# Is there any toy for learning algebraic manipulation of fractions?

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.

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Your question prompted me to enter fractions toy into Google Images. I suppose I shouldn't be surprised that most of these toys involve shapes cut up into some number of congruent pieces... –  Guess who it is. Sep 15 '11 at 8:59
This is an interesting question. I've always felt algebraic manipulation like this was mechanical-feeling enough it could be embodied in a toy's admissible maneuvers. Although it wouldn't help understand why these moves are valid, it should at least help in the memorization department as far as pedagogy is concerned. @J.M.: Incidentally, google searching fraction manipulation toy gives this page as the first result, even though it's only existed a few minutes. –  anon Sep 15 '11 at 9:00
anon, I'm not sure it wouldn't help understand why these moves are valid, unless you are saying that nobody understands why. Its constraints should play the role of an axiom, right? I agree that the pie pieces and such aren't much help in understanding algebra though. –  Dan Brumleve Sep 15 '11 at 9:12
I suspect you want four controls (button/lever/knob etc) each one of which switch a variable from one side of the equals sign to the other, and 16 different results. –  Henry Sep 15 '11 at 10:06
A pencil and lots of paper did the trick for generations... –  Mariano Suárez-Alvarez Nov 1 '11 at 0:06

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).

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In software, you may have a button for each of the items, $a, b, c, d$. Pressing that button would move it to the other side. –  Shaun Ault Sep 15 '11 at 12:22

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.

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