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

marilynburnsmathblog.com

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

#71

Post by qubex »

mkl wrote: post #67
qubex wrote: post #60

Earlier quoted context omitted.

This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f ( x )= fx = f × x while f [ x ] means “apply the function f to the argument x ”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.

Not at all. f[x] is clearly the xth component of the array f :-D

On the blackboard in the mathematics department it’s not.

But I get your point.

I have also heard the objection that [x] is a 1×1 matrix whose only element is ‘x’.

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

#73

Post by uryga »

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?…

> (1,2) + (2,4) - can't be done.

are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors

> a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.

what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)

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

#74

Post by Denvercoder9 »

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…

> \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.

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

#75

Post by pwdisswordfish2 »

qubex wrote: post #60
wycy wrote: post #47

Earlier quoted context omitted.

I think x(3) would more commonly imply a function of 3, since you'd virtually always otherwise just write 3x.

This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f ( x )= fx = f × x while f [ x ] means “apply the function f to the argument x ”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.

It's not devoid of ambiguity, because f[A] usually means the image of the set A under f.

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

#76

Post by forbiddenvoid »

mywittyname wrote: post #49
forbiddenvoid wrote: post #40

The units are missing, and I think that's a key factor here. Both of the equations up on the board at the end are correct because they are counting different things. This is a huge miss if you use a numeric only approach to fractions. The student came up and wrote 1/3 + 1/3 = 2/6. What they meant by that is 1/3 (of the students at a table) + 1/3 (of the students at a different table) = 2/6 (of the students at those t…

This was what I came up with in the moment, but with drawing. Draw two circles on the board, each divided into thirds and compare it with a single circle divided into sixths. It demonstrates that the total has grown.

Also a fantastic way to represent this. The assumption in fractional arithmetic is that you're always performing arithmetic on things with the same type/unit.

The two formulas on the board are essentially:

1/3x + 1/3y = 2/6z

1/3x + 1/3x = 2/3x

Both are correct, but without units labeled you wouldn't know that.

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

#77

Post by wcarey »

uryga wrote: post #73
wcarey wrote: post #37

Earlier quoted context omitted.

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?…

> (1,2) + (2,4) - can't be done. are you talking about points or open intervals? the notation is ambiguous ;) but addition of points is fine if you look at them as vectors > a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope. what's wrong with the 2nd and 3rd ones? the 3rd one looks like half of Pythagoras theorem which is widely considered "fine" afaik ;)

Indeed. And if they're cartesian vectors, you're good. But if that second number is an angle measured in radians, you use yet a different set of rules for the addition.

The conversation with 16 year olds when you explain that their previous teacher who told them that you couldn't add points wasn't lying, but was, perhaps oversimplifying things to make their life easier, is a fun and fraught one.

I've had to reason kids through the fact that a^2 + b^2 is not equal to (ab)^2 or even (a+b)^2 more times than I can count. What's particularly difficult is that, confronted with the fact that 25 and 49 are manifestly different numbers, many still cling to the rule that a^2 + b^2 = (a+b)^2, because of the "law of distribution", which they haven't learned as the "law of distribution of multiplication of monomials over addition, and only that".

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

#78

Post by dr0nkprogrammer »

Javascript:

console.log((0.1 + 0.2) === 0.3) // => false

C#:

(0.1m + 0.2m) == 0.3m // => true

see here: https://dotnetfiddle.net/zVnrNQ

Javascript is such a poor language, there is no even a good way to work with currencies or exact numbers and you have to multiply numbers with a factor to avoid such problems. The worst language I ever used.

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

#79

Post by wcarey »

Denvercoder9 wrote: post #74
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…

> \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?

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

#80

Post by rocqua »

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.

It's not because 'function powers' make sense, yhey are just iterated function application. That's how they work for the natural numbers and when you extend that to the integers, you immediately get f^-1 for the inverse.

Notation in higher level maths is almost always very ambiguous. Because many concepts are analogues of each other and to reflect that notation is just taken from the analogue. Within a single domain, (like high-school arithmetic) you will usually not have this ambiguous problem. But once you move past that, it is something to get used to.

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