Math Antics - Mixed Numbers - Free Educational videos for Students in K-12

Math Antics - Mixed Numbers - By Mathantics

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00:03	Uh huh . Hi . I'm rob . Welcome to
00:07	Math Antics in this lesson , we're going to learn
00:09	about mixed numbers and how they relate to improper fractions
00:13	. Mixed numbers . I love mixed numbers . So
00:17	full of protein and vitamins . You want some uh
00:20	those are mixed nuts and no thanks . I'm trying
00:23	to explain mixed numbers . Oh mixed numbers . You
00:26	mean like when I get a whole bunch of numbers
00:29	and put them in a bag and shake it up
00:31	, pull out one to see who wins . You
00:33	want to play ? That's a raffle and that's not
00:36	what mathematicians mean when they say mixed numbers . Well
00:40	you sure are hard to please today . As I
00:43	was saying , we're going to learn how mixed numbers
00:45	relate to improper fractions . But first we need to
00:48	make sure that you understand when an improper fraction really
00:51	is to do that . Let's start with a short
00:54	number line that counts whole numbers from 0-3 . And
00:57	let's subdivide the spaces in between each whole number into
01:01	four equal parts or fourths . So a block .
01:04	This size represents the amount one because it covers the
01:07	distance from 0-1 . Well a block this size represents
01:11	the fractional amount 1/4 because it only covers the distance
01:15	from zero to the first subdivision that we made ,
01:18	which is one out of four or 1/4 . But
01:21	suppose we add another fourth to the fourth . We
01:23	already have that would give us 2/4 and adding another
01:27	fourth would give us 3/4 and adding another fourth would
01:31	give us 4/4 . What's 4/4 , yep . It's
01:35	what I like to call a whole fraction because its
01:38	value equals one whole . Oh and by the way
01:42	we're going to be using the term whole fraction a
01:44	lot in this video . It's not an official math
01:47	terms so your teacher might not use it . But
01:49	just remember that whenever I say whole fraction . I'm
01:53	talking about any fraction that has the exact same number
01:56	on the top and on the bottom to over 28
02:00	over 800 . Over 100 . Those are all what
02:03	I call whole fractions . Since all whole fractions equal
02:06	one . You can replace any whole fraction with one
02:10	and you can replace one with any whole fraction And
02:14	you can see how that works on our number line
02:17	. The 4/4 all combined to cover the same distance
02:20	as one . The fraction for over four is equivalent
02:24	to one and Vice Versa . But now what if
02:27	we get a little crazy and add one more fourth
02:30	now we have 5/4 and we've gone past one on
02:33	the number line 5/4 or 5/4 is what we call
02:37	an improper fraction Because the numerator , the top number
02:41	is greater than the denominator . The bottom number ,
02:44	that means it's value is greater than one in this
02:48	particular case . How much greater than one is it
02:51	? Well on the number line we've gone past one
02:53	by the fractional amount . 1/4 . So 5/4 turns
02:57	out to be equivalent to one and 1/4 . Have
03:01	a good look at this diagram for a minute because
03:03	it shows us something really important about how proper fractions
03:07	, improper fractions , whole fractions and mixed numbers all
03:10	relate to each other . First on this side of
03:13	the number line we have 5/4 which we know is
03:16	an improper fraction , but notice how we got it
03:19	. We added a proper fraction to a whole fraction
03:22	. When we just had 4/4 we had a whole
03:25	fraction , but when we added one more fourth it
03:28	became the improper fraction . 5/4 . So one way
03:31	to think of an improper fraction is that it's a
03:34	combination of one or more whole fractions and a proper
03:37	fraction . That's helpful because we know that all whole
03:41	fractions can be simplified to the whole number one .
03:44	And if we did that , we'd get what's shown
03:46	on the other side of the number line , we'd
03:48	get the combination of the whole number one and the
03:51	proper fraction 1/4 . In other words , we would
03:54	get a mixed number . That's all the mixed number
03:57	is it's the sum of a whole number and a
04:00	proper fraction . It's an alternate way to write an
04:03	improper fraction where all the whole fractions that are inside
04:06	the improper fraction have been simplified out into a whole
04:10	number to help that sink in . Let's look at
04:13	another improper fraction . 8/3 or eight thirds . We'll
04:18	use the same number line that goes from 0-3 .
04:20	But this time let's subdivide each whole number into three
04:24	parts . To make county thirds easier . So this
04:27	would be one third . This is two thirds and
04:29	this is three thirds . Oh there's a whole fraction
04:32	already . Let's make note of that . Well we
04:34	continue adding thirds . Next we have four thirds ,
04:37	five thirds . Six thirds . Oh we just formed
04:40	another group of three thirds . That would equal a
04:43	whole fraction also , Let's note that . And continue
04:46	7/3.8 . As we noted , eight thirds actually contained
04:52	two whole fractions . Each of those whole fractions simplifies
04:55	to one . So that means are mixed number form
04:58	would have one plus one or two as the whole
05:01	number part And then the fraction that's left over .
05:04	After simplifying all the whole fractions is 2/3 . So
05:08	the improper fraction eight thirds is equivalent to the mixed
05:12	number two and two thirds . Pretty cool . Huh
05:16	? And to show you that you can go back
05:17	and forth between these two forms . Check this out
05:20	in the mixed number two and two thirds . The
05:23	whole number two and the fraction two thirds are being
05:26	added together . That's really important to know . Even
05:29	though the plus sign isn't usually shown . So two
05:32	and two thirds is really two plus two thirds .
05:36	And if we wanted to we could expand the two
05:39	into one plus one . Right ? That gives us
05:41	one plus one plus two thirds . And then we
05:45	could replace each of those ones with the whole fraction
05:48	3/3 . Since 3/3 equals one . That gives us
05:53	3/3 plus 3/3 plus 2/3 . Since these fractions all
05:58	have the same denominator , we can add them easily
06:01	. The denominator of the answer will stay the same
06:03	three and the numerator will be the some of the
06:06	other enumerators , three plus three plus two equals eight
06:10	. And there we are back to our original improper
06:13	fraction . 8/3 . You can convert any mixed number
06:17	into an improper fraction . Using that procedure you can
06:21	change the whole number part into a sum of whole
06:23	fractions and then add everything up . For example ,
06:27	two and 1/8 could be changed into 8/8 plus 8/8
06:31	plus 1/8 . Which all add up to 17/8 and
06:36	three and 4/5 could be changed to 5/5 plus 5/5
06:41	plus 5/5 plus 4/5 . Which all add up to
06:45	19/5 and four and two thirds can be changed to
06:49	3/3 plus 3/3 plus 3/3 plus 3/3 plus 2/3 ,
06:55	which all adds up to 14/3 . Notice that we
07:00	always chose whole fractions with the same denominator of the
07:03	fraction part of the mixed number so that they're all
07:05	like fractions that can be added easily . And some
07:09	of you might see the shortcut here in each case
07:11	. Did you notice how many of the whole fractions
07:14	we needed to add together , yep , it's the
07:16	same as the whole number part of the mixed number
07:20	. If it's too then we need to add to
07:22	whole fractions . If it's three we need to add
07:25	three whole fractions . If it's for we need to
07:28	add four whole fractions and so on . The whole
07:31	number tells us how many times to repeat the addition
07:35	. And since multiplication is repeated addition , we can
07:38	multiply , instead of adding two times 8/8 , gives
07:41	us 16/8 and then we add that result to 1/8
07:45	to get 17/8 . Three times 5/5 gives us 15/5
07:51	. And then we had that result to 4/5 to
07:53	get 19/5 . Four times 3/3 gives us 12/3 .
07:58	And then we had that result to to over three
08:01	to get 14/3 . That shortcut really helps when the
08:05	whole number part of the mixed number is big .
08:08	Like what if we needed to convert the mixed number
08:10	15 and 1/4 into an improper fraction . Instead of
08:14	having to add 15 whole fractions together , we can
08:17	just multiply 15 by the whole fraction 4/4 . Which
08:21	gives us 60/4 . Then we had that result to
08:24	the fraction 1/4 to get 61/4 . That's the improper
08:29	fraction form of 15 and 1/4 . All right then
08:33	. But what if we need to go the other
08:35	way ? What if we start with an improper fraction
08:37	and need to convert it into a mixed number ?
08:40	Well , whenever we have an improper fraction , we
08:43	know there's at least one whole fraction hiding in there
08:45	that we could simplify out . The question is how
08:48	many to see what I mean ? Let's try converting
08:51	the improper fraction 7-2 into a mixed number . Using
08:55	a little trial and error . First let's try subtracting
08:58	out just one whole fraction . 7/2 minus 2/2 equals
09:03	5/2 . That means we can write 7/2 as the
09:06	mixed number one and 5/2 . Since we subtracted out
09:10	one whole fraction and had 5/2 left over . And
09:13	even though that's true , it's bad form because 5/2
09:16	is still an improper fraction , which means that there's
09:19	at least one more hole fraction hiding in there that
09:22	we could have subtracted out . So let's try again
09:24	. But this time let's subtract out to whole fractions
09:28	. 7/2 minus 2/2 minus 2/2 equals 3/2 . That
09:33	means we could write 7/2 as the mixed number two
09:36	and 3/2 since we subtracted out to whole fractions and
09:40	had three over to left over . But that's still
09:43	bad form because the fraction part is still improper .
09:46	We could have subtracted out another whole fraction . So
09:49	let's try again subtracting three whole fractions . This time
09:53	7/2 minus 2/2 minus 2/2 minus 2/2 equals one half
09:58	. That means we could write 7/2 as the mixed
10:01	number three and one half since we subtracted out three
10:04	whole fractions and had one half left over and that's
10:07	the proper mix number form of 7/2 because it's a
10:10	whole number and a proper fraction . So there's no
10:13	more whole fractions that we could simplify out that process
10:16	makes sense . But it's kind of messy having to
10:18	subtract out so many whole fractions . It turns out
10:22	there's a shortcut we can take here to just like
10:24	multiplication is repeated addition , division is basically repeated subtraction
10:30	. That means we can figure out how many whole
10:32	fractions we can subtract out of an improper fraction by
10:35	just dividing the top number by the bottom number .
10:39	Let's do that with our example , 7/2 . If
10:42	we divide seven by two , we find out that
10:45	too will divide into 73 times . Leaving a remainder
10:48	of one . That remainder is actually important . As
10:51	we'll see in a minute notice that the answer to
10:54	our division problem is exactly how many whole fractions we
10:58	were able to subtract out of the improper fraction ,
11:00	three . So the answer to the division tells us
11:04	what the whole number part of the mixed number will
11:06	be . And here's the really cool part . The
11:08	remainder of the division tells us what the leftover fraction
11:11	will be . The remainder is the numerator , the
11:14	top number of the leftover fraction in this case ,
11:17	since the remainder is one will have one over to
11:21	left over in our mixed number . Let's do one
11:24	more example to make sure you've got that . Let's
11:26	convert 22/5 into a mixed number . If we divide
11:30	22 by five , we see that five will go
11:32	into 22 4 times with a remainder of two .
11:36	That means that the whole number part of the mixed
11:38	number will be four and the fraction part will be
11:41	to older five because the remainder was too . That's
11:44	how many 5th will be left over . So 22/5
11:48	is the same as four and 2/5 . All right
11:52	. So now you know what mixed numbers are there
11:54	a combination of a whole number and a proper fraction
11:57	? And you know that those two parts are actually
11:59	being added together , even though the plus sign is
12:02	usually not shown . You also know that a mixed
12:05	number is basically a simplified form of an improper fraction
12:09	and that you can use the procedures we learned to
12:11	convert back and forth between the two forms . But
12:14	the way to make sure that you really understand mixed
12:16	numbers improper fractions and how to convert between them is
12:20	to practice . You need to do some exercises on
12:22	your own and check your answers to make sure you're
12:24	doing the procedures right ? Oh , I'm doing it
12:27	right . In fact , I found an even quicker
12:29	shortcut . Wow . How is that for mixed numbers
12:38	? As always . Thanks for watching Math Antics and
12:40	I'll see you next time . Learn more at Math
12:44	Antics dot com .

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