Практика Shadowing: Fractions Made EASY! - Изучайте разговорный английский по видео

Okay, let's talk about fractions.

And of course, this is everyone's favorite topic.

But in actuality, most math students, when they see fractions, they have one or two expressions.

Either they're very angry, they're like, fractions, just give me regular numbers like two and three.

You know, I don't mind doing math when the numbers are nice and easy.

But, you know, some people get upset, then others are just totally just lost.

They're like, oh my goodness, fractions, anything but fractions.

I'll do push-ups, but not fractions.

But here's the deal.

If you stick with me for a couple minutes, I'm going to give you some outstanding little ways to remember how to deal with fractions.

You're going to be very, very happy.

And what we're going to focus on is how to do the following operations with fractions.

We're going to learn how to add, subtract, divide, and multiply these primary operations.

Of course, most of you out there probably already learned this, but here's the deal.

Most students that have learned fractions don't really work with fractions that often, and they get confused, and they struggle because, you know, there's things involved,

like finding the lowest common denominator, et cetera, et cetera.

But again, we're going to go ahead and give you a nice quick power lesson in this video.

So you're going to find fractions much easier by the time you finish this video.

But I'm going to get to all of this in just one second.

But first, let me quickly introduce myself.

My name is John.

I'm the founder of Tablet Class Math.

I'm also a middle and high school math teacher.

I'm going to leave a link to all my information in the description of this video.

I've been teaching math for decades, and I've come to the conclusion that all students, every single student out there, can be successful in mathematics.

But it requires two things.

One, you have to be willing to do the work, practice, take notes, study.

But that's the first thing.

The second thing you need is clear and understandable instruction.

So that's what I can offer you.

So if you're at the middle school, high school, or even college level, definitely check out my math help program if you need assistance.

By the way, if you're preparing for any test that has a math section, I'll talk about things like the SAT, ACT, maybe a teacher certification exam, GED.

I can help you out there.

If you homeschool, I have fantastic homeschool, middle, and high school math courses you may want to check out.

And if you need some math notes, I'm going to leave links to my math notes in the description of this video as well.

Okay, so let's get into fractions.

And again, if you've been having difficulty with fractions, that's kind of normal.

A lot of students struggle with this, but we're going to make it easy for you here.

So let's go ahead and first of all, set this up for us.

Okay, so I want you to think of the world of fractions in kind of two rules.

I'm going to break it up as one rule, and then we're going to have another rule.

So the first rule is super easy.

We're talking about multiplication and division.

When I show you how to multiply and divide fractions, you're going to be like, that is so easy.

So we're already 50% there.

So think about this.

Four operations we're talking about, multiplication, division, and then over here, addition and subtraction.

So I'm telling you right now, you're going to see how we multiply and divide fractions.

You're going to be like, that is really, really easy.

So that's like one rule.

OK, so that's just half of what you need to know about fractions.

Now, the other half is addition and subtraction.

And this is where most people, when they think of fractions, they don't like dealing with fractions because this requires a little bit more work.

This is when you have to do like the LCD, the lowest common denominator and all that kind of good stuff.

So if you remember that and you're like, oh, I hate finding the LCD.

Well, listen, I'm going to give you a nice little shortcut.

I'm going to call it a hack that can bypass you having to find the LCD.

Now, you still need to know how to find the LCD.

But if you're confused about this, if you'll get these problems right, adding and subtracting problems every single time by using this nice little shortcut,

you're going to want to know this for sure.

And here's the great news.

Adding and subtracting fractions is basically the same procedure.

Okay, so here's the deal.

If you know how to multiply fractions, then you're going to know how to divide.

It's effectively the same thing.

If you know how to add fractions, it's the same thing as subtracting.

So again, just think of the world of fractions as two rules.

Rule one, rule two.

So let's get on to this first rule, multiplying and dividing fractions, because it's super, super easy.

Let's get into it right now.

So our first rule, okay, multiplying and dividing fractions, again, very easy.

So let's go ahead and take a look at a couple examples.

So here we have 2 thirds times 1 fifth.

Now, just a quick review.

The top number in a fraction is called the numerator.

The bottom number is called the denominator, okay?

But here's what you need to know.

When we're multiplying fractions, okay, The way we multiply fractions is we simply multiply the respective numerator.

So in this case, it's going to be 2 times 1.

And then we're going to multiply the respective denominators.

That's 3 times 5.

I mean, it doesn't get much easier than that.

So we're just going to multiply across.

So 2 times 1 is 2.

3 times 5 is 15.

You are done.

There is the answer.

And let's go ahead and give ourselves a nice little happy face for, you know, feeling good about that.

Like, hey, I know how to multiply fractions now.

It's literally as easy as that.

Okay, so let's talk about how to divide fractions.

Now, dividing fractions, what we're going to do is take division problems and we're going to convert them into multiplication problems.

So we don't really divide fractions per se.

I mean, technically, yes, that's what we're doing, but we're going to change the problem into a multiplication problem.

And being that you already know how to multiply fractions, you're going to see how easy this is.

So let's take a look at this problem, two thirds divided by one fifth.

Okay, so we're going to change this problem from division to multiplication.

Okay, but how do we do that?

Well, you have to pay attention to the fraction to the right of the division symbol.

100

Okay, this division operator right there, the fraction to the right of it is one-fifth.

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Okay, now here's how you change this problem to multiplication.

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You simply take the fraction to the right of the division symbol and you flip it upside down.

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So if this is 1 over 5, we're going to flip it upside down to 5 over 1.

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That's called the reciprocal.

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So that's all you do.

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And now this becomes a multiplication problem.

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So again, we're going to go from division to multiplication by flipping the fraction to the right of the division symbol.

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And now you already know how to multiply fractions.

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We're simply going to multiply across.

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So that's going to be 2 times 5, of course, is 10.

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And 3 times 1 is 3.

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And we are done.

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OK, so we'll give ourselves a nice little happy face for this.

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So here you know how to multiply.

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Here you know how to divide.

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Now, of course, you're going to want to practice this, but literally this is how easy this is.

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OK, so I told you this first rule, rule one, which covers multiplication and division, is very easy.

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OK, let's go ahead and add in something here.

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OK, just to make sure we're kind of complete.

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because some of you might be thinking, well, what about when we're dealing with mixed number fractions?

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Something like 2 and 1 5th, and we're going to multiply it by 1 and 1 3rd.

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Well, how does that work?

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Well, what you're going to want to do is rewrite these fractions as improper fractions.

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So, for example, 2 and 1 5th, I can write that as a single fraction by going 5 times 2.

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That's what?

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That's 10.

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Then I add 1.

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So that's going to be 10 plus 1 or 11 5ths.

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OK, and then one

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and one third as an improper fraction is three times one is three plus one is four or four thirds.

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So I'm going to rewrite this problem with mixed number fractions as eleven fifths times four thirds.

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OK, so now I'm just going to simply multiply across.

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So eleven times four, eleven times four is 44 and five times three is 15.

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And we are done.

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OK, again, very, very easy.

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and now we're 50% done with learning fractions.

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Okay, you already know how to multiply and divide.

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Okay, so let's get into our second rule here.

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And our rule number two is going to be focusing in on how to add and subtract fractions.

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And again, this is where you have to be thinking about the lowest common denominator.

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But I am going to give you a lovely hack here in just one second.

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it's like a shortcut okay

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that you don't even have to do fraction problems using the lowest common denominator you still need to understand it

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but i'm going to give you a technique that

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if you're struggling with fractions you can do addition

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and subtraction problems right correct every single time okay

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so let's go ahead and first just review some basic concepts about adding

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and subtracting fractions again when you're adding fractions or subtracting fractions, you're doing the exact same thing.

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Okay, that's why I'm going to just classify this as one rule.

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We'll call it rule number two.

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All right, so here's the deal.

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You can add or subtract fractions if you have the exact same denominator.

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Okay, the denominators must be the same.

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So let's take a look at this problem here.

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I have two sevenths plus one seventh.

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So I'll look at the denominators.

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I'm like, are they the same?

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This is seven, that's seven.

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So our answer is going to have a 7 as a denominator.

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So if the denominators are the same, we're going to keep that denominator in our answer, and then we're going to add the numerators.

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So this is 2, and this is 1.

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So that's going to be 2 plus 1.

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Okay, so we're going to add the respective numerators if the denominators are the same.

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If this was a subtraction problem, I would subtract the numerators.

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But in this case, 2 plus 1 is 3, and that would be 3 over 7, and that is it.

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Okay, so pretty easy to add and subtract fractions, especially if the denominators are the same.

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OK, so if the denominators are the same, we simply add or subtract the respective numerators, and we'll get our final answer.

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So again, that's not that difficult.

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But here is where the fun starts.

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Okay.

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So what happens when the denominators are not the same?

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Well, this is when most people go, oh, this is where I had to think about the LCD and everything else.

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That is true.

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Okay.

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You do need to think about what the LCD is and you have to learn how to find the LCD.

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This is very, very important.

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However, what I'm doing in this little lesson is giving you a shortcut.

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All right, so if I have this fraction, 2 fifths plus 1 third, I want to add these together, but they don't have the same denominator.

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So I'm going to have to rewrite this fraction and this fraction such that they do have the same denominator.

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Okay, so I'm going to have to write 2 fifths as, I'm going to have to find an equivalent fraction of 2 fifths.

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Okay, it's the same thing as two fifths, but it has a denominator that's going to be the same as an equivalent fraction here that has the same denominator.

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In other words, I've got to match up these denominators, and we're trying to find that same denominator that these two fractions have in common.

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That is the lowest common denominator.

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Okay, so if I, most people would say, okay, here, what is the LCD?

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Most of you would say it's 15.

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Okay, you would probably know that, and you would be correct.

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Now, I have additional videos on how to find the LCD.

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It's important that you understand that, but for this particular video, let's just assume that you know that, okay, yeah, this is the lowest common denominator is 15.

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So that means I'm going to rewrite each fraction such that the denominator is 15, okay?

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So here, how do I turn this 5 into 15?

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Well, I have to multiply it by 3, okay?

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So if I multiply the denominator by 3, I also have to multiply the numerator by 3, and that's how I get 3 times 2.

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This is 6 over 15.

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So that's how this first fraction right here I rewrite as 6 over 15.

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And then a 1 third, how do I get this to be a 15?

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I multiply by 5, so I'm going to multiply the numerator by 5.

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So I get 5 over 15.

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So I'm showing you kind of the long way that we deal with fractions.

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Okay, this is probably the way most of you have learned this problem or learned how to add or subtract fractions, and that's perfectly fine.

200

Again, I'm going to show you a nice little shortcut here in a second.

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But now I rewrote these fractions such that they have a common denominator, the lowest common denominator.

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So remember the rule, when the denominators are the same, I simply go ahead and add or subtract the respective numerators.

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So this is 6 plus 5.

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Of course, that's 11.

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So I got 11 15ths.

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Okay, so that is the answer.

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And this is the way most of you have learned this using the LCD method.

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Okay, so we'll go ahead and put this here, LCD.

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But hey, what if you don't want to deal with LCD?

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Well, you're in luck because you're watching this video.

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I want you to remember this little symbol right here.

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Okay, it's called a bow tie.

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Okay, just like you wear a little bow tie.

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I don't think too many people wear bow ties anymore.

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But this is the general pattern I want you to remember.

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Okay, this is going to be super, super easy.

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This works with variables.

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It works with any addition or subtraction fraction problem.

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You will get the answer right 100% of the time.

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Okay, so here we go.

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I want to show you how easy this is.

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Here's the pattern.

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You start in the bottom right and you go this way.

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Okay, you always start from the bottom right and you go this way first.

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Okay, so we're going to do 3 times 2.

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We're going to put our answer there.

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And then you're going to go from the bottom left and you're going to go this way.

228

So you can see it has like a little crisscross pattern.

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We're going to put that answer there.

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So let's do that now.

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3 times 2 is what?

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6. Now, this is an addition problem.

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That's going to be plus 5 times 1 is what?

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5, okay?

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That's our numerator.

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So this times this will always start from here, okay, because this will make an impact if you're subtracting fractions.

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So 3 times 2 is 6, plus 5 times 1 is 5, over our denominator is going to be 5 times 3.

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So here is our bow tie.

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5 times 3 is 15.

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Now we simplify that.

241

6 plus 5 is what?

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11 over 15.

243

It is exactly the same thing we have right here.

244

But look, I didn't have to mess around and change these fractions, even think about the LCD.

245

I simply just thought about my little bow tie method.

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I went this times this plus this times this over this times this.

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Okay.

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This is the bow tie method.

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You're going to want to know this not only for arithmetic, but for variable fractions as well.

250

So let's go ahead and practice this right now.

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Okay.

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So let's take a look at this problem here.

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Four ninths minus one half.

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Okay.

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So what is the LCD of this problem right here?

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Okay.

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Well, everyone, you know, we're looking at this video right now.

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You can see I kind of wrote it.

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But most of you, hopefully, could say, well, the LCD is 18.

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So that's good.

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If you know that the LCD is 18, that's excellent.

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So you can, you know, write each of these denominators as 18.

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You're like, well, I would just multiply this by 2 and this by 9.

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And now I'm going to go ahead and fix this thing up right here and, you know, use the LCD method.

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So some of you, you know, may be non-believers in the LCD, this little bow tie method I'm actually saying.

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You're like, no, I'll just stick with LCD.

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And that's good.

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You need to know that.

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But what about this situation?

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Okay.

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What's the LCD between these two denominators?

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Okay.

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Now this is where it gets fun because most people are going to be like, all right, I'm not watching your video anymore.

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I'm not doing this problem.

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Okay.

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So to find the LCD of these two denominators, this gets much more interesting.

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All right.

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So a lot of you are like, oh, okay.

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Okay.

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I'll maybe I'll just use your bow tie method because to do this problem, it's super easy.

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I would just go this times this plus this times this over this times this.

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Now I'm going to have big numbers, but I will add these fractions and I will have the exact right answer.

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The only thing with the bow tie method is that oftentimes, well, not oftentimes, sometimes you're going to need to reduce, okay, your final answer.

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But your answer will be correct, but you may need to reduce your final answer.

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But you will have an accurate answer, okay?

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So that's the main idea.

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All right, so let's go ahead and practice this bow tie method right now.

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Okay, so again, let's take a look at this first problem.

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We're not going to be thinking about the LCD.

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We're going to go, okay, I'm going to start from the bottom right.

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I'm going to multiply it this way.

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Okay, so 2 times 4, that's 8.

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This is a subtraction problem, so I have a subtraction operator there.

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And then 9 times 1 is 9.

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And then my denominator is going to be 9 times 2, which, of course, is 18.

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Now, when you're subtracting fractions especially, that's why the order is very, very important.

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And that's why you have to start this way, okay, because we have our 8 there.

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If you put your 8 right here, you would have a different sign.

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So when you're subtracting fractions or subtracting any number, you've got to be very, very careful that you have the correct sign.

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So 8 minus 9 is the same thing as 8 plus negative 9, which is negative 1, okay?

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So the final answer here is negative 1 over 18, okay?

302

Negative 1 over 18, that is the answer.

303

But again, we're using that bow tie method.

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I don't have to think about it.

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I just have to follow the procedure and make sure I do the arithmetic correctly.

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All right, so how does this work with mixed number fractions?

307

So basically the same way we did with multiplication and division, we want to rewrite each of these respective mixed number fractions as improper fractions.

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So 3 and 1 third is the same thing as 10 thirds.

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And then 2 and 3 fifths is the same thing as 13 fifths.

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So now I can go ahead and just do the bow tie method.

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So that's 5 times 10 is 50.

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3 times 13 is 39 over 3 times 5, which is 15.

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And I'm just going to go ahead and add my numerators.

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50 plus 39, that's 89 over 15.

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And we are done.

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Okay, so hopefully your facial expression about fractions went from here, or maybe like it was total confusion from here to here to here,

317

to a big, lovely, happy face, okay?

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This is the whole idea behind this video, right?

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I'm giving you years and years and years of experience.

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Not that it takes years and years of experience or years to learn fractions, but being a math teacher,

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I'm an observer of where students make a lot of mistakes.

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And I'm telling you right now, students make a ton of mistakes when dealing with fractions

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because they forget how to deal with fractions because we have this little thing called a calculator that we use, and we don't really do a lot of,

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we don't do arithmetic as much as hand arithmetic as much as we should, okay?

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So when you get away from working with fractions, you're going to forget, but hopefully this video you'll be, you know, you'll remember it.

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You'll be like, oh, I remember that.

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There's like two rules.

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Remember that little bow tie method.

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And, of course, it doesn't mean that you don't have to learn the LCD because you do, but again you're going to have to know how to deal with fraction problems

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because they're everywhere and obviously any level of math whether it's elementary math middle school math

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and high school math and beyond okay fractions are everywhere

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so hopefully this little video helps you out and if

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that is the case go ahead and consider smashing that like button

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and i want to strongly suggest that you follow through in practice okay so i'm going to give you a couple suggestions.

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I definitely would check out my pre-algebra course.

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I have a whole chapter on fractions in my math program.

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And then I have additional videos on my YouTube channel about fractions.

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I have a ton of videos actually about fractions as well.

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But if you're new to my YouTube channel, hopefully you'll consider subscribing.

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I've been on YouTube for 10 plus years.

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I have over a thousand plus math videos from basic math to advanced math like calculus and everything in between.

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So if you like my teaching style, please take advantage of my content.

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I make it for you, but my best math help will always be within my math help program.

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Okay, so with that being said, I definitely wish you all the best in your mathematics adventures.

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Thank you for your time and have a great day.

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Что такое техника Shadowing?

Shadowing — это научно обоснованная техника изучения языка, изначально разработанная для подготовки профессиональных переводчиков и популяризированная полиглотом доктором Александром Аргуэльесом. Метод прост, но эффективен: вы слушаете аудио на английском от носителей языка и немедленно повторяете вслух — как тень, следующая за говорящим с задержкой в 1–2 секунды. В отличие от пассивного прослушивания или грамматических упражнений, Shadowing заставляет мозг и мышцы рта одновременно обрабатывать и воспроизводить реальные речевые паттерны. Исследования показывают, что это значительно улучшает точность произношения, интонацию, ритм, связную речь, понимание на слух и беглость речи — что делает его одним из самых эффективных методов для подготовки к IELTS Speaking и реального общения на английском.

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