Is there any toy for learning algebraic manipulation of fractions?

Question

Is there any toy for learning algebraic manipulation of fractions? If you don't know of any, how would you design one?

What I'm imagining is something similar to a Rubik's cube whose manipulation produces only true equations in some number of variables, for example:

$\frac{a}{b} = \frac{c}{d}$

(turn a knob)

$a = \frac{b c}{d}$

(twist a handle)

$a d = b c$

(push a button)

$\frac{a d}{b c} = 1$

(flip a switch)

$\frac{a}{b c} = \frac{1}{d}$

(touch a screen)

$\frac{1}{b c} = \frac{1}{a d}$

As the last manipulation implies, I'm also wondering about how this could be done in software, as well as a mechanical toy.

Answer 1

This is just a hastily-drawn idea. The rule for moving each of $a, b, c,d$ across the $=$ sign is that it switches position in the fraction -- numerator becomes denominator and vice versa. So there are simple rods in the figure, and each of $a, b, c, d$ are beads on the rods (with a bit of friction, so they don't perpetually live in the $\frac{1}{bc} = \frac{1}{ad}$ configuration).

Answer 2

Since you gave Rubic's Cube as an example, this reminded me of a square. We may think the numbers $a, b, c$ and $d$ as the vertices of a square such that $a$ and $c$ (top vertices) represent the numerators and $b$ and $d$ (bottom vertices) represent denominators. We think that the vertices of the square gives us the equality top left / bottom left = top right / bottom right, i.e., $a/c = b/d$.

Also, instead of turning a knob or twisting a handle etc., when we move a number, it moves two vertices counterclockwise. For example, if we move $a$, then we get $1/c=b/ad$. Then the vertices of the square are 1, c, b, ad (starting from the top left vertex and continuing counterclockwise).

I do not know if it is worth considering, but it is just an idea.