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CCSS.Math:

when we were first exposed to multiplication and division we saw that they had an inverse relationship or another way of thinking about is that they can undo each other so for example if I had two times four one interpretation of this is I could have four groups of two so that is one group of two two groups of two three groups of two and four groups of two and we learned many many videos ago that this of course is going to be equal to eight but we could express a very similar idea with division we could start with eight things so let's start with one two three four five six seven eight things so now we're going to start with the eight and we could say well let's try to divide that into four groups four equal groups well that's one equal group two equal groups three equal groups and four equal groups and we see when we take start with eight divided into four equal groups each group is going to have two objects in it so you probably see the relationship 2 times 4 is 8 8 divided by 4 is 2 and actually if we did 8 divided by 2 we would get 4 and this is generally true if I have something times something else is equal to whatever their product is if you take the product and divide by one of those two numbers you will get the other one and that idea applies to fractions and it actually makes a lot of sense with fractions so for example let's say that we started off with one-third one-third and we wanted to multiply that times 3 x 3 well there's a couple of ways we could visualize it actually let me just draw a diagram here so let's say that this block represents a hole and let me shade in 1/3 of it let me shade it in 1/3 of it so that's 1/3 we're going to multiply by 3 so we're going to have 3 of these 1 thirds or another way of thinking about is going to be 1/3 x or 1/3 plus another 1/3 plus another 1/3 that's our first third our second third and our third third and we get the whole this is 3/3 or one so this is going to be equal to one so you use the exact same idea if one-third times three is equal to one then that means that 1/3 that 1/3 must be equal to one-third must be equal to one-third and this comes straight out of how we first even thought about fractions the first way that we we ever thought about fractions was well let's start with a hole and that hole would be our 1 and let's divide it into 3 equal sections the same way that we divided this eight into four equal groups so if you divide this into three equal sections the size of each of those sections is going to be exactly is going to be exactly 1/3 now this leads to an interesting question that might be popping in your brain notice we have 1 as the numerator 3 is a denominator and we just said that this is equal to the numerator divided by the denominator 1 over 3 is the same thing as 1 divided by 3 is this always true for a fraction well let's just do the same thought experiment but let's do it with a different fraction let's try let's try let's take three-fourths and multiply it by 4 so multiply it by 4 so once again let's see if I could draw a fourth here let me do this in a new color so I'll take let's say that this block right over here is a whole I'm going to divide it into four equal sections so now I've divided into four it's and let me copy and paste it so I can use it multiple times so copy all right now 3/4 that's going to be why we could assume I didn't draw it perfect actually I could draw it a little bit better than that just to make the three the four equal sections actually look equal so that looks like a little bit better of a job I'm trying to make them four equal sections and let me copy that one so let me use it for later now 3/4 this is four equal sections and three fourths represents three of them one two three but now we're going to multiply it by four so we're going to have three fourths or times so we're going to need some more holes here so let's throw in another hole so this is one 3/4 now let me do the next 3/4 in another color so this is another that's 1/4 that's the second fourth that's a third fourth that's another 3/4 and now let's do so we've done two 3/4 just now let me make it clear this is the first 3/4 and then this plus this is the second 3/4 now let's do a third 3/4 and we're going to have to neat we're going to have to use a few we're going to have to use another hole right over here and I will do that in this color so my third 3/4 so here's a fourth here's my second fourth here's a third fourth so in green I have another 3/4 and now we need for 3/4 so let's do that in a color I have not used yet maybe white so that's 1/4 that's two forts and that is 3/4 so notice now I have now I have one 3/4 to 3/4 three 3/4 and for 3/4 and what did I do when I got those for 3/4 well it's pretty clear this is turned into three wholes so this is equal to three wholes well if 3/4 times four is equal to 3 that means that 3/4 3/4 is equal to 3/4 is equal to 3/4 so the same idea again 3 over 4 is the same thing as 3/4 and in general this is true the the fraction symbol here can be interpreted as division and looking at this diagram right here it make complete sense if you started with three wholes and you want to divide it into four equal groups one group two groups three groups four groups each group is going to be 3 is going to have three Forte's in it