Multiplying fractions and whole numbers

let's think a little bit about what it means to multiply 2/3 times 6 one way to think about it is to literally take six two-thirds and add them together this is six two-thirds right over here and if we wanted to actually compute this this would be equal to well we're going to take these six twos and add them together so we could view it as 2 times 6 2 times 6 over 3 2 times 6 over 3 which is the same thing of course as 2 4 6 8 10 12 12 thirds 12 thirds and what is 12 thirds equal to well we could rewrite 12 as so this is equal to we could rewrite 12 as 3 plus 3 plus 3 plus 3 over the yellow 3 and this is going to be the same thing as I'm going to do it like this so I don't have to keep switching colors this is going to be the same thing as 3/3 plus 3/3 plus 3/3 plus 3/3 and each of these are obviously a whole each of these equal 1 that's 1 that's 1 that's 1 and that's 1 so this is going to be equal to 4 so that's one way to conceptualize 2/3 times 6 another way to think of it is as 2/3 of 6 so let's think about that let me draw a number line here and I'm going to draw the number line up to 6 so what I care about is a section of the number line that goes to 6 so that looks pretty good so this is 1 2 3 4 5 & 6 so if we want to take 2/3 of 6 we can think of this whole section of the number line between 0 & 6 as the hole and then we want to take 2/3 of that so how do we do that well we divide it into thirds to three equal sections so that's one equal sections two equal sections and three equal sections we want two of those thirds so we want 1/3 and 2/3 now where does that get us that gets us that gets us to 4 so we get obviously to the same answer would be in a tough situation if somehow we got 2 different answers either way 2/3 times 6 or 6 times 2/3 either way that is going to be equal to 4 but there are two different ways of viewing this this first way is literally viewing it as 2/3 6 times and this way is we're taking a fraction of the number 6 we're going two-thirds of the way to 6 which would get us to 4