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3charting the course Re: -Teaching Calculus to 15 year olds, freshman High School, sequence plan

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  3charting the course Re: -Teaching Calculus to 15 year olds, freshman  High School, sequence plan Archimedes Plutonium <plutonium.archimedes@gmail.com> - 2018-06-26 16:43 -0700
    Re: Archimedes "just not bolted down too tight in the first moroney@world.std.spaamtrap.com (Michael Moroney) - 2018-06-27 00:34 +0000

#416469 — 3charting the course Re: -Teaching Calculus to 15 year olds, freshman High School, sequence plan

3Newsgroups: sci.math
Date: Mon, 25 Jun 2018 23:43:36 -0700 (PDT)

Subject: charting the course Re: -Teaching Calculus to 15 year olds, freshman
 High School, sequence plan
From: Archimedes Plutonium <plutonium....@gmail.com>
Injection-Date: Tue, 26 Jun 2018 06:43:37 +0000

charting the course Re: -Teaching Calculus to 15 year olds, freshman High School, sequence plan

Now this is all part of being a Logical Person, having a logical mind and thinking and doing things, logically. Is that we plan ahead in a Logical Sequence of events or actions. We plan ahead step by step, logically. 

And this planning ahead to teach 15 year olds Geometry and Calculus in Geometry class has focused my attention upon Y = mx + b. That formula is not only the formula of a line but is the very best "first function" to teach. 

So, the first day of Geometry class, we go ahead and teach one of the most important of all theorems in mathematics, Pythagorean theorem and it is important because it connects numbers with geometry. Rarely are numbers connected to geometry in such a big way. 

And the next biggest connection of numbers with geometry is Calculus. 

So we teach Freshman High School Geometry, and open it with Pythagorean Theorem , then, Y = mx + b, which soon leads into Calculus. 

Now, I am going to do something very different from all other Geometry classes in High School. I am not going to spend much time or emphasis on the Euclid axioms. Only briefly and cursory, on Euclid axioms, enough to allow students to do congruence proofs of triangles. 

Instead, I am going to teach Geometry with a emphasis on geometry figures, and proofs not using axioms but like the Pythagorean theorem, a proof in pictures. Picture proofs are stronger proofs than are those using a table of axioms. 

Now consider axioms or postulates, in the first place, such as Euclid's (1) two points determine a unique line (2) three points, not on a line determine a unique plane, (3) Ruler Axiom, points on a line can be numbered, etc etc. 

But now consider that the proof of the Pythagorean theorem needed no axioms or postulates whatsoever, for it was just a picture proof, needing no words. 

So, are postulates and axioms algebra and numbers, and not geometry? The proof of the Fundamental Theorem of Calculus is a picture proof, needing no words, no axioms. The proof of the area of many geometry figures needs no axioms, no words, just a few pictures. 

So we have to ask ourselves, can we do all of Geometry by staying inside of geometry and not having to resort to axioms or postulates? If not, how far can we go in Geometry with picture proofs. Here I think I know a clue as to the answer of this. The reason a picture proof is available for Pythagorean theorem for FTC, for area of parallelogram, trapezoid is because they are full of algebra already, with numbers, that they need no axioms to do a proof. Whereas something like triangles being congruent, they are not injected with numbers and so, need algebra axioms to help them get a proof. 

So instead of leaning so heavily on postulates and axioms, I want to strike a new course in Geometry that favors "pure picture proofs over proofs by axioms" 

And the way to do this, is to enlist our famous function Y = mx + b which covers all lines that are slanted and all flat lines like Y = 0x + 3 which is Y = 3. 

But, I need the perpendicular line to x-axis and that be X = 0y + b such as the perpendicular line X = 2. We just, in our sloppiness seldom write our functions fully. 

So instead of a lot of postulates that most Geometry books start with. 

We start with Pythagorean Theorem then turn to 

Y = mx+b and Y = 0x + b and X = 0y + b 

Then we make a study of the areas of figures leading to the trapezoid area, in order to do Calculus. 

The lines of Y = mx+ b give us the formulas or functions that are many figures of geometry. We use intervals and make a well defined concept of INTERVALS. So that we can list figures such as squares, rectangles, trapezoids. The integral of calculus depends on interval. 

I am looking at Jacobs book Geometry, 2nd ed, 1987. About 1/2 is good, the other 1/2 is poor and fake. 

What we dire need in 15 year old Geometry class, is to fit in the program, the teaching of Calculus. And that program cannot be done with so much emphasis on axioms and postulates. 

The two greatest geometry feats were Pythagorean theorem and Calculus, and thus, Geometry for 15 year olds needs those two taught. 

AP 

Newsgroups: sci.math
Date: Tue, 26 Jun 2018 09:01:45 -0700 (PDT)

Subject: attention to the concept of interval Re: -Teaching Calculus to 15
 year olds, freshman High School, sequence plan
From: Archimedes Plutonium <plutonium....@gmail.com>
Injection-Date: Tue, 26 Jun 2018 16:01:46 +0000

attention to the concept of interval Re: -Teaching Calculus to 15 year olds, freshman High School, sequence plan

On Tuesday, June 26, 2018 at 1:43:43 AM UTC-5, Archimedes Plutonium wrote: 
> Now this is all part of being a Logical Person, having a logical mind and thinking and doing things, logically. Is that we plan ahead in a Logical Sequence of events or actions. We plan ahead step by step, logically. 
> 
> And this planning ahead to teach 15 year olds Geometry and Calculus in Geometry class has focused my attention upon Y = mx + b. That formula is not only the formula of a line but is the very best "first function" to teach. 
> 
> So, the first day of Geometry class, we go ahead and teach one of the most important of all theorems in mathematics, Pythagorean theorem and it is important because it connects numbers with geometry. Rarely are numbers connected to geometry in such a big way. 
> 
> And the next biggest connection of numbers with geometry is Calculus. 
> 
> So we teach Freshman High School Geometry, and open it with Pythagorean Theorem , then, Y = mx + b, which soon leads into Calculus. 
> 
> Now, I am going to do something very different from all other Geometry classes in High School. I am not going to spend much time or emphasis on the Euclid axioms. Only briefly and cursory, on Euclid axioms, enough to allow students to do congruence proofs of triangles. 
> 
> Instead, I am going to teach Geometry with a emphasis on geometry figures, and proofs not using axioms but like the Pythagorean theorem, a proof in pictures. Picture proofs are stronger proofs than are those using a table of axioms. 
> 
> Now consider axioms or postulates, in the first place, such as Euclid's (1) two points determine a unique line (2) three points, not on a line determine a unique plane, (3) Ruler Axiom, points on a line can be numbered, etc etc. 
> 
> But now consider that the proof of the Pythagorean theorem needed no axioms or postulates whatsoever, for it was just a picture proof, needing no words. 
> 
> So, are postulates and axioms algebra and numbers, and not geometry? The proof of the Fundamental Theorem of Calculus is a picture proof, needing no words, no axioms. The proof of the area of many geometry figures needs no axioms, no words, just a few pictures. 
> 
> So we have to ask ourselves, can we do all of Geometry by staying inside of geometry and not having to resort to axioms or postulates? If not, how far can we go in Geometry with picture proofs. Here I think I know a clue as to the answer of this. The reason a picture proof is available for Pythagorean theorem for FTC, for area of parallelogram, trapezoid is because they are full of algebra already, with numbers, that they need no axioms to do a proof. Whereas something like triangles being congruent, they are not injected with numbers and so, need algebra axioms to help them get a proof. 
> 
> So instead of leaning so heavily on postulates and axioms, I want to strike a new course in Geometry that favors "pure picture proofs over proofs by axioms" 
> 
> And the way to do this, is to enlist our famous function Y = mx + b which covers all lines that are slanted and all flat lines like Y = 0x + 3 which is Y = 3. 
> 
> But, I need the perpendicular line to x-axis and that be X = 0y + b such as the perpendicular line X = 2. We just, in our sloppiness seldom write our functions fully. 
> 

It has always bothered me, ever since I learned a circle has a formula, but a square or rectangle does not. So there must be something totally wrong in Old Math, that a formula is apparent for the circle 

(x-u)^2 + (y-v)^2 = r^2 where r is radius and (u,v) center 

A formula for circle yet nothing for the square or rectangle in Old Math 

So, Old Math must really be screwed up for it lacks formula for square and rectangle. And the remedy, I believe is intervals coupled with Y = mx+b. And which will do trapezoids, parallelograms, even polygons. The key is "intervals", something that Old Math, for whatever reason seemed to shy away from intervals. 

They could not shy away from interval in Calculus integral, but then teachers of Old Math Calculus, seem to think they get their hands dirty when doing intervals. So, maybe it is a matter of laziness that intervals are shyed away from. Just too much darn work. 


> So instead of a lot of postulates that most Geometry books start with. 
> 
> We start with Pythagorean Theorem then turn to 
> 
> Y = mx+b and Y = 0x + b and X = 0y + b 
> 
> Then we make a study of the areas of figures leading to the trapezoid area, in order to do Calculus. 
> 
> The lines of Y = mx+ b give us the formulas or functions that are many figures of geometry. We use intervals and make a well defined concept of INTERVALS. So that we can list figures such as squares, rectangles, trapezoids. The integral of calculus depends on interval. 
> 
> I am looking at Jacobs book Geometry, 2nd ed, 1987. About 1/2 is good, the other 1/2 is poor and fake. 
> 
> What we dire need in 15 year old Geometry class, is to fit in the program, the teaching of Calculus. And that program cannot be done with so much emphasis on axioms and postulates. 
> 
> The two greatest geometry feats were Pythagorean theorem and Calculus, and thus, Geometry for 15 year olds needs those two taught. 
> 
> 

So, I think Geometry needs to have a increased attention and focus on interval. 

AP

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#416483 — Re: Archimedes "just not bolted down too tight in the first

Archimedes "Failure" Plutonium fails:

>AP writes: Google has abuse, but what good is that when it cannot

Fail, fail, fail. That's all Archimedes "Failure" Plutonium can do is
fail. Fail at math. Fail at physics. Why do you insist on attacking me,
failure?

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