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# Multiplying with area model: 16 x 27

CCSS.Math:

## Video transcript

so I'm going to multiply 16 times 27 and I'm going to do it using something called an area model and the whole point of an area model is to really understand what's going on in the multiplication process so 16 you can represent literally as 10 plus 6 this one is in the tens place it represents 1/10 so we can represent that as 10 10 plus 10 plus 6 let me do the 6 in that same green color 10 plus 6 and let me mark off 10 and 6 here so this is 1 2 3 4 5 6 7 8 9 10 so this part in blue that's the 10 so I'm representing I've gone 10 slashes or I'm representing 10 boxes right over there and then the 6 actually I wanted to do in that green color and still did in that the 6 let's mark off 6 boxes 1 2 3 4 5 6 so 16 is this whole length the blue part is the 10 the green part is the 6 the 10 comes from the 1 in the tens place it's already 1/10 and this is the 3 6 ones now let's think about 27 27 well we already know that the 2 in the tens place is representing 20 so let's count off 20 so it's 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 so up until that point right over here this line right over here has a length of 20 this has a length of 20 but we're not just talking about 20 we're talking what 27 so it's 20 plus 7 so 20 plus 7 so let's count off 7 here 1 2 3 4 5 6 7 now if we look at the total number so if we have 16 times 27 the product 16 times 27 would give you the area of a 16 by 20 rectangles let me draw that so it's 16 this is a width of 16 let's bring let's bring a little line down right over here and then the height of my rectangle the way I've drawn it is 27 it is 27 so let me 27 this is 1 2 3 4 5 6 7 so the area of this rectangle is going to be 16 times 27 the area of this entire rectangle now we could do is break it up by parts because it'll be easier to compute it we can see what those with those it what part of the area those different products represent so for example we can think about what 10 so let me draw all that let me do it make this let me separate out the sections so let me draw so this section is 10 wide and then let me draw the section like this let me draw a section like this so we could figure out the areas of each of these sections and then the area of the entire rectangle which is going to be this product is going to be the area of all of these rectangles combined so we could first think about it well let's think about what 20 times 10 is well 20 times 10 is a fairly straightforward thing it's going to be it's going to be 200 you can already think of it as 2 times 1 and you have 2 zeroes there or you probably know in your head 20 times 10 is just going to be you can add a 0 here so it's going to be 200 so 20 times 10 is 200 and let me highlight that in so you have that's not the color I wanted to use let me use this blue right here so that's the blue from the 10 let me put some orange in there from the 20 to make it clear that this is the product of both of those numbers so 20 times 10 20 times 10 is 200 now what's 20 times 6 well 2 times 6 is 12 plus you have this zero here so this right over here is going to be 120 and it has the orange in it it has the orange in it so this is the 20 and then let me put some green in for the 6 20 times 20 time's the six now what's the area of this section right over here well it's seven high and it is 10 wide so it's going to be 7 times 10 or 70 and I'll have an area of 70 square units so we do it in this purple do purple and I'll throw some blue in there too it's kind of a fun art project I'll throw some blue in there too and then finally what's the area of this little section right over here it's 7 high and it's 6 wide so it's going to be 7 times 6 or 42 square units of area 42 square units of area and let me color it in so I got some magenta in there and then I got some green in there so I have some green in there right over there so what's the area of this entire thing going to be well it's going to be the 200 it's going to be the 200 plus the 120 plus let me do this way it's going to be 200 plus 120 Plus 70 plus 42 and when you add that up you get let's see in the ones place you get a 2 and you get 2 tens plus 7 tens is 9 tens plus 4 tens it's 13 tens which is the same thing as three tens and 100 and then this is a 4 did I add that up right let's see this will be 11 yep that looks right so this is 432 so this is going to be equal to this is equal to 432 and you might be saying hey Sal why did we go through all of this business you know I've seen before that if I take 16 I could take 16 times 27 like this you probably learned this type of this type of a process 16 times 27 and you say okay let's start with the 7 in the ones place and you do 7 times 6 is 42 is 42 you write the 2 down and you carry the 4 you're really just putting the 4 in the tens place because it's a 40 but they're right when you write when you did that 7 times the 6 we essentially calculated this right over there and then when you multiply the 7 times the 1 you're really multiplying 7 times 10 and then you're adding the 4 from the 42 you so when you do the seven times this one this is you're actually calculating this area and then when you carry when you add this carried four and put it right down here your takis entually taking the sum of both of these things because you're multiplying 7 times the 16 to get this area seven times the sixteen to get this area so let's just do it 7 times 1 is 7 plus 4 is 11 11 so 112 what you just figured out right over here is this area right over here so 7 times 1 or the 70 or 7 times the 10 or the 70 plus 7 times 6 the 42 now when you go into the tens place you've probably always said oh well you know I just throw a I just throw a zero down there but why do you throw a zero down there because this 2 is representing 10 so if you multiply a 2 times the 6 and you put it you know 12 and we put a 2 down and carry the 1 that wouldn't be right this is a 20 so that's why you put the zero there so let's scratch this out so we don't get confused 2 times 6 is 12 so we're used to writing the 2 down here to down here and carrying the 1 but notice we're really thinking about 20 times 60 is 120 just doing that we just calculated that right over there and then we do 20 times 10 is 200 but that we had that one from the 120 so this is going to be 300 so this is going to be 300 so what we just did when we multiply the 2 times the 16 we just calculated this total area we do the 2 times 20 times the 6 to get that and then we have the 20 times the 10 to get that when we did the 20 times the 6 we carried this one into the hundreds place and so we added it all together and got 320 and then this step right over here you're literally just finding the your the combined area the area of this plus the area of that the area of this plus the area of that so that's going to get us to we deserve a drum roll now 2 plus 0 is 2 1 plus 2 is 3 1 plus 3 is 4