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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

#111

Post by Wowfunhappy »

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 about the two tables together."

Now, I think something closer to what you're suggesting is, the teacher could have written the following two equations on the board:

"1/3 of the students at the first table + 1/3 of the students at a second table = 1/3 of the students at both tables"

"1/3 of the students at the first table + 1/3 of the students at the first table = 2/3 of the students at the first table"

Accompanied by some drawings, maybe that would have worked. But I think it could just as easily end up confusing everyone—you've made the concept of addition much more complicated! And sure, the real world is more complicated too, but you've got to learn the basics first.

---

The more I think about it, the more I think the best response might have been: "No, you can't do that, because those kids are at a different table. If we added another third of the kids at the same table...", and move on. Ignore the confusing example and refocus on the simple one.

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

#112

Post by gowld »

mkl wrote: post #67
qubex wrote: post #60

Earlier quoted context omitted.

This is one of the reasons I annoy people by following Wolfram’s convention in Mathematica of using square braces to denote arguments passed to a function: f ( x )= fx = f × x while f [ x ] means “apply the function f to the argument x ”. An unusual convention it may well be, but at least it’s one devoid of ambiguity.

Not at all. f[x] is clearly the xth component of the array f :-D

An array is merely a function from N to some set.

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

#113

Post by CydeWeys »

tsimionescu wrote: post #99
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.

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 becomes 1/3), but you can't do that here and still support the mediant operation. https://en.wikipedia.org/wiki/Mediant_(mathematics)

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

#114

Post by virgilp »

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.

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

#115

Post by ken »

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 can't see the page (slashdotted), but that sounds just like the puzzle from an old kids' math show we used to watch:

https://www.youtube.com/watch?v=bCoGMYV3UPk&t=2m29s

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

#116

Post by 6gvONxR4sf7o »

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. But 1/3 = 2/6, so that should be the same as 2/6 @ 2/3 = 4/9. But 3/6 isn't 4/9. You end up with this issue of a/b @ c/d = p a / p b @ q c / q d = (p a + q c) / (p b + q d), which can be anything. So we have the conclusion that if this operator makes sense, then all numbers are equal. You end up with a notion of numbers that is useless for the original problem of combining groups of people of different genders.

What we should do here is define it on pairs of numbers rather than fractions. A fraction and a total (p, N), or the number of boys and number of girls (b, g). The latter is super straightforward: (paul, jenny) + (bob, alice) = (paul and bob, jenny and alice), so numerically, it's (b1, g1) @ (b2, g2) = (b1+b2, g1 + g2), but (1, 3) is not (2, 6) here, unlike with fractions. Real simple. If we want to connect this back to the world of fractions, (p1, N1) @ (p2, N2) = (number of boys / total, total) = ( (p1 N1 + p2 N2)/(N1+N2), N1+N2 ). It's just a weighted average, so you need to keep track of the weights.

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

#117

Post by gowld »

quietbritishjim wrote: post #51
didgeoridoo wrote: post #13

Earlier quoted context omitted.

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…

I think this just adds to the confusion. Is "adding" ratios really the same as averaging them? (I would just say that adding ratios is simply not defined.) Can you only average things by moving from fractions to ratios and then back again? I think better to address the problem directly in fractions by saying that they're two different ways of combining them, addition and averaging, like the parent comment says. Feel…

There's a lot more than 2 ways to combine things!

Arithmetic average or geometric average, to start.

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

#118

Post by OscarCunningham »

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…

> 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 surprised there is no such operator for averaging.

This would also cause confusion, because combining 1/3 with 1/2 and 2/4 in this way would yield different results, even though 1/2 = 2/4.

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

#119

Post by joshocar »

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…

This. What the student said was 'wrong' and they need to look at the whole. This "1/3 + 1/3 = 2/6" problem would be a great example to use for a lesson on units though.

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

#120

Post by albedoa »

ColinWright wrote: post #102
SV_BubbleTime wrote: post #33

Earlier quoted context omitted.

Unlike 95% of the other answers here, this may have been received And been processed by children who are just learning fractions. I think the most basic answer to the “what do you do next” is unfortunately to explain they can’t do 1/3+1/3=1/6 but that why will be a future lesson. The article gets a “fair” stopping point for the day at being mindful of the whole. And that you just can’t always add them.

> ... to explain they can’t do 1/3+1/3=1/6 ... Except that 1/3 + 1/3 = 2/3, not 1/6. You probably just mis-typed, but in this context it's an interesting mistake.

>> they can’t do 1/3+1/3=1/6

> Except that 1/3 + 1/3 = 2/3, not 1/6.

That's why they can't do it.

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