By Sean Alexander
YouTube video

This lesson we'll be covering fractions which are a math subject that causes the most trouble with students going all the way from junior high, all the way to the end of high school. Many students don't ever master the fraction. And what happens is, later in their high school career, it really comes to bite them. By the time you get to pre calculus, if the student is scared of fractions, you're going to start getting long multi-step problems of which a fraction is just a small part, but not the main part. And that's when students tend to melt down or get overwhelmed. So I just want to go over this once and for all. If you can learn what's in this lesson now you'll never have to think about it again.

So we're actually going to start with multiplying fractions because that's the easiest thing you can do with them. When it comes to multiplying, all you got to do is multiply across the top and multiply across the bottom. You don't need common denominators and none of that stuff. Simply two times four is eight and three times five is 15.

Now you'd always want to check to see if you can reduce the fraction. In this case you can not. So that's all there is to it. So for multiplying, you should multiply across the top and the bottom. One thing that you might see that causes students some grief sometimes is if you have a fraction multiplied by a whole number, like a non fraction. So two thirds times five.

Well, all you got to do in that case is turn the non fraction, the five in this case into a fraction by putting a one underneath, because five divided by one is five. Didn't change the problem. Then we just proceed as above. Two times five is 10 and three times one is three. Again, we want to look for reducing whenever possible, but again in this case, it is not. Multiplying is easy. Next we'll move on to division, which is only slightly more difficult. All you got to do... If you had say two thirds divided by four fifths, is we keep the first fraction the same. We just recopy it down.

We switch the division to a multiplication, and then we flip the second fraction over. So instead of four fifth, I'm going to write five fourth. And that's it. Basically so you flip the second fraction and now we're multiplying and we just learn how to multiply. That's easy, right? So in this case we have two times five is 10 and three times four is 12. Now, in this case, you can reduce... Basically when reducing, you just want to try and think of a number that divides evenly into both the numerator and denominator. Well, if they're both even numbers, we know that two definitely goes in. So two goes into 10, five times and two goes into 12, six times. So we get five, six.

So if you can multiply, you can divide. Now, before we move on, I always like to... At Alexander Tutoring, it's important that we know why the rules are the way they are. So it might seem a little arbitrary that we just flipped the second fraction and multiplied, why would we do that when we're dividing? So that's a question I'd like to briefly address now. And so I'm going to do it just by throwing something out there. A question I like to ask is forget fractions, forget everything we're doing, just use common sense, what is one divided by a half? That is how many times would a half fit into one? Wait for it. Most students can figure out that a half would fit into one, two times. Just common sense, right? If I have two cups, one's half full, I would have to pour two half full cups into the empty one to get a full cup.

So let's see how the math keeps track of that. So I just said one divided by one half. So we just used our logic to understand that half fits into one twice, but let's see how the rule keeps track of that. So I'm going to change the one into a one over one. We're going to change the division sign to multiplication. And then we flip the second fraction over to get to over one. Multiply across the top, multiply across the bottom. Two over one is two, which is the answer that we got just from using our logic. You can kind of see why we're flipping that second fraction. Let's do another one. I like to ask, how many times would one third fit into one? Just use logic, no math. How many times would one third fit into one? Pretty, obviously the answer is three, right? You could take... If you had a glass one third full of water, it would take three of those to fill up an empty glass. So let's just take a look at that.

So we're going to make the one, one over one. We change the divisions sign into a multiplication sign. Flip the second fraction over to get three over one equals... Multiply across the top to get three. Multiply across the bottom to get one, equals three. So you can see the math does the work for us. It gets us a result that we thought about. Now, what I'd like to ask is, one third fits into one three times, how do many times you think one third fits into two? Just using logic. Most people can use their logic to figure it out, that will be twice as many as before. One third should fit into to six times. Let's see how the math keeps track of that. So we have two divided by one third. We turn the two into a fraction by putting it over one. Change the division sign to multiplication and flip the second fraction as before.

Two times three is six. One times one is one to get six. The answer that we're looking for. So I hope you can see now why the rule is flip the second fraction and multiply. It's very important at Alexander Tutoring that we know the why behind everything that we do. It's not enough to just memorize a rule. Finally, let's move on to ironically, the most difficult thing you can do with fractions, which is add and subtract them. You think it'd be the easiest, but it's not. Because this is the case where we do need the common denominator. So before we jump into that, I want to just get into some logic here. I'm going to draw some pies or pizza or whatever. The first pie here, I'm going to cut into thirds. So both pies we'll say are the same size. The circle is the same size, but the first one I'm going to cut into thirds. And the second one I'm going to cut into fourths or quarters.

Now, if I said that I wanted to add two thirds... So what's two thirds. It would be the pink region, right? And I want to add that to say one fourth... One fourth would be this. So that's the visual. Let me write it in math terms. Let's see. So what I want to try and figure out is two thirds as diagrammed by the pie above, plus one fourth as illustrated by this guy. Now looking at the pies, it's pretty clear that we don't have three of anything, right? We can't just add the two pink ones to the one blue one because they're different. It's like saying I have two pencils and one pen, I don't have three of anything, right? So what we need to do is basically cut the pie so that the slices are the same size. And that way we can add same sized slices together. How do we do that? I'll show you.

So, you see the three here in the denominator, what we're going to do is steal that, take it to the other side. And we're going to multiply the second fraction by three over three. What is three over three? Well, it's just one. What is one times one fourth? Well, it's just one fourth. Point being the maneuver I just did does not change the original problem. We're still going to get the same two thirds plus one fourth. Anyway, now I'm going to take the four, the denominator from the other fraction, and I'm going to take it over to the first guy and I'm going to multiply him by four over four. You can see what that did is it guarantees that we get a common... The same thing in the bottom, right? Because you can see, we have a four times three in the first set of fractions and a four times three in the bottom.

So pulling the denominator from the opposite fraction guarantees what we call a common denominator. Now we're just multiplying fractions, which was where we started, right? That's the easy thing to do with them. And so that's no problem. So for the first one we have four times two is eight over four times three on the bottom is 12 plus, for this second guy we have one times three is three over four times three is 12. And now the last step, what we do is we add the top numbers eight plus three is 11, and we keep the denominator the bottom the same. So that's our final answer. Now I just want to try and do an explanation of this last step here. So what did we do? Let's go back to our pies.

We started with two thirds plus one fourth. And then once we did a little work here, we ended up with eight twelfths plus three twelfths. Now, in theory, those are the same sizes as the original pie. So let's just take a look at that real quick. So what I'm going to do is break the new pie into chunks of 12. So what's going to roughly look something like this. So there's 12 chunks and I am to break the other guy into chunks of 12 as well. It's not my prettiest work, but you get the idea.

So eight twelfths. So I'm going to color that in on pie number one. If we had one, two, three, four, five, six, seven, eight. Now, if you compare these two guys, look, you'll notice that it's the same amount of pie, right? Even though we chop it up into smaller slices, it's still the same quantity of pie. And then similarly for the other pie... We'll get our blue back here. Now we're looking at three twelfths instead of one fourth. Let's see what that gives us. So I'm going to count up three of these 12 slices... It's twelfth slices. One, two, three, and you'll notice it's the same amount of pie as one fourth as we just decrease the slice size. What's the point? You'll notice that in our two pies that we're trying to add now, we can add the slices together because all the slices are the same size. So in the first pie we have eight twelfth slices. In the second pie we have three twelfth slices. Those are the same size slices, and we are welcome to add them together, to get a grand total of 11, twelfth slices.

A lot of students like to add the bottom number as well. But you can see that that would be wrong because the bottom number says the size of the slice, the top number says how many of them we get. If you can remember everything that we did and learn how to do everything we did on this little lesson, you will be in so much better shape for your entire high school career and you'll just feel way better and things will go way smoother. Anyway, thanks for watching


  • Sean Alexander

    COMMAND PILOT, OWNER Sean has been a professional educator for 15 years and has taught math, physics, and astronomy at all levels.  His experience ranges from working at a high school for severe learning differences to teaching advanced physics at Stanford.  After completing his graduate work in theoretical physics Sean founded Alexander Tutoring, with the mission of revealing the deep connections between math and nature to as many students as possible. 

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