I was trying to create a video illustrating "one divided by four" by showing a circle being cut into four slices. However, I ran into a problem--namely once the circle had been cut into four slices, there were 4 one-fourths left on the screen.
My child says, "but why are there four one-fourths left if the answer is one-fourth?"
What has gone wrong in my illustration?
Here's a video I made to illustrate my problem if it's not clear.
On
Visually, you took one circle and cut it up into $4$ equal slices, so in your child's point of view he or she could interpret this as $1\times 4 = 4$. You start with one piece, then after cutting it into four slices now you have a total of $4$ pieces. You could even extend it to do other multiplicative illustrations.
If you wanted to make an illustration for division, what I would do is this: say we have one circle, then we want to divide it equally among four people. So after you split it apart into fourths, you need to then make sure each person gets the same amount, and the answer to the division problem would be the number of pieces that any one person has.
For this example, each person would have $1/4$ of a piece, thus $1÷4=\frac14$
This is the best way I can think of to describe it in a way to be seen visually.
On
This is a language problem. When you cut up the cake you have not yet "divided it by four", though you have "divided it into fourths" Dividing it into four pieces is a step you take to prepare for dividing it by four.
Here's my suggestion for your video: include pictures of your kid and three of their friends. After you cut it into four (equal) pieces, give each kid one of the slices -- that's when the dividing by four happens, and each kid has one fourth of the cake.
Now take a new cake, and cut it into eight equal pieces. At that point we've got eight eighths, which is one full cake -- we haven't "divided by four yet". Now share it equally -- that's when "divide by four" takes place. How much does each kid get? In this case it's two eights, so you can also say that one divided by four is two eighths. It's the same amount of cake for each kid, it's just cut up a bit differently, so $$\dfrac{2}{8} = \dfrac{1}{4}$$and you can explain that the same fractions can be expressed in more than one way.
And if your kid is following this just fine you can give a more complicated example. Draw one and a half cakes and share it among six kids. To do this you have to cut the full cake into four pieces, and the half cake into two pieces ("You see, a half of a half is a fourth."), and then you can share it evenly and each kid gets one quarter. You can write this for them as $1 \dfrac{1}{2} \div 6 = \dfrac{1}{4}$, which is pretty advanced math for a kid just learning fractions.
(P.S. What a nice thing to do for your kid.)
When your 'cake' is divided into $4$ parts, $\color{red}{one}$-fourth is how large $\color{red}{one}$ piece is, the key word being each. Similarly, $\color{red}{two}$-fourths is how large the $\color{red}{two}$ pieces are combined.