The Distributive Property In Arithmetic - Free Educational videos for Students in K-12

The Distributive Property In Arithmetic - By Mathantics

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00:03	Uh huh . Hi , I'm rob . Welcome to
00:07	Math antics . In this video , we're going to
00:09	talk about a really important math concept called the distributive
00:12	property . Well at least that's what it's called .
00:15	Sometimes you may hear referred to as the distributive law
00:19	or even the distributive property of multiplication over addition by
00:23	people who want to sound really technical , but no
00:26	matter what it's called , the concept of the distributive
00:28	property is the same . Before we actually dive into
00:32	that concept . There are two quick things that will
00:34	help make it easier to understand . The first is
00:36	simply knowing what the word distribute means to distribute .
00:40	Something means to give it to each member of a
00:42	group , like an old fashioned paper boy delivering newspapers
00:46	to each house in a neighborhood . Mhm . One
00:50	for you , one for you , I'm sorry ,
00:54	one for you And one for you . The second
01:00	thing you need to know is the order of operations
01:02	rules of arithmetic , which we cover in another video
01:05	. So you might want to watch that if you're
01:07	not familiar with those rules already . That's because the
01:10	distributive property is actually a way of allowing us to
01:13	change the order of operations . We do in certain
01:16	types of problems to see what I mean . Have
01:18	a look at this simple arithmetic expression three times the
01:22	group . Four plus six . We're going to simplify
01:25	this expression in two different ways . The first way
01:28	we'll just use the basic order of operations rules that
01:31	you already know . But the second way we use
01:33	the distributive property and if we do the arithmetic right
01:36	, both ways will give us the same answer .
01:39	So for the first way , our order of operations
01:41	rules tell us that we need to do any operations
01:44	in sight of groups first . And since these parentheses
01:47	form a group , we first need to add the
01:50	four and six , which gives us 10 . Next
01:53	we can multiply that by three , which gives us
01:55	a final answer of 30 . Now , let's use
01:58	the distributive property . The distributive property allows us to
02:02	change this expression into a different form instead of multiplying
02:05	three by the whole group at once , we can
02:08	distribute that factor of three and multiply it by each
02:11	member of the group individually . That means we'll make
02:13	a copy of the three times for each member of
02:16	the group , the four and 6 . So after
02:19	applying the distributive property , our expression looks like this
02:23	three times four plus three times six . Because we
02:26	distributed the multiplication to each member of the group ,
02:30	the group isn't needed anymore . So the parentheses can
02:32	go away . Now we can continue to simplify this
02:35	new form using our order of operations rules , those
02:39	rules tell us to do multiplication before addition . So
02:42	three times four is 12 and three times six is
02:45	18 . The last step is to add those two
02:48	results together 12-plus 18 equals 30 . Well , look
02:52	at that . We got the same answer in both
02:54	cases , which means our original expressions are equivalent ,
02:57	even though they have different forms . In the first
03:01	form , the factor three is being multiplied by the
03:03	entire group all at once . So we needed to
03:06	do the addition inside that group 1st . But in
03:09	the second form we use the distributive property to rearrange
03:12	the expression so that the factor of three is multiplied
03:15	by each member of the group individually , instead of
03:18	the whole group , all at once , distributing that
03:21	factor made the group go away . So we didn't
03:23	have to do the addition inside that group first .
03:26	So the distributive property is basically a way of getting
03:29	rid of a group that is being multiplied by a
03:31	factor . If you distribute the factor to each member
03:35	of the group , you'll get the same answer .
03:37	You would If you calculate what's in the group first
03:39	and then multiply and it works no matter how many
03:42	members are in the group . Like in this problem
03:45	, we have to multiply four by the group ,
03:47	one plus two plus three again , let's try simplifying
03:51	this both ways . In the first way we start
03:53	by simplifying what's in the group ? One plus two
03:56	plus three equals six . And then we multiply four
03:59	times six , which gives us 24 . Now let's
04:03	use the distributive property . We distribute a factor of
04:06	four to each member of the group , which makes
04:08	the group go away and allows us to do those
04:11	multiplication is individually four times one is 44 times two
04:15	is eight and four times three is 12 . Finally
04:18	, we add up those three individual answers . Four
04:21	plus eight is 12 and 12 plus 12 is 24
04:25	. See , the distributive property gave us another way
04:28	to arrive at the same answer . It's like the
04:30	distributive property is an alternate path that you can take
04:33	to arrive at the same point . Mhm . Yeah
04:42	. Well what are you doing here ? I'm always
04:45	here . Okay , great . We have two ways
04:48	to get to the same answer , but why do
04:50	we need two different ways to do the same calculation
04:54	? And it seemed like the distributive property way was
04:57	even more complicated than the regular way . Why would
05:00	we ever want to use it ? That's a good
05:02	question . And it's true . There are times when
05:05	the distributive property way is harder . Like in our
05:07	first problem , it was easier to just go ahead
05:10	and simplify the group first because it's easy to multiply
05:14	three times 10 mentally . But there are also times
05:17	when the distributive property way is easier , like in
05:20	this case eight times the group 50-plus 3 . If
05:24	we decide to simplify the group first in this problem
05:27	, we end up needing to multiply eight times 53
05:30	, which is not so easy to do mentally .
05:33	But if we apply the distributive property instead , we
05:35	can change the expression into eight times 50 plus eight
05:38	times three . And that's easier to do mentally .
05:41	Eight times 50 is 408 times three is 24 ,
05:45	so the answer is 424 , realizing that the distributive
05:50	property can make some calculations easier to do mentally .
05:53	Can come in really handy for certain basic multi digit
05:56	multiplication problems . That's because you can break up the
05:59	multi digit factor into a group , some , you
06:02	know like expanded form and then distribute the other factor
06:06	to the members of that group . Sound confusing ?
06:09	Here's what I mean . Let's say you need to
06:11	multiply five times 47 . Well you could just use
06:14	the multi-digit multiplication procedure . Or you could change this
06:18	into a problem where the distributive property will make it
06:21	a little easier to do . The key is to
06:23	realize that you can replace the 47 with 40-plus 7
06:28	. Then the problem becomes five times the group 40-plus
06:31	7 . And the distributive property lets us change that
06:34	into five times 40 plus five times seven . Those
06:38	two multiplication czar easy to do Five times 40 is
06:41	205 times seven is 35 . So our answer is
06:46	200 plus 35 or 235 . Want to see another
06:50	example . Let's apply that same idea to this multiplication
06:54	problem three times 127 . But instead of 127 ,
06:59	let's change that into the group 100 plus 20-plus 7
07:04	. We need to multiply that by three . And
07:06	the distributive property lets us distribute that multiplication to each
07:10	member of the group three times 100 plus three times
07:14	20 plus three times seven . That helps because we
07:17	can do those mentally three times 100 is 303 times
07:22	20 is 60 and three times seven is 21 .
07:25	All that's left to do is add those three products
07:28	up , which is not too hard to do mentally
07:30	, either . 300 plus 60 plus 21 gives us
07:34	381 as our final answer . Now , before we
07:38	wrap up , there is one more important thing that
07:40	you should know about the distributive property . You already
07:43	know that the distributive property works when the members of
07:46	a group are being added . But it works the
07:48	same way for members of a group that are being
07:51	subtracted . Like in this problem , seven times the
07:54	group 10 -4 . You could do this problem the
07:58	typical way and simplify the group . 1st 10 -4
08:01	is six , and then seven times six gives us
08:03	42 or you could use the distributive property . You
08:07	distribute the seven times to both members of the group
08:11	to get seven times 10 minus seven times four ,
08:15	Seven times 10 equals 70 and seven times 4 is
08:18	28 and 70 -28 equals 42 . Again , both
08:24	ways are equivalent . So the distributive property works for
08:28	groups of any size and it works the same for
08:30	group members that are being added or subtracted even if
08:34	there's a mixture of addition and subtraction in the group
08:37	. But the distributive property doesn't work when the members
08:40	of a group are being multiplied or divided . For
08:42	example , if you have five times the group ,
08:45	two times three , you can't distribute a copy of
08:48	the factor five to each member of the group without
08:51	getting a completely different answer . And the same goes
08:54	for division . If the members of a group are
08:56	being divided like four times the group , six divided
08:59	by two , you will not get the right answer
09:02	. If you distribute the factor four to each member
09:05	, that's why the technical name is the distributive property
09:08	of multiplication over addition . You're distributing the multiplication over
09:13	all of the members of a group that are being
09:15	added . And the reason that it also works for
09:18	subtraction is that subtraction is really just a negative form
09:21	of addition , since subtraction and addition are inverse operations
09:25	. All right , So the distributive property is a
09:28	handy way to rearrange arithmetic expressions . It's like a
09:32	tool that you can use in certain situations if you
09:35	think it will make a particular calculation easier to do
09:38	. And even if you don't end up using the
09:40	distributive property a whole lot for arithmetic problems , it's
09:44	still a really important math concept that will be even
09:47	more useful when you get to algebra until then be
09:50	sure to practice what you've learned in this video by
09:52	trying some of the exercise problems . Practice is the
09:55	best way to make sure that you really understand .
09:58	As always . Thanks for watching Math Antics and I'll
10:01	see you next time learn more at Math Antics dot
10:05	com .

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