See algebra as learning to describe the world using math language.

Imagine describing a painting with English. You’d label the main items (two people, walking) and maybe the background environment, etc.

Algebra is the art of describing a scenario with math. Distilling it down to the essence.

Like a mystery novel, you can look at the relationship in the painting (walking towards each other? away?) and make other deductions (they like each other, or don’t). With algebra, you can express some relationships and then draw other conclusions that might not have been apparent at first.

What are the essential insights for algebra?

  • Naming things. Why? Have the name, can see the connection. Otherwise it’s a bunch of “stuff”.
  • A function f(x) is very generic. Can’t see the parts. f(x) = x^2 + 2x + 3 shows you the stuff inside. Can see how each part interact.
  • Algebra is about the parts and the interactions between them.
  • Get good at labeling the parts
  • Get good at labeling the interactions
  • With a description, you can make predictions. If f(x) is setup like so (x^2 + 2x + 3) then when x=3 we’ll have the following result…
  • With a description, you can simplify. Sometimes you have a jumble of parts. Can you organize them to better understand the big picture? (When organized, you know when it reaches different positions. Now you can make better predictions).
  • Algebra is about naming parts, and describing the connections between them. “The algebra of X” or “The algebra of Y”. Isn’t that most of science? Or much knowledge? What are the pieces, and how do they work together?

That’s the essential insight. As we practice, we can get really good at seeing the pieces and organizing them. Or making predictions based on these relationships. “Given these connections, if x becomes 3, then the entire system will become 15 (or whatever the value is).”

Being “good with algebra” means you can:

  • Label the parts
  • Describe the connections
  • Predict what will happen
  • Simplify the description into something easier to understand. We took a jumble of parts and factored (organized) them into TWO parts. Each of those parts has 2 subparts. So, a bunch of loose pieces get tidied up. It’s hard to understand what’s happening when everything is scattered on the floor. (The legos which are spilled everywhere, put them together and see the shape you have.)

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Also, functions can be viewed as generalised versions of equations: you specify the structure of your system but not the desired value, just like the slope formula is a more dynamic version of the Pythagorean theorem.This is also why the zeros of a function are so important, along with the domain (which values make my system go phut?).


I like that. If we have a table of data (input/output pairs, or sets of many relationships) then a function is a refactoring to pull out the common connection.

Often we present functions without the alternative (we can write f(x) = 2x, or we can give you this giant lookup table…).