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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

#101

Post by CydeWeys »

bwestergard wrote: post #62
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…

Adding numerators and denominators to find the "mediant" is sometimes called "doing a freshman sum". In some countries I gather this is introduced as a separate operation (see page 23 in the translation of a Soviet primary school textbook by Gelfand linked below). https://en.wikipedia.org/wiki/Mediant_(mathematics) https://www.cimat.mx/ciencia_para_jovenes/bachillerato/libro...

Thanks for putting a name on this specific operation. It sounds like it definitely needs its own specific operator.

And hi Bjørn!

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

#102

Post by ColinWright »

SV_BubbleTime wrote: post #33
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…

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.

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

#103

Post by soperj »

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 can. It's what you do in fruit salad.

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

#104

Post by bradlys »

p4bl0 wrote: post #63
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…

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...

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

#105

Post by dejj »

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…

They sneakily replaced scalar addition by point-wise vector addition. [1,3] [1,3] = [2,6]

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

#106

Post by OscarCunningham »

qubex wrote: post #71
mkl wrote: post #67

Earlier quoted context omitted.

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

On the blackboard in the mathematics department it’s not. But I get your point. I have also heard the objection that [x] is a 1×1 matrix whose only element is ‘ x ’.

Square brackets can also denote the floor function (especially in old works where '⌊' and '⌋' were typographically unavailable).

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

#107

Post by tsimionescu »

forbiddenvoid wrote: post #76
mywittyname wrote: post #49

Earlier quoted context omitted.

This was what I came up with in the moment, but with drawing. Draw two circles on the board, each divided into thirds and compare it with a single circle divided into sixths. It demonstrates that the total has grown.

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 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) ...).

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

#108

Post by mamcx »

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…

So, have been any sufficient effort in design a new syntax of maths that could help the regular folk?

If we learn new syntax(langs) all the time then push for a new syntax(math) could not be that bad of a idea...

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

#110

Post by gowld »

wcarey wrote: post #79
Denvercoder9 wrote: post #74

Earlier quoted context omitted.

> \sin{30}. Radians or degrees? Probably the writer means degrees, but there's no way to tell. I once had a professor that insisted that sin(30) meant sin(30 pi) in radians, with the pi being implicit. Unsurprisingly, it was the worst class I've ever taken.

Absolutely terrifying. How did he communicate with other peer mathematicians?

Mathematicians understand how to read definitions of notation.

If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.

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