Hall Passap Calculus

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  1. Hall Pass Ap Calculus Ab Exam
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Powered by Create your own unique website with customizable templates. Calculus is designed for the typical two- or three-semester general calculus course, incorporating innovative features to enhance student learning. The book guides students through the core concepts of calculus and helps them understand how those concepts apply to their lives and the world around them. Games by HallPass - Armor Games.

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Has the United States become too dumb to survive? Not yet, but some aim to take us in that direction.

A recent piece on Slate argues that we shouldn't send our children to private school, no matter what. Sure, the local schools may be bad. But that's apparently just the price we all need to pay. In fact, the lousier the public school is, the more important it is that your children go there.

No, not just to be indoctrinated into believing whatever the government wants you to believe. Allison Benedikt has an even more frightening reason for insisting on universal education.

She begins by bragging about her own ignorance. She says she read a total of one book in high school, and brags she didn't learn much on her way through the University of Michigan, either. She insists she's “doing fine,” but how would she know? To paraphrase Don Rumsfeld: education, to her, is an unknown unknown. She's never studied, clearly doesn't care to, and would prefer that nobody else do so either.

Hall Pass Ap Calculus Ab Exam

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'I’m not saying it’s a good thing that I got a lame education. I’m saying that I survived it, and so will your child, who must endure having no AP calculus so that in 25 years there will be AP calculus for all,' Allison Benedikt writes. There’s a rallying cry: If one kid can’t pass AP classes, get rid of them for everyone. Talk about a race to the bottom.

Sir Isaac Newton declared that if he saw far (and he did) it was only because he stood on the shoulders of giants. Those educated people went before him, wrote down what they learned and passed it on. Benedikt would prefer that we, instead, raise a generation of midgets, who won’t understand the world even if they bother to look around it.

The fact is, as the world has become more complicated, there’s more we don’t understand. Doctors and lawyers specialize in particular areas. Few really know how a cell phone works, or how a PC connects to the internet. We can’t change our car’s oil or build our own homes.

And that’s all right, as long as some people understand each of these things. That’s what we need for civilization to endure. Yet Benedikt has fallen for the illusion of permanence. She assumes that since we have smart phones, and airplanes, and safe drinking water, and flat screen TVs we will always have those things, if not better ones.

But it doesn’t work that way. When the Roman Empire collapsed, it wasn’t replaced by a civilization that built larger buildings and better roads. Instead, it gave way to centuries rightly called the dark ages, when people struggled just to survive. Most forgot how to read and write. Rome’s aqueducts and walls endured for centuries. Some remain today. But it was centuries before people could start building anything like them again.

President Obama is more careful with his language, but he’s pushing the same ideas as Benedikt. When he says he’ll “ask for the wealthy to pay a little bit more” and his administration imposes regulations under Dodd-Frank and ObamaCare, he’s assuming that things will continue exactly as they are. Despite the tide of taxes and debt and red tape, he assumes everyone will keep on inventing, innovating and creating jobs at the exact same pace they always have.

However, if people respond to the incentives Obama’s giving, they’ll stop being as productive. Where will our economy go then? Downhill, fast.

Let’s take Bebedikt’s ideas to their logical conclusion. Instead of putting education on a glide path to destruction, why not just shutter all the schools today and send the kids off into the streets? There’s no reason to waste their time for the next couple of generations until we all forget everything we know. Instead, let’s just forget everything and move on.

Despite what Benedikt thinks, private schools are a great way to improve public ones. They introduce a level of competition that public schools have to respond to. You can’t fight City Hall. But if you make it irrelevant, it may change itself. Then we can all keep learning.

I have a love/hate relationship with calculus: it demonstrates the beauty of math and the agony of math education.

Calculus relates topics in an elegant, brain-bending manner. My closest analogy is Darwin’s Theory of Evolution: once understood, you start seeing Nature in terms of survival. You understand why drugs lead to resistant germs (survival of the fittest). You know why sugar and fat taste sweet (encourage consumption of high-calorie foods in times of scarcity). It all fits together.

Calculus is similarly enlightening. Don’t these formulas seem related in some way?

They are. But most of us learn these formulas independently. Calculus lets us start with $text{circumference} = 2 pi r$ and figure out the others — the Greeks would have appreciated this.

Unfortunately, calculus can epitomize what’s wrong with math education. Most lessons feature contrived examples, arcane proofs, and memorization that body slam our intuition & enthusiasm.

It really shouldn’t be this way.

Math, art, and ideas

I’ve learned something from school: Math isn’t the hard part of math; motivation is. Specifically, staying encouraged despite

  • Teachers focused more on publishing/perishing than teaching
  • Self-fulfilling prophecies that math is difficult, boring, unpopular or “not your subject”
  • Textbooks and curriculums more concerned with profits and test results than insight

‘A Mathematician’s Lament’ [pdf] is an excellent essay on this issue that resonatedwithmany people:

“…if I had to design a mechanism for the express purpose of destroying a child’s natural curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being done — I simply wouldn’t have the imagination to come up with the kind of senseless, soul-crushing ideas that constitute contemporary mathematics education.”

Imagine teaching art like this: Kids, no fingerpainting in kindergarten. Instead, let’s study paint chemistry, the physics of light, and the anatomy of the eye. After 12 years of this, if the kids (now teenagers) don’t hate art already, they may begin to start coloring on their own. After all, they have the “rigorous, testable” fundamentals to start appreciating art. Right?

Hall

Poetry is similar. Imagine studying this quote (formula):

“This above all else: to thine own self be true, and it must follow, as night follows day, thou canst not then be false to any man.” —William Shakespeare, Hamlet

It’s an elegant way of saying “be yourself” (and if that means writing irreverently about math, so be it). But if this were math class, we’d be counting the syllables, analyzing the iambic pentameter, and mapping out the subject, verb and object.

Math and poetry are fingers pointing at the moon. Don’t confuse the finger for the moon. Formulas are a means to an end, a way to express a mathematical truth.

We’ve forgotten that math is about ideas, not robotically manipulating the formulas that express them.

Ok bub, what’s your great idea?

Feisty, are we? Well, here’s what I won’t do: recreate the existing textbooks. If you need answers right away for that big test, there’s plenty of websites, class videos and 20-minute sprints to help you out.

Instead, let’s share the core insights of calculus. Equations aren’t enough — I want the “aha!” moments that make everything click.

Formal mathematical language is one just one way to communicate. Diagrams, animations, and just plain talkin’ can often provide more insight than a page full of proofs.

But calculus is hard!

I think anyone can appreciate the core ideas of calculus. We don’t need to be writers to enjoy Shakespeare.

It’s within your reach if you know algebra and have a general interest in math. Not long ago, reading and writing were the work of trained scribes. Yet today that can be handled by a 10-year old. Why?

Because we expect it. Expectations play a huge part in what’s possible. So expect that calculus is just another subject. Some people get into the nitty-gritty (the writers/mathematicians). But the rest of us can still admire what’s happening, and expand our brain along the way.

It’s about how far you want to go. I’d love for everyone to understand the core concepts of calculus and say “whoa”.

So what’s calculus about?

Ap classroom

Some define calculus as “the branch of mathematics that deals with limits and the differentiation and integration of functions of one or more variables”. It’s correct, but not helpful for beginners.

Here’s my take: Calculus does to algebra what algebra did to arithmetic.

Ap classroom
  • Arithmetic is about manipulating numbers (addition, multiplication, etc.).

  • Algebra finds patterns between numbers: $a^2 + b^2 = c^2$ is a famous relationship, describing the sides of a right triangle. Algebra finds entire sets of numbers — if you know a and b, you can find c.

  • Calculus finds patterns between equations: you can see how one equation ($text{circumference} = 2 pi r$) relates to a similar one ($text{area} = pi r^2$).

Using calculus, we can ask all sorts of questions:

  • How does an equation grow and shrink? Accumulate over time?
  • When does it reach its highest/lowest point?
  • How do we use variables that are constantly changing? (Heat, motion, populations, …).
  • And much, much more!

Algebra & calculus are a problem-solving duo: calculus finds new equations, and algebra solves them. Like evolution, calculus expands your understanding of how Nature works.

An Example, Please

Let’s walk the walk. Suppose we know the equation for circumference ($2 pi r$) and want to find area. What to do?

Realize that a filled-in disc is like a set of Russian dolls.

Here are two ways to draw a disc:

  • Make a circle and fill it in
  • Draw a bunch of rings with a thick marker

The amount of “space” (area) should be the same in each case, right? And how much space does a ring use?

Well, the very largest ring has radius “r” and a circumference $2 pi r$. As the rings get smaller their circumference shrinks, but it keeps the pattern of $2 pi cdot text{current radius}$. The final ring is more like a pinpoint, with no circumference at all.

Now here’s where things get funky. Let’s unroll those rings and line them up. What happens?

  • We get a bunch of lines, making a jagged triangle. But if we take thinner rings, that triangle becomes less jagged (more on this in future articles).
  • One side has the smallest ring (0) and the other side has the largest ring ($2 pi r$)
  • We have rings going from radius 0 to up to “r”. For each possible radius (0 to r), we just place the unrolled ring at that location.
  • The total area of the “ring triangle” = $frac{1}{2} text{ base} cdot text{height} = frac{1}{2} (r) (2 pi r) = pi r^2$, which is the formula for area!

Yowza! The combined area of the rings = the area of the triangle = area of circle!

(Image from Wikipedia)

This was a quick example, but did you catch the key idea? We took a disc, split it up, and put the segments together in a different way. Calculus showed us that a disc and ring are intimately related: a disc is really just a bunch of rings.

This is a recurring theme in calculus: Big things are made from little things. And sometimes the little things are easier to work with.

A note on examples

Many calculus examples are based on physics. That’s great, but it can be hard to relate: honestly, how often do you know the equation for velocity for an object? Less than once a week, if that.

I prefer starting with physical, visual examples because it’s how our minds work. That ring/circle thing we made? You could build it out of several pipe cleaners, separate them, and straighten them into a crude triangle to see if the math really works. That’s just not happening with your velocity equation.

A note on rigor (for the math geeks)

I can feel the math pedants firing up their keyboards. Just a few words on “rigor”.

Did you know we don’t learn calculus the way Newton and Leibniz discovered it? They used intuitive ideas of “fluxions” and “infinitesimals” which were replaced with limits because “Sure, it works in practice. But does it work in theory?”.

We’ve created complex mechanical constructs to “rigorously” prove calculus, but have lost our intuition in the process.

We’re looking at the sweetness of sugar from the level of brain-chemistry, instead of recognizing it as Nature’s way of saying “This has lots of energy. Eat it.”

I don’t want to (and can’t) teach an analysis course or train researchers. Would it be so bad if everyone understood calculus to the “non-rigorous” level that Newton did? That it changed how they saw the world, as it did for him?

A premature focus on rigor dissuades students and makes math hard to learn. Case in point: e is technically defined by a limit, but the intuition of growth is how it was discovered. The natural log can be seen as an integral, or the time needed to grow. Which explanations help beginners more?

Hall Passap Calculus Pdf

Let’s fingerpaint a bit, and get into the chemistry along the way. Happy math.

(PS: A kind reader has created an animated powerpoint slideshow that helps present this idea more visually (best viewed in PowerPoint, due to the animations). Thanks!)

Hall Pass Ap Calculus Frq

Note: I’ve made an entire intuition-first calculus series in the style of this article:

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