Why is the number of threes in a number related to the number divided into three equal parts?

Question

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?

Answer 1

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.

Answer 2

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 $51$.