01 - What are Equivalent Fractions? - (Calculate & Find Equivalent Fractions) - Part 1 - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . The title of this lesson | |
| 00:02 | is called finding equivalent fractions . This is part one | |
| 00:05 | . I'm actually really excited to teach this lesson because | |
| 00:08 | here we're starting on a new sequence of lessons covering | |
| 00:12 | all manner of information about the really important concept of | |
| 00:15 | what we call fractions . Right fractions give students in | |
| 00:19 | many cases a lot of problems . But by going | |
| 00:22 | through these lessons and by walking with me step by | |
| 00:24 | step , I promise that not only would you understand | |
| 00:27 | what fractions are , but you'll understand in your bones | |
| 00:31 | what they actually mean , you'll understand what fractions really | |
| 00:34 | represent , you understand as we go through the lessons | |
| 00:37 | , how to add and subtract fractions and do other | |
| 00:39 | operations with fractions . Uh in fact this one here | |
| 00:42 | is called finding equivalent fractions . So we'll be starting | |
| 00:45 | our journey there . But at the end of it | |
| 00:47 | , what I want you to really understand is that | |
| 00:48 | fractions can be worked with and be used just like | |
| 00:53 | everyday numbers that you understand . But the reason why | |
| 00:56 | fractions give some students problems is because you have a | |
| 00:58 | number on the top and the number on the bottom | |
| 01:01 | , like one half is one with a fraction bar | |
| 01:04 | in the number two . And when you start manipulating | |
| 01:06 | and working with fractions , it can get confusing . | |
| 01:08 | So let's hear , start our journey and break down | |
| 01:11 | the barriers and make it all very easy to understand | |
| 01:14 | . So we have this idea of a fraction , | |
| 01:15 | let's work with a fraction that we all understand one | |
| 01:18 | half if you have a pizza or a peanut butter | |
| 01:21 | and jelly sandwich or whatever , and you cut it | |
| 01:23 | in half . Everybody here knows what half of a | |
| 01:26 | pizza looks like , cut the thing in half , | |
| 01:28 | and when I take half , one out of those | |
| 01:30 | two pieces away , that's what we call the fraction | |
| 01:32 | one half . So let's take a look at that | |
| 01:35 | . We talk about the idea of one half as | |
| 01:38 | being one with a fraction bar here and then the | |
| 01:41 | number two . Now in the past we've talked about | |
| 01:43 | this but we haven't really labeled very much . So | |
| 01:46 | I want to spend a minute to understand and label | |
| 01:48 | some things . When we have the number one half | |
| 01:51 | , we ? Re read it as one out of | |
| 01:53 | two pieces anytime you see a fraction , I want | |
| 01:56 | you to think about it as the top number and | |
| 01:58 | the bottom number , you read it as one out | |
| 02:00 | of two pieces . So if you consider this being | |
| 02:04 | a whole pizza , right ? This this circle here | |
| 02:07 | being a whole pizza , then this is one . | |
| 02:10 | And if you cut this guy in half so you | |
| 02:12 | have two pieces , the bottom number tells you how | |
| 02:14 | many pieces you have total . And the top number | |
| 02:17 | is how many pieces you have taken away to eat | |
| 02:19 | . So this is one out of two pieces , | |
| 02:22 | that's what fraction one half . And this is also | |
| 02:24 | one out of two pieces as well . So this | |
| 02:27 | is half of the pizza and this is half of | |
| 02:29 | the pizza , one out of two pieces . So | |
| 02:32 | actually if this were the whole pizza then if I | |
| 02:35 | only have one of those two pieces this is what | |
| 02:38 | we call one half of the pizza . This is | |
| 02:39 | what you all think about when you think about pizzas | |
| 02:42 | and think about cutting it in half it's one out | |
| 02:44 | of two pieces . Cut the thing in two pieces | |
| 02:46 | , take one piece away . Now the top number | |
| 02:48 | here is a one and we have a word for | |
| 02:51 | that in math . This is called the numerator numerator | |
| 02:57 | right ? It's a big fancy word in math . | |
| 03:00 | Sometimes we have big fancy words but I don't want | |
| 03:03 | you to get scared of . The of the word | |
| 03:05 | . All it means is top number . So when | |
| 03:07 | you think of the numerator just think of the top | |
| 03:09 | number . All right . The bottom number here is | |
| 03:11 | a two and this has a special name also . | |
| 03:14 | It's called the denominator denominator . So we have to | |
| 03:21 | talk about these labels because in math sometimes they'll say | |
| 03:26 | you know , multiply the numerator by three . Or | |
| 03:28 | what is the numerator of this fraction ? What is | |
| 03:31 | the denominator of this fraction ? Don't let the words | |
| 03:34 | scare you . Numerator means top number , denominated , | |
| 03:37 | denominator means bottom number . All right . So the | |
| 03:40 | numerator is one , denominator is to in this fraction | |
| 03:43 | . So we know that this fraction is equal to | |
| 03:46 | half of the pizza . We know that that's what | |
| 03:48 | it is . The topic of this lesson is finding | |
| 03:50 | equivalent fractions , right ? So we can actually make | |
| 03:54 | the fraction look and appear to be different . But | |
| 03:57 | in fact it's exactly the same amount of pizza as | |
| 04:00 | one half as we have talked about here . So | |
| 04:03 | one with a fraction bar to we can change it | |
| 04:06 | to look different but actually it represents exactly the same | |
| 04:09 | thing and I'm gonna show you exactly what we're talking | |
| 04:11 | about uh in just a moment as we actually solve | |
| 04:14 | a bunch of problems . So it turns out that | |
| 04:16 | with fractions you can think of this bar that goes | |
| 04:19 | here , you can think of it as like a | |
| 04:20 | balance . So think of like a seesaw like you | |
| 04:23 | know , is in balance right ? If everything is | |
| 04:25 | in balance , I can multiply the top and the | |
| 04:28 | bottom of this fraction , the numerator and the denominator | |
| 04:31 | by any number I want . As long as I | |
| 04:34 | do that to both the top and the bottom of | |
| 04:37 | the fraction the numerator and the denominator . If I | |
| 04:39 | multiply the top by two , I must also multiply | |
| 04:43 | the bottom by two to keep it balanced right ? | |
| 04:46 | I can multiply the top by five . As long | |
| 04:48 | as I also multiply the bottom by five . I | |
| 04:50 | can multiply the numerator by 1000 . I can multiply | |
| 04:54 | the numerator by 10 million . As long as I | |
| 04:56 | also multiply the denominator by the same number by 10 | |
| 05:00 | million . So I can multiply the top and the | |
| 05:02 | bottom by any number I want . And I mean | |
| 05:05 | any number as long as I do it to the | |
| 05:07 | top and the bottom and that keeps everything balanced . | |
| 05:10 | So let's see what happens if I actually do that | |
| 05:13 | . Let's say that I take the fraction one half | |
| 05:17 | and I'm going to extend the little bar here . | |
| 05:19 | And let's say that I actually take and I multiply | |
| 05:23 | the top of this fraction by two . But I'm | |
| 05:26 | gonna multiply the bottom by the fraction also by two | |
| 05:28 | because it's like a balance right ? If I multiply | |
| 05:31 | the top by to its imbalance . As long as | |
| 05:33 | I also multiply the bottom by two , what would | |
| 05:37 | I have as an answer on the top ? What | |
| 05:39 | is one times two ? And by the way in | |
| 05:42 | math now we have to start using the dots , | |
| 05:44 | right ? Because in the past you've been thinking of | |
| 05:46 | multiplication with exes . But that gets confusing because later | |
| 05:50 | on we'll be using X is for other things in | |
| 05:53 | math , we'll be using them for what we call | |
| 05:55 | variables . So we're gonna stop using excess . You | |
| 05:58 | can if you want put an X . There . | |
| 06:00 | Ok . I'm not gonna like count it off or | |
| 06:02 | say it's wrong . I'm just saying we're gonna start | |
| 06:05 | using these dots to mean multiplication because excess can get | |
| 06:08 | confusing . So here we have one times to what | |
| 06:11 | is one times two that's equal to two On the | |
| 06:14 | bottom , we have two times two . What does | |
| 06:16 | that equal to four ? What I am saying is | |
| 06:20 | that the fraction one half here ? I'll put an | |
| 06:22 | equal sign here and I'll put one half . The | |
| 06:25 | fraction one half is exactly the same as the fraction | |
| 06:28 | 2/4 because I've taken the numerator , the top number | |
| 06:32 | , and I've multiplied by two and I've also taken | |
| 06:34 | the denominator and I've also multiplied by two . So | |
| 06:37 | because I did the same thing to both , I | |
| 06:40 | actually get something that looks different , but it's actually | |
| 06:43 | the same thing . Let's see if it makes sense | |
| 06:45 | here . So here I'm going to bring up what | |
| 06:48 | I have down here at the bottom to help us | |
| 06:50 | here . So now the pizza is not cut into | |
| 06:53 | two pieces . The pizza is cut into four pieces | |
| 06:57 | . So you can see that right here we have | |
| 06:59 | uh the pizza cut into 1234 pieces . So this | |
| 07:03 | is one out of four pieces , two out of | |
| 07:05 | four pieces , three out of four pieces , four | |
| 07:07 | out of four pieces . This is 1/4 another 1/4 | |
| 07:10 | another 1/4 another 1/4 because it's one out of 41 | |
| 07:13 | out of 41 out of 41 out of four . | |
| 07:15 | So all together they make a whole pizza . But | |
| 07:18 | the fraction here is saying I have two out of | |
| 07:20 | four pieces . I've cut the pizza into four pieces | |
| 07:23 | , but I actually have two of them . So | |
| 07:26 | I'm going to kind of tilt this or I should | |
| 07:28 | say take away these two pieces , I'll kind of | |
| 07:30 | remove them because I only have two out of four | |
| 07:33 | pieces . I'm gonna pull these down and look at | |
| 07:36 | how much pizza I have . I have this much | |
| 07:38 | pizza , right ? And if I were to eat | |
| 07:41 | this much pizza all together , these two out of | |
| 07:43 | four pieces , it will be exactly the same amount | |
| 07:45 | of pizza is if I had eaten this much you | |
| 07:48 | see one half represents this amount of pizza to force | |
| 07:53 | represents this amount of pizza , it's actually two pieces | |
| 07:55 | , but we put them together and it's the same | |
| 07:57 | amount This you can see is exactly the same as | |
| 07:59 | this . In fact I can lift it up and | |
| 08:01 | put it right on top and I can show you | |
| 08:03 | that it's exactly the same amount of pizza . So | |
| 08:06 | I want you to burn in your mind this idea | |
| 08:09 | that I can take a fraction , I can multiply | |
| 08:11 | the top and the bottom by any number I want | |
| 08:14 | and I will change the way the fraction looks . | |
| 08:18 | But I don't change what the fraction means . That's | |
| 08:21 | really important . I'm gonna say it again , I | |
| 08:24 | can change the way the fraction looks . But that | |
| 08:26 | doesn't change what it actually means right . The reason | |
| 08:30 | these are the same thing is because here when I | |
| 08:34 | went from one half to 14 to 2/4 I have | |
| 08:37 | had in this case I had the pizza cut into | |
| 08:39 | two pieces . In this case . Over here I | |
| 08:43 | had the pizza cut into four pieces . So I | |
| 08:45 | doubled the amount of pieces I have over here . | |
| 08:49 | So I doubled the number of slices , right ? | |
| 08:51 | But at the same time I actually doubled the amount | |
| 08:54 | of slices that I took away . So of course | |
| 08:57 | I'm going to eat the same amount of pizza if | |
| 08:59 | I double the amount of slices . But I also | |
| 09:01 | double the amount of slices I take away from the | |
| 09:04 | pizza . that's why they are the same thing , | |
| 09:06 | that's why they're equivalent because I've increased the amount of | |
| 09:10 | slices total . But I've also increased the number of | |
| 09:13 | slices that I took away by the exact same factor | |
| 09:16 | by the exact same amount I doubled in both cases | |
| 09:19 | . Why did I double because I multiplied by two | |
| 09:22 | . The number of slices and the number of slices | |
| 09:24 | that I took away To eat . It's really important | |
| 09:27 | for you to understand that . And that's why I | |
| 09:28 | repeated about three times . So we multiply the numerator | |
| 09:33 | and the denominator of this fraction by two . And | |
| 09:36 | we get a new fraction called 2/4 and it's exactly | |
| 09:40 | the same amount that one half also represents . Now | |
| 09:45 | I'd like to play with the fraction one half a | |
| 09:47 | little bit more because it's really really , really important | |
| 09:50 | . Let's instead slide are one half down here and | |
| 09:53 | let's say also what would happen if we take one | |
| 09:55 | half and we multiply it by instead of multiplying by | |
| 09:59 | two ? Let's multiply top and bottom by a different | |
| 10:02 | number . Let's multiply top and bottom by the number | |
| 10:04 | three multiple . Top by three in the bottom by | |
| 10:06 | three . Remember I said you can multiply a fraction | |
| 10:10 | by anything . You want any number as long as | |
| 10:14 | you multiply the top and the bottom of the numerator | |
| 10:16 | and the denominator by the same number by the same | |
| 10:20 | number here . What do we have ? We have | |
| 10:24 | to have a different color here . What do we | |
| 10:26 | have on the top one times three is three , | |
| 10:29 | two times three is six . And what I'm claiming | |
| 10:32 | is that one half looks totally different than 36 but | |
| 10:35 | it actually represents the same amount of pizza . So | |
| 10:39 | let's go and grab this up here and let's take | |
| 10:42 | a look here , I have now a pizza cut | |
| 10:44 | into six slices because that's what the denominator represents , | |
| 10:48 | it's how many slices I cut the pizza into 123456 | |
| 10:53 | . But Notice that this is 16 , this is | |
| 10:56 | once this is one out of six pieces , one | |
| 10:57 | out of six pieces , one out of six pieces | |
| 10:59 | , one out of six pieces , one out of | |
| 11:00 | six pieces , one out of six pieces . But | |
| 11:02 | in this fraction I have three out of six pieces | |
| 11:05 | . So what do I have ? I have , | |
| 11:07 | I have these three pieces , I have 123456 but | |
| 11:11 | I'm only taking away three of those six pieces . | |
| 11:14 | So I have this amount of pizza so I will | |
| 11:17 | slide that away , that's gone . So what we've | |
| 11:19 | said now is that one half is exactly the same | |
| 11:22 | as 3/6 . And you can see why it's exactly | |
| 11:26 | the same . It's because this time I had the | |
| 11:28 | pizza into two pieces , cut into two pieces . | |
| 11:31 | Here , I triple the amount of pieces I cut | |
| 11:33 | into . And I tripled it so that now I've | |
| 11:35 | cut the pizza into six pieces , but at the | |
| 11:38 | same time , I triple the amount of slices that | |
| 11:41 | I took away . So I take a pizza I | |
| 11:44 | triple the amount of slices . But then I triple | |
| 11:47 | the amount of slices I'm taking away . So I | |
| 11:49 | have exactly the same amount of pizza . If I | |
| 11:51 | eat this amount of pizza or I eat this amount | |
| 11:53 | of pizza , I'm actually eating exactly the same amount | |
| 11:56 | of pizza . These two fractions are equivalent . They're | |
| 11:59 | exactly the same thing . All right , we're gonna | |
| 12:02 | play with us a little bit more , and eventually | |
| 12:04 | we're gonna drop the uh we're gonna drop the the | |
| 12:08 | magnets here in a second . But let's take a | |
| 12:09 | look at what if we do instead of one half | |
| 12:12 | here . Let's move our guy down here , our | |
| 12:14 | little one half . Let's , instead of multiply by | |
| 12:17 | three . Let's multiply by . What do you think | |
| 12:20 | we're going to do ? Let's try multiplying by four | |
| 12:23 | And we'll multiply by four right here . What do | |
| 12:27 | we get on the top ? Multiply the top and | |
| 12:30 | bottom by the same thing . One times four is | |
| 12:32 | four . What do we get right here ? Two | |
| 12:34 | times four is eight . So what we're claiming is | |
| 12:37 | that the fraction 484 out of eight pieces of something | |
| 12:41 | is exactly the same amount as one out of two | |
| 12:44 | pieces . Let's see if that is actually true . | |
| 12:47 | Let's see if that's actually true . So we need | |
| 12:50 | a pizza cut into eight pieces . Here's a pizza | |
| 12:52 | cut into eight pieces , 12345678 Actually to give myself | |
| 12:57 | room , I'm gonna slide these over like this and | |
| 13:00 | slide these over like this . Just give me a | |
| 13:01 | little more room to think I'll turn this one sideways | |
| 13:05 | later . Um What do we have here ? Four | |
| 13:07 | out of eight pieces , 12345678 We have 1/8 1 | |
| 13:12 | 8th , 1/8 1 eighth all the way around . | |
| 13:13 | But I have four out of eight pieces so I'm | |
| 13:17 | gonna take away these four , I have 1234 out | |
| 13:20 | of 123456784 out of eight pieces . So I'm gonna | |
| 13:24 | take this and slide it away because it isn't going | |
| 13:27 | to affect the rest of this . Sorry about that | |
| 13:29 | little distraction down there but you can see that one | |
| 13:32 | half is exactly the same as 123 4/8 because this | |
| 13:37 | pizza is exactly the same amount as this pizza . | |
| 13:40 | So again the fraction one half , it's the same | |
| 13:42 | as 4/8 . And I'm only gonna do one more | |
| 13:44 | here and I promise we won't do quite as many | |
| 13:47 | for the next problem . But I do want to | |
| 13:49 | um I want to drive the point home . Let's | |
| 13:52 | say we have one half And let's multiply it by | |
| 13:56 | five . Uh Let's multiply it by five . So | |
| 14:00 | we'll multiply the numerator by five and will also multiply | |
| 14:03 | the denominator by five . Remember we can do whatever | |
| 14:07 | we want , multiplying a fraction as long as we | |
| 14:09 | do it to the top and the bottom , what | |
| 14:11 | do we have ? One times five is five and | |
| 14:15 | two times five is 10 . So what we're claiming | |
| 14:19 | is that the fraction one half , one half of | |
| 14:21 | a pizza is exactly the same as the fraction three | |
| 14:25 | or 5/10 . Let me turn that sideways and getting | |
| 14:27 | up out of my way and now I'm gonna grab | |
| 14:29 | a pizza that's cut into you guessed at 10 pieces | |
| 14:32 | . Let me get some of these other ones out | |
| 14:34 | of our way here . So this one's cut into | |
| 14:37 | 10 pieces , 123456789 10 . Each of these slices | |
| 14:43 | is one out of 10 pieces each of these slices | |
| 14:45 | of 1 10 . So I have 12345 maybe double | |
| 14:50 | check 12345 out of 10 . So these pieces go | |
| 14:54 | away . This is what I actually have five out | |
| 14:57 | of 10 pieces . If I cut a pizza into | |
| 15:00 | 10 equal slices , but I take five , it's | |
| 15:02 | exactly the same amount of pizza as the one half | |
| 15:06 | , which is exactly the same amount of pizza as | |
| 15:08 | the 4/8 which is exactly the same amount of pizza | |
| 15:11 | as the 36 which is exactly the same amount of | |
| 15:14 | pizza as the 2/4 . They're all the same . | |
| 15:17 | You see I can take a fraction and I can | |
| 15:19 | multiply it by anything I want and I will change | |
| 15:22 | the way the fractions look right , I will change | |
| 15:24 | the way they look , but that won't change what | |
| 15:26 | they mean . That's what equivalent fractions are multiply numerator | |
| 15:31 | and denominator by six if you want , no problem | |
| 15:33 | , want to multiply top and bottom by 17 if | |
| 15:35 | you want , no problem , want to multiply top | |
| 15:37 | and bottom by 27 if you want . No problem | |
| 15:40 | . You will change the numerator and the denominator but | |
| 15:43 | you will not change what the fraction represents . That | |
| 15:46 | is the point of this example . So what I'm | |
| 15:48 | gonna do is a few more with the magnets and | |
| 15:50 | then we're going to drop the magnets completely . Hopefully | |
| 15:53 | you'll kind of believe me that this is the way | |
| 15:55 | it works . And then going forward will just multiply | |
| 15:58 | top and bottom to get an equivalent fraction . All | |
| 16:02 | right , So let's go over to the next board | |
| 16:04 | . Let's start with the fraction 3/4 . And I | |
| 16:11 | want to change this fraction . I want to multiply | |
| 16:14 | the numerator and denominator by the number two to find | |
| 16:17 | an equivalent fraction because I can I can find any | |
| 16:19 | equivalent fraction I want by multiplying by anything but what | |
| 16:22 | I want . But in this problem let's try to | |
| 16:24 | change it by multiplying the top and bottom by two | |
| 16:27 | . So what does that mean ? I have to | |
| 16:29 | make a longer fraction bar And I'm going to multiply | |
| 16:33 | The top of the fraction by two in the bottom | |
| 16:35 | of the fraction by two to try to change it | |
| 16:39 | . What am I going to get ? Three times | |
| 16:40 | two is six , and four times two is eight | |
| 16:44 | . Four times two is eight . So what we're | |
| 16:46 | claiming is that 3/4 is exactly the same fraction as | |
| 16:50 | 6/8 . They look different , but they're the same | |
| 16:53 | thing because I've doubled the amount of slices I've cut | |
| 16:56 | the pizza into , but at the same time I've | |
| 16:58 | doubled the amount of slices that I take away to | |
| 17:00 | actually eat . So I have exactly the same amount | |
| 17:04 | of pizza . Let's see if this is correct . | |
| 17:06 | So let me grab My fourths here , this is | |
| 17:09 | 1/4 , this is another 14 so that's 2/4 . | |
| 17:14 | This is 3/4 and this is 4/4 . So that | |
| 17:17 | means a whole pizza here . Right ? So three | |
| 17:19 | forces three out of four pieces , 123 out of | |
| 17:22 | four pieces . So this can go away , This | |
| 17:25 | is three out of four pieces . 3/4 that's what | |
| 17:28 | represents here . Uh up above now , I have | |
| 17:32 | six out of eight pieces of something . So I | |
| 17:34 | need a fraction with the denominator of eight . So | |
| 17:37 | I'm gonna grab my blue ones here and we'll take | |
| 17:39 | me a second to transform over . So let me | |
| 17:42 | let me do it right , let me move it | |
| 17:43 | all over here . And what we're going to find | |
| 17:46 | out is that even though they look different , they're | |
| 17:48 | actually exactly the same thing . So here are my | |
| 17:50 | eighth . Right ? Let me go in and line | |
| 17:53 | them up first . I want to show you that | |
| 17:55 | it makes a whole pizza . So you can see | |
| 17:57 | you have the whole right here and it's cut into | |
| 18:00 | eight pieces . The bottom number . The denominator means | |
| 18:03 | how many pieces I'm cutting into . How many pizza | |
| 18:06 | pieces am I going to actually eat ? Six pieces | |
| 18:08 | ? So that's gonna be 123456 So , if make | |
| 18:13 | sure you agree 123456 out of eight pieces . This | |
| 18:17 | is the amount of pizza I have . So I'm | |
| 18:19 | gonna take and put those away . What we're claiming | |
| 18:21 | is that these two fractions are the same . It | |
| 18:24 | may not look like it , but let me kind | |
| 18:26 | of tilted like this and then let me move them | |
| 18:28 | close together or kind of I guess put them in | |
| 18:31 | the middle . Let me do it like this . | |
| 18:33 | Let me put it like this and I'll move this | |
| 18:34 | one a little bit closer . Do you agree that | |
| 18:36 | those are the exact same amount , That is the | |
| 18:38 | exact same amount of pizza . If I eat this | |
| 18:41 | much pizza and if I eat this much pizza , | |
| 18:43 | it's exactly the same amount of pizza . So I | |
| 18:44 | can multiply a fraction by any number . I want | |
| 18:48 | to change the way it looks . But they're actually | |
| 18:50 | exactly equivalent . All right ? So let me move | |
| 18:53 | this down here And I want to do one more | |
| 18:56 | with the fraction 3/4 . Let's say what if I | |
| 18:59 | have 3/4 ? See ? And I want to multiply | |
| 19:04 | numerator and denominator instead of buy to let me multiply | |
| 19:07 | top and bottom by three . I can multiply by | |
| 19:11 | whatever I want . But what am I going to | |
| 19:13 | get on the top ? I'm going to have three | |
| 19:15 | times three is nine and four times three is 12 | |
| 19:18 | . So what I'm claiming is that the fraction 9 | |
| 19:21 | , 12 is exactly the same thing as 3/4 . | |
| 19:24 | So what I need to do is let's move this | |
| 19:26 | under the 3/4 like this . We all know that | |
| 19:29 | this is 3/4 . Now I have a pizza cut | |
| 19:32 | into 12 pieces , 12 equal slices . Let's double | |
| 19:35 | check this . 123456789 10 , 11 , 12 pieces | |
| 19:41 | . That's how many pieces I've cut this pizza into | |
| 19:43 | but I only have nine of them . So 123456789 | |
| 19:50 | Make sure you agree that this is nine out of | |
| 19:54 | 12 , 123456789 out of 10 , 11 , 12 | |
| 19:59 | pieces . So I'm gonna take and put these away | |
| 20:01 | . This is the amount of pizza I'm actually going | |
| 20:03 | to eat . Now , if I rotate it like | |
| 20:05 | this and kind of bring it closer , you can | |
| 20:07 | see that this is exactly the same amount of pizza | |
| 20:09 | as this , right ? So 3/4 is exactly the | |
| 20:13 | same thing as 9 12 . And you can always | |
| 20:16 | know that it is the case by using these fraction | |
| 20:19 | magnets . But after a while we're going to stop | |
| 20:22 | using the magnets and you'll just have to believe that | |
| 20:24 | when you multiply the top and the bottom by any | |
| 20:26 | number you want , no matter what the end result | |
| 20:28 | looks like . It's going to represent the same amount | |
| 20:32 | of material , of pizzas , of pies , of | |
| 20:35 | whatever it is you're talking about . That's how fractions | |
| 20:37 | work . All right , we're gonna do one more | |
| 20:41 | with magnets and then we're going to call it a | |
| 20:43 | day and just start solving some more problems . Let's | |
| 20:46 | take the fraction 2/3 and let's multiply the fraction two | |
| 20:51 | thirds by the number five . Let's actually let's multiply | |
| 20:56 | by the number two first . Let's multiply by the | |
| 20:58 | number two because I'm gonna get an equivalent fraction no | |
| 21:01 | matter what . Let me pick the number two to | |
| 21:03 | multiply by . What do we get here ? Two | |
| 21:05 | times two is four and three times two is six | |
| 21:08 | . So what we're claiming is that the fraction two | |
| 21:11 | thirds is exactly the same as the fraction 4/6 . | |
| 21:15 | So let me go ahead and grab my magnets and | |
| 21:17 | let's prove to ourselves that that is the case . | |
| 21:20 | All right . The fraction two thirds means I have | |
| 21:23 | a pizza cut into 123 pieces , but I only | |
| 21:26 | actually have two of them to eat . So here | |
| 21:29 | is two out of three pieces . I'll take that | |
| 21:31 | and throw that away . This is how much pizza | |
| 21:33 | I actually have to try to eat . I'll kind | |
| 21:36 | of arrange it like , you know , like like | |
| 21:37 | this right ? And then I have another pizza that's | |
| 21:41 | cut into six pieces , but I only have four | |
| 21:43 | of them , 123456 pieces total , but I only | |
| 21:47 | have four of them . Make sure you understand that | |
| 21:50 | . This is 1234 out of 56 pieces total , | |
| 21:54 | so I'll throw that away . What we're saying is | |
| 21:56 | we think that this is both exactly the same amount | |
| 21:58 | of pizza . Let's see if we agree it looks | |
| 22:01 | a little bit different , but when I arrange it | |
| 22:03 | like this , I think you can convince yourself that | |
| 22:05 | it's the same . If I could put this on | |
| 22:08 | top , you can see it covers that perfectly and | |
| 22:10 | then another equivalent piece is going to go down here | |
| 22:13 | and cover this one perfectly as well . So that | |
| 22:15 | is exactly the same amount of pizza . So we've | |
| 22:19 | used the magnets to to prove to you , you | |
| 22:21 | know , I don't like just telling you to do | |
| 22:23 | things . We're going to do one more with the | |
| 22:26 | magnets and then after that we're going to drop them | |
| 22:28 | all together and stop using them and you'll just have | |
| 22:31 | to at that point , hopefully believe that this is | |
| 22:34 | the way it works . Let's say we have 2/3 | |
| 22:40 | And let's multiply the top and bottom instead of by | |
| 22:43 | two . Let's multiply by four . What do we | |
| 22:48 | get ? Two times four is eight and three times | |
| 22:50 | four is 12 . So here we have 2/3 and | |
| 22:59 | we just take and multiply the top and the bottom | |
| 23:01 | by four . Now this is two thirds . This | |
| 23:04 | is the amount that it covers . And now I | |
| 23:06 | need to grab my pieces of 12 over here and | |
| 23:09 | I'm gonna I'm gonna build a pizza out of 12 | |
| 23:11 | over here . So there's three there's four there's 56 | |
| 23:19 | Yeah 78 Here's nine There's 10 there's 11 , there's | |
| 23:26 | 12 . They put them all together . Mhm . | |
| 23:29 | And you can see that this pizza is now cut | |
| 23:32 | into 12 equal pieces 123456789 10 , 11 , 12 | |
| 23:37 | . But I only have eight of them . So | |
| 23:39 | what do I have ? 12345678 out of 9 , | |
| 23:42 | 10 , 11 , 12 . Let me throw those | |
| 23:44 | away . This is the amount of pizza , I | |
| 23:46 | have eight pieces out of 12 , and whenever I | |
| 23:48 | rotate them and put them uh next to each other | |
| 23:51 | , you can see that this is also exactly the | |
| 23:54 | same amount of pizza there . So two thirds is | |
| 23:56 | exactly the same thing as 8 12 . So now | |
| 24:01 | we have used our magnets , we have learned how | |
| 24:05 | to make equivalent fractions . And now for the rest | |
| 24:09 | of the problems , we're not going to use any | |
| 24:10 | magnets , it's going to go much faster . We're | |
| 24:13 | just going to multiply the top and the bottom by | |
| 24:15 | whatever number I tell you to and get an equivalent | |
| 24:18 | fraction . And now we are going to understand that | |
| 24:20 | they really are the same thing , even though they | |
| 24:22 | look different . So let's take our 2/3 And let's | |
| 24:27 | multiply that by five . So if we have 2/3 | |
| 24:32 | and we multiply it by five , we have to | |
| 24:35 | multiply the numerator and the denominator by five . What | |
| 24:39 | am I going to get ? Two times five is | |
| 24:40 | 10 and three times five is 15 . So what | |
| 24:44 | we're saying here is the fraction 25th is exactly the | |
| 24:48 | same as two thirds . Now , I don't have | |
| 24:49 | a magnet cut into 15 pieces , But if you | |
| 24:53 | make a circle and you cut it into 15 pieces | |
| 24:55 | but only take 10 of them , that's gonna be | |
| 24:57 | exactly the same amount as the two thirds that we | |
| 25:00 | have been playing with this entire time . All right | |
| 25:04 | now the rest of these problems are going to go | |
| 25:06 | way , way , way faster because we're not going | |
| 25:08 | to be dealing with the actual magnets here . Let's | |
| 25:13 | take the fraction 1/5 . And let's change it into | |
| 25:17 | an equivalent fraction by multiplying the top and the bottom | |
| 25:19 | by the number three , you have to multiply the | |
| 25:22 | top and the bottom by the same exact number . | |
| 25:25 | So we'll take the 1/5 . Let's change it by | |
| 25:28 | multiplying the top and the bottom , numerator and denominator | |
| 25:31 | by the number three . What are we going to | |
| 25:32 | get ? one times 3 is three , five times | |
| 25:36 | three is 15 . So now we know that the | |
| 25:39 | fraction 3 , 15 is exactly the same thing as | |
| 25:42 | 1/5 . And now we can cruise along because we're | |
| 25:46 | not we don't have to use the magnets for every | |
| 25:48 | time we now know that these are equivalent and why | |
| 25:50 | they are . Let's take a look at 3/12 . | |
| 25:53 | Let's find an equivalent fraction where we multiply numerator and | |
| 25:56 | denominator of this fraction by the # two . So | |
| 26:00 | we're going to multiply 3,53 , 12 Numerator and Denominator | |
| 26:05 | by the # two . What do we get ? | |
| 26:09 | Three times two is six and 12 times to from | |
| 26:12 | your multiplication tables is 24 . So if I had | |
| 26:16 | a pizza cut into 24 pieces , but I only | |
| 26:18 | took six of those pieces , it would be the | |
| 26:20 | same amount of pizza is if I cut a pizza | |
| 26:23 | into 12 pieces but only took three . And the | |
| 26:26 | reason is because I doubled the amount of pieces I | |
| 26:28 | cut it into but at the same time I doubled | |
| 26:31 | the amount of pieces that they took away . That's | |
| 26:32 | why they're the same thing . All right . See | |
| 26:36 | they're going much faster now Let's take a look at | |
| 26:39 | the fraction 1/3 . And let's now multiply the top | |
| 26:44 | and the bottom of this fraction by the # nine | |
| 26:46 | . And let's get an equivalent fraction . Let's multiply | |
| 26:49 | top and bottom numerator and denominator by the number nine | |
| 26:53 | . What would be the equivalent fraction we would get | |
| 26:55 | one times nine is nine , three times nine is | |
| 26:59 | 27 . The fraction 9 , 27 9 out of | |
| 27:02 | 27 pieces is exactly the same as one out of | |
| 27:05 | three pieces . Okay , What about the fraction to | |
| 27:13 | fifth ? Let's change it . Find an equivalent fraction | |
| 27:17 | by multiplying top and bottom by five . So we'll | |
| 27:21 | take 2/5 and will multiply top and bottom by the | |
| 27:24 | number five multiply the top by five , multiply the | |
| 27:27 | bottom by five have to do it to the top | |
| 27:29 | and the bottom two times five is 10 , 5 | |
| 27:33 | times five is 25 . And now we know that | |
| 27:36 | the fraction 10 , 25th is the same thing as | |
| 27:40 | the fraction 2/5 . If I have a pizza cut | |
| 27:43 | into 25 pieces , but I only take 10 of | |
| 27:45 | them , I'll have exactly the same amount of food | |
| 27:48 | is if I cut a pizza into five pieces , | |
| 27:50 | but only take two of those . All right , | |
| 27:54 | we're actually very close to being done . Let's take | |
| 27:57 | the fraction 1/8 . And let's find an equivalent fraction | |
| 28:01 | by multiplying top and bottom by the number two . | |
| 28:04 | So let's go ahead and say uh what's not one | |
| 28:07 | half ? Let's take the fraction 1/8 . And let's | |
| 28:12 | multiply numerator and denominator by the number to multiply top | |
| 28:17 | and bottom by the number two . What do we | |
| 28:18 | get on the top one times two is two and | |
| 28:21 | eight times two is what ? 16 ? So , | |
| 28:24 | the fraction to 16th , two pieces out of 16 | |
| 28:28 | total is the same thing as the fraction 1/8 , | |
| 28:31 | 1 out of eight total . All right , now | |
| 28:35 | , we're gonna go do our final problems on this | |
| 28:36 | board over here . We only have two more . | |
| 28:39 | So , we're almost done . Let's take a look | |
| 28:42 | at 36 and let's change it . Let's find an | |
| 28:46 | equivalent fraction by multiplying top and bottom by the number | |
| 28:50 | four by the number four . So let's multiply the | |
| 28:53 | top by four in the bottom by four . We'll | |
| 28:56 | get an equivalent fraction three times four Is 12 , | |
| 28:59 | 6 times four is 24 . So the fraction 12 | |
| 29:04 | out of 24 , 12 , is the same as | |
| 29:06 | the fraction 36 . Even though they look different . | |
| 29:10 | All right . And finally , our very last problem | |
| 29:14 | , let's take the fraction 1/6 . And let's multiply | |
| 29:17 | the top and bottom by the number six . So | |
| 29:19 | , we'll take the 1/6 multiply the top by six | |
| 29:22 | and the bottom by six . What do we get | |
| 29:24 | ? One times six is 66 times six is 36 | |
| 29:30 | . So the fraction six , is the same as | |
| 29:34 | the fraction 16 . So if I cut a pizza | |
| 29:36 | in the 36 pieces , but take six of those | |
| 29:38 | pieces , I'll have exactly the same amount of food | |
| 29:41 | is if I take another pizza cut it into six | |
| 29:43 | pieces and only take one of them . So this | |
| 29:46 | was a long lesson , but it's actually a really | |
| 29:48 | , really important lesson . This idea of equivalent fractions | |
| 29:51 | . We're going to use it like a lot , | |
| 29:54 | right ? We're going to use it here for these | |
| 29:56 | problems , but we're also going to use it to | |
| 29:57 | add and subtract fractions later on . I don't want | |
| 30:00 | to get into why we need to worry about that | |
| 30:02 | , but just trust me , we're going to use | |
| 30:04 | this process to compare fractions among each other and also | |
| 30:09 | to add and subtract fractions . So it's never going | |
| 30:11 | to go away . That's why I really want you | |
| 30:13 | to understand that you can have two fractions that look | |
| 30:15 | different , but they actually can represent the same exact | |
| 30:18 | thing . And the way that you can find an | |
| 30:20 | equivalent fraction is by taking any fraction you want and | |
| 30:23 | multiplying by Top and bottom numerator and denominator by any | |
| 30:28 | number you want . You will get a new looking | |
| 30:30 | fraction but they'll actually mean exactly the same thing and | |
| 30:33 | be equivalent . It's extremely important . I'd like you | |
| 30:36 | to go through this again so you fully understand , | |
| 30:38 | get the correct answers to all of these . Follow | |
| 30:40 | me onto part two . We'll get a little bit | |
| 30:41 | more practice with equivalent fractions . |
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01 - What are Equivalent Fractions? - (Calculate & Find Equivalent Fractions) - Part 1 is a free educational video by Math and Science.
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