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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

#151

Post by prmph »

virgilp wrote: post #114
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…

Or a different way to look at it is that if you put two things together, the mathematical operation is not always "+"; it totally depends on the things (and how you put them together). You use plus if you put together fractions of the same thing (e.g. fractions of the same box of crayons), but potentially some totally other operation if you put together different things.

Someone mentioned on this thread using CS to teach math. What you have just described here is the fact that the + operator can be overloaded.

Wonder if the kids can grasp that concept though.

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

#152

Post by tener »

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…

One could explain the mistake to have different units on both side without fractions at all:

1+1=1

One shoe plus one shoe equals one pair of shoes.

Once they grasp that explaining the fraction issue should be easier.

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

#153

Post by bonoboTP »

abdullahkhalids wrote: post #136
billfruit wrote: post #92

Earlier quoted context omitted.

Another is sin^2(x) meaning (sin(x))^2, but by a more intuitive reading it should mean sin(sin(x)). I don't know what exactly is being gained by the usual notation: surely better clarity should be preferred over the time/effort saved in writing the extra pair of paranthesis, and I would prefer it be written as (sin(x))^2 always. Thing is mathematics, and mathematical pedagogy seems hardly concerned with such rampant…

Does f^2(x) in any scenario mean (f(x))^2? Usually the square of a function, with named arguments is written as f(x)^2. So, not clear on the confusion.

Yes, in trigonometry sin^2(x) is commonly written to mean (sin(x))^2

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

#154

Post by 7786655 »

Spivak wrote: post #46
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…

I think it's important to know that this really isn't true. Maths at all times is a subjective language. Maths notation is imprecise "intentionally confused", or just ad-hoc defined all the time. When you see something like. f(x) = summation(x^n, n=0, 10) We conveniently ignore that this polynomial is defined at x=0 despite 0^0 not making any sense by ad-hoc defining 0^0 = 1 in this context.

It makes perfect sense, everyone knows:

0^0 = 0

1^0 = 1

0^1 = 1

1^1 = 0

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

#155

Post by macspoofing »

jancsika wrote: post #133
macspoofing wrote: post #65

Earlier quoted context omitted.

>We didn't invent fraction out of nowhere. But we did. When mathematicians provided a rigorous definition of fractions (rational numbers) they separated them from the real world. Rational numbers do not exist in the real world. Real-world does not have infinities. It does not have negative values. In the real world 1/3+1/3 does equal 2/6 in the way that the fourth-grader applied the analogy. >The moment where you hav…

> You can't do "1/3+1/3=2/6" not because it doesn't make sense for tables of boys and girls (because it does) but because it's against the rules for adding fractions. It is pedagogically superior to choose the route implied by the comments about this being a type error. That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.), you gift them a tool they can use to map betwe…

>That is, if you teach the students to "type" all those fractions (e.g., 1/3 of this blue table, etc.),

Sure. You can certainly ad hoc extend the analogy to bring it in line with the mathematical rules. But at this point, you do hit a higher level of complexity. Your simple analogy is gone and you're slogging through the weeds. That was my point. Analogies are flawed. The teacher started with a very simple rule that worked well and communicated the ideas under certain constraints and then those rules were extended in a logical but incorrect manner by a fourth-grader ... and now the complexity that was hidden in the abstraction is leaking out.

Your explanation is more confusing to me and wouldn't be grasped by the vast majority of fourth-graders. At some point, simply stating that fractions have different rules is the most simple (and correct) explanation.

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

#156

Post by tsimionescu »

p4bl0 wrote: post #128
tsimionescu wrote: post #99

Earlier quoted context omitted.

It's not really a typing problem, it's a problem of references. You can't add 1/3 apples and 1/3 oranges. However, you can add 1/3 * x people with 1/3 * y people and get 2/6 * z people, if x people +y people =z people.

abstract class Fruit: prop name: string prop rotten: bool class Apple (Fruit): prop name: string = "apple" prop rotten: bool = False class Orange (Fruit): prop name: string = "orange" prop rotten: bool = False func add (a: List , b: List ) -> List : return foldl(lambda l, f: l.append(f), a, b) func filter (p: Func , l: List ) -> List : return [f for f in l if p(f)] func is_rotten (f: Fruit) -> bool: return f.rotten a…

This does nothing to elucidate the problem. 1/3 apples + 1/3 oranges could be 2/6 fruits, or 2/781 fruits, or any other number you want. In normal mathematical notation, 1/3 is interpreted as 1/3 of 1, and in that case 1/3 apple + 1/3 orange is 2/3 fruit.

And of course this only works if 1 apple = 1 orange = 1 fruit, which means that they are the same unit of measure, or, equivalently they are of substitutable types. It's even debatable of it's correct to say that 'if I have 1/3 of an apple and 1/3 of an orange, I have 2/3 of a fruit', so depending on what you want to do with your apples, oranges and fruit, your hierarchy may in fact break Liskov substitution. For example, if 1 apple + 1 orange = 2 fruit, so 2 fruit - 1 apple = 1 orange, so 1 apple = 1 orange, which is not true.

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

#158

Post by wcarey »

CydeWeys wrote: post #97
wcarey wrote: post #77

Earlier quoted context omitted.

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 ki…

How did I go all of my life so far without hearing the super useful word "monomial"? It's such an obvious concept to have a meaningful name, and yet I don't recall anyone ever having said that word.

I also didn't encounter it until I was relatively older. Our school has been pushing to introduce more rigorous language and definition in our 7-12th math program. For some students, it really seems to help. For others, it's really hard.

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

#159

Post by prvc »

Her first example, "proportion of a six-pack of water" is actually very complex, with a lot of potentially distracting details. Nowhere does she mention that's what her fractions are measuring, either! The whole lecture was simply a mass of detail with no reference to the underlying concept it was putatively intended to explain. No wonder the children were confused. Why not start with cutting a pie? Guess that doesn't lead to nearly as much busy-work.
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