Understand Fractions & Their Meaning - Multiplying Fractions & Adding Fractions - [31] - By Math and Science
Transcript
00:00 | Hello . Welcome back . The title here is called | |
00:02 | fractions as multiples . This is part one . The | |
00:05 | driving purpose of this section is for you to understand | |
00:08 | that the fractions that we have been working with , | |
00:10 | all of the fractions that we've been working with , | |
00:12 | you can think of them as you can break apart | |
00:15 | the fractions and express them as multiples of a of | |
00:18 | a smaller fraction , and that's a little hard to | |
00:21 | understand by words , but it'll be really simple once | |
00:23 | they get it on the board . Actually , I've | |
00:25 | been emphasizing this whole concept from the very first fraction | |
00:29 | lesson we've done . So when you look at this | |
00:31 | , you might think you already know that . Well | |
00:32 | , that's because I've been emphasizing it from the beginning | |
00:35 | to try to make it easier to understand . Let's | |
00:37 | just jump right in . Let's take the fraction 2/8 | |
00:41 | . I've been telling you since we started that what | |
00:44 | this means is two pieces out of eight of a | |
00:46 | pizza . Cut a pizza into eight slices . That | |
00:50 | means every slice would be 1/8 but you don't have | |
00:52 | 1/8 you have 2/8 . So I've been teaching you | |
00:56 | to think of these slices of the pizza like this | |
00:58 | is a slice of the pizza . To think of | |
01:00 | these slices as things that you count , you count | |
01:03 | them as you go around . This is 1/8 . | |
01:05 | If you have two of them it's to eight and | |
01:07 | then 3/8 and then 48 So you've been counting uh | |
01:10 | these slices of the pizza , counting in eighth the | |
01:12 | whole time . So what I want to do to | |
01:14 | to illustrate the point here is I want us to | |
01:17 | write this fraction two eights . I want us to | |
01:19 | write it two different ways . The first way I | |
01:21 | want to write it as a multiplication equation or something | |
01:25 | involving multiplication . And then the second way we're going | |
01:28 | to write it in terms of addition . So we're | |
01:30 | gonna write two different ways . We're going to write | |
01:32 | an equation with multiplication and an equation with addition . | |
01:36 | It's going to be very very simple . Let me | |
01:38 | ask you to 8th . It's going to be easier | |
01:41 | for us to get to the punch line and then | |
01:43 | go from there for this one for our multiplication equation | |
01:46 | we can say that 2/8 is equal to what ? | |
01:50 | 1/8 but times two , that's what 1/8 is times | |
01:56 | two you get to eight . So you can kind | |
01:58 | of like rip this apart and the two can go | |
02:01 | over as a whole number and the 1/8 can come | |
02:03 | over here because remember there's like an invisible being this | |
02:07 | over one . I can write this as to over | |
02:09 | one times 18 and when I multiply one times two | |
02:13 | is 21 times eight is eight . So it makes | |
02:16 | perfect sense that I can say that this fraction is | |
02:19 | equal to the fraction 18 times two . That's why | |
02:22 | it's called multiples . I want you to think of | |
02:25 | 28 and 3/8 and 4/8 is just like multiplying 1/8 | |
02:30 | times two or multiplying diffraction 18 times three or whatever | |
02:34 | this is called a multiple . And so I can | |
02:37 | show you that with a uh with a fraction or | |
02:40 | with a with a model we can say this is | |
02:43 | the fraction two eights . That's what this is . | |
02:45 | This is the fraction to AIDS . But I can | |
02:46 | write that as the fraction 1/8 times to write . | |
02:52 | And that's another way of of of writing that 18 | |
02:55 | the fraction 18 times . To the other way we | |
02:57 | want to write it is in terms of addition , | |
02:59 | we can say that 2/8 is equal to just like | |
03:02 | two times 1/8 . We can write it as 1/8 | |
03:04 | plus 1/8 because remember all multiplication can be written as | |
03:09 | addition . So these are the two answers I'm going | |
03:11 | to give you a fraction . And our goal is | |
03:14 | to write it as the multiplication of something and then | |
03:16 | also to write it as the addition of something . | |
03:18 | And every problem will be the same . This two | |
03:21 | eights means the fraction 1/8 times to just like this | |
03:25 | . And then we can also think of these as | |
03:26 | being just added together . 1/8 plus 1/8 gives you | |
03:29 | 2/8 . Because if you add these you keep the | |
03:32 | same denominator , add enumerators , you get the 2/8 | |
03:35 | . If you multiply this , you multiply the tops | |
03:38 | , multiply the bottoms . Of course the bottom of | |
03:39 | the one here and so you get the same thing | |
03:41 | back . All right now that all of that talking | |
03:44 | is out of the way , It's gonna be much | |
03:47 | , much easier for us to just jump in and | |
03:49 | understand the rest . Let's write the fraction . 4 | |
03:53 | 5th . We're gonna write it two different ways . | |
03:56 | We're gonna write it first . It's multiplication , And | |
03:59 | then we're gonna write it as edition . So , | |
04:01 | 4/5 . What does this mean ? It means we | |
04:03 | have four pieces out of five . It means we | |
04:06 | have four slices , each of which is 1/5 . | |
04:09 | It means that we can take the fraction 1/5 which | |
04:12 | is a small fraction . And we can multiply it | |
04:14 | times four , Right ? Four times 1/5 . Because | |
04:18 | that's what we're doing . We're basically going and saying | |
04:20 | we have a fraction is 1/5 and we replicate it | |
04:22 | . And we multiply it times four and that's what | |
04:24 | 4/5 is . So , let's take this . We | |
04:28 | have uh here we have the fraction 4/5 there's 1/5 | |
04:33 | there's 2/5 there's 3/5 There's 4/5 , there's a fraction | |
04:37 | 4/5 . But that's the same thing as just taking | |
04:40 | the fraction 1/5 and multiplying it times four . I | |
04:44 | guess I can flip this over like this 1/5 times | |
04:47 | for the representing it . This way , it's the | |
04:49 | same amount of pizza as uh when you when you | |
04:54 | kind of wrap it around like this , it's the | |
04:55 | same . Exactly the same thing , whether you spread | |
04:57 | it out or you put it into a circle is | |
04:59 | the same thing . This front , 4/5 is the | |
05:01 | fraction 1/5 times four . That's what 4/5 is actually | |
05:05 | equal to . And we can also then write it | |
05:08 | . In terms of edition , we can say that | |
05:10 | 4/5 is going to be equal . What ? Thinking | |
05:12 | about this , we can say 1/5 plus 1/5 plus | |
05:17 | 1/5 plus 1/5 +1234 times . Because multiplication is the | |
05:24 | same thing as adding the thing that many times . | |
05:27 | So this whole fraction which is 4/5 can be thought | |
05:30 | of as 1/5 plus 1/5 plus 1/5 plus 1/5 . | |
05:33 | That's all we're saying . It can also be thought | |
05:36 | of as 1/5 times . four , same thing . | |
05:39 | So , we circle all of these . All right | |
05:43 | , Let's take a look at problem # three . | |
05:47 | What about the fraction 56 ? How do we write | |
05:49 | that as multiplication . And how do we write it | |
05:52 | as edition ? Well , what we're saying here is | |
05:55 | that we have the fraction 56 can be written as | |
05:58 | five times the fraction 1/6 . And if you can | |
06:02 | think about it , if you think about five being | |
06:04 | 5/1 , then five times one is five and the | |
06:08 | one that's down here , one time six is six | |
06:10 | . So this equation is true and we can also | |
06:13 | write that 56 is equal to 1/6 plus 1/6 plus | |
06:18 | 1/6 plus 1/6 plus 1/6 +12345 times . So we're | |
06:25 | just adding them together . So let's see if that | |
06:28 | makes sense in terms of six , let's put the | |
06:31 | fraction on the board . There's 1/6 there's +26 we'll | |
06:34 | add that , there's 3/6 there's 46 and then of | |
06:39 | course we have 56 All we're saying here is whether | |
06:43 | or not I keep it tucked into a circle like | |
06:45 | this , or if I spread it out , if | |
06:48 | I add all of these sixes together , it equals | |
06:51 | the 56 Or if I take 16 times five It | |
06:55 | equals what we already know 56 to be . So | |
06:57 | I want you to start thinking of fractions As yes | |
07:01 | it's five out of six slices but it's also like | |
07:03 | taking 1/6 and just adding it to itself that many | |
07:07 | times . Or multiplying times that many numbers there . | |
07:11 | So that's all we're trying to do in this lesson | |
07:15 | . Let's take a look At the fraction . 5/12 | |
07:20 | . Let's write it as a multiplication first . Well | |
07:23 | we can say 5/12 is the same thing as 1/12 | |
07:29 | Right ? Multiplied by five . Right ? Because if | |
07:32 | I put it put this over one it would be | |
07:34 | five times one is five and then one down here | |
07:36 | times 12 is 12 . So this is how to | |
07:38 | write it as a multiplication . But also we can | |
07:41 | write it as what ? 1/12 plus 1/12 plus 1 | |
07:48 | 12 Plus 1 12th . That's four times plus 1 | |
07:54 | 12 . So we're basically doing the multiplication , we're | |
07:56 | representing it as addition . And if you add these | |
07:59 | fractions together , the denominator will be the same and | |
08:02 | you'll add all the ones and you'll get 5/12 . | |
08:05 | So we can do the same thing . Here's 1/12 | |
08:09 | . 2 12 , 3 12 , 4 12 5/12 | |
08:13 | . That's what we It's five out of 12 pieces | |
08:16 | , but we can also represent it as being 1 | |
08:18 | 12 , just just multiplied by five . It's a | |
08:22 | multiple This fraction 5 12 is a multiple of the | |
08:25 | fraction . 1 12 . It's just a smaller fraction | |
08:28 | . 1/12 replicated and multiplied times five . So , | |
08:31 | it's a multiple of the smaller fraction . It's like | |
08:34 | the smaller fractions the core fraction . And then we | |
08:38 | make it a multiple of this by multiplying by five | |
08:42 | and it becomes then 5 12 . Or you can | |
08:43 | think of it as addition . 1 12 plus 1 | |
08:45 | 12 plus 1 12 plus 1 12 plus 1 12 | |
08:51 | . All right . How many more do we have | |
08:53 | ? Only two more ? How do we write the | |
08:56 | fraction ? 2/4 . In terms of multiplication ? We | |
09:02 | can write that as the fraction 1 4th times two | |
09:07 | or two times 1/4 . However , you want to | |
09:08 | write it because this too is really to over one | |
09:11 | and then two times one is two and then the | |
09:13 | one times four is four . So it's like taking | |
09:16 | the smaller fraction 1/4 and just multiplying by two . | |
09:19 | That's what the larger fraction is . And we can | |
09:22 | also represent that as simple . In addition we can | |
09:26 | say its 1/4 plus 1/4 . We do it two | |
09:31 | times and adam together . So the fraction 2/4 is | |
09:34 | this this is the fraction to fourth , two out | |
09:37 | of four pieces . It's the same thing as saying | |
09:40 | , here's the fraction 1/4 times to that equals to | |
09:44 | fourth or the fraction 1/4 plus 1/4 That equals the | |
09:48 | 2/4 . So this concept should be pretty familiar now | |
09:54 | because as we've been going through this , I've been | |
09:55 | kind of emphasizing this without really saying it . Last | |
09:59 | problem . 4/10 . Let's write it as a multiplication | |
10:04 | . Well , we can say that that's 1/10 times | |
10:07 | four , because if this were 4/1 , 4 times | |
10:11 | one is 41 times 10 is 10 . We can | |
10:13 | also represent 4/10 as 1/10 since it's times 4 . | |
10:18 | 1/10 we'll just add it four times plus 1/10 plus | |
10:22 | 1/10 plus one 10th . Let's see if that makes | |
10:28 | sense . What I'm saying here is that we have | |
10:30 | 1/10 from here , 1/10 from here and 1/10 from | |
10:33 | here and 1/10 from here . And if we add | |
10:36 | all of these together we get the fraction 4/10 that's | |
10:39 | all we're saying . Same thing is thinking about it | |
10:42 | in multiplication . This larger fraction 4/10 is just equal | |
10:45 | to one of these small ones and multiplying it times | |
10:49 | the top number times for so it's not 1/10 it's | |
10:53 | 4/10 because we've multiplied the small fraction times four . | |
10:57 | So here in this lesson we're just trying to get | |
10:59 | you to think about fractions in different ways . Yes | |
11:01 | it is four out of 10 slices of a pizza | |
11:04 | but it can also be thought of as 1/10 multiplied | |
11:08 | by four or 1/10 plus 1 10 plus 1 10 | |
11:11 | plus 1/10 . We can think about fractions in different | |
11:13 | ways and sometimes the different ways we think about them | |
11:16 | are helpful in different in different problems . So I'd | |
11:19 | like you to practices yourself when you feel like you | |
11:22 | understand it . Follow me on to the next lesson | |
11:23 | . We're gonna give you a little bit more practice | |
11:25 | with this concept |
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