00:06 hi i'm rob welcome to math antics in
00:09 this video we're going to talk about a
00:10 really important math concept called the
00:12 distributive property
00:14 well at least that's what it's called
00:15 sometimes you may hear it referred to as
00:18 the distributive law or even the
00:20 distributive property of multiplication
00:22 over addition by people who want to
00:24 sound really technical but no matter
00:26 what it's called the concept of the
00:28 distributive property is the same
00:31 before we actually dive into that
00:32 concept there are two quick things that
00:34 will help make it easier to understand
00:36 the first is simply knowing what the
00:38 word distribute means
00:39 to distribute something means to give it
00:42 to each member of a group like an
00:44 old-fashioned paperboy delivering
00:45 newspapers to each house in a
01:00 the second thing you need to know is the
01:02 order of operations rules of arithmetic
01:04 which we cover in another video so you
01:06 might want to watch that if you're not
01:07 familiar with those rules already
01:10 that's because the distributive property
01:12 is actually a way of allowing us to
01:14 change the order of operations we do in
01:16 certain types of problems to see what i
01:18 mean have a look at this simple
01:19 arithmetic expression three times the
01:22 group four plus six
01:24 we're going to simplify this expression
01:26 in two different ways the first way
01:28 would just use the basic order of
01:30 operations rules that you already know
01:32 but the second way we use the
01:34 distributive property and if we do the
01:36 arithmetic right both ways will give us
01:39 so for the first way our order of
01:41 operations rules tell us that we need to
01:43 do any operations inside of groups first
01:46 and since these parentheses form a group
01:49 we first need to add the 4 and 6 which
01:53 next we can multiply that by 3 which
01:55 gives us the final answer of 30.
01:58 now let's use the distributive property
02:00 the distributive property allows us to
02:02 change this expression into a different
02:05 instead of multiplying 3 by the whole
02:07 group at once we can distribute that
02:09 factor of 3 and multiply it by each
02:11 member of the group individually that
02:13 means we'll make a copy of the 3 times
02:16 for each member of the group the 4 and
02:19 so after applying the distributive
02:21 property our expression looks like this
02:23 three times four plus three times six
02:26 because we distributed the
02:28 multiplication to each member of the
02:29 group the group isn't needed anymore so
02:32 the parentheses can go away
02:34 now we can continue to simplify this new
02:36 form using our order of operations rules
02:39 those rules tell us to do multiplication
02:41 before addition so 3 times 4 is 12 and 3
02:45 times 6 is 18. the last step is to add
02:48 those two results together 12 plus 18
02:51 equals 30. well look at that we got the
02:53 same answer in both cases which means
02:56 our original expressions are equivalent
02:58 even though they have different forms
03:00 in the first form the factor 3 is being
03:03 multiplied by the entire group all at
03:06 so we needed to do the addition inside
03:09 but in the second form we use the
03:11 distributive property to rearrange the
03:13 expression so that the factor of 3 is
03:15 multiplied by each member of the group
03:17 individually instead of the whole group
03:20 distributing that factor made the group
03:23 go away so we didn't have to do the
03:24 addition inside that group first
03:27 so the distributive property is
03:28 basically a way of getting rid of a
03:30 group that is being multiplied by a
03:33 if you distribute the factor to each
03:35 member of the group you'll get the same
03:37 answer you would if you calculate what's
03:38 in the group first and then multiply
03:41 and it works no matter how many members
03:43 are in the group like in this problem we
03:45 have to multiply four by the group one
03:48 plus two plus three again let's try
03:50 simplifying this both ways
03:52 in the first way we start by simplifying
03:54 what's in the group one plus two plus
03:57 three equals six and then we multiply 4
04:00 times 6 which gives us 24.
04:03 now let's use the distributive property
04:05 we distribute a factor of 4 to each
04:07 member of the group which makes the
04:09 group go away and allows us to do those
04:13 four times one is four four times two is
04:16 eight and four times three is twelve
04:18 finally we add up those three individual
04:20 answers four plus eight is twelve and
04:23 twelve plus twelve is twenty four
04:25 see the distributive property gave us
04:27 another way to arrive at the same answer
04:30 it's like the distributive property is
04:31 an alternate path that you can take to
04:33 arrive at the same point
04:42 oh but what are you doing here i'm
04:46 okay great we have two ways to get to
04:49 the same answer but why do we need two
04:51 different ways to do the same
04:54 and it seemed like the distributive
04:56 property way was even more complicated
04:58 than the regular way why would we ever
05:02 that's a good question and it's true
05:04 there are times when the distributive
05:06 property way is harder like in our first
05:08 problem it was easier to just go ahead
05:10 and simplify the group first because
05:12 it's easy to multiply 3 times 10
05:16 but there are also times when the
05:18 distributive property way is easier like
05:20 in this case 8 times the group 50 plus
05:25 if we decide to simplify the group first
05:27 in this problem we end up needing to
05:28 multiply 8 times 53 which is not so easy
05:32 to do mentally but if we apply the
05:34 distributive property instead we can
05:36 change the expression into eight times
05:38 fifty plus eight times three and that's
05:40 easier to do mentally
05:42 eight times fifty is four hundred and
05:44 eight times three is twenty-four so the
05:46 answer is four hundred twenty-four
05:49 realizing that the distributive property
05:51 can make some calculations easier to do
05:53 mentally can come in really handy for
05:55 certain basic multi-digit multiplication
05:58 that's because you can break up the
06:00 multi-digit factor into a group sum you
06:02 know like expanded form
06:04 and then distribute the other factor to
06:06 the members of that group sound
06:08 confusing here's what i mean
06:11 let's say you need to multiply 5 times
06:13 47 well you could just use the
06:15 multi-digit multiplication procedure or
06:17 you could change this into a problem
06:19 where the distributive property will
06:21 make it a little easier to do
06:23 the key is to realize that you can
06:24 replace the 47 with 40 plus 7. then the
06:28 problem becomes 5 times the group 40
06:31 plus 7 and the distributive property
06:34 lets us change that into five times
06:36 forty plus five times seven
06:38 those two multiplications are easy to do
06:41 five times forty is two hundred and five
06:43 times seven is thirty five so our answer
06:46 is two hundred plus 35 or 235
06:50 want to see another example let's apply
06:52 that same idea to this multiplication
06:55 three times 127 but instead of 127 let's
07:00 change that into the group 100 plus 20
07:04 we need to multiply that by 3 and the
07:07 distributive property lets us distribute
07:09 that multiplication to each member of
07:12 3 times 100 plus 3 times 20 plus three
07:17 that helps because we can do those
07:19 mentally three times one hundred is
07:21 three hundred three times twenty is
07:23 sixty and three times seven is twenty
07:26 all that's left to do is add those three
07:28 products up which is not too hard to do
07:30 mentally either 300 plus 60 plus 21
07:34 gives us 381 as our final answer
07:38 now before we wrap up there's one more
07:40 important thing that you should know
07:41 about the distributive property
07:43 you already know that the distributive
07:44 property works when the members of a
07:46 group are being added
07:48 but it works the same way for members of
07:50 a group that are being subtracted like
07:53 seven times the group ten minus four
07:57 you could do this problem the typical
07:58 way and simplify the group first 10
08:01 minus 4 is 6 and then 7 times 6 gives us
08:05 or you could use the distributive
08:07 property you distribute the seven times
08:10 to both members of the group to get
08:12 seven times ten minus seven times four
08:15 seven times ten equals seventy and seven
08:18 times four is twenty-eight and seventy
08:21 minus twenty-eight equals forty-two
08:24 again both ways are equivalent
08:26 so the distributive property works for
08:28 groups of any size and it works the same
08:30 for group members that are being added
08:32 or subtracted even if there's a mixture
08:35 of addition and subtraction in the group
08:37 but the distributive property doesn't
08:39 work when the members of a group are
08:41 being multiplied or divided for example
08:44 if you have five times the group two
08:46 times three you can't distribute a copy
08:48 of the factor five to each member of the
08:50 group without getting a completely
08:52 different answer and the same goes for
08:55 division if the members of a group are
08:56 being divided like 4 times the group 6
08:59 divided by 2 you will not get the right
09:02 answer if you distribute the factor 4 to
09:05 that's why the technical name is the
09:07 distributive property of multiplication
09:11 you're distributing the multiplication
09:13 over all of the members of a group that
09:15 are being added and the reason that it
09:17 also works for subtraction is that
09:19 subtraction is really just a negative
09:21 form of addition since subtraction and
09:23 addition are inverse operations
09:26 all right so the distributive property
09:28 is a handy way to rearrange arithmetic
09:32 it's like a tool that you can use in
09:34 certain situations if you think it will
09:36 make a particular calculation easier to
09:39 and even if you don't end up using the
09:41 distributive property a whole lot for
09:43 arithmetic problems it's still a really
09:45 important math concept that will be even
09:47 more useful when you get to algebra
09:50 until then be sure to practice what
09:51 you've learned in this video by trying
09:53 some of the exercise problems practice
09:55 is the best way to make sure that you
09:59 as always thanks for watching math
10:00 antics and i'll see you next time
10:03 learn more at mathantics.com
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