This is exactly why third-level teaching is so much easier.
Can 1/3 and 1/3 = 2/6? It seemed so
31–40 of 308 posts
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#32No cat has nine tails. Every cat has one more tail than no cat. Ergo, every cat has 10 tails.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#33I 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…
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.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#34Proportions are tricky to introduce since they are the first obvious move away from absolute quantities. We're taught that division is just fancy subtraction, but it's actually the more subtle idea of proportionality. Similar with multiplication as dimensionality.
From here, it feels like the natural setup to show that you can't just 'combine' proportionalities without accounting for what portion these proportions contribute to the new whole.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#35Earlier 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...
It's a bit more complicated than "debunked/retracted", as that second document explains:
> Dehnadi, to his credit, stuck to his guns and did the meta-analysis that showed that he’d discovered a phenomenon and that his test was a worthwhile predictor.
The original paper contained several linked claims: that there is an ability to make consistent mental models, that it's intrinsic and fixed, that it predicts ability to program, and that few people have it, and hence few people can learn to program. AIUI, the debunked/retracted claims are that it's intrinsic and fixed, and that few people have it. It looks like the ability exists, but it can be learned, and it is linked to programming ability.
Which i think does line up with wcarey's point:
> 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. [...] The answers they wrote didn't matter; what mattered was whether their answers evinced consistent meaning applied to terms and operations.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#36I would try to explain semantics of addition (without the word “semantics” of course). I would say, in the first group 1/3 is the girl so if we take out one boy and replace it with another 1/3 (another girl), the resulting proportion is 2/3. Once we mix both groups, that is not what “plus” is.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#37Earlier 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…
Yeah that's a good approach. The problem remains though that if you use the + operator on ratios you're still overloading it to mean something different in a way that doesn't retain its meaning when you start expressing things as fractions instead. So 1:2 + 1:2 works, but 1/3 + 1/3 doesn't. I think you still want a different operator for this. Maybe ⊕ or ⋃ or ⋓ ? I'm just spitballing here. There's definitely enough o…
And + is overloaded in a bunch of ways students encounter in high school, and much time is spent talking about when you're allowed to add and when you're not and which rules apply when. Examples:
1 + 2 - fine. 1/2 + 1/2 - one set of rules. 1/2 + 1/3 - a subtly different set of rules.
1:20 + 0:45 - yet another set of rules. Modular. 30° + 350° - fine? But maybe modular.
15% + 20% - who knows? 15% of what? 20% of what?
(1,2) + (2,4) - can't be done.
a^2 + a^2 - fine. a^2 + a^3 - nope. a^2 + b^2 - nope.
It would be lovely if mathematics were taught as a strongly typed language without overloaded operators, alas all our corpus is in the language it's in.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#38A 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. Th…
I think this one of those situations where you do a bunch of working out and get to the end and see that, mathematically, the problem is fixed, but in your heart it still feels like the original problem is still there. (1/2) * (1/3) + (1/2) * (2/3) might seem like a small calculation to us, but if you've just encountered fractions for the first time I think that is a huge amount of abstract notation.
Instead, I think it's better to follow CydeWeys's suggestion of saying that both are correct results of combining 1/3 with 1/3, but they're two different ways of combining them. Say that when you combine two fractions within the same group we call it "addition" and use a plus, but when we combine two fractions from different groups we call it "averaging" (and maybe make up your own symbol for it).
Once you've talking about averaging a bit you can move on to multiplication, which in some ways is a more basic concept but, for fractions, is actually a bit less intuitive.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#39A 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. Th…
I would suggest to replace 1/2 with 3/6 since the number of students at each table is what matters, not the number of tables, which only yields the correct result if each table has the same number of students.
Re: Can 1/3 and 1/3 = 2/6? It seemed so
#40The 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 tables).
The teacher then demonstrates an entirely different formula: 1/3 (of the students at a table + 1/3 (of the students at a table) = 2/3 (of the students at a table).
The confusion comes because no one calls out that they're talking about fractions of different things.
Edit: There are a whole range of exploratory questions you can follow on from here as well.
Imagine if the tables have different numbers of students or if there are more than two tables. Helping students navigate these types of ratio transformations is why keeping track of units is so important. Otherwise, things can get hairy for the students very quickly.