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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