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

181–190 of 308 posts

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

#181

Post by 08-15 »

That's what happens when you try to "build an intuition" for a concept instead of defining it.

Okay, I get what a third of a sixpack is, it's two bottles. And half a sixpack is three bottles. I can even add them: one third (of a sixpack) plus one half (of a sixpack) is five bottles, hence five sixths! Now what's a quarter of a sixpack? That don't make no sense!!

The whole point of common fractions is to close integer arithmetic over division by introducing a new kind of numbers, precisely those fractions that don't correspond to integers. The question isn't so much "What is half a bottle of water?", it's "How do you calculate with one half (a bottle or whatever)."

Math is beautiful, but only if built up from simple rules. It's amazing how much you can make from counting and a handful of convenient notations. Even children can recognize that beauty. But high school instead teaches math as a jumbled mess of things you have to memorize, without structure, without rhyme and reason. Needless to say, I hated it. (School, not math.)

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

#182

Post by NiceWayToDoIT »

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…

You are right, I think before jumping to different things, kids should learn on simple examples, that do not include different types, and then when they grasp one concept they should be gradually introduced to the other concept. table/tables would just confuse them even more.

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

#184

Post by tsimionescu »

soperj wrote: post #103
tsimionescu wrote: post #99

Earlier quoted context omitted.

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.

Well, in fruit salad 1/3 apple + 1/3 orange = 1/3 apple + 1/3 orange. I can say 3 * (1/3 apple + 1/3 orange) = 1 apple + 1 orange.

And then by way of analogy of course, (e apples)^(pi oranges) + 1 apple = 0.

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

#185

Post by rovolo »

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…

The core concept kids need to know when adding fractions together is that the denominators need to be the same before you can add the numerators. Your use of units is on the right track, but I think you need to use units for both the top and bottom of the fraction:

(1 girl at table A)/(3 students at table A) + (1 girl at table B)/(3 students at table B) = (2 girls at tables A & B) / (6 students at tables A & B)

vs

(1 boy at table A)/(3 students at table A) + (1 boy at table A)/(3 students at table A) = (2 boys at table A) / (3 students at table A)

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

#187

Post by Wowfunhappy »

forbiddenvoid wrote: post #174
joshocar wrote: post #119

Earlier quoted context omitted.

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.

What they said wasn't wrong. Their mental model was absolutely correct. "One third of this and one third of that is two-sixths of everything" is absolutely right. Telling them "No, you're wrong" is counter-productive. "You have to look at the whole" isn't a helpful statement because, in this case, there are THREE 'wholes.' Their written representation of the mental model was incorrect because their instruction was fo…

Separate from the discussion of what is or is not counterproductive / educative, the student's mental model was wrong.

The student was taking the model for ratios and applying it to fractions. If I need to add 2 + 2, and I multiply instead, I did the wrong calculation. It does not matter whether I multiplied correctly, nor does it matter that, in this case, both operations equal four.

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

#188

Post by Phlogistique »

adrianmonk wrote: post #176

The difficulty isn't with fractions. It's about understanding what "+" means. Words have multiple meanings/senses. Addison knows the word "plus" already and knows it can mean summing up numbers ("2 plus 2 is 4") or it can mean combining things in other ways ("tonight, we'll eat pizza plus see a movie"). His teacher has introduced "+", and pronounced it "plus", so it's reasonable for him to apply what he knows about t…

That's my favourite explanation so far. That's how I would put it to the student:

Yes, that's a perfectly correct operation. When you take 1/3 of the first table and you put them toghether with 1/3 of the second table, you get 2/6 of both tables. However, that's not what mathematicians mean when they use the sign "+". Let's explain the difference with examples:

* At your table, Bob is 1/3 of the table, and Sandra is 1/3 of the table. Bob PLUS Sandra equals 2/3 of the table.

* Sandra is 1/3 of the table. Alice is 1/3 of the other table. When you put the two tables together, Alice and Sandra are 2/6 of both tables.

The first operation is what mathematicians call "+". They write "1/3 + 1/3 = 2/3"

The mathematicians do not have a good name for the second one, so let's invent one: "1/3 1/3 = 2/6"

What's better with this explanation, compared to the "ratios vs. fractions" thing, or the units thing, is that you do not have to introduce a separate category of numbers that sound very similar but act differently.

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

#189

Post by Wowfunhappy »

rovolo wrote: post #185
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…

The core concept kids need to know when adding fractions together is that the denominators need to be the same before you can add the numerators. Your use of units is on the right track, but I think you need to use units for both the top and bottom of the fraction: (1 girl at table A)/(3 students at table A) + (1 girl at table B)/(3 students at table B) = (2 girls at tables A & B) / (6 students at tables A & B) vs (1…

Ah, yes, that would definitely be a better way to write it if you were to go forward with the unit approach.

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

#190

Post by boomlinde »

Of course 1/3 + 1/3 = 2/6, and in this case you can make an error in reasoning that is probably common among children who are just learning fractions that will still arrive at the same answer.

In this case I would have demonstrated a case where the error in reasoning that was possibly made does not lead to the correct answer, for example 1/8 + 1/4. The erroneous intuition of simply adding the groups and their subsets together breaks down when you can show that 1/8 + 1/4 ≠ 2/12 but 3/8

Post reply on HN