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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

#11

Post by barbegal »

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…

And the "missing dollar riddle" can be exploited as the "change raising scam" https://www.youtube.com/watch?v=1Uaw88H0AmE

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

#13

Post by didgeoridoo »

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…

Someone in the comments makes a good point that the best thing to do here may be to introduce ratio notation for proportions (e.g. 2:4) which CAN be added/combined according to the kids’ intuitions — 1:2 combined with 1:2 does indeed equal 2:4, which reduces back to 1:2.

You could then teach how to go from ratios to fractions by adding the ratio sides together and putting that in the denominator for each side... poof, you’ve invented averages!

No way someone would have come up with that approach on the fly, though.

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

#14

Post by wcarey »

mkl wrote: post #7
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 x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.

Right.

Of course, if you write x(3), other mathematicians should frown at you because you're making bad notational choices.

It's a bit like explaining to students that the real way to know which 3rd declension nouns are i-stems in Latin is to say the genitive plural both ways. The one that doesn't sound wrong is correct. But you have to have a lot of time in the language for that to work.

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

#15

Post by yamrzou »

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…

The confusion comes from the fact that we attach the same label "3" to two different sets in the case of bottles, so we end up saying "one out of three plus one out of three". It should be "one out of group 1 + one out of group 2", and then as you said, we'll need a special operator for combining groups.

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

#16

Post by jerf »

mkl wrote: post #7
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 x(3) would more commonly mean x times 3. f(3) would be function application. Even more context dependence.

x(3) is a function "x" being applied to the number 3, because "three times x" is 3x.

Of course context would help resolve this if you have a function named x or not, or a variable named x or not, and if you have both a function and a variable named x, well, you worked for your confusion and you have obtained it; congratulations. :)

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

#17

Post by ColinWright »

wcarey wrote: post #10
ColinWright wrote: post #5

Earlier quoted context omitted.

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…

The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced…

As I recall, that study was debunked/retracted, so you may have trouble finding it.

The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf

Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...

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

#18

Post by twic »

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.

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

#19

Post by wcarey »

ColinWright wrote: post #17
wcarey wrote: post #10

Earlier quoted context omitted.

The more I've taught math, the more convinced I am that getting people to "think about what they mean", and to think about what mathematical words mean is 90% of the project. I remember reading long ago (I'd love to find it again) about a CS department that gave a quiz to incoming students that was very predictive of their success. The answers they wrote didn't matter; what mattered was whether their answers evinced…

As I recall, that study was debunked/retracted, so you may have trouble finding it. The paper: http://eis.mdx.ac.uk/research/PhDArea/saeed/paper1.pdf Retraction: http://www.eis.mdx.ac.uk/staffpages/r_bornat/papers/camel_hu...

Thanks! I hadn't seen the retraction. That's really interesting.

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

#20

Post by JBorrow »

A great opportunity to introduce multiplications of fractions!

1/3 of the students at table A are girls. 1/3 of the students at table B are girls. What fraction of the tables does table A represent? A is one out of two tables, so A is 1/2 of the tables. Likewise, B is 1/2 of the tables, too.

When we want to consider the whole here, we need to take into account what fraction of the whole each proportion represents.

The question that we want to answer is 'what proportion of _all students_ at _all of the tables_ are girls?'. This is a combination of the question 'what proportion of students at table A are girls, and what proportion of students at table B are girls', and 'what proportion of all of the tables does each table represent'? That second question might seem quite convoluted but it is important!

To do this, we need to multiply the fractions together like so:

(Fraction of tables that A represents) * (Fraction of students at table A that are girls) + (Fraction of tables that B represents) * (Fraction of students at table B that are girls) = (Fraction of students at tables A AND B that are girls).

So in this case we would have:

(1/2) * (1/3) + (1/2) * (1/3) = (1/6) + (1/6) = (1/3).

This is even clearer when we consider the case where there are two girls at table B. There, we can do the same thing:

(1/2) * (1/3) + (1/2) * (2/3) = (1/6) + (2/6) = (3/6) = (1/2).

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