Same question, but 6 different pieces of student work.

What is a lesson that you can glean from this question and these student responses? Say you’re the teacher of this class. What do you do next? (For full context: the submitted pictures are from a 9th grade Algebra 1 classroom.)

• Cry?

• crazedmummy

I thought, “run.”

• On a more serious note, I suggest revisiting fraction models to re-teach the meaning of equivalence and possibly the meaning of a fraction. It seems like the students have been immersed in symbolic notation of fractions for so long that they’ve forgotten the meaning behind it. Luckily there are a lot of free tools on the internet to model and play with fractions.

• Students seem to have some confusion with this and algebra (what you do to one side, do to the other all remains equal). Not much math training experience but I’d be curious as to how common this mistake is in 6-7th grade before most students see algebra. Just a thought.

• I would get some colored chips (or whatever manipulatives you can get) and set up a “game board” with four squares and an equal sign in the middle like ( )/( ) = ( )/( ). Then start them off with some leading set-ups like 2 reds / 3 blues = 4 reds / ???. Once they do a few of those, they can start coming up with their own equivalencies. Maybe get them to try to stump each other?

• How about squares on paper to color in, above and below a line?

• Aaron F.

In tropical arithmetic, the “product” of two numbers is their sum, so the first response given is totally correct: the “quotients” 4/5, 3/4, 5/6, and 6/7 are all equal to -1. 😉

I think this joke may actually have a grain of truth in it. There’s a very precise analogy between fractions and negative numbers, although this analogy is hidden by the way we usually talk about the latter. Maybe our confused student has accidentally grasped this analogy! In case you’re wondering, the analogy goes like this…

If you’re working with the whole numbers, {1, 2, 3, …}, you soon run into the problem that not every number has a multiplicative inverse. You fix the problem by introducing fractions, like 5/6 and 1/9. Fractions that can be reduced to the form n/1 correspond to ordinary whole numbers. Fractions that can be reduced to the form 1/n are the multiplicative inverses of whole numbers, and they have a special name: they’re called “unit fractions.”

If you’re working with the natural numbers, {0, 1, 2, …}, you soon run into the problem that not every number has an additive inverse. You fix the problem by introducing “additive fractions,” like 4 – 5 and 0 – 8. Additive fractions that can be reduced to the form n – 0 correspond to ordinary natural numbers. Additive fractions that can be reduced to the form 0 – n are the additive inverses of natural numbers, and they have a special name: they’re called “negative numbers.”

As it turns out, every additive fraction is either a natural number or a negative number, so we don’t have to talk about additive fractions at all if we don’t want to. And we don’t—but maybe we should.