Math Relearning/Progressions/6-8 Expressions and Equations

These are my comments on the Common Core Math Progressions document Expressions & Equations: Grades 6-8 (PDF).

This is one of the topics I've been looking forward to, though I didn't really realize it. It feels like home base for me, a central stopping point for math skills. Earlier concepts were leading to it, and it leads to a bunch of others. Plus it's what I normally think about when I think of math, and I feel like I'm pretty good at this part.

Overview
"a series of nested or parallel operations" - This reminds me that I want to find various ways to represent complex structures like math expressions, ways like parse trees.

Grade 6
"objects in their own right" - They must've read Tall. This sounds like procepts.

"what is the meaning of the 7?" - Interesting, I hadn't really thought of coefficients as having their own meaning, but obviously it makes sense.

"Looking for structure ... sequence of operations" - Structure vs sequence is a good pair of terms for the relation-operation dichotomy. Also, I've been privileging structure because Dijkstra did, but sequences have their own logic that's worth studying, and everything we do comes down to them anyway. The way we use it, at least in programming, structure is partly just a translation of sequence meant to hide (from) its difficulties. But sequence is unavoidable in many cases (microprocessors?, games, music), and it's long been mysterious to me. It's often a magical process I can't quite follow of transitioning from one state to a quite different one.

"any order, any grouping" - I hadn't heard of that one. It seems like it should have a more formal name. I wonder if EngageNY will cover it.

"hold numerical expressions unevaluated" - This seems related to my idea of translations as ways of making expressions more powerful for particular purposes.

"does not necessarily dictate how to calculate them" - Another expression of relation vs operation, and an interesting one. The operation properties give you ways to regroup the quantities.

"The distributive law is of fundamental importance" - Good to know.

"accustomed to solving such problems by division" - It occurs to me that I should learn how the rules of transforming expressions work. This might mean tracing the meanings back to the basics, and for doing this, it might be good to have a list of concepts and skills and the visual models that represent them.

"solving equations of the form" - It's interesting and helpful that we teach equations by grouping them by their form. It's worth asking why these groupings.

"the number satisfying the equation" - I'm kind of impatient to get to graphing because I feel like it makes it easier for me to work out these solutions. Visualizations are reassuring and satisfying at least.

"Analogous arithmetical and algebraic solutions" - I wonder why parents don't complain that algebra overcomplicates arithmetic. If they did, how would the teacher argue for algebra?

Grade 7
"two different possible next steps" - I hadn't thought that there would be more than one way to simplify an expression. Reversing the distributive property is new to me.

Grade 8
"Properties of Integer Exponents" - I'll need to see how these work.

"we define 10^0 = 1 because" - Ah, a different explanation from the one I'd seen, which was based on division. Very interesting. And I used to think these questions were mysteries I'd have to dig for the answers to. Of course, maybe some of them still are.

"sqrt p is defined to mean" - Why?

"Know that sqrt 2 is irrational" - How did we find that out? How do we calculate it? I've seen an algorithm online somewhere.

"visual representation of the relationship" - What does graphing do for us exactly? How does a visual representation help?

"geometry of similar triangles" - Interesting! I'm looking forward to relating geometry and algebra.

"easier for students than reasoning through a numerical solution" - Ah, good, an example of how algebra helps.

"Linear equations also arise" - How did we get the idea to graph equations? What other kinds of coordinate systems are there, and how would the same equations look in each?

"problems that lead to simultaneous equations" - The theater problem is actually an interesting one. It's a good example of a situation where math would give me a handle on a question that's too complex for me to guess at. I might like simultaneous equations.