A challenge: JNewman85 claims there are half a dozen ways to solve this equation in his post. How many can you find?

Which would you expect students to gravitate towards? Which would you like students to gravitate towards?

Note: mentioning bad words and linking the corresponding document doesn’t guarantee a link back from this blog, but it sure doesn’t hurt!

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Thanks for the shout out! I’d like to place the disclaimer that many of the half-dozen use ridiculously specific strategies that would not be good for students to learn or use (one of which I saw today while observing a geometry class!).

My pre-algebra daughter solved this by thinking of equivalent fractions. To go from the known fraction to the unknown equivalent, you have to do the same thing (that is, multiply or divide by the same number) to the numerator and the denominator. Since 3 must be divided by 3 to get 1 (the numerator), then 2 must be divided by 3 to get x.

I solved it by flipping (which is probably one of your “bad words”?): If two fractions are equal, then their reciprocals must also be equal.

I can also think of two completely different ways to use a common denominator to figure this out, but I’m curious to know what JNewman’s “ridiculously specific” strategies might be — I haven’t come up with any methods that aren’t generally applicable.

Yeah, those two of the three main ways I’d solve this problem (I think the “non-bad word” for flipping they like to use are “take the reciprocal” of both sides, but yes, I’ll sometimes resort to saying “flipping” to help students see what I mean). I like how your daughter solved the problem in such an intuitive way–I’ll bet she’s comfortable with fractions unlike most students.

Since I’m pretty sure nobody would share this as a way to solve this fraction (I hope), I’ll share the “ridiculously specific” strategy that I saw another teacher show students today:

https://docs.google.com/drawings/d/1WKA6rfOWFfFINyBQQ_X19_NJZu0SxPdrR17lcO11Gjo/edit?usp=sharing

Explanation if the link doesn’t work: you take the number above the variable in the fraction, multiply it by the other denominator, then follow that by dividing by the numerator of that denominator. Besides being totally counter-intuitive (to me, at least), this would be a complex rule to memorize in such a specific case, that it wouldn’t be worth it.

Ugh! That’s like cross-multiplication, but you’ve killed it and it’s come back as a zombie.

I wish there were a like button on comments! It being my blog I could probably make that happen, but it’s easier to just say LIKE to the zombie reference.