Learn the Meaning of Dividing Decimals - [17] - Free Educational videos for Students in K-12 | Lumos Learning

Learn the Meaning of Dividing Decimals - [17] - Free Educational videos for Students in k-12


Learn the Meaning of Dividing Decimals - [17] - By Math and Science



Transcript
00:00 Hello . Welcome back . The title of this lesson
00:02 is called dividing decimals using models . This is part
00:06 one . A better title or another title might be
00:08 understanding what decimal division really means using pictures . So
00:13 what we want to do in these lessons is really
00:14 understand what we're doing . When we divide one decimal
00:17 by another decimal . We all kind of understand what
00:20 regular division means and we'll review that in just a
00:23 second . But when we get to something like 1.6
00:26 divided by 1.4 or 3.2 divided by 2.7 we start
00:31 to lose a little bit of the meaning of what
00:33 it is . So for instance the problems that we're
00:35 going to be doing doing in this lesson , in
00:37 fact the first problem that we will do will be
00:39 something like this . 1.6 divided by 0.4 and we'll
00:44 talk about this exact problem as our first problem here
00:47 but we want to understand what does it mean to
00:49 take 16 of something and divide it by 0.4 .
00:53 Of course we will learn how to calculate the answer
00:56 , but we want to know what we're doing .
00:57 So before we really draw a picture of this ,
01:00 we want to go back and review regular division because
01:04 well we understand what regular division is doing . The
01:06 understanding the decimal division is very simple . Okay ,
01:09 so what we're going to do , we're gonna put
01:11 this on the back burner , we'll talk about this
01:13 in just a minute . So instead of talking about
01:15 that , I want to first talk about a much
01:19 simpler problem , the number six divided by two ,
01:23 six divided by two . Now I we're going to
01:26 use some pictures here in a second to understand this
01:28 , but I think you all probably know what the
01:29 answer to this is . Six divided by two is
01:32 three . How do we know this ? Because three
01:35 times two is six . Another way of writing ,
01:37 that would be something like we take six and we
01:39 divided by two . That's all we're doing here .
01:41 Six divided by 26 divided by two . The answer
01:44 is three Why ? Because two times three is six
01:47 , you should multiply these numbers together and get what's
01:50 underneath the division kind of symbol there . Because division
01:53 and multiplication are opposites of each other really in math
01:57 , we say that they're in verses of each other
01:58 , but whatever , we can use the word opposites
02:00 , they're basically opposites of each other . But even
02:03 though we know the answer to this , let's review
02:05 a little bit what it actually means . Okay ,
02:07 so we were saying that we have six objects ,
02:10 right ? This is uh the uh the number that
02:13 we're dividing , there is six objects , so we're
02:15 going to represent that by six objects . Now we
02:18 can think of this division in two different ways and
02:21 they both have their uses . So let's first talk
02:23 about the first way when we say six objects divided
02:27 by 21 way of thinking about it is we take
02:29 the six objects and we put it into two equal
02:32 piles . We divide the number of objects into that
02:35 many piles . And let's see what happens . We
02:37 have six objects . The only real way to make
02:40 two equal sized piles is to do something like this
02:45 . Here's the first pile in here is the second
02:47 pile , right ? And so when we do division
02:49 , what we're trying to do is make equal piles
02:51 of things . And so that's what we have done
02:53 . Notice what happens ? We have two piles ,
02:55 but in each of those piles we have three objects
02:57 . That's why we say six divided by two is
03:01 equal to three because six objects divided into two equal
03:05 piles means there are three objects in each pile .
03:09 So that's the first way of thinking about it .
03:11 And that's a good way of thinking about it .
03:12 I like that . But there's another way of thinking
03:14 about division that helps us a little bit more when
03:17 we're talking about decimal division . So what we have
03:20 is six objects , we're dividing by two . Instead
03:23 of thinking about dividing it into two piles , I
03:25 want you to think about this . Okay we have
03:29 six objects and we're going to divide it by two
03:32 . Here is two objects . Here is six objects
03:35 , six objects on the top , two objects on
03:37 the bottom . So if you want I can put
03:38 like a little division symbol right here because we have
03:41 six objects divided by two objects . So what we're
03:44 trying to do when we divide by two objects .
03:46 The other way of thinking about it is we want
03:48 to see how many times we can fit two objects
03:52 into our original amount . So what we're kind of
03:56 doing is we're saying , look , we have six
03:57 objects . We want to divide two objects and see
04:00 how many times the two objects can fit in to
04:04 the six objects . I'll say it one more time
04:07 . How many times can the two objects fit into
04:10 the six objects ? So we're dividing by two means
04:13 that we're dividing and seeing how many times these two
04:16 objects can fit in here . Well , here's one
04:18 time it can fit in here . Here's another time
04:21 , number two times it can fit in there .
04:23 And here is the third time it can fit in
04:24 there . So six divided by two objects can fit
04:28 three times . Because here I'm fitting at one time
04:31 here , I'm fitting at the second time here ,
04:33 I'm fitting at the third time so we can divide
04:35 in three times . Now . This little picture of
04:39 division is what I want you to have in your
04:41 mind when we do decimal division because it's a little
04:43 easier to think about what we're dividing is the first
04:46 number , what we're dividing by is we're just trying
04:49 to see how many times it fits in there .
04:51 In this case , if it's three times so the
04:53 answer is three . Now let's go and turn this
04:56 idea to talking about decimal division here , we have
05:01 the same problem . 1.6 divided by 0.4 . The
05:04 thing I want you to know is that decimal division
05:07 ? It really means the same thing as whole number
05:10 division . The idea is the same . All we
05:13 have to do is say , well we have 16
05:15 of something And we're dividing it by .4 . We
05:18 want to see how many times this .4 can fit
05:21 into this . Just like we over here , we're
05:24 trying to see how many times the two can fit
05:26 in one time , two times three times . So
05:29 it can go three times their six , divided by
05:32 two is three . All right . Um so the
05:36 first number , 1.6 , how do we represent that
05:39 ? Well , on the board we have this giant
05:43 square here represents one whole the one hole is we
05:47 have it subdivided into 10 slices . 123456789 10 .
05:53 But 10 out of 10 slices means we have one
05:55 hole . So really this thing right here is just
05:59 , I'll just say one hole . It means one
06:03 whole thing . Right ? So if this thing is
06:06 one whole thing and then this thing is of course
06:09 less than that . This is less than one whole
06:11 . What do we think this is going to be
06:12 ? This is going to be uh 10.6 because it's
06:15 1/10 . 2/10 3 10th , 4/10 5 , 10
06:19 6/10 . Notice that the width of this little rectangle
06:22 represents 1/10 . We have 10 of those 10th right
06:25 there . So that means the whole thing is right
06:26 here , that's the one . And then the 0.6
06:29 is this 1 10 to 10 ? Three tens .
06:31 Four tens , 5/10 16th . You see this is
06:33 less than a whole . So this represents one whole
06:38 . This part right , right over here represents 0.6
06:42 we put these guys together and we get 1.6 .
06:45 So this whole thing represents the number 1.6 . This
06:50 part represents one whole and this part represents a little
06:54 bit more than half because remember , 0.5 is a
06:57 half . So 0.6 would be a little bit more
06:59 than half . And so we have one whole plus
07:02 a little bit more than a half . Okay ,
07:05 now we're dividing it by this which is a little
07:08 smaller . 4/10 . So , you see we have
07:10 the one the to the three and the four here
07:13 . All right , So this is 4 10 .
07:15 So , again , if you were to extend this
07:17 , you would have a hole over here , but
07:18 we only have 4/10 . If you would extend this
07:21 , you would have a whole . We only have
07:22 6/10 . This actually is a whole . So it's
07:25 1 6 divided by less than half . 0.4 .
07:30 Now , in order to do the division , I
07:32 guess I'll right over here . This represents right here
07:34 , 0.4 . In order to do the division .
07:38 All we need to do is just like before six
07:44 divided by two . We're dividing by two . How
07:46 many times can to fit ? There's one , there's
07:48 two . There's three . The answer is three .
07:50 So , If this is the amount 1.6 and we're
07:55 dividing by 0.4 , how many times can this thing
07:59 , this amount of a sandwich or whatever can fit
08:02 inside of what we're dividing uh into . All right
08:07 . And so what we want to do is shave
08:09 that to get to the answer . So , you
08:11 see what we have here , we're dividing by 4/10
08:13 . So we have four little rectangles . So ,
08:16 let's shaded here . Here's one , here's four rectangles
08:19 . So what I'll do is I'll just kind of
08:21 like do a big shading like this . There's 1
08:26 , 2 , 3 , 4 . So here it
08:27 fits one time , let's switch colors here , it's
08:32 going to go to 34 to all the way to
08:34 their . So here is it fitting a second time
08:41 that second time goes to right there . Now to
08:44 get to the third time notice we have two slices
08:47 here and two more here , that would be the
08:49 four . So really we have to get to our
08:52 third time that it fits in here , we have
08:54 this and then we have two rectangles on the other
08:56 side , that's the third time that it can fit
08:59 into their and then the fourth time we have four
09:03 little rectangles on the end here is the fourth time
09:06 right here . So if we start with 1.6 and
09:11 we divided by 0.4 , how many times can this
09:13 chunk of stuff fit in there ? One time ,
09:17 Two times three times four times The answer is four
09:22 . The answer is four . It's really important for
09:25 you to watch this a few times so that you
09:28 understand what's happening . The idea of dividing a decimal
09:31 by another , decimal confuses . A lot of people
09:34 even confuses me sometimes , but you're doing the same
09:37 thing that we do when we divide whole numbers ,
09:40 the number you start with is what you start with
09:42 . When you divide by something , you're trying to
09:44 see how many times can this thing equally fit in
09:48 ? In this case it fit in three times .
09:50 We're doing the same thing here . We represent the
09:53 decimal as a large amount and we slice it up
09:56 so that we can figure out exactly what 1.6 or
09:59 something is . It's 10 out of 10 , that's
10:01 the one and then the 10.6 is six out of
10:03 10 right there . And we divide of course by
10:06 the four out of 10 right there , How many
10:07 times can this fit ? 123 We had to kind
10:11 of spread across there and then four . It goes
10:13 four times . All right . So that all of
10:17 that talking was problem number one . All right .
10:20 Let's move on to problem number two . Now we
10:22 have something less than one . We're dividing it by
10:27 0.3 . But the same concept applies 0.9 of something
10:30 . Just means you have less than a whole .
10:32 If this piece of paper represents the whole thing ,
10:36 0.9 is not quite the whole thing , but almost
10:39 all of it because of course you go 0.60 point
10:42 70.80 point nine . Then you roll over to one
10:46 0.5 of course is just half of something . So
10:49 when we say 0.9 of something , what we're really
10:52 saying is you have 1/10 . 2/10 . Because this
10:54 is the 10th place . Three tens , four tens
10:56 , 5 , 10 , 6 , 10 , 7
10:58 , 10 8/10 9 10th . The entire object would
11:02 be if you extend this rectangle out one more ,
11:05 then you would have one whole thing . But when
11:07 we , when we have almost the whole thing ,
11:09 we have 9/10 and we're dividing it by 3/10 how
11:13 many times can this fit into this ? All we
11:16 have to do is count . This is three little
11:18 rectangles . So we'll go to here . So there
11:20 is the first time it goes in , it goes
11:21 in one time , right there and then number two
11:26 again , three more rectangles . So there's number two
11:30 , Right ? And then the third one is of
11:33 course the final three . So we can just shade
11:35 that one , then this is number three right here
11:37 . So this amount of stuff can go in 123
11:40 It can fit three times and it can fit three
11:43 times . So this is how we use models ,
11:45 we have two more problems to do but I want
11:48 to point something out very important for you before we
11:51 move on to the next couple of problems . Remember
11:54 when we did multiplying decimals ? So here we're doing
11:56 dividing decimals . But remember when we did multiplying decimals
11:59 , we basically said you can ignore the decimal point
12:02 and you can just do the math . And then
12:04 at the end of the problem we ignore the decimal
12:06 completely . We just pretend it wasn't even there .
12:08 And then at the end of the problem we just
12:10 moved our decimal into place . So really the decimal
12:13 point when we multiply decimals doesn't really matter until you
12:17 get to the answer . All of the numbers are
12:19 the same . The decimal point just gets placed in
12:22 the answer . The same thing happens for division .
12:25 What if you ignore the decimal then you would have
12:27 16 . Right ? What if you ignore the decimal
12:30 and the zero ? You would have a four .
12:31 What is 16 divided by four ? Think about 16
12:34 divided by four has to be four . Right ?
12:36 Because four times four is 16 . Notice the answer
12:39 is four . All right . What about nine ,
12:42 ignore the decimal divided by three , ignore the decimal
12:45 so ignore him , ignore what is nine divided by
12:47 three ? Nine divided by three is three . That's
12:50 the answer . So they all worked out to be
12:53 the exact same answer because these problems are a little
12:55 simpler . Sometimes the decimal and we'll have to move
12:57 it in the right spot . But I'm trying to
13:00 let you know that when you multiply decimals or divide
13:02 decimals , you can kind of ignore the decimals and
13:06 then calculate the answer and the numbers will be the
13:09 same . Sometimes this the decimal point will move around
13:12 a little bit and when we get to the problems
13:14 , I'll show you how to do that for these
13:16 . The answer that we get is actually exactly the
13:18 same as if we just ignore the decimal . All
13:22 right , so let's move on just for giggles .
13:25 Let's move on to problem number three . And let's
13:29 figure out what would happen if we just , we
13:32 just kind of guess here . What about 1.2 ?
13:35 Let's ignore the decimal . 0.4 , ignore the decimal
13:38 . So what if you had 1.2 ? Which means
13:40 12 ? If you ignore the decimal , 12 divided
13:43 by four because you ignore the decimal and everything .
13:45 What would you get ? 12 divided by four is
13:47 three right ? Because three times four is 12 .
13:50 So we know that our answer has to have a
13:51 three in it , write the decimal might move around
13:54 a little bit , depending on on as we get
13:56 to the problems . I'll show you how to handle
13:57 that , but the number three will be in the
13:59 answer . What about this ? One ? 1.5 divided
14:01 by 0.3 . Take away the decimal ? You have
14:04 15 and take away this decimal in the zero you
14:06 have three . What's 15 divided by three ? 15
14:09 divided by three is five . Why five times three
14:13 is 15 ? So we know that a five has
14:15 to be in the answer somewhere . We will move
14:17 the decimal where it needs to go as we solve
14:19 the problem . So now that you wanted to kind
14:21 of mention that to you , let's use pictures to
14:23 figure this out . Here . We have 12 ,
14:26 which is this is one whole 123456789 20th means a
14:32 whole point to is the 2/10 right here and we're
14:36 dividing by 4/10 . 1234 out of 10 . So
14:40 all we need to do is figure out how many
14:41 times this fits in . So it's four little columns
14:44 here . Here's 1 , 2 , 3 , here's
14:47 four . So I'll try to do it like this
14:49 . Here is the first time that it fits in
14:53 there . There's number one . Next time will be
14:58 something like this , number two . So if it's
15:02 two times going all the way to their now for
15:06 the third time , let's see what happens over here
15:09 . We have two and two . So we can
15:10 say that if it's right here And right here ,
15:15 these kind of go together to make the 14th .
15:18 So if it's one time two times these together make
15:21 three times the answer is three . The answer is
15:24 three . What did we say ? The answer would
15:26 be if we ignore the decimals anyway , we get
15:28 12 divided by four is three . So I don't
15:30 want you to think that you can just ignore the
15:33 decimals and always get the answer . We will have
15:35 more complicated problems where the decimal point may have to
15:38 move around , but the number the three has to
15:41 be in the answer because the decimal point comes later
15:44 at the end of the problem . Yeah . All
15:46 right . Let's take a look at our last problem
15:48 . 1.5 divided by 0.3 . This rectangle here represents
15:52 one whole , it's 10 slices out of 10 and
15:55 then this is another half 100.5 is a half ,
15:57 12345 10 . So we have one whole plus another
16:01 half of a sandwich And we're dividing by 3/10 which
16:04 is 0.3 . How many times can this fit into
16:07 here ? Well , let's just shade it . We
16:09 have three rectangles right here fits one time . Right
16:12 here , there's one time we have three more rectangles
16:17 right here , that will be number two . Okay
16:21 , there's number two . Number three would be three
16:24 more rectangles right here . Okay , there's three .
16:30 So that's that's three times number four , we're gonna
16:33 take one from here , one little rectangle from here
16:37 and two from over here . So that's going to
16:38 be another one . That means if it's four times
16:42 and then finally , this will be the fifth time
16:45 for the last three . So the this amount of
16:48 a sandwich or whatever fits 12345 goes five times five
16:56 times . So I've been trying to draw it to
16:59 something that you already understand which is the division of
17:03 whole numbers , We take six divided by two .
17:05 It means we take six objects and we try to
17:07 see how many times the two will fit in .
17:09 There fits once fits twice . It's three times .
17:13 Same thing is true with decimals 1.6 divided by 0.4
17:18 . It fits 1234 times 12340.9 divided by three .
17:23 This fits 123 times 1.2 divided by 0.4 . It
17:30 fits 123 times 1.5 divided by 0.3 . It fits
17:34 12345 times . So division of decimals fundamentally is the
17:42 same thing as regular division that we've done . It's
17:45 just a little harder for us to picture because we're
17:46 not so good at picturing what decimals look like ,
17:49 but we're drawing these pictures to give us a little
17:51 practice with that . So I'd like you to At
17:54 least sketch these out yourself and see if you can
17:56 understand why these work and then follow me onto part
18:00 two . We'll get a little more practice with understanding
18:03 decimal division using pictures and models .
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