You are about to discover why
A = πr²

ENTER

MIV

Module IV

The page on the other side of this only has videos on it. But just two of them individually are worth the price of a password. 

No kidding.

This derivation of πr² is rarely taught to students, or even to teachers.

Most of the time when i show this to teachers they have never seen it. i have never shown this to a student who had already seen it in 35 years of doing this.

Not once.

Here is more study of triangles. When finding the area of a circle we use triangles. Then we can make triangles using squares and study pythagoras...and then we can study special triangles and put those triangles on the unit circle.

You'll see wooden blocks in my videos cut into triangles, chances are you have these blocks, especially if you are Montessorian. i show you how to use them to help understand simple trig concepts. Starting with wooden blocks. Triangles they can get their hands on. Then we can turn triangles into rectangles. Those lessons get us ready for these lessons. There is no correct order, when you teach by concept.  Then it becomes degree of difficulty. 

i need to make a more step by step video showing all the different things you need to do to get to this point but for now these short vids will have to do. You should be able to make simple lesson plans depending on the age of your students. i can tell you from experience 3rd graders have no problem mastering this information if you present it at their level. Start with rectangles; then squares, then triangles.

Then find the area of those triangles. Then cut hexagons into triangles. Then cut up octagons. This could be one lesson or several different lessons depending on age and aptitude and time alloted.  

The few minutes you spend here may well clear up a lot of confusion and give you a few "a-ha" moments.  If you are a teacher these vids should give you a few ideas. Again, i have taught this to 3rd graders without much difficulty, but as the video says it may take you hours of preparation to get here. Have to talk about areas of rectangles and triangles first.

These lessons are also designed to teach 45 addends and 400 multiplication facts... 

Peek behind the paywall.

Here is a Pre-view, a short peek behind the paywall, i gloss over the bonuses:

Here is the original video from way back when, just so you can see how much the page has improved. Way more stuff has been added to the site in ten years and this page is no exception.  And now i explain it the same way several times AND give you a completely different explanation which will lead you down the road to calculus. "Here's another way of looking at it..." ~often quoth a 12 year old Genius i had the privilege of tutoring.

i have the unique experience of having tutored geniuses using base ten blocks over the years. They taught me as much or more than i taught them.  Check out the logic page for a great (fully binge-watchable) video series on Genius. It's Yale. Don't judge me.  Not aging well if you've woken or have been rudely wakened by recent events. It's laced with propaganda. Still worth your time. 

ENTER

MIV

i also put bonus videos on the page, from a YouTube channel that makes that have animations that very much resemble base 10 blocks. He mentions several times in his video series that a good way to visualize multiplication is by thinking about area.  He also shows you the derivation of πr² with cool animation.  His derivation is completely different than mine. i've always said you should know how to do something more than one way. On the next page you get two great explanations that should help you attain greater understanding...And get you ready for calculus.

i hope you support his channel "3Blue1Brown."   He has over 1 million subscribers. [Edit many millions more since i first wrote this.] Many of his videos are priceless. But he has many, many videos now, and the one that's on the next page might be difficult to find...also my TikTok videos are so numerous the ones on the next page might be hard to find, so you are also paying for me aggregating the knowledge for you, that you won't find anywhere else all onto one website. 

It's never about storage, it's always about retrieval.

Here is another page of priceless information for homeschoolers that will put you ahead of most public and private schools when it comes to understanding not just memorizing basic mathematics.