Math Relearning/Progressions/6-7 Ratios and Proportional Relationships

These are my comments on the Common Core Math Progressions document Ratios & Proportional Relationships: Grades 6-7 (PDF).

Overview
"Ratios have associated rates." - So a rate is a ratio where one side is set to one. And a proportion is an equality between two ratios? I couldn't quite tell if they were distinguishing between proportions and proportional relationships. And a percent is a ... rate, I think.

Ratios and proportions are another area that will be hard to wrap my mind around. I think it's because, like fractions, they involve relating two or more numbers. Of course, every number is related to a reference point, if only 0. But fractions, ratios, and so on add numbers to relate to each other. My mind gets quickly overwhelmed with the moving parts. But I can think of it like a game. Actually I don't know how much that helps. Games can be pretty confusing too. But it's at least an interesting way to look at it. One thing that I think would help is translating the math into English.

So what's involved in these relationships, and why do mathematicians pick these to study and teach?

"collection of equivalent ratios" - So a coordinate can be seen as a ratio, and a line with certain conditions (e.g., through the origin) marks a collection of ratios that are equivalent and represents a proportional relationship between the two components of the ratios (right terms?).

Grade 6
"Solving a percent problem" - This is an example of how falling behind in math causes problems when trying to learn later math. I'm not fluent with my fraction problem solving strategies, so I feel like I'm lagging when I follow the solution to this problem. Especially I haven't grasped how the fraction concepts relate to the rules I learned, so I fumble around when I try to connect the concepts to these ratio problems. I can imagine what it might be like for a student struggling with their homework or a test without the time needed to truly grasp what they're doing, feeling desperation or despair or resignation. Learning the concepts is supposed to help with problem solving, so to explore how the concepts help to think flexibly it would be good to concentrate on how the concepts translate into the various representations and skills and connect them.

Grade 7
"Ratio problem specified by natural numbers" - These solutions all have implicit steps, which will confuse people like me who don't have a firm grasp on them. Some parts of the solution seem to work backward from the step's destination, like solving a maze from the other end. You have to know you can do this and how. Why make 6 batches, for example? Of course, hopefully by grade 7 students will have gotten enough instruction and practice to know these things.

"what fraction of the paint is blue" - There's some ambiguity in using fractions in cases like this, if you're not thinking carefully. There's a fraction already involving blue paint in the problem, but you have to ask, a fraction of what? What's given is a fraction of a cup, but what you're looking for in this step is the fraction of the total paint.

"not the case that for every 10 years" - At least at this stage, math is largely tied to application, and you have to understand the domain you're applying it to, in this case aging.

"Correspondence among a table" - Even this helpful diagram has at least one key implicit step: deducing the 2/5 increase in y for every 1 increase in x. Obviously it comes from the 2 cups peach for every 5 grape, but it would make things crystal clear if we could see how that transformation happens via another diagram or two and the symbolic manipulation involved.

"rationale from cross-multiplying" - Ah, finally. And it makes sense to me. Except that now I need the rationale for canceling out.

"obscured by the traditional method" - I'm glad they acknowledge that some ways of expressing things can obscure more helpful methods of solving a problem.

"Skateboard problem 1" - I dismissed tape diagrams at first, but I keep seeing how they're really useful for breaking down certain kinds of problems and translating them into other problems. In this case 80% becomes four of 20%, which highlights that you can divide the cost by 4 to give you a basis for the solution, which is to find 100%. That's only one possible method, of course. Tape diagrams are also useful for showing the difference between the first and second skateboard problems in terms of what wholes the 20% are referring to.

The factoring and canceling of 140 * 100 / 80 in the second method is interesting. Explore that.

"Using percentages in comparisons" - "25% more" sounds ambiguous to me. Does it mean 25% of the smaller or larger number? The smaller, as it turns out. This highlights the need to understand certain details of English grammar and usage.

"I used percentages" - How do the multiplicative ratio table and the percentages relate? I'm always interested in how it is that different problem solving methods are equivalent.

Appendix
"Definitions and essential characteristics" - Lists like this of definitions and such are helpful for analyzing without extra, distracting explanations and comments on things like educational concerns.