Why is the number of threes in a number related to the number divided into three equal parts? Like here's the video:
https://www.khanacademy.org/math/arithmetic-home/multiply-divide/division-intro/v/division-1?modal=1
Explanation 1 is at 4:08
Explanation 2 is at 5:31
How are the two related?
 A: Okay you are asking why splitting $51$ into $3$ groups will give you $17$ in each group and why splitting $51$ into groups of $3$ will give you $17$ groups of $3$. an both of them are the two different meanings of $51\div 3$?
Well that is because if you have $3$ groups of $17$ watermelons you will have $51$ watermelons and if you have $17$ groups of $3$ watermelons you will have $51$ watermelons.
So your question is why is $N$ groups of $k$ things the same thing as $k$ groups of $N$ things.
Well, the reason that is is:
Suppose you have $17$ groups of $3$ things.  So you have $17$ bags of watermelons each with $3$ watermelons.  If you take one watermelon from each bag and put them in a crate, you will have a crate of $17$ watermelons (one from each bag).  Each bag will have $2$ watermelons left.  You can do that again and have a second crate of $17$ watermelons.  Now each bag has $1$ watermelon in it.  Do it one last time and you get a third crate of $17$ watermelons and all the bags are empty.
So you have $3$ crates with each crate having $17$ watermelons.
So $17$ bags of $3$ watermelons each $=$ $3$ crates of $17$ watermelons each.
So $N$ groups of $k$ things = $k$ groups of $N$ things = $N\times k$ things.
And the value $51 \div 3$ can be thought of as either:  If I split into $3$ groups how much is in each group.  Or it can be thought of as: If I split this into groups of $3$ how many groups will I have.
These are the same because 
$3$ groups with $?????$ each adding up to $51$
is the same thing as 
$?????$ groups with $3$ each adding up to $51$.
A: The answer as a slogan is "Because multiplication is commutative." In other words, you can multiply numbers in any order.
For example, if you say that there are five threes in $15$, then we write that as the equation
$$15=3+3+3+3+3=5\times3$$
But $5\times3=3\times5$, so we also know:
$$15=3\times5=5+5+5$$
And so $15$ divided into three equal parts is
$$15\div3=5$$
This also works with fractions. You can say that $1=\frac13\times3$, meaning that there is one-third of a $3$ in a $1$. This is equivalent to $1=3\times\frac13$, meaning that there are three one-thirds in a $1$.
Now, if you want to know why order doesn't matter in multiplication, that's another topic. But as a hint, it has to do with rotating rectangles.
