All right
This is fourth-grade module five lesson three and in this lesson students are going to be taking just fractions
And they're going to be representing them not only as a decomposition of unit fractions
but they're also going to be taking that repeated addition concept and turning it into
multiplication
Let's get started on this
so here it says decompose each fraction
modeled by the tape diagram as a sum of unit fractions
So there's that unit fraction thing and then turn it into multiplication and so our first example here. We have a fraction
That's clearly 2/3 right cuz the whole thing is one whole it's been chopped up into 3 equal sized pieces
there's our 2
So our sum of unit fractions is 1/3 plus 1/3 and since we have two of them that could be 2 times
1/3 all right, so that's our kind of our leave in so let's take a look at what we've got here. So this is 3/4
And we know it's 3/4 because our hole has been cut into four pieces three of them are shaded
so as a sum of unit fractions that's going to be 1/4 plus 1/4 plus 1/4 and
Just like like if you were to say you know you know 4 Plus 4 plus
4 is the same thing as 3 Times 4
Here you've got 1/4 plus 1/4 plus 1/4, so this is the same thing as 3 times
1/4 and there we go
more of the same only this time we can see we have 4 pieces shaded in and
It takes three pieces to make one whole so our fraction is
4/3 and
if we want to write it as a sum of two
products
So there's a lot of ways we could do it, but I'm gonna say ok well right here is
3/3 plus we have 1/3 right so I'm going to take this fourth the and I'm gonna move it down here
Right here, and I'm gonna say well that is
let's see I could think of it as
1/3 Plus 1/3 Plus 1/3
That's this first part
plus this last third, which is this guy over here now this by the way is not a
You know expected as part of Eureka math, but I want to kind of make it obvious here
And so that makes this 3 times 1/3
plus and this is 1 times 1/3, so there's one way to
List it as a sum of two
Products now. Why are we doing that well we're doing that because down the road
we want students to be able to see this as one whole and
This is one-third so that we want students to see that eventually this is one and one-third
Kind of guiding our students towards a standard algorithm for turning
Improper fractions into mixed numbers
but really it's also nice because
this leads our students towards the idea of
Multiplying fractions which they will do in fifth grade now
There is no magic for what the answer could be I mean really if you think about it
We could say well 4/3 is
1/3 Plus 1/3 plus
1/3
Plus 1/3 and we could put our parentheses right here, and we can change this to
2 times 1/3
plus
2 times 1/3 I suppose. There's no rule
Right now at this point for how we use a sum of two products and so there's another all option
Now here. We're just kind of taking away some of the scaffolding. We're supposed to draw the tape diagram and then record
A couple of things so first let's draw the tape diagram, so in this case 3/5
here's our one whole
Five pieces and we're gonna shade in three of those five pieces
now if we're Gonna record it as a
Let's see and record the given fractions
as A
Decomposition into unit fraction so okay, so that means three fifths is going to be
1/5 Plus 1/5 Plus 1/5
and
Then if we want to record that as a multiplication, we'd say well, that's 3 times
1/5
so moving to 8/5 a
little bit more Tricky because our
Numerator 8 is larger than our denominator 5 but basically that 8 means we have 8 pieces
So I'm gonna Draw 8 pieces and I'm Gonna cut it into 4 pieces
then each one of those I'm going to cut into half so now we have 8 pieces and the fact that the
denominator is a 5 tells us that 1 2 3
4 5 these 5 pieces right here is one whole so that is
one whole right there and
Then we still have
These 3 pieces remaining over here, and I kind of overlapped a little bit there. We still have those 3 remaining pieces
so first off, what is our sum of
unit fractions so eight fifths is going to be
1/5 Plus 1/5 plus 1/5 plus 1/5
Plus 1/5 plus 1/5 1 2 3 4 5 6
7 8 so we have
1/5 8 times now. We are going to record it as a multiplication sentence, and so that's going to be 8 times
1/5
Now parents and teachers if you want with your high-flying students or just to enrich your students expiry
You could say well
8/5 is
You know you could think of it as these five fifths
Put together is five fifths
plus the remaining three fifths that would be these three fifths over here and
That leads us to one plus
three-fifths
Which is one and three fifths. It's kind of a foreshadowing of turning a mixed. I'm sorry an improper fraction
Into a mixed by the way, there's nothing
Improper about this fraction as in wrong with it
Improper makes it sound like it's wrong and really improper. Just means the numerator is bigger than the denominator
and
that wraps up
Fourth-grade module five lesson three where we're taking fractions representing it as a multiplication problem for example
3/8 is three Times 1/8