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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.

Dan Brumleve
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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... – J. M. ain't a mathematician Sep 15 '11 at 08:59
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    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 09:00
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    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 09:12
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    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
  • Henry, if it is possible to include some additive terms as well, that would be even better. Ideally there should be a simple way to read the equation from the state of the device, i.e. I can look at it and say "a divided by b equals c divided by d.". And how could I forget to mention the abacus as a related example? – Dan Brumleve Sep 15 '11 at 18:07
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    A pencil and lots of paper did the trick for generations... – Mariano Suárez-Álvarez Nov 01 '11 at 00:06

2 Answers2

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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).

Fraction Sliders

Shaun Ault
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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
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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.

Karatuğ Ozan Bircan
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