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Can 1/3 and 1/3 = 2/6? It seemed so

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Re: Can 1/3 and 1/3 = 2/6? It seemed so

#81

Post by MaxBarraclough »

twic wrote: post #18
wcarey wrote: post #4

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conce…

> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.

A good example of this can be found in an incredible algorithm with a terrible name: Fast inverse square root.

It sounds like an algorithm for computing the inverse of the square root function, so one might think it's for squaring non-negative numbers, or something along those lines. Not so. It's for computing the reciprocal (the 'multiplicative inverse') of the square root of a number. [0]

Related to this: the way a superscript '2' next to a function means the function shall be applied twice (that is, composition with itself)... unless it's a trigonometric function, in which case it means the square of the result. [1] [2]

[0] https://en.wikipedia.org/wiki/Fast_inverse_square_root

[1] https://en.wikipedia.org/wiki/Function_composition#Functiona...

[2] https://en.wikipedia.org/wiki/List_of_trigonometric_identiti...

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#82

Post by slantyyz »

ValleZ wrote: post #22

I don't understand why this simple 2nd grade math is featured on hacker news. I guess the next topic will be a long discussion what would be 1/2 divided by 2/3.

After I read the article, I thought the article had less to do with the math involved than the way kids see things compared to adults. So I thought it was pretty interesting.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#83

Post by pdkl95 »

As others have already pointed out, the question is ambiguous without more information. Always include the unit; bare magnitudes could mean anything! The student was describing the mediant[1], which was the correct solution when the problem is interpreted with forbiddenvoid'a units[2].

"Adding" fractions with the mediant leads to fun things like the Farey sequence[3] (related to Ford circles[4]) and the very interesting Stern–Brocot tree[5]. (Numberphile has a nice introduction[6] to the fun properties of the Farey sequence)

[1] https://en.wikipedia.org/wiki/Mediant_%28mathematics%29

[2] "1/3 (of the students at a table) + 1/3 (of the students at a table) = 2/6 (of the students in the room)"

[3] https://en.wikipedia.org/wiki/Farey_sequence

[4] https://en.wikipedia.org/wiki/Ford_circle

[5] https://en.wikipedia.org/wiki/Stern%E2%80%93Brocot_tree

[6] https://www.youtube.com/watch?v=0hlvhQZIOQw

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#84

Post by simiones »

twic wrote: post #18
wcarey wrote: post #4

This is a lovely (edit: having been in similar shoes, also terrifying-in-the-moment) example of a broader problem in teaching mathematics: the language we use to describe mathematical reasoning is a natural language, like English or Latin, and therefore full of the sorts of bizarre irregularities you'd find in a natural language. Mathematics is also a language about rigorously and precisely defined objects. The conce…

> x^{-1} means \frac{1}{x}, but f^{-1}(x) doesn't mean \frac{1}{f(x)}. That notation for inverse functions is truly appalling. I don't know how the first mathematician to think of that didn't immediately discard it as nonsensical and misleading.

To me it always seemed pretty natural. \frac{1}{f(x)} would be (f(x))^{-1}, by analogue to x^{-1}+1 vs (x+1)^{-1}. I would have been very surprised if raising just the 'f' part of the function application expression to some power were to mean raising the whole expression to that power.

I also would expect f^{2}(x) to mean f(f(x)), not (f(x)) * f(x)) (which would be (f(x))^2).

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#86

Post by jessermeyer »

wcarey wrote: post #50
jessermeyer wrote: post #34

Proportions are tricky to introduce since they are the first obvious move away from absolute quantities. We're taught that division is just fancy subtraction, but it's actually the more subtle idea of proportionality. Similar with multiplication as dimensionality. From here, it feels like the natural setup to show that you can't just 'combine' proportionalities without accounting for what portion these proportions co…

Perhaps an argument for arithmetic followed by geometry using Nicomachus and Euclid. We sit on that until 9th grade, but I wonder how young you could go with it?

Introducing line segments as alternative representations of numbers at this point feels very natural, and is already implied by most circulum with the standard 'number line'. As you say, we don't do anything with that until much later.

Let's leverage that early on!

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#87

Post by pdevr »

The moment you add three more students, the first girl is no longer one girl out of a group of three students, she is one girl (out of the two girls) out of a group of six students.

Same with the second girl.

So, once you add 3 more students, 1/3 becomes 1/6.

1/6 + /6 = 1/3.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#88

Post by OJFord »

wcarey wrote: post #37
CydeWeys wrote: post #26

Earlier quoted context omitted.

Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough o…

And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples: 1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules. 1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular. 15% + 20% - who knows? 15% of what? 20% of what?…

> 15% + 20% - who knows? 15% of what? 20% of what?

That's just notational sugar on fractions though (15/100, 20/100) - so why is that less troubling than the general case?

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#89

Post by Denvercoder9 »

wcarey wrote: post #79
Denvercoder9 wrote: post #74

Earlier quoted context omitted.

> \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell. I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.

Absolutely terrifying. How did he communicate with other peer mathematicians?

I honestly don't know - it was an intro level physics class.

Re: Can 1/3 and 1/3 = 2/6? It seemed so

#90

Post by knappa »

From a 'not currently in front of the class' perspective, it's pretty clear what the student meant by 1/3 + 1/3 = 2/6. They were taking + to mean something like a general 'and' or combining action, not in the strict sense of standard fraction addition. It even has a name in mathematics, 'Farey addition'.

The kids clearly want to write it in shorthand, so maybe the thing to do is to come up with another symbol for this similar but distinct operation. For example, ⊕.

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