What I like about Algebra 2

I worry about Algebra 2 from time to time. Its the class that is mostly likely to be target for reform, usually complete replacement. But  my older son is currently going through it and that has given me a chance to look at it more slowly and deeply,  More than any other class Algebra 2 is denigrated in current math articles.  See: Example1, Example2 etc.  There are a couple of different attacks taken:

• Its irrelevant given the ascendancy of numerical computing.
• Its hard to learn and acts like a gatekeeper for many non mathematical fields.

 So is it all destined for the waste bin of history and would that be a great loss?

Let’s dispel with an often mentioned myth about the class - that its arcane symbolic manipulation.  For the most part the algebraic techniques taught in Algebra 2 are straightforward and have geometrical interpretations. They’re neither arcane nor purely symbolic.

• Think completing the square.
• Or the binomial theorem
• Or even polynomial division.

Moreover, although it been recast the topics covered roughly trace the areas investigated on the path towards calculus. We’ve lost a lot of that coherence nowadays via tacked on  subjects like matrix math or statistics and the often followed emphasis on function families.   The taxonomy approach is really common and while its a straightforward system of categorization I think its often not as interesting as the reasons why Mathematicians turned to these problems in the first place.  I also question whether the sort of familiarity with functions behaviors that such approaches emphasize is that important moving forward.   I.e. you see a lot of classes spending a lot time understanding the shape of various curves without using them for anything.

• I really like Polynomial Division and the Rational Roots Theorem. [Link] Its a super convenient short hand for skipping fancy factoring with polynomials.  Is it superseded by technology?  I’m thinking now that since it links so well to gradients that there is still intrinsic value here.
• Conics are super interesting. I’m still fascinated by the tilt factor the generalized equation introduces and how it relates to trig and matrix rotational vectors. I also find a whole class of interesting relationships between the conics and geometric figures.
• Vieta Formulas  / Discriminants are super powerful if you haven’t seen them yet in Alg I.
• Also inequalities like the AM-GM  or Cauchy Schwartz are fun.

So here’s a list of essential topics via Matt Enlow:  (and translated to text with google lens)

1. What are the simplest kinds of functions?

2. How can more complicated functions be “built” from simpler functions?

3. What can I determine about a function from its equation?

4. What can I determine about a function from its graph?

5. What kinds of equations can I solve, and how?

6. How is solving an inequality different from solving an equation?

7. Which processes can be easily reversed, and which cannot?

8. How can I use what I know about equations and functions to solve “real-world” problems?

9. What are the proper roles/uses of technology when doing mathematics?

10. What do you do when you don’t know what to do?

Based on above how would I organize topics to emphasize the narrative I think would be valuable:

1. When was this developed, why and by whom? i.e a nod towards the 15th-16th century

2. What’s the link between Geometry and Algebra?  Can we reframe everything in terms of alegraba and why is that useful?  Conics in  particular seem like a bridge topic since they produce many of the families of equations that are normally studied.

3. How do we manipulate functions and what kinds of new questions do we arrive at thinking about them?    (Emphasize link towards calculus like slopes and asymptopes)

4. What kind of equations can we solve and are there ones we can’t? Does this matter? What new ideas did we need to handle things like cubics? Why were they so vexing?

5. Can we approximate solutions?  How do series play a role in this?  How do we handle these in real life nowadays?

6.  What new questions in Science pushed things forward and why limits etc. became more pressing.