Properties of multiplication

so if you look at each of these 4x6 grids it's pretty clear that there's 24 of these green circle things in each of them but what I want to show you is that you can get 24 is the product of three numbers in multiple different ways and it actually doesn't matter which products you take first or what order you actually do them in so let's think about this first so the way that I've colored it in I have these three groups of four if you look at the blue highlighting this is one group of four two groups of four three groups of four actually let me make it a little bit clearer one group of four two groups of four and three groups of four so these three columns you could view as three times four now we have another 3 times 4 right over here this is also 3 times 4 we have one group of 4 two groups of four and three groups of four so you could view these combined as two times three times four we have one three times four then we have another three times four so the whole thing we could view as the whole thing we could view as let me get myself some more space as two times let me do that in blue two times three times four that's the total number of balls here you can see it based on out with how it was colored and of course if you did three times four first you get twelve and then you multiply that times two you get 24 which is the total number of these green circle things and I encourage you now to look at these other two pause the video and think about what these would be the product of first looking at the blue grouping then looking at the purple grouping in the same way that we did right over here and verify that the product still equals 24 well I assume that you've paused the video so you see here in this first in this first I guess you could call it a zone we have two groups of four so this is 2 times 4 right over here we have one group of 4 another group of 4 that's 2 times 4 we have one group of 4 another group of 4 so this is also 2 times 4 if we look in this purple zone one group of 4 another group before so this is also 2 times 4 so we have 3 2 times fours so if we look at each of these are all together this is 3 times 2 times 4 so 3 times 2 times 4 2 times 4 notice I did a different order and I'm and I'm here I did 3 times 4 first here I'm doing 2 times 4 first but just like before 2 times 4 is 8 8 times 3 is still equal to 24 as it needs to because we have exactly 24 of these green circle things once again pause the video and try to do the same here look at the groupings in blue then look at the groupings in purple and try to express these 24 is a product of some kind of product of 2 3 & 4 well you see first we have these groupings of 3 so we have one grouping of three in this purple zone two groupings of three in this purple zone so you could do that as 2 times 3 and we have 1 3 and another 3 so in this purple is oh this is another 2 times 3 we have another 2 times 3 whoops I wrote 2 times 2 2 times 3 we have another 2 times 3 2 times 3 and then finally we have 1/4 2 times 3 so how many 2 times 3 do we have here well we have 1 2 3 4 2 time threes so this whole thing could be written as could be written as 4 times 2 times 3 2 times 3 now what's this going to be equal to well it needs to be equal to 24 and we can verify 2 times 3 is 6 times 4 is indeed 24 so what the whole idea of what I'm what I'm what I'm trying to show here is that the order in which you multiply does not matter how you associate it this is some let me make this very clear so whether you do let me pick a different example a completely new example so let's say that I have let's say that I have 4 times 5 times 6 you can do this multiplication in multiple ways you could do 4 times 5 first or you could do 4 times five times six first and you can verify that I encourage you to pause the video and verify that these two things are equivalent and this is actually called the associative property it doesn't matter how you associate these things which which of these that you do first also order does not matter we've seen that multiple multiple times whether you do this or you do five times four times six notice I swap the five and four this doesn't matter whether you do this or six times five times four doesn't matter here I swap the six and the five times four all of these are going to get the exact same value and I encourage you to pause the video so when we're talking about which one we do first where we were we which whether we do the 4 times 5 first or the 5 times 6 that's called the associative property it's kind of a fancy word for for a reasonably simple thing and when we're saying that order doesn't matter whether we're done it doesn't matter whether we do 4 times 5 or 5 times 4 that's called the commutative property and once again fancy word for a very simple thing just saying it doesn't matter what order I do it in