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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

#191

Post by rcxdude »

ColinWright wrote: post #5
CydeWeys wrote: post #2

I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fractions, i.e. when you're totaling subgroups into a larger group. It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surpris…

I never understood why the Missing Dollar Riddle ever confuses people. As soon as they say "Add the $2 to the $27" I say, "But why are you adding something someone has to a total that people paid?" That, in turn, is like the "Age of the Shepherd" problem[0] ... people just add/subtract/multiply/divide things randomly without thinking about what they mean. [0] https://mystudentvoices.com/how-old-is-the-shepherd-the-pr…

This reminds me of this problem which made the rounds recently[0], where a word problem essentially makes a lot of people commit a fencepost error

[0] https://math.stackexchange.com/questions/379927/how-long-wil...

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

#192

Post by OJFord »

wcarey wrote: post #148
OJFord wrote: post #88

Earlier quoted context omitted.

> 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?

I mean, the general case is certainly trouble for kids. But 15% and 20% look very similar to 15x and 20x. That the two (sometimes!) operate using different rules causes confusion for many students. The implicitness of the fractions conceals something important that's explicit if you write out the fractions. For example, multiplying percents does not do what most students intuitively think it does. 15% of 8 times 20%…

The main thing I find myself explaining with percentages (IANA educator) is that 'grossing up' != adding the same percentage back, e.g. 80% * 1.2 != 100%, which looks pretty obvious like that, but it's a common mistake among adults talking about real life percentages like taxes.

Another common, and I suppose related, one (but that I don't bother correcting often) is 'percent' != 'percentage point'. Talk of 47% of something being '3% less than' half of it really winds me up - and it's stupidly common among journalists - but it's too common to bother pointing out IMO. Live and let get wrong.

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

#193

Post by walshemj »

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 if students are in both tables

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

#194

Post by renewiltord »

munificent wrote: post #163
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…

> You're right, but the tricky part is, how do you explain that to a group of children who is just being introduced to the concept of adding fractions, without leading them off track? That's hard! She's already introduced bottles of water and pencils. So one interesting question is, "If you take three bottles of water, and add three pencils, do you now have all of the bottles of water?"

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.

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

#195

Post by 0xfaded »

Not that I would have thought of this on the spot, but ...

The numerator and denominator represent different things. The numerator is "how many things", and the bottom is "of how many spaces". Asking what fraction of the table is girls is dangerous, because the answer is (1 girl) / (1 girl + 2 boys). Still a stretch, rephrasing as "what fraction of seats are filled by girls" might have helped show the idea that the number of spaces is fixed.

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

#197

Post by hathawsh »

This is fantastic material. I'm going to present this quandary to my kids. Some of my kids struggle at math, so it will help them understand that math is a language and, like any language, it has abbreviations that sometimes lead you down a confusing path until you spell out more. Some of my kids are good at math and it will help them relate with others.

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

#198

Post by sdenton4 »

6gvONxR4sf7o wrote: post #116
CydeWeys wrote: post #2

I guess you'd have to come up with a way to explain that adding numbers (which is what you're doing with 1/3 + 1/3) is not the same as combining/averaging fractions, i.e. when you're totaling subgroups into a larger group. It's almost like we need a different "combining" operator for the latter that means to add both the numerator and denominator, because + isn't right for this. Now that I think about it, I'm surpris…

> I'm surprised there is no such operator for averaging. You can't have an operator for combining portions of groups with fractions alone, because 1/3 = 2/6. Combining groups of B boys and N people total, you get B1+B2 boys and N1+N2 people. Let's use @ for that operator, just to not distract from the usual addition. a/b @ c/d = (a+c)/(b+d). Let's combine a group of 1/3 boys with a group of 2/3 boys. 1/3 @ 2/3 = 3/6.…

Well, you're right so long as 1/3 = 2/6. This is a mathematical equivalence... Fractions are a 2d projective space: they are pairs (n, d) where (mn, md) is equivalent to (n, d). The suggested operation is totally fine if you are working in ambient (that is, non-projective) space... ie, if you continually keep track of the 'size' of the thing being measured.

The equivalence operation is hard to master, and makes the arithmetic complicated.

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

#199

Post by wcarey »

bradjohnson wrote: post #139
wcarey wrote: post #69

Earlier quoted context omitted.

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.

This is definitely a good explanatory strategy (and one that I'd use). I might label the units of the right side as well. Would you attempt to explain the abstract rules of fraction arithmetic as an system?
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