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CCSS.Math:

we want to figure out how many balloons we have here and obviously we could just count these but now we have other ways of thinking about it especially because they're arranged in this nice in this nice array this nice grid pattern here and the reason why it's useful to not just always have to count it but to be able to use a little bit of multiplication with the number of rows and the number of columns is that you might run into things and you will run into things where it's very hard to count each of the objects individually but it's not might be a little bit easier to count the rows and to count the columns so for example right over here we see that we have one two three four rows and we have and we have 1 2 3 4 5 6 7 columns so this is a you could view this as a 4 you could view this as an array of objects where we have four rows let me write that down we have four rows and we have seven columns seven columns and you might already remember that we can calculate the total number of objects by multiplying the rows times the columns four rows times seven columns now why does this work why do we why does this why will this give us the actual the actual number of objects well we could view this we have four rows so we have four groups of things and how many things are in each of those in each of those rows well the number of columns we have seven things in each of those four rows so for four groups of seven or you could view it the other way around you could view that you each column is a group so then you have seven groups and how many objects you have in each well that's what the rows tell you you have four things in each of those in each of those columns and we already know that both of these both of these quantities are going to find the exact same number the number of things that we have right over here so these two things are equivalent 4 times 7 is equal to 7 times 4 and there's a bunch of ways that we can calculate either one of these we can skip count by four we say 4 times 4 8 12 16 20 24 28 and let's see let me make sure that's 7 so 4 times 1 2 3 4 5 6 7 so we get 28 we could just calculate their 28 objects here and likewise we could skip out skip count by sevens we could go 7 7 times 2 is 14 times 3 is 21 times 4 is 28 just adding 7 every time and so we could get 28 the other way let's do that in that same color we could get 28 the other way but what if you had a situation where you didn't know where you either didn't want to do these techniques it would have been hard to do these techniques or you didn't know what 4 times 7 was off of the top of your head which you should know at some point in the very very near future is there any way to break this down into something that maybe you do know or maybe that's a little bit easier to compute well you could realize that 7 columns is the same thing as 5 columns and then 2 columns so you could view 7 columns as you could view it as 5 columns so this is 5 columns right over here plus 2 columns plus juice in another color plus 2 plus 2 columns so that's just like saying that 4 times 7 4 times 7 we do the 7 in that purple color I want to stay color-coded is the same thing as 4 4 times 5 plus 2 5 5 plus 2 and all I do is I replace the 7 with a 5 + 2 7 has been replaced with the 5 plus 2 now why why is this interesting well now I can break this up into two separate arrays so I could say well look there's a 4 there's the array that has 4 rows and 2 columns right over here and then there's the array that has 4 rows and 5 columns right over here so how many objects are in this one in the yellow one right over here well there's 4 times 5 four times five objects so there's four times five objects in the yellow grid or yellow array and how many and how many in the in this orangish looking thing well there's going to be 4 times 2 4 times 2 4 times 2 and if we take the sum of the 4 times the 5 and the 4 times the 2 what are we going to get well we're going to get the 4 times the 7 we're going to get the 4 times the 5 plus 2 so if we take the sum of these things and we want to do the multiplication first so I'll just put a parentheses around that to emphasize that that this is going to be the same thing as these things as these things up here and so you might say well I know what 4 times 5 is 4 times 5 is 20 4 times 5 is 20 4 times 2 is 8 20 plus 8 is 28 and you might say okay Sal you know that's that was kind of a you know I get it 4 times 7 is 28 4 times which is the same thing as 4 times 5 plus 2 and I see that that's the same thing as 4 times 5 2 plus 4 times 2 and actually this is called the distributive property that 4 times 5 plus 2 is the same thing as 4 times 5 plus 4 times 2 but you know I could just I could just do with one of these first techniques you talked about why is this distributive property that you just showed me why is this useful for computing well or doing multiplication up problems well let me give you a a a slightly more difficult a slightly more difficult one let's imagine you wanted to multiply you wanted to multiply 6 times 6 times let's write it as 6 times 36 make sure I don't need to write that parenthesis 6 times 36 so how could you do this well you could decompose 36 into 2 products or into two numbers where it's easier to find the product of that and 6 so for example 36 is the same thing as 30 plus 6 so this is going to be equal to 6 times 30 plus 6 6 times 30 plus 6 and what's this going to be what we just saw 6 times these two things added together first this is going to be equal to this is going to be equal to 6 times 36 times 36 times 30 plus 6 times 6 plus 6 times 6 6 times 6 notice we distributed this 6 6 times 30 plus 6 times 6 now why is this useful why was this useful at all I'm going to put parentheses to emphasize we're going to do the multiplication first in general when you see multiplication in addition in a row like this or end division you wonder your multiplication and division first then do your addition and subtraction so what's 6 times 30 well this is easier to calculate 6 times 3 we know to be 18 so 6 times 30 is going to be 180 and 6 times 6 well we know that's going to be 36 so this is going to be 180 plus 36 and what's that going to be 180 plus 36 well 0 plus 6 is 6 8 plus 3 is 11 1 plus 1 is 2 so you just figure it out that 6 times 36 is equal to 216 and what we just did what we just did with the distributive property this is actually going to be how you're going to multiply all sorts of larger numbers way larger than what we just saw so the distributive property which hopefully you're pretty convinced by by based on how we broke things up is a super useful thing as you want to compute larger and larger numbers and you're going to find it even more useful when you go even further in your mathematical career