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let's look at the the common core standard from grade four that's part of
extend understanding a fraction equivalence and ordering
explain why a fraction a over b is equivalent to a fraction n times a
over n times b by using visual fashion models
with attention to how the number and size of the parts differ even though the two
fractions themselves are the same size
use this principle to recognize and generate equivalent fractions
now this one is a little hard to understand primarily because of the uh...
variables that are used so i think the main thing to clarify would be
what happens with n times a over n times b
and also uh...
this additional idea
that the two factions themselves being the same size even though the numbers and size of
the parts... differ
so
let's look at the these pieces here to clarify what uh... the standard is
really getting to
if
we take the expression
and try to make some sense out of it
uh... let's rewrite it
to where
we have uh...
have this in the form where it's vertical
and it
still doesn't make a whole lot of sense so let's keep going
and let's split that up into two fractions
now it's starting to get a little bit more clear
because now what you really have equal to since n over n
now represents some number divided by itself that's always going to be equal
to one
so by
getting to this point
it should be a little bit more clear that here's the reason why
there's no actual change why they would be equivalent fractions
when you multiply by n over n because you really multiplying
by one so then
you do have a over b still equal to itself
so let's take a visual model and let's uh... suppose that ah...
we have two thirds
as our fraction
and so we take our uh... two-thirds or we split up our rectangular model here
now let's suppose
that the n over n is three over three
so this is what happens
to your visual model
uh... we're going to take each of those one-thirds and split them up
into three additional pieces each
so that now what we have
is a total of six
smaller pieces but it's still equivalent to the original two larger ones
so we have six pieces
uh... and then as far as the total
that'll be out of a total of nine
okay let's go ahead and uh... take another situation
and uh...
continue to uh...
you make some sense out of this
now let's go with a circular model this time
and let's suppose
that our fraction is three over four
and this time
let's let our n over n the say it's two over two
so now what we have is
one two three four five six pieces
but again notice that nothing changed I still have the same whole
and
even though i've chopped it up into more pieces I now have six it's still the
same size out of now a total of
eight pieces
so the six-eights is still equivalent to the original three over four
but again notice what we're doing with the two over two in this
case is we're cutting up
each of the original three pieces into smaller ones
now we can also use a distance
model uh...
for this
okay let's rewrite this
to where it makes a little bit more since
so now
let's let
our original fraction be two over three again
but this time we are using a number line so now what we're dealing with is two-thirds of the
distance from zero to one
now let's go ahead and let our n over n be two over two
so again what happens
is that each of those two uh... pieces is divided up into two smaller ones
so that now we have a total of
four
out of
the original
well now six pieces
so again it's still the same distance
uh...
but it's still equivalent
i'd be sort of like if you divided something up into feet and then you split
those up into inches
uh... but you're still dealing with the same distance