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Click on the links below to familiarize yourself with various strategies.

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2-Digit by 1-Digit Multiplication (Area Model)

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Multiplication Area Model 4 digit by 1 digit

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Multiplication Area Model 4 digit by 1 digit

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Multiplying: 4 digits times 1 digit (using grid) | 4th grade | Khan Academy

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Multiplication Area Model2 Digit by 2 Digit 

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Multiplication Area Model2 Digit by 2 Digit 

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I'm going to multiply 78 times 65 in a little less than standard way, but hopefully it'll make some sense, and you'll realize that there's multiple ways that you can multiply. And this is actually the way that I multiply numbers in my head. So 78 times 65. And then we're actually going to think about what the different parts of this process represent on this area model. So 78 times 65. So I'm going to start just the way we normally start when we multiply. I'm going to start with this 5 in this ones place, and I'm going to say 5 times 8 is 40. And instead of just writing a 0 and carrying a 4 right over here, I'm just going to write the number 40. So this was the 5 times the 8. Now, I'm going to multiply the 5 times the 7. And we have to be a little bit careful here because 5 times-- this isn't just any 7, this is a 70. So what is 5 times 70? Well, 5 times 7 would be 35, so five times 70 is 350. So I'll write that down, 350. So just as a review, 5 times 8 is 40, 5 times 7 is 350. If you add these two together, this is going to be 5 times 78. Now let's go over to the 6. So let's multiply the 6 times the 8. Now we have to be careful again. This 6 is not just a regular 6, it's in the tens place. This is a 60. 60 times 8. Well, 6 times 8 is 48, so 60 times 8 is going to be 480. So it's going to be 480. And then 6 times 7. Well, that would be 42, but we have to be careful. This is 60 times 70, so we're going to have two zeroes at the end. This is 4,200, not just 42. So 6 times 7 is 4,200. And now we can add everything together. And this is a very similar process to what we do when we do the traditional method of multiplying. I just made it a little bit more explicit what parts are from multiplying which digits. But we can add everything together. In the ones place, we have a 0. In the tens place, we have 4 plus 5 is 9. 9 plus 8 is 17. And now we can carry a 1. 1 plus 3 is 4. 4 plus 4 is 8. 8 plus 2 is 10. Carry a 1, regroup of 1 even, and then you have a 5 right over there. So you get 5,070. Now, I want to think about-- I want to visualize what was going on here using this area model. So once again, we had 78. So I'm going to make this vertical length represent 78. So this distance right over here represents the 70. That's the 70, and then we'll make this distance right over here represent the 8. Let me make that a little bit cleaner so you see what I'm talking about. So this distance right over here represents the 8 and then we're going to multiply that times 65. So this distance right over here-- it's not drawn perfectly to scale, but it gives the idea-- this is 60. And then this distance right over here is the 5. So this whole distance is 65. This entire distance is 78. So when you multiply-- if you had a rectangle that's 65 units wide and you multiplied it, and it had a height of 78 units, then its area is going to be 78 times 65. It's area is going to be 5,070. Now, each of these parts we can map to one part of this area model right over here. When we multiplied the 5 times the 8 and got the 40, that was this section right over here, 5 times 8 is equal to 40. When we multiplied 5 times 70, and got 350, that's this right over here. 5 times 70, and we got 350. When we multiplied 60-- the 6 in the tens place-- 60 times 8 and got 480-- that's this right over here-- this is 60 times 8 is equal to 480. And then finally when we multiplied 60 times 70 and got 4,200-- that's this area right over here-- this area is 60 wide, 70 tall. So this is 60 times 70, which is equal to 4,200. And then when we added everything up to get 5,070, we were essentially just adding up the areas of each of these tiles. This big one is 4,200, that makes up most of the area, and we get 480 from this magenta one, then 350 from this greenish-yellow one, and then 40 from this greenish-blue one to get 5,070. So the area of this entire thing is 5,070 square units.

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Multiplying with area model: 78 x 65 (video) | Khan Academy

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I'm going to multiply 78 times 65 in a little less than standard way, but hopefully it'll make some sense, and you'll realize that there's multiple ways that you can multiply. And this is actually the way that I multiply numbers in my head. So 78 times 65.

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Subtraction Algorithm 4th Related to 3rd Grade

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Subtraction Algorithm 4th Related to 3rd Grade

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Let's try to subtract 659 from 971. And as soon as you start trying to do it, you face a problem. You go to the ones place, and you say, how am I going to subtract a 9 from a 1? And the answer lies in regrouping, taking value from one of the other places here and giving it to the ones place. And to understand that a little bit better, let me rewrite these two numbers. Let me expand it out. So this 9 is in the hundreds place, so it represents 900. The 7 is in the tens place, so it represents 7 tens. And then, this 1 is in the ones place, so it just represents 1. And then down here, this 6 represents 600. This 5 represents 5 tens, or 50. And then, this 9-- well, it still just represents 9 ones, or 9. And we're subtracting this. We're subtracting 600 plus 50 plus 9. Or another way of thinking about it, we're subtracting 600, we're subtracting 50, we are subtracting 9. So let's work it out over here. So this is the exact same problem, just written a little bit differently. And we still have the same issue. How do we subtract a larger number from a smaller number? And the solution lies in trying to take value from one of the other places. And the easiest place to go is-- look, we've got 70 here. Why don't we take 10 from here, and we'll be left with 60, and give that 10 to the ones place. So if you add 10 to 1, what do we have? Well, then we're going to have 11. Notice, I have not changed the value of the number. 971 is the same thing as 900 plus 60 plus 11. It's still 971. And now we can actually subtract. 11 minus 9 is 2. 60 minus 50 is 10. And 900 minus 600 is 300. So this subtraction should result in 300 plus 10 plus 2, which is 312. Now, let's do the exact same thing here, but we're going to do it without expanding it out. So same issue-- how do we subtract a 9 from a 1? Well, let's take a 10 from the tens place. We're going to regroup. So we're going to get rid of one of these tens, so we're only going to have 6 tens left in the tens place. And we're going to give that 10 to the ones place. So 10 plus 1 is 11. Now we are ready to subtract. 11 minus 9 is 2. 6 minus 5 is 1. 9 minus 6 is 3. We get-- let me do that same color-- 312.

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Worked example: Subtracting 3-digit numbers (regrouping) (video) | Khan Academy

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Let's try to subtract 659 from 971. And as soon as you start trying to do it, you face a problem. You go to the ones place, and you say, how am I going to subtract a 9 from a 1?

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No Remainders

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- [Instructor] In this video, we wanna compute what 833 divided by seven is. So I encourage you to pause this video and see if you can figure that out on your own. All right, now let's work through it together. And you might have appreciated, this is a little bit more difficult than things that we have done in the past. And in this video, I'm going to show you a method that your parents have probably not seen, but you'll see that it's kind of fun, and it's called division with partial quotients, which is a very fancy word, but as I said, it'll be fun. So, the first thing I will do is I will rewrite this as 833 divided by, divided by seven. So you can view these as the same expression. The reason why we do it this way is it formats it so it's a little bit easier to do our division with partial quotients. So the way that division with partial quotients works, and once again, it's not the way that your parents probably learned how to do it, is you just say, hey, how many times can seven go into 833? I don't have to get it exactly. I just wanna go under 833. And so my brain immediately thinks, well, seven hundreds is less than 833, so we're going to go into 833 at least 100 times. And so what we would do is we would write that hundred up here. We'd wanna be very careful about our place value. You can view this column as the hundreds column, this is the tens column, this is the ones column, and then we wanna see, how much do we have left over? How close did seven times 100 get us? So what we do is we multiply 100 times seven to get 700, and then we can subtract that 700 from 833 to figure out how much more we have left. And so 833 minus 700 is 133. And so we can then say, all right, we still have another 133 to go. So how many more times can seven go into this? Well, seven goes into 133, once again, you don't have to have it exactly. If you know seven times 10 is equal to 70, actually, let's go with that. We know we go at least 10 times. So let's write that up here. We're going at least 10 times, and to figure out how much more we have left, let's multiply 10 times seven to get 70, and then we can subtract, and we see that we have, let's see, three minus zero is three. 13 tens minus seven tens is six tens. So we have 63 left. So seven definitely can go into 63. We're gonna keep doing this until we have a number less than seven over here. So let's see, seven, how many times does seven go into 63? You might know from your multiplication tables that seven times nine is 63. So you could get it exactly. So you could just write that up here. We have nine more times to go into the number, and then you would say nine times seven is 63. You could say, hey, we got exactly there. We have nothing left over. And as long as this number is less than seven, you know that you can't divide seven anymore into our original number. And so, you're done. And so, how many times does seven go into 833? Well, we said it went 100 times, and then we were able to go another 10 times, and then we were able to go another nine times. And so what we wanna do is add these numbers. So you wanna add 100 plus 10 plus nine. When you add up all of them, what do you get? You get nine ones, one 10, one hundred. You get 119, so this is equal to 119. All I did is I added these up. Now, I wanna be very clear that you could do division with partial quotients and not do it exactly like this. That's kind of why it's fun. So let's do it another way. So let's say we wanna figure out again how many times does seven go into 833, 833? We could have said, maybe it goes 150 times. So what you could have said is you could have said, all right, my current guess or estimate is 150 times, that I could multiply 150 times seven. How would I do that? Let's see, zero times seven is zero, five times seven is 35. You can carry the three, so to speak. One times seven is seven plus that three is going to be 10. And so that gets us to 1050. Well, over here, we just finished overshooting. It doesn't go 150 times. There's nothing left over. So 150 is too high. So we would wanna backtrack that. And then you would go, well maybe I could go to 110. So let's try that out. So 110, and now let's multiply. Zero times seven is zero. One times seven is seven. One times seven is seven. So, 110 times seven is 770. So that works. It's less than 833, but let's see what we have left. So we subtract, we get a three here, and then, let's see, 83 tens minus 77 tens, that's six tens, and actually, that got us there a lot faster. So then you could just know that hey, seven goes into 63 times. But let's say we didn't know that. We could say, all right, let's say I'm gonna estimate it goes eight times. So you would put an eight up here. And then you say, how much did we have left over? Eight times seven is 56. You subtract, and then 63 minus 56 is exactly seven. And you say, okay, look, I can go one more time. So I'd write that up there. So one times seven is seven, and then you see we have nothing left over. So we are done. So how many times did it go in? One plus eight is nine plus 110 is 119. So hopefully, you find that interesting, and I really want you to think about why this is working. We're just trying to see how many times can we go in without overshooting it, and then what's left over? So how many more times can we go in, and then what's left over, and then how many more times can we go in, until what we have left over is less than seven, so that we can't go into it any more times.

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Introduction to division with partial quotients (no remainder) (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

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Multiplying fractions is about combining parts of a whole. When you multiply two fractions together, you're taking a part of a part. When you multiply a fraction by a whole number, you're taking multiple copies of that fraction. In both cases, the result is a new fraction that represents a different part of a whole. Created by Sal Khan.

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Video transcript

Let's think a little bit about what it means to multiply 2/3 times 6. One way to think about it is to literally take six 2/3 and add them together. This is six 2/3 right over here. And if we wanted to actually compute this, this would be equal to-- well, we're going to take these six 2's and add them together. So we could view it as 2 times 6 over 3. 2 times 6 over 3, which is the same thing, of course, as 2, 4, 6, 8, 10, 12, 12/3. And what is 12/3 equal to? Well, we could rewrite 12 as-- so this is equal to-- we could rewrite 12 as 3 plus 3 plus 3 plus 3 over the yellow 3. Let me do it like this so I don't have to keep switching colors. This is going to be the same thing as 3/3 plus 3/3 plus 3/3 plus 3/3. And each of these are obviously a whole. Each of these equal 1. That's 1 and that's 1, so this is going to be equal to 4. So that's one way to conceptualize 2/3 times 6. Another way to think of it is as 2/3 of 6. So let's think about that. Let me draw a number line here. And I'm going to draw the number line up to 6. So what I care about is the section of the number line that goes to 6. So that looks pretty good. So this is 1, 2, 3, 4, 5, and 6. So if we want to take 2/3 of 6, we can think of this whole section of the number line between 0 and 6 as the whole. And then we want to take 2/3 of that. So how do we do that? Well, we divide it into thirds, to three equals sections. So that's one equal section, two equal sections, and three equal sections. And we want two of those thirds. So we want 1/3 and 2/3. Now where does that get us? That gets us to 4. So we get, obviously, to the same answer. We would be in a tough situation if somehow we got two different answers. Either way, 2/3 times 6 or 6 times 2/3, either way, that is going to be equal to 4. But there are two different ways of viewing this. This first way is literally viewing it as 2/3 six times. And this way is we're taking a fraction of the number 6. We're going 2/3 of the way to 6, which would get us to 4.

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Multiplying fractions and whole numbers (video) | Khan Academy

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Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.