Thread by @normalsubgroup

(1/11) I've seen this posted a lot, but usually not with many comments. This thread has:

1. A version for teachers (ready to go for the class)

More importantly:

2. A view that promotes a richer, more human view towards algebra.
nytimes.com/2020/02/05/sci…

(2/11) As a disclaimer, I didn't read the article carefully. I do what I normally do, look just enough until I see an idea, and then go and recreate a version. This is how I (and probably most) mathematicians operate.

Therefore, this may look nothing like what's in the article.

(3/11) Here's a version for teachers. First an example, then (in the next tweet) a general description, then another example.

The main idea seems to be to better take advantage of the symmetry. The roots are the same "distance" from the vertex.
This example is golden, literally.

(4/11) Here's how the method always works. This means the quadratic formula is accidentally derived:

(5/11) One more example, for kicks:

(6/11) I have taught students who were unsuccessful with factoring previously and had some success with this:

If asked to factor 6x^2+7x-5, then:

1. Find roots (previous tweet) -5/3, 1/2.

2. Factored version: 6(x+5/3)(x-1/2).

They used the quadratic equation to factor.

(7/11) Note the article promotes the "no guesswork" angle. This provides that!

A point coming up, is understanding that we know that 2 polynomials are equal by comparing coefficients.

If you insist that (3x+5)(2x-1) is somehow better than
6(x+5/3)(x-1/2) then go ahead, @ me.

(8/11) Now for the *good* part.

I teach diverse students from many countries and many backgrounds and almost all intro calculus students benefit from this basic algebra question, with these specific instructions:

(9/11) The whole point is to have an awareness of how the solution will look before solving it; it will be a polynomial in x of degree 2. Therefore, it's best to answer 3 questions.

1. How much x squared is there?
2. How much x is there?
3. How much constant is there?

(10/11) Ask a class if the 3s cancel *without expanding*. Ask them if they're sure. Ask them why. Do not progress in the course until a good answer appears. Here's more to consider:

(11/11) I'm also challenging the interpretation of the ubiquitous "show *all* your work" slogan.

In particular, I *encourage* my students to go directly from

[(x+h)^2+3]-[x^2+3] to:

2hx+h^2

It's more fun that way!
It's about awareness and richness.

(11+1)/11. Either I never really counted how many tweets were in the thread or else I'm making a bizarre statement that fractions have to be limited by n/n. You choose. Two more in the thread:

(13/11) In tweet 3/11, I wondered if there would be objections to stating, "the u-coefficient will be zero".
Maybe there's a fear of bad habits; but can we look at it, and know how to ask "how much u is there?" and just *see* it? That's what I'm suggesting. It's very natural.

(14/11) This approach involves using more of the brain, not just one hemisphere. It aligns better with how I see math.

If this seems possibly correct, or if it's totally incorrect but possibly useful to consider, then that's great!
Thanks and best wishes.

END.