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CCSS.Math:

you have two fractions one four and five six and you want to rewrite them so they have the same denominator and have whole number numerators what numbers could you use for the denominator so here's our fractions 1/4 and 5/6 and we want to rewrite these fractions to have new denominators so we currently have a 4 and a 6 as our denominator and can we just be any newton thing like maybe 5 could we say let's change them both to have 5 is the denominator the answer is no we have to pick a multiple of 4 and 6 a multiple some number that we can multiply 4 and get this number as an answer so for example 4 for some multiples of 4 would be 4 times 1 is 4 4 times 2 is 8 4 times 3 is 12 and so on those are multiples of 4 and let's just pause here and look at why we have to pick a multiple of 4 and 6 why we can't just pick any number but we have to pick a multiple of our denominators so the fraction we were just talking about was 1/4 we could look at either one but let's look at 1/4 here we have a picture showing fourths and to show 1/4 we shade one of these 4 equal sized pieces well maybe I want to change this and I want to say I want to I want to is my numerator so to have a numerator of 2 I'm going to need to split this fourth up here into two pieces now I have two shaded pieces so can I say this is 2 1 2 out of 1 2 3 4 5 pieces it's not 2/5 because these are not equal size pieces so if I split this forth right here in half I need to split all of them in half and what I'm doing is doubling the amount of pieces so now this is 2 pieces this is 2 and this is two because we need equal sized pieces so now this is one two pieces out of one two three four five six seven eight equal sized pieces so two eight and you can see eight is a multiple of four because we multiplied by two and that's what we did we multiplied each of our pieces by two we also multiply our numerator by 2 because that was also doubled the amount of shaded pieces doubled when the entire amount of pieces doubled and we don't have to do this just with two we could do any multiple for for example we can do one more here if we again let's shade one-fourth one of the four pieces and maybe this time we want to split it into three equal sized pieces so to have a new numerator of three and these should be equal sized here's a numerator of three but it's we can't do our denominator yet because we don't have equal sized pieces so to get equal sized pieces we'll need to split each of these fourths into three so we are tripling the amount of pieces so now we have three shaded pieces out of a total of 1 2 3 4 5 6 7 8 9 10 11 12 out of 12 and I could have figured that out without even counting because I knew that we tripled this time we multiplied our original denominator times 3 we tripled and we also multiply our numerator times 3 so these are the multiples 8 12 and so on and so those are the denominators we can pick something that we can multiply our denominator by so we can multiply the entire amount of pieces and again so this is super clear one fourth and two eighths and three twelfths they all represent the same amount whether we had 1/4 the original here's to AIDS 312 they're all equivalent they all represent the same amount they're just different ways of writing the same number back to our original question what denominators can we use 4/4 and 4 sixths well we know we need to use multiples so let's look at the multiples for 4 we've already gone through some of these the first multiple of 4 is 4 times 1 which is 4 second multiple of 4 is 8 4 times 2 is 8 so we could split our fourth Cinda in half and get eighths or we could say 4 times 3 is 12 which we showed again where we split our fourth seat each fourth into three equal pieces or we could do 4 times 4 which is 16 4 times 5 is 20 4 times 6 is 24 and so on the reason I'm stopping at 24 is I've looked at my answer choices and I can see the largest possible answer is 24 so I don't need to write any larger multiples there are many many many more multiples of 4 but we don't need to list them all because the largest number we're going to can have to consider is 24 let's do the same for 6 we could leave our sixths alone 6 times 1 and keep 6 pieces or we could double our 6 6 times 2 would be 12 if we doubled the pieces we would have 12 pieces we could say 6 times 3 which is 18 or 6 times 4 we could divide each of our sixths into 4 pieces and we'd have 6 times 4 which is 24 or twenty-fourths and so on again I'll stop at 24 since it's the largest number we need to consider so down to our answer choices what numbers could we use for the denominator could we use 8 let's look at these lists 8 is a multiple of 4 so we could definitely split fourths into eighths but 8 is not a multiple of 6 so we cannot split 6 into eighths so 8 will not work as a denominator for both actions how about 12 12 we can see is a multiple of four and we showed that we drew that already and 12 is a multiple of six we could split our six into two equal pieces each and we would have twelfths so twelve does work twelve is a denominator a common denominator 4/4 and six 18 18 is here on the sixth we could split six into 18 because 18 is a multiple of six but it is not a multiple of four so we can rule out 18 18 is not a common denominator and 24 you may remember was the last number we wrote on both of them so yes 24 could be a denominator 4/4 and 6 so we could use either 12 or 24 and there's a lot more numbers we could use two as common denominators but from these choices we could use 12 or 24 as a common denominator 4/4 and 6 it's just a note lots of times people like to use the smallest one the least common denominator which in this case is 12 and it makes a lot of sense because it's easier to do computation with smaller numbers but you don't absolutely have to use the smallest one you could use 12s or 24th soar or lots of other options but again 12 says probably the simplest one to work with just because generally it's easier to work with smaller numbers but for this question the common denominators we can use from these choices 4/4 and 6 are 12 and 24