying that a half is bigger than a quarter, but two is smaller than four.
00:03:22.270 --> 00:03:27.790
But what you gotta remember that a denominator is telling us how many people were sharing the cake between.
00:03:28.090 --> 00:03:32.410
And if we share a cake between more people, we’re getting a smaller amount each.
00:03:34.380 --> 00:03:45.250
So if we’ve got one piece of a cake that we shared between four people, that’s gonna be smaller than we would’ve got if we would’ve shared that- if we have got that one piece of a cake shared between two people.
00:03:45.620 --> 00:03:50.980
Remember this quarter is smaller than this half of the same cake.
00:03:52.920 --> 00:04:02.180
So to summarize that, with fractions if the numerators are equal, then the fraction with the larger denominator represents the smaller proportion.
00:04:02.650 --> 00:04:06.020
We can see these have all got one as their numerator.
00:04:06.380 --> 00:04:11.230
A sixth represents a smaller proportion than a half.
00:04:11.590 --> 00:04:20.150
It’s got a bigger denominator than half, but it represents a smaller piece of the cake we’re sharing it out between more people.
00:04:22.380 --> 00:04:26.230
Now let’s compare some more pieces cut from identical cakes.
00:04:27.480 --> 00:04:31.610
This time we’ve cut them into halves, quarters, and eighths.
00:04:33.180 --> 00:04:42.170
And using what we just learned, we can see that because they’ve all got one as their numerator, as the denominator increases the pieces of cake get smaller.
00:04:42.210 --> 00:04:46.980
So a half is bigger than a quarter and a quarter is bigger than an eighth.
00:04:48.520 --> 00:04:51.460
But what if I took more than one piece from some of the cakes?
00:04:51.810 --> 00:04:59.770
If I took two of the quarters or four of the eighths, then I’d have an equal amount of cake in each case.
00:05:00.100 --> 00:05:05.340
And obviously we use the equal sign to represent when we’ve got equal bits of cake.
00:05:05.630 --> 00:05:12.670
So a half is equal to two- quarters or two-quarters is equal to four-eighths.
00:05:14.220 --> 00:05:16.960
In fact in this case we can do these all three ways.
00:05:16.990 --> 00:05:20.220
And we can also say that a half is equal to four-eighths.
00:05:22.470 --> 00:05:30.550
Now in all the examples that we’ve looked at so far in this video, the cakes we’ve been cutting up were the same size, so we can compare fractions directly.
00:05:31.100 --> 00:05:34.690
But it’s important to remember that you have to be careful when comparing fractions.
00:05:35.010 --> 00:05:38.690
If there’re fractions of different wholes, then you can’t easily compare them.
00:05:39.090 --> 00:05:43.910
So a quarter of a small cake is not equal to a quarter of a large cake.
00:05:45.920 --> 00:05:49.140
Okay back to two cakes which are the same size.
00:05:50.470 --> 00:05:55.060
And we’ve cut these equal sized cakes into eight equal sized pieces.
00:05:55.440 --> 00:06:01.860
And the cake on the left, we’ve selected one of those eighths and the cake on the right, we’ve selected three of the eighths.
00:06:02.770 --> 00:06:06.540
So now we got two fractions in which the denominators are the same.
00:06:08.130 --> 00:06:17.400
So given that the cakes are the same size and we’ve cut them up into the same number of pieces, eight, then the more pieces of that cake that we have, the more cake we’ve got overall.
00:06:18.890 --> 00:06:22.140
So three-eighths is gonna be bigger than one-eighth.
00:06:22.310 --> 00:06:28.490
So we can draw the sign in with the big end of the sign against the three-eighths and the small end of the sign in against the one-eighth.
00:06:29.990 --> 00:06:33.680
Or we could’ve written the fraction the other way round and written the sign the other way around.
00:06:33.680 --> 00:06:36.060
So three-eighths is greater than one-eighth.
00:06:37.990 --> 00:06:43.910
So for fractions with the same denominator, then a higher numerator means a larger proportion.
00:06:44.250 --> 00:06:49.780
If we’ve got more pieces of the same size cake, then we’ve got a larger proportion of that same size cake.
00:06:51.870 --> 00:06:56.660
So moving away from cake for a moment, we can also make comparisons on the number line.
00:06:57.320 --> 00:07:09.060
So if we have a half and compare that to a third, we can see that a half is bigger than a third.
00:07:09.060 --> 00:07:15.610
Because if I just draw a little dotted line coming down here, this bottom red arrow is slightly smaller than the top red arrow.
00:07:17.040 --> 00:07:18.720
And that ties in with what we’ve just learned.
00:07:19.010 --> 00:07:24.520
If we’ve got the same numerator, then the bigger denominator will be the smaller fraction.
00:07:24.820 --> 00:07:28.030
The bigger denominator means we’re sharing the cake out between more people.
00:07:28.200 --> 00:07:30.930
So each person is gonna get a smaller piece of cake.
00:07:32.440 --> 00:07:37.460
And don’t forget we could also write that round the other way: a third is less than a half.
00:07:39.280 --> 00:07:41.270
Now let’s think about a quarter.
00:07:44.560 --> 00:07:52.230
Again if we compare a third and a quarter, the arrow representing a quarter is shorter than the arrow representing a third.
00:07:52.260 --> 00:08:00.600
So a quarter is less than a third or a third is greater than a quarter.
00:08:03.070 --> 00:08:11.440
And again because we’ve got the same numerator in each case, the larger the denominator the smaller the proportion that represents.
00:08:11.470 --> 00:08:15.690
Remember we’re sharing that cake between more people, so everyone’s gonna get a smaller piece.
00:08:17.170 --> 00:08:19.630
Okay it’s time to test yourself on what we’ve learned.
00:08:20.010 --> 00:08:26.390
I want you to write those symbols: less than or greater than or equals to to complete these statements.
00:08:26.700 --> 00:08:31.010
Now we’re assuming that these are fractions of the same overall sized whole.
00:08:32.090 --> 00:08:36.410
So just pause the video now and then check your answers in a moment.
00:08:41.490 --> 00:08:44.790
In number one then, we’ve got the same numerator.
00:08:45.310 --> 00:08:49.760
So the bigger denominator means that we’re sharing that out between more people.
00:08:49.760 --> 00:08:54.510
So it’s gonna be the smaller fraction, so a third is greater than a ninth.
00:08:54.510 --> 00:08:57.270
The small end of the sign points to the small fraction.
00:08:58.490 --> 00:09:03.310
With number two, we’ve also got the same numerator, but hold on!
00:09:03.310 --> 00:09:05.820
We’ve also got the same denominator.
00:09:06.370 --> 00:09:09.850
So three-fifths is the same as three-fifths in this case.
00:09:09.850 --> 00:09:12.970
So we put our equal sign; those two fractions are the same.
00:09:14.540 --> 00:09:17.670
For number three, we’ve got the same numerator again.
00:09:18.030 --> 00:09:23.220
And the larger denominator means we’re sharing that out between more people.
00:09:23.220 --> 00:09:25.060
That’s gonna be the smaller fraction.
00:09:25.060 --> 00:09:30.490
So the small end of the sign goes against that and the large end of the sign goes against the other fraction.
00:09:32.070 --> 00:09:34.900
For number four, we’ve got the same denominator.
00:09:35.130 --> 00:09:37.750
So all these little pieces are the same size.
00:09:37.900 --> 00:09:46.360
Now for the first fraction, we’ve got one of those fifths of a cake and in the second fraction we’ve got three of the fifths of a cake, so we’ve got more in the second fraction.
00:09:46.650 --> 00:09:49.490
So the big end of the sign goes against the bigger fraction.
00:09:49.700 --> 00:09:52.020
The small end of the sign goes against the smaller fraction.
00:09:53.110 --> 00:09:56.120
In number five, we’ve also got the same denominator.
00:09:56.370 --> 00:10:04.180
So again we’re looking for which is the biggest numerator; that’s gonna be the biggest fraction cause we’re gonna have more of those sevenths of cake.
00:10:04.470 --> 00:10:08.980
So it’s gonna go this way around: four- sevenths is more than only two-sevenths.
00:10:09.780 --> 00:10:16.290
And the last question here number six, we don’t have the same numerator and we don’t have the same denominator.
00:10:16.490 --> 00:10:19.270
So what we gotta remember is our equivalent fractions thing.
00:10:19.980 --> 00:10:24.680
So look — two — if I multiply two by two, I get four.
00:10:24.900 --> 00:10:28.960
And if I multiply three by two, I get six.
00:10:29.290 --> 00:10:38.170
So we’ve multiplied the numerator by two and the denominator by two, the same number in each case, so that means that they’re equivalent fractions.
00:10:38.200 --> 00:10:40.250
So in fact these are equal.