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### Course: Arithmetic > Unit 7

Lesson 5: Multiply 2-digit numbers with area models# Multiplying with area model: 78 x 65

Sal uses an area model to multiply 78x65. Created by Sal Khan.

## Want to join the conversation?

- if you do 78 times 65 and then 65 times 78 will you get the same answer?(58 votes)
- 78*65=5070

65*78=5070

78*65=65*78

You will get the same answer because of the commutative property of multiplication.(114 votes)

- at2:00he says "i just made it a little more explicit" what does that mean in math?(i know it does not mean what it normally means)(34 votes)
- According to the Merriam-Webster dictionary, explicit means "fully revealed or expressed without vagueness, implication, or ambiguity : leaving no question as to meaning or intent".

Basically, he's saying that he made it easier to see; in this use, it's just a general phrase, not a math one.(32 votes)

- This is confusing me, please help me, smart people of the world.(31 votes)
- all you have to do is multiply the numbers then you add then all up and you get your answer its that easy(12 votes)

- sal how do this stuff are you really good at it(10 votes)
- Yah, i pretty good at it.(1 vote)

- my mom said please make this easily pls(12 votes)
- yes make it more easy(0 votes)

- me just wondering what he just did(10 votes)
- Say you have 3 × 25 on a rectangle. Add 3× 2 which would be 6 and then add 3× 5 which would be 15 add 6+ 15 which would be 21. That is your answer. I hope this helps!💖(0 votes)

- What are factors(6 votes)
- Factor, in mathematics, a number or algebraic expression that divides another number or expression evenly—i.e., with no remainder. For example, 3 and 6 are factors of 12 because 12 ÷ 3 = 4 exactly and 12 ÷ 6 = 2 exactly. The other factors of 12 are 1, 2, 4, and 12.(4 votes)

- Isn't the shape a square? If not I'm confused.(2 votes)
- No it isn't. It's a rectangle.

So basically we represent this multiplication with 4 major squares.

70 * 60

1. Tens Digit * Tens Digit (The largest one)

70 * 5

2. Tens Digit * Ones Digit (The light green)

8 * 60

3. Ones Digit * Tens Digit (The purple)

8 * 5

4. Ones Digit * Ones Digit (The blue)

Then the sum of all areas are equal to 78 * 65.

Let's observe 70 * 60 and 70 * 5.

When we add those areas together, we are basically saying

70 * 60 + 70 * 5

Factor the 70 out

70 * (60 + 5)

= 70 * 65

Now, let's observe 8 * 60 and 8 * 5.

8 * 60 + 8 * 5

= 8 * (60 + 5)

= 8 * 65

Last, we add these 2 areas together.

70 * 65 + 8 * 65

= (70 + 8) * 65

= 78 * 65(8 votes)

- 0:47is he doing partial products?(5 votes)
- what is another name for numerater(3 votes)
- I don't have another name, but I have a way of thinking about it: it "numerates" (or counts) how many of a fraction you have. So if I have 2/3, or "two thirds," the numerator is two (2) because I have two of a fraction, which is "a third." So maybe another unofficial name for the numerator would be "the counter" while another name for the denominator could be "the divider" or "the divisor." These are just names that I made up for these words, and are not commonly used by other mathematicians.(3 votes)

## Video transcript

I'm going to multiply 78
times 65 in a little less than standard way, but
hopefully it'll make some sense, and you 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 8 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 40, 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.