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Topic: Non-Euclidean Arithmetic
Replies: 108   Last Post: Sep 13, 2012 3:39 PM

 Messages: [ Previous | Next ]
 kirby urner Posts: 3,690 Registered: 11/29/05
Re: Non-Euclidean Arithmetic
Posted: Sep 10, 2012 6:45 PM

On Mon, Sep 10, 2012 at 2:03 PM, Paul Tanner <upprho@gmail.com> wrote:
>
>

> > Scaling can also be about volume as volumes scale as surely as lengths
> > do, and so multiplication, presented as scaling, could show a sphere
> > getting bigger and smaller.
> >

>
> The set of volumes that are bigger or smaller is still a set under a
> total or linear order.
>

The intermediate value theorem or something like it might be used.
For the volume to go from 3 to 4, it must go through pi along the way,
i.e. all real numbers between 3 and 4 are hit, as the volume increases
from 3 to 4.

>
> You will agree that the radius, not just

> > the circumference could be pi, but so could the volume in the
> > continuum of algebra, i.e. we can set the volume to pi and compute the
> > radius accordingly (some irrational number). In this way, a growing
> > sphere might represent two irrationals being multiplied. No need to
> > get hung up on length.
> >

>
> No, but I was using "length" because it's not easy to get away from a
> set under a total or linear order when we are talking about scaling.
> If it's only under a partial order, then how does scaling work? Sure
> one could perhaps come up with a whole lot of definitions for
> "scaling" when the set is only partially ordered, but at least for
> kids I would think that that would be out.

It's very easy to use volume or area in place of length. Just show it
growing and shrinking with no change in surface and central angles
i.e. the *shape* is held fixed while the "size" (sometimes related to
"frequency" in STEM) is a variable.

We are keen to keep the 1, 2, 3 powering of line, surface, volume in
view, i.e. take a shape as complicated as a sewing machine and scale
it up, linearly, by 3.445.

The volume will increase by a factor of 3.445**3 (i.e. "to the third
power") exactly. We don't even need to know the initial volume to
know what scale factor applies.

We might want to (definitely want to) use a growing / shrinking
tetrahedron in some segments.

Since you agree scaling covers the Reals (R) as well as the Rationals
(Q), this kind of scaling (using volume) should be no problem.

But then we also have our discrete units of volume, and our factional
quantities.

STEM chart (shape : volume pairs):
Tetrahedron : 1
Octahedron : 4
Cuboctahedron : 20

(all with same edge length)

A lot of the "repeated addition" model will propagate over, i.e. when
we show a shape growing and shrinking, we can talk about how much
volume is being added or taken away, with repetition involved.

>
>

> >> The lack of fluent proportional reasoning in a large percentage of
> >> students all the way up through adulthood is one of big problems that
> >> must be addressed, and a least including this scaling way of modeling
> >> multiplication using equivalent or equal ratios or fractions from day
> >>

> >
> > My pet peeve is to always cast multiplication in terms of numbers,
> > always using a number set. That's terribly unimaginative and not
> > worthy of a second look. You need to see the addition operator used
> > like this: "ABC" + "RFP" == "ABCRFP" (concatenation), and the
> > multiplication operator used like this: "TA" * 3 == "TATATA".

>
> But this multiplication in terms of numbers, using a number set as the
> domain of the one of the factors.

Yes, an integer appears. But so does a character string. It's an
actual use of the multiplication operator that is common in many
notations and should be mixed in as one of the many examples of
multiplication in the wild.

What I was saying is that text books which are exclusively numeric in
their presentations of maths, and not alphanumeric, are deficient, not
STEM-worthy, not STEM-compliant. If we're to align with STEM (as all
but the Americans might be doing), then we need to expand our horizons
beyond use numbers exclusively, even with respect to "the four
operations". Another reason why scientific calculators are not
suitable.

Kirby

Date Subject Author
9/1/12 Jonathan J. Crabtree
9/1/12 Paul A. Tanner III
9/2/12 kirby urner
9/3/12 Paul A. Tanner III
9/3/12 kirby urner
9/3/12 Paul A. Tanner III
9/4/12 kirby urner
9/4/12 Paul A. Tanner III
9/4/12 kirby urner
9/5/12 Paul A. Tanner III
9/5/12 Robert Hansen
9/6/12 kirby urner
9/1/12 kirby urner
9/1/12 Joe Niederberger
9/1/12 Wayne Bishop
9/1/12 Joe Niederberger
9/2/12 Robert Hansen
9/3/12 Paul A. Tanner III
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/3/12 Joe Niederberger
9/3/12 Paul A. Tanner III
9/4/12 Joe Niederberger
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Paul A. Tanner III
9/5/12 Joe Niederberger
9/5/12 kirby urner
9/5/12 Joe Niederberger
9/5/12 Robert Hansen
9/5/12 Joe Niederberger
9/6/12 Joe Niederberger
9/8/12 Robert Hansen
9/7/12 Jonathan J. Crabtree
9/8/12 kirby urner
9/8/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/10/12 Paul A. Tanner III
9/10/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Robert Hansen
9/8/12 kirby urner
9/8/12 Joe Niederberger
9/8/12 Jonathan J. Crabtree
9/9/12 kirby urner
9/8/12 Clyde Greeno @ MALEI
9/8/12 Jonathan J. Crabtree
9/8/12 Jonathan J. Crabtree
9/8/12 Joe Niederberger
9/8/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/9/12 Paul A. Tanner III
9/9/12 Robert Hansen
9/10/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/9/12 Joe Niederberger
9/9/12 Paul A. Tanner III
9/9/12 Wayne Bishop
9/9/12 Paul A. Tanner III
9/10/12 Wayne Bishop
9/10/12 Paul A. Tanner III
9/9/12 Paul A. Tanner III
9/9/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/10/12 Paul A. Tanner III
9/10/12 Joe Niederberger
9/10/12 Clyde Greeno @ MALEI
9/10/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/11/12 Paul A. Tanner III
9/11/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/13/12 Paul A. Tanner III
9/13/12 kirby urner
9/13/12 Paul A. Tanner III
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 kirby urner
9/11/12 Joe Niederberger
9/11/12 Joe Niederberger
9/11/12 Paul A. Tanner III
9/11/12 israeliteknight
9/11/12 Joe Niederberger
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/12/12 Paul A. Tanner III
9/12/12 kirby urner
9/11/12 Jonathan J. Crabtree