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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

#131

Post by scarmig »

bradlys wrote: post #104
p4bl0 wrote: post #63

Earlier quoted context omitted.

Yup, it's a typing problem. Do it with different things in the two sets and it becomes clearer: 1/3 apples + 1/3 oranges = 2/6 fruits . This is another instance of situations where teaching compsci or at least programming could help teaching maths rather than the other way around as it is traditionally thought.

Interesting - I never thought of CS as being something to help with this. Instead, I’d think of other engineering fields where there is a heavy reliance on correct units and unit conversion. Pound of feathers vs pound of gold , etc...

F# offers a nifty "unit of measure" language feature to help with that. You might have

  [] type lbFeather 
  [] type lbGold
  [] type lbLead

  let convertGoldToLead ( weight : float ) = 1.0 / 1.0 * weight

  // works
  let barOfGold = 1.0
  let barOfLead = convertGoldToLead barOfGold
  // doesn't work
  let bagOfFeathers = 1.0
  let leadDuck = convertGoldToLead bagOfFeathers
  // Error reported: mismatched units.

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

#132

Post by [deleted] »

pdkl95 wrote: post #83

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 interes…

[deleted]

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

#133

Post by jancsika »

macspoofing wrote: post #65
bsaul wrote: post #58

Earlier quoted context omitted.

Except mathematical concepts emerged from real-world observation and problems. We didn't invent fraction out of nowhere. The moment when you have to stop relying on intuition is a pretty delicate matter, but it's still interesting to try to rely on metampho, and then understand when and why a particular metaphor stops working. Much better (imho) than teaching math as a purely formal and transcendent topic that happen…

>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 between the real world and basic unitless mathematical notation. (I'd even add explicit operator definition to that.)

For example-- such an educated student could hear your ascetic declaration that "it's against the rules" and quickly grasp something like the following:

1. "1/3+1/3=2/6" doesn't have any units, but it must somehow map to operations with units.

2. If unitless math can be applied regardless of units, then perhaps "1/3+1/3" may mean "1/3 blue table + 1/3 of the red table, where + means joining the two tables." That would equal 2/6 of the joined tables. But "1/3 blue table + 1/3 (same) blue table" would give 2/3 of that blue table, with + mapping to adding those two fractions of the same table.

3. 2/3 does not equal 2/6, so unitless math can't map to both operations.

4. macspoofing said that 2/6 is wrong.

5. Therefore, unitless fraction addition implies addition of things of the same units, and not joining two different things together and finding the new fraction of the new joined unit thingy.

If on the other hand a student of your apparent method of declaring rules for unitless math came to a class that had practiced explicitly mapping unitless unit math, they wouldn't have any tools to understand the mapping. (Well, at least if the teacher made a similarly ascetic declaration regarding mapping.)

I offer into evidence this very article to show what happens when a student of your apparent method becomes the teacher and encounters the most trivial of unit -> unitless mapping errors.

Edit: clarification

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

#134

Post by nofinator »

CydeWeys wrote: post #30
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.

A lot of people are at home right now trying to teach their kids who are also at home.

Bingo. To me, this is the key point of the article:

> It’s hard to think and teach at the same time!

This teacher seems pretty bright. I'm sure if she had a few quiet minutes to think and regroup, she could come up with a stellar explanation. But instead, she has to proceed with the lesson, manage the classroom, and tackle an interesting puzzle at the same time.

At home, merely trying to manage my work and two kids with their remote learning often puts me in the same spot. I will often do some tutoring with them during a work break. But my brain power is already silently depleted, so if any kind of obstacle or creative question arises, I struggle to even begin thinking of an approach to it.

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

#135

Post by DavidSJ »

lisper wrote: post #93
DavidSJ wrote: post #48

Earlier quoted context omitted.

They’re both inverses: one with respect to multiplication, the other with respect to function composition. In abstract algebra, we observe that there are many types of “products”: multiplication, addition, function composition, composition of rotations, matrix multiplication, etc. A common, unified “power” notation for repeatedly taking the product of a single element, or of its inverse, has some value. There is some…

> one with respect to multiplication, the other with respect to function composition And, of course, multiplication is a function so one is just a specific instance of the other. Infix notation is another blight on the mathematical notation landscape. The amount of human effort that has been put into figuring out how to parse a+b*c is staggering. All of this confusion could have been avoided if we'd just started with…

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).

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

#136

Post by abdullahkhalids »

billfruit wrote: post #92
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…

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.

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

#137

Post by Chathamization »

Wowfunhappy wrote: post #111
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…

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

I agree. I think many people are thinking the main obstacle here is the mathematical mistake, when the main obstacle here seems to be explaining the mathematical mistake to 4th graders who are learning fractions for the first time. I'm also not sure what the best way to do that would be. If you start with one type of fraction (say, half circles), it's probably easier to intuit things like 5 halves and how much that is, but other things are probably going to be more difficult.

So yes, telling kids that there are more complications when approaching certain problems that you can explain later might be a good approach.

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

#138

Post by Wowfunhappy »

forbiddenvoid wrote: post #127
Wowfunhappy wrote: post #111

Earlier quoted context omitted.

You're right, but the tricky part is, how do you explain that to children who are just being introduced to the concept of adding fractions, without leading them off track? That's hard! > The confusion comes because no one calls out that they're talking about fractions of different things. But she did: "When thinking about fractions, it’s important to keep your attention on what the whole is. [...] you’re thinking abo…

I don't think I agree with everything here. Glossing over critical mental models while explaining new concepts is what creates the confusion. Absolutely using drawings would have made things easier. That's a good way to establish those correct mental models. "No you can't do that" is probably not the right approach. The kid's instinct was spot on. The calculation they did was correct, just out of context. Rather than…

I'd absolutely agree if you were tutoring the child one-on-one. My concern is about leading the other 19+ kids astray with a confusing example before they're ready for it.

Ignoring it does feel wrong, but I don't know that it is. There's only so much time in a class period, etc.

That said, I really liked where mnsc went with it: https://news.ycombinator.com/item?id=23312266

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

#139

Post by bradjohnson »

wcarey wrote: post #69
macspoofing wrote: post #43

Earlier quoted context omitted.

>The fourth grader is saying "1/3 + 1/3 = 2/6" but the idea she's trying to get across is that "avg(1/3,1/3) = 1/3 but the size of the whole has doubled." I disagree. The fourth-grader is perfectly correct within the context of the analogy that the teacher used. The problem is the analogy is wrong which is a general problem of relying on metaphors and analogies to explain rigorous technical concepts. Fractions are no…

How would you explain that successfully to fourth graders? (Or, for that matter, seventh graders?)

I would avoid multiplication entirely and say that if you are combining tables, you need to update the bottom with the new number of total students. So something like:

  (T1 girls) + (T2 girls)   1 + 1
  _______________________ = _____
  (T1 total) + (T2 total)   3 + 3
I think that's more intuitive than averages, and would still work if the tables were of different sizes.

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

#140

Post by jrumbut »

CydeWeys wrote: post #113
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.

You definitely can add 1:2 apples and 1:2 oranges, you just need to do so using a common base type (such as fruits or objects). Note that I'm using ratio notation because the answer for the above is not the same as adding 5:10 apples and 1:2 oranges; in other words, the exact numerator and denominator both matter, so it's not really a simple fraction; a simple fraction can be reduced to its lowest terms (e.g. 5/15 be…

The grandparent's note about units and your mention of ratios get at the same problem for me:

The lesson mixes two of the physical world concepts that are often modeled by fractions.

One is an incomplete whole (the six pack) and the other is the proportions of a mixture (table demographics).

Fractions always have the same rules, but the rules map differently to the different territory being modeled.

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