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Studying for a test? Prepare with these 6 lessons on Modul 5: Brüche als Zahlen auf der Zahlengeraden.

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# Ganze Zahlen als Brüche

Video-Transkript

Let's say that this circle right
over here represents one whole. And we've divided this circle
into one, two, three, four, five equal sections. So each of these sections
represents 1/5 of the circle. And we've seen this already. 1/5, 1/5, 1/5, and 1/5. And if we were to then
color in some of this-- So, let's say we were to color
in three of these sections. So, that's one of the
sections right over there, another one right over here. So, we've colored
in two of the fifths and then three of the fifths. Notice, you put those
three 1/5's together, how much have we now shaded in? We've shaded in
3/5 of the whole. So the fraction that's
actually shaded in now is 3/5. 3/5 is what's shaded in. Now let's do something
in some ways a little bit simpler but also in some
ways kind of interesting. Let's start with
the whole again. So, once again,
this is one whole. Let me label it. One whole. And instead of dividing it
into five equal sections, I'm just going to divide
it into one equal section. So, if I were to shade
this in just like that-- so I'm shading in my one
whole, my one equal section, how many of the equal
sections are now shaded in? Well, just to remind ourselves,
there is one equal section, and I have shaded in exactly
one of those one equal section. I've shaded in the whole thing. Or I could say that 1/1--
which you'll never hear someone actually say-- is
shaded in, or I could say that the whole
thing is shaded in. So this is equal to one whole. So that's a whole. That's interesting. And I want you to keep
in mind, remember, look. We literally have one,
two, three fifths, and we literally call that 3/5. Now, this is one whole. Now, what happens if we were
to do this multiple times? Let me copy and paste that. So now I have another
one whole and then another one whole
right over here. So now, in total, how
many wholes do I have? Well, I have three. One, two, three wholes. And I've actually
shaded in three wholes. So this one right over
here is equal to-- Let me make sure I label it right. This right over
here is equal to, if I were to take
the combination, this is equal to 3. 3 wholes. Or if I were to think of it in
terms of numbers, just a number line, literally, this would
represent the number 3. But what's another way
I could represent it? Notice when I took a 1/5,
another 1/5, and another 1/5, I could call that 3/5. So now if I take one 1/1,
another 1/1, and another 1/1, well, I should be able to
call this 3/1, or 3 firsts, or however you want to call it. So I could call this 3 firsts. So this is interesting. Now we're seeing where the
top number on a fraction is larger than the bottom one. But another way of thinking
about this fraction symbol is that it's division. So you could view this as 3
divided by 1 is equal to 3. Or you could say, well,
look, 1 over 1 is a whole, and I now have three of them,
so this is equal to 3 wholes. So 3/1 is the same
thing as the number 3. And let me make sure,
let me emphasize that. Let me draw this
on a number line. So once again, let me
go all the way to 3. So 0, 1, 2, and 3. So one whole gets us exactly
one jump on the number line. So this right over here,
that gets us to 1/1. We do another jump, now
we've gotten 2 firsts, I guess you could say. We've essentially taken
two of these jumps. Each jump is 1/1. Now we are at 2/1, which
is the same thing as 2. You take another jump,
and we essentially get to, or we do get to 3/1, which
is the exact same thing as 3.