Understand The Quadratic Equation.

Enter
Module IV

Rather than just memorize a formula how about understanding Problem-Solving well enough to derive it. Because that's basically all you're doing...You are solving for x where

ax² + bx +c  = 0

You're going to use your two problem-solving tools, Hero Zero and No Fun Get Back To 1.  You are employing concepts four & five of the five basic concepts.

No one ever took the five minutes to show me this. This is the quick and dirty derivation without lots of explanation, a couple f steps skipped because completing the square and factoring a square number should be "understood", (they're not). When you click enter you will get much, much more explanation than this...PLUS a few examples putting the Quadratic Formula into action. You get to see it multiple ways. 

Inside i use the blocks
...and drawings.
Symbols Only Presentation:

i left out a step, which where you factor the expression to get the square root. But you can SEE it in the graphic below. The sides of the square are x + b/2a as any child can plainly SEE. And they can get their hands on it. They know what it is. More than just memorizing. 

Enter
Module IV

After you're done with the next page you will understand how we get from this: 

ax² + bx +c  = 0

to this:

i don't just teach it to you,
i teach it to you
so you can teach others

Once you understand it, you can explain it. And when you explain it, the people you are explaining it to understand it too. This is the final stage rhetoric, get there starting at grammar, then logic then rhetoric: being able to express an idea mathematically using economy of symbol.

The explanations on my quadratics page are better than this graphic. However, this is superior in every way to MUS which uses symbols only at the algebra level...this guy is great but he fumbles on the one yard line.

Just a teeny tiny little things like hero zero and isolating x...this one of the ways to do it...i show you three...and finish the job. Notice lack of -4ac. i may not have all the cool graphics but at leaste i show you how to do it right.

This isn't wrong, it is just incomplete. But i hope by now you see the same blocks we just call them different things...and manipulate them 

On the other side i show you this and more. CORRECTLY. The units should be in the upper right not lower, in his case they aren't units they are (b/2a)² 's.

i hope by now you can figure out that if you've been playing with these blocks since you were four this is no big deal, we're just calling the pieces different names. Grammar. 

When we are doing (allegedly advanced) algebra we call them this, when we were doing base ten mathematics we call them that..."that" being hundreds, tens, and units.  With algebra we can call them all kinds of different things, depending on what we want to do. 

Again: there is no labor saving device created by the mind of man equal to that of the algebra. AI is algebra...combined with probability.

Here they are for calculus...

Same blocks different lessons, different names. It is/was important to some people to make sure you that believe math is hard. That wasn't your idea, somebody put that belief in your mind. And that somebody then became you, once you believed it. That person might be a 7 year old child, who was not understanding whatever it was, or a teen ager not understanding algebra in the ridiculous way it is presented. But now you are an adult. You can't hardly believe your own eyes.  Belief has effects on biology. The belief is erroneous, put there by a child; time for the man or woman to identify the belief, when it was put there and by whom (meditate on it for minute: you know) and then change the belief. 

For Sarah it was well meaning teachers from the time she was five. i wish i had these lessons with Sarah on video. She thought it was pretty simple. She decided she wasn't bad at math, it was just nobody taught her because they didn't know how. i show you how. 

The information has just been occulted. i reveal an oral tradition and have fun doing it. i use the classical method of presenting it apply the trivium and the pedagogical concepts of scaffolding, degree of difficulty, vertical alignment, and directed discovery and the five mathematical concepts particularly three, four and five. And when it comes to three we are going to use a particular kind of rectangle, a square, and then we're only interested in one side. employing the methodology, in other words. With proper use of socratic method your students will tell you the formula is, not the other way around. Starting with the formula is teaching backwards.

Whether you are here as a student or teacher or both you will SEE how easy this derivation is...  

In the trivium the quadratic formula is simply mathematical rhetoric, using economy of symbol we express precisely what x is given ax² + bx +c  = 0 

Module two parent teacher training will help you make sense of this diagram. But we start with very broad concepts and small numbers and work our way up to formulae.

Home

Allegedly Advanced Algebra
Higher Powers x³ x⁴ 
The Trivium

"Imagination is more important than knowledge or information" ~some guy named One Stone.

"Sometimes it is useful to know how large your zero is." ~Author Unknown

"Mathematics is the only good metaphysics." ~Lord Kelvin

“The difference between the poet and the mathematician is that the poet tries to get his head into the heavens while the mathematician tries to get the heavens into his head.” ~G.K. Chesterton

More math quotes