Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

marilynburnsmathblog.com

Post reply on HN
Print view View original

241–250 of 308 posts

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

#241

Post by biddlesby »

I would say it was a mistake to introduce the concept of adding fractions from two different "wholes". Instead, teach that in order to add fractions, you first have to get everything into the same "whole".

Like, you can add Jack / 3 to Ben / 3 because they were at the same table. But adding the boys from one table to another is quite a different thing.

Instead, you should teach that you should first make the fractions with both tables as the whole. Only then are you allowed to add. This could come as a later concept

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

#243

Post by saagarjha »

Spivak wrote: post #235
saagarjha wrote: post #232

Earlier quoted context omitted.

That's just lazy, and I haven't ever read a good text use summation where any of the terms would be outright undefined if evaluated.

You've never seen a textbook present the series form of e^x? exp(x) = summation(x^n/n!, n=0, inf) exp(0) = 1 is perfectly valid but the identity above only holds for x=0 only if we define 0^0 = 1. And we do! Fair, infinite series are pretty esoteric. How about derivatives. The power rule: d/dx x^n = n x^(n-1) This identity doesn't hold for n = 1 and x = 0 unless 0^0 = 1. Eh maybe not, programmers don't use calculus t…

Ok, fair, I think I have seen those. Personally I would prefer noting the special case even when using the fairly standard degenerate case of 0^0=1, but I agree that a lot of people don't.

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

#244

Post by GuB-42 »

benlivengood wrote: post #237

I love this discussion and want to throw in my two cents. Why not teach beginning students group theory as the foundation to addition and subtraction? While it wouldn't solve this particular problem since fractions require fields, it would give teachers the tools to explain precisely why addition of two tables is not a group or field operation; they are from different sets and require measurement with units to define…

Bad idea!

They tried exactly that in France in the 60s and 70s. It was called Maths modernes (New maths). I think it was the same in most of the western world. It was a disaster and they abandoned it in the 80s. The problem was that while it may have helped the best students understand advanced concepts later on, it produced a generation of people who lacked practical counting skills.

EDIT: By "bad", I don't mean "stupid". The people who came up with new maths were experts in their field, with the goal of having a more scientifically literate population. No doubt very smart people, but maybe a little too smart for their own good. Turned out these concepts were too much for most young kids, and too remote from the way math is used on a day to day basis.

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

#245

Post by gowld »

saagarjha wrote: post #230
gowld wrote: post #110

Earlier quoted context omitted.

Mathematicians understand how to read definitions of notation. If all of your trig is fractions of pi, writing pi redundantly everywhere is not useful.

…but that's not a fraction of pi?

30pi is a fraction of pi.

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

#246

Post by RHSeeger »

PaulDavisThe1st wrote: post #216
inimino wrote: post #208

Earlier quoted context omitted.

There is no track!!! Of course there is, because what we call mathematics education is broken beyond repair, but that's a crime. Shutting down the student's curiosity about perhaps the only thing of mathematical interest that happened all day is not the best you can do. Do you think you know what the basics are? Are you sure you know which example is confusing and which simple? Are you sure that ignoring confusing an…

there's not much evidence that most humans would ever discover the concept of fractional arithmetic by themselves. this is a generic mathematics class with young children who by themselves are not likely to walk or even stumble into fractional arithmetic by themselves. there's also no evidence that teaching people complex sophisticated stuff before they have grasped basic concepts enhances their curiosity or learning…

I have a hard time accepting this. My daughter is in 3rd grade and seems to have a reasonable grasp of fractions. They teach her about them at (public) school and we've discussed them at home. Sure, "1/3 + 2/7" is outside her reach, but simpler stuff, things she can visualize, are well within her grasp.

> why is 2/3 different from 2/6 anyway

Because 2 pieces of a pie that you cut into 3 pieces is more than 2 pieces of a pie that you cut into 6 pieces. She understands that 4/8 == 2/4 == 1/2. Sometimes she needs to think about it a bit, but she does "get it".

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

#247

Post by chx »

I happen to have a math teacher masters though I myself do not teach (but I do help with the program of a tiny, tiny reform school). This teacher here had painted themselves into a corner and it's hard to get out of it. Do not explain fractions with things you can't change the denominator of.

Rather tell it with money. Say, 1/3 means if the table has 3 coins , one kid gets 1 coins. If there are 6, 9, 12, 15, one kid gets how many? If the other table has five kids and 5 coins then 1/5 means when splitting five coins a kid gets one. Figure out together what happens with splitting 10, 15, 20 coins. Now putting together the two tables we want to calculate 1/3+1/5, how many coins can we do that with? Step through it, we practice 1/3 with 3, 6, 9, 12 ... but can you tell what the fifth of nine coins are? You can't ... Then find 15 and then show them 1/3+1/5 means 8/15. This is all play and very smooth.

To answer the question posed in the blog post: I would plan the class carefully to avoid the entire situation. But if I must, I'd point out 1/3 is a mere shorthand for division, 1:3 and writing "1:3 + 1:3" is adding two operations together and it does not even make sense. We can restore sanity but that has its own rules.

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

#248

Post by amluto »

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…

The units aren’t the issue IMO. You said:

> 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 tables).

That’s not what ‘+’ means. Addition doesn’t mean “I have this thing and the other thing; please describe the result”; addition means a specific operation on numbers (or on elements of an additive group, or on numbers with units, etc). But you cannot fully describe 1 student at table of three people as 1/3. Sure, 1/3 of the students at that table are that one student, but if you want to add across tables, you need more information and a better description.

Explaining this in a classroom setting may be quite challenging indeed.

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

#250

Post by jtsiskin »

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…

Yes, 1/3x + 1/3x = 2/6(x + x).

Units are absolutely necessary, and 'fractions' don't make sense without them.

'1/3' without units is not a 'fraction' but instead is 0.3333, a number on the number line, which is why it looks so wrong to see 1/3 + 1/3 = 2/6.

Post reply on HN