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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

#261

Post by TeMPOraL »

veilrap wrote: post #236
tener wrote: post #152

Earlier quoted context omitted.

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.

This is a great baseline for the units answer.

I think it then needs an addendum on things that are the same dimension but expressed in different units.

Like, "students at table A" and "students at table B" are both of the same dimension as "students", but are different units. Like meters and miles. You can add them together, but only if you know the factors needed to convert them to a common unit. In this case, the conversion factors are, how many students are at table A and how many at table B.

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

#262

Post by TeMPOraL »

tsimionescu wrote: post #107
forbiddenvoid wrote: post #76

Earlier quoted context omitted.

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.

As I said elsewhere, this is not a problem of units or types. If it were, then the computation wouldn't make sense. It is a problem of implicit refernces. The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other). The correct equation would have been 1/3 * 3 + 1/3 * 3 = 2/6 * 6. Note that 3, 3 and 6 have the same unit. If they didn't, then this would be meaningless. 1/3 o…

> The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other).

That is, precisely, the problem of units. Each object is its own unit here.

> The correct equation would have been 1/3 3 + 1/3 * 3 = 2/6 * 6.*

That's not how you add fractions. The correct equation would have been, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 (or, 1+1=2), if these 3 all truly had the same units. But they don't, so you can't add like that.

> 1/3 of a meter + 1/3 of a Pascal does not equal 2/6 of anything (or maybe it does equal 2/6 of (2 meters + 2 Pascals)

That's the point (but it's 1/3, not 2/6). Also, 1/3 of a meter, + 1/3 foot = 1/3 (1 meter + 1 foot). Different units, but same dimension, so if you know the conversion factor (here, 1 meter = 3.3 feet), you can change it into (1/3 meter * 3.3 feet/meter) + 1/3 foot = 1.1 foot + 1/3 foot = 33/30 feet + 10/30 feet = 43/30 feet = 1.43(3) feet.

You can do the same math with students at tables.

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

#263

Post by DavidSJ »

lisper wrote: post #143
DavidSJ wrote: post #135

Earlier quoted context omitted.

Multiplication is a function of numbers, and function composition is a function of functions, so neither is an instance of the other (unless you take the unusual perspective of thinking of numbers themselves as functions, but I don’t think that’s what you meant).

> I don’t think that’s what you meant Why not? Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.

I didn’t think you meant that numbers are functions because you said: “multiplication is a function so one [multiplication] is just a specific instance of the other [function composition]”.

Thus I thought your point was that, since multiplication is a function, it’s a form of function composition. But that wouldn’t follow, for the reasons I said.

As for multiplication being “composed” addition, addition being “composed” succession, etc.: The multiplication function (at least of integer arguments) is composed of addition. Function composition is a function that returns a composition of two other functions. The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing.

Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)). The function which does function composition is not itself a composed function, and the act of multiplication is the act of a composed function, but not an act of function composition.

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

#264

Post by lisper »

DavidSJ wrote: post #263
lisper wrote: post #143

Earlier quoted context omitted.

> I don’t think that’s what you meant Why not? Multiplication is a composition of addition, and addition is a composition of the successor function. The successor function is not a composition of anything, but you can define it in terms of set theory if you don't want to simply accept it as a primitive. But sets are functions too.

I didn’t think you meant that numbers are functions because you said: “multiplication is a function so one [multiplication] is just a specific instance of the other [function composition]”. Thus I thought your point was that, since multiplication is a function, it’s a form of function composition. But that wouldn’t follow, for the reasons I said. As for multiplication being “composed” addition, addition being “compos…

> The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing.

Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. That is the whole point.

Numbers and addition and multiplication happen to be particularly important functions, but they are not structurally different from any other functions. Giving them special notation, especially when you are first introduced to them, obscures this fact. This kind of mental damage is very hard to recover from in later life. I believe it's one of the reasons so many people think they hate math. Math can be beautiful and elegant, but the standard notation used for school-book arithmetic is arbitrary and perverse, a bizarre accident of history with no actual merit.

IMHO of course.

[UPDATE:]

> Function composition is a specific operation that takes two functions, f and g, and returns the function f∘g defined by the behavior (f∘g)(x) = f(g(x)).

Function composition is a function, no different from any other function. There is no more reason to use infix notation for it than there is for any other function. In fact, if you drop the infix notation it immediately becomes obvious how ubiquitous and non-special function composition actually is:

  compose(f,g)(x) = compose(f)(g)(x) = f(g(x))
On that view, the COMPOSE function is actually the identity function!

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

#265

Post by isanybodythere »

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…

I'm not sure we can really guess what the student meant, but I do know for sure that she was wrong.

If you add up a sixth of a six-pack and a sixth of another six-pack you get a sixth of two six-packs -- two twelveths.

The student's misunderstanding comes from being taught fraction addition in terms of items in a collection -- which only holds if you keep to the same set (what you called scale).

This is a common choice -- "students already know how to add integers, so let's start from there", but as it did in this case, it doesn't always work as intended.

This is a great example of taking an analogy so far that the student didn't learn anything new at all. Everyone feels happy -- teacher's teaching, student's learning -- until you test what your knowledge on outside the domain of the analogy.

Fraction and integer addition are one and the same, yes -- but from the point of view of fractions, wherefrom integer addition is a special case. It remains challenging to teach and understand from the point of view of the integers, which is where the student stands.

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

#266

Post by romwell »

TeMPOraL wrote: post #259
DigitallyFidget wrote: post #150

Earlier quoted context omitted.

The issue is that it's ratios, not fractions. 1:3. You take 1:3 and 1:3 and it's still 1:3 or 2:6. You haven't changed the ratio for this at all by showing it as fractions, it's simply being represented and presented without correct context.

Isn't this why you end up seeing these "silly" units like kg/kg in chemistry? So that, while the value is technically dimensionless, it doesn't get added to another dimensionless value (e.g. l/l) that's a ratio of values of a different dimension?

This hits the nail on the head:

1 (person at table A) / 3 (people at table A) can't be added to 1 (person at table B) / 3 (people at table B) without conversion of units.

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

#267

Post by [deleted] »

lisper wrote: post #264
DavidSJ wrote: post #263

Earlier quoted context omitted.

I didn’t think you meant that numbers are functions because you said: “multiplication is a function so one [multiplication] is just a specific instance of the other [function composition]”. Thus I thought your point was that, since multiplication is a function, it’s a form of function composition. But that wouldn’t follow, for the reasons I said. As for multiplication being “composed” addition, addition being “compos…

> The way you’re using the term blurs the distinction between the thing that is composed and the thing that does the composing. Yes, that is intentional. Both of these things are functions. Even numbers are functions, they just happen to be functions of zero arguments, or functions that ignore their arguments, or functions whose value is constant regardless of what the argument(s) is(are). It's all the same thing. Th…

[deleted]

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

#268

Post by tsimionescu »

TeMPOraL wrote: post #262
tsimionescu wrote: post #107

Earlier quoted context omitted.

As I said elsewhere, this is not a problem of units or types. If it were, then the computation wouldn't make sense. It is a problem of implicit refernces. The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other). The correct equation would have been 1/3 * 3 + 1/3 * 3 = 2/6 * 6. Note that 3, 3 and 6 have the same unit. If they didn't, then this would be meaningless. 1/3 o…

> The two 1/3 fractions refer to different objects ('wholes') than the 2/6 fraction (and from each other). That is, precisely, the problem of units. Each object is its own unit here. > The correct equation would have been 1/3 3 + 1/3 * 3 = 2/6 * 6.* That's not how you add fractions. The correct equation would have been, 1/3 * 3 + 1/3 * 3 = 2/3 * 3 (or, 1+1=2), if these 3 all truly had the same units. But they don't,…

You are trying to look at a different problem. It was absolutely correct that 1/3 of the students at one table of 3 plus another 1/3 of the students at another table of 3 is the same number as 2/6 of the 6 students sitting at the two tables. This is not disputable.

The way you can write this observation mathematically is as I did: ((1/3) × 3) + ((1/3) × 3) = ((2/6) × 6), or 1 + 1 = 2, after computing the fractions. The student's observation was perfectly correct, but he was missing the proper explanation, as it is not about the addition of fractions (it is almost a coincidence that the fractions used on one side of the equation happen to have the sum of their numerator and the sum of their denominators equal to the numerator and denominator of the fraction on the other side - this only happens because we are multiplying the fractions by their denominators).

Sure, you can express this in terms of units and dimensions of you really choose to. You can also express it in terms of different definitions of +, or even of =. It is pretty unnatural to me to invent an ad-hoc measurement unit N1, "number of people at 1 table" and a different measurement unit, N2, "number of people at 2 tables", with the relation 1N2 = 2N1, and then correct the student's formula to 1/3N1 + 1/3N1 = 2/6N2. It is correct, but it is extremely artificial to me.

By far the most natural way to explain it is using the correct mathematical interpretation of the phrase "one third of the 3 people" - (1/3) × 3.

Inventing measurement units to describe exact quantities reminds me of a silly joke from Portal: "computer: 2 + 2 = 10 ... in base 4". You can always find a way to make the formula direct by adding assumptions.

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

#269

Post by kortilla »

TeMPOraL wrote: post #258
renewiltord wrote: post #194

Earlier quoted context omitted.

This is interesting. I wonder if kids can then take that and transfer it over. Bottles and Pencils are obviously different but kids at table A and kids at table B are both kids. Even more problematically, the equation appears to work. I suspect there'll be some amount of dissatisfaction here and the kids won't understand.

Not sure if this is at the level of comprehension of these kids, but I'd explain it this way: "kids at table A" and "kids at table B" are both "kids", but different amount of kids. You can treat them as the same only if you have a conversion factor. So, if you know that there are 12 kids at table A, and 20 kids at table B, you can multiply your variables by these amounts, and now both expressions have the unit "kids"…

> you can multiply your variables by these amounts, and now both expressions have the unit

In grade school where these kids are, they won’t understand the concept of an “expression”, a “variable”, and barely the importance of “units”.

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

#270

Post by wjmao88 »

adrianmonk wrote: post #176

The difficulty isn't with fractions. It's about understanding what "+" means. Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie"). His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about t…

I would say the issue with symbols is why I had trouble with linear algebra in college, specifically dot and cross products. The fact that they used the same symbols as scaler multiplication and are also called products confused my mind so much.
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