Math Relearning/Progressions/3-5 Fractions

These are my comments on the Common Core Math Progressions document Number & Operations-Fractions: Grades 3-5 (PDF).

Grade 3
"The meaning of fractions" - I like this, but what about Chapin and Johnson's other meanings? Of course, that's too much to cover in grade 3. I'd also like to explore the complex nature of the fraction representation. It's a single number that's represented using more than one number, to start with.

"same shape and size" - Isn't size the only real requirement?

"The importance of specifying the whole" - It seems like a lot of things (everything?) in math are identified in relation to something else.

"basic building block of fractions" - This is the kind of thing I would come up with.

"Estimate lengths" - This reminds me about a difference I observed between measurement and fractions. Fractions divide a whole into equal parts. Measurement adds up equal parts to reveal the amount of a whole, which may not be a multiple of a single part. I'm sure there's a better way to say that.

Grade 4
(4 * 7)/(4 * 9) = 7/9 - Very interesting! There are some subtleties I want to understand here. Fractions are an interesting representation, since they have multiple parts, and their parts don't work quite like regular integers.

"no mathematical reason" - Aha! Some things we do in math are for convenience, not for mathematical necessity. Take that, Jeremy!

"multiplying by 1" - This is an example of a seemingly useless move that is very useful in a particular context. It works in this case because you're not just multiplying by 1 but by an equivalent in another representation. You've translated the 1 into a more powerful form.

"2/3 + 5/8 as a length" - I notice the diagram doesn't mark the length of the whole. Instead it seems to base the fractional lengths on each other. This made me realize you can view two fractions as a ratio or proportion.

"decimal as a fraction generalizes" - I don't see how a visual fraction model wouldn't generalize. Maybe I don't know what cases they mean.

With these fraction examples I'm finding myself taking the symbol-manipulation shortcuts. I'm going to have to slow down when I'm doing the lessons and think about the problems conceptually. But as the concepts build on each other, I'll have to find summary models that'll remind me of what the more advanced concepts mean so I don't have to think through the whole chain of concepts for each problem.

Conceptual understanding isn't just about being able to picture the meaning of the procedures but about being able to think about the concepts flexibly to solve problems, which means breaking down the situation and transforming it into more suitable forms for solving, perhaps largely by mixing and matching the conceptual pieces. So that's a skill to concentrate on.

Grade 5
"least common denominator" - Interesting. I always thought that's just what you did, but I see that its non-necessity is an extension of the fact that simplifying fractions isn't mathematically necessary.

fractions as division - Explore the relationship between this and fractions as addition or multiplication of parts.

"contribute 1/3 of itself" - I usually think of division linearly as the objects being grouped as wholes except where the dividing lines cut through them, sort of like the second example solution with the sack of rice.

"general formula for the product of two fractions" - The formula reminds me of how detailed math is and how long it would take to list all the concepts and formulas at a fine grained level. I sometimes think that would be a good thing to do, I think to give me a more concrete basis for mathematical problem solving, but the prospect is intimidating. This makes me think being decent at problem solving with math takes commitment! There are a lot of details to know and interrelate.

"reason out many examples" - I might still have gaps to fill with my own investigation as I'm going through Common Core conceptual math. One type of question I still might ask is how these concepts might have developed in the first place, or at least how someone might develop them from experience and necessity, even if we don't know how they actually came about.

"same as multiplying the number by a unit fraction, 1/3 x 5" - Wouldn't that be 5 x 1/3? I don't quite know how to conceptualize 5 x 1/3 = 5 ÷ 3. I only know that's the rule, and following it comes very easily to me. However, I do see that when you divide a part into parts, you get more (and smaller) parts, and maybe that's the key.

area model for 3/4 x 5/3 - I wonder how people came up with these explanations for math concepts. They seem clever. That's another type of question worth exploring. Explanations of fraction operations would also be a good place to experiment with rewording the explanations to answer my own difficulties in understanding. Maybe translating the operations into English that makes sense to me would help. To take a later example, 3 ÷ 1/6 means "1/6 goes into 3 how many times?"

"Multiplication as scaling" - I'm glad they're covering this. Keith Devlin is happy about it too, I'm sure. But I would've thought the scaling factor would come second in the expression, so 3 x 5 and 3 x 1/2 in the examples. You present what you have and then how you're changing it. I'm curious how they're conceptualizing the mathematical statement they're expressing.