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Topic: Common Core snippet a little distressing
Replies: 73   Last Post: Jul 26, 2013 6:27 PM

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 kirby urner Posts: 3,690 Registered: 11/29/05
Re: Common Core snippet a little distressing
Posted: Jul 16, 2013 2:36 PM
 att1.html (4.5 K)

On Tue, Jul 16, 2013 at 10:36 AM, Robert Hansen <bob@rsccore.com> wrote:

>
> On Jul 16, 2013, at 12:04 PM, Joe Niederberger <niederberger@comcast.net>
> wrote:
>
> Or is there a nice one for the tetrahedrons that I'm just not seeing?
>
>
> The problem I see there is that the big tetrahedron is not solid. It has
> voids, unless I am mistaken. Thus, it isn't a mapping of volume like the
> cube. It is the count of its pieces that follow the cubes of 1, 2, 3, ....
> While there is of course a mathematical reason for this, it isn't intuitive.
>
> Bob Hansen
>

No, you got lost with those holes.

They were just to show the anatomy of the bigger tetrahedron which consists
entirely of smaller tetrahedrons AND octahedrons.

*All* the volume gets filled in (counted) at every layer, there are no
holes of any kind.

That's why I switched to talking about pouring water, to get away from your
fixation on "holes".

Here's another related demo. All it really requires is being able to
imagine a tetrahedron "upside down" i.e. with its base pointed skyward, so
we can pour water into it, and yet it doesn't tip over and spill. Think of
how TV towers are anchored by guide wires and maybe that'll work.

Here's a picture:
http://www.flickr.com/photos/kirbyurner/4902706131/in/set-72157624750749042/

Start with an upside down tetrahedron with edges 1 foot. That's what we
can call our "cup of water". Now there's a completely empty tetrahedron
next to it with edges 2 feet. All the edges are twice as long. What to
say about volume? We know "one cup" of water fills the first. It turns
out eight cups of water fill the second.

And if we make the edges three feet? Then 27 cups.

Four feet? 64 cups.

And so on.

What's true is *any* shape that grows in a self similar fashion i.e. all
surface and central angles held invariant, grows as a 3rd power of the
change in linear dimensions. True of a complicated shape too, like a
sewing machine.

However the tetrahedron is the shape we focus on because:

(a) it's topologically simpler than the cube, so a logical candidate for a
unit of volume and
(b) the tetrahedron plays well with others, in terms of giving students a
lot of whole number
volumes to play with, for other related shapes (like the octahedron of
volume 4)

Indeed, there's a large and growing accessible literature (branch in math)
that just assumes the regular tetrahedron is a unit volume.

This used to be something you'd only learn at MIT, like in Dr. Loeb's
class, but now, in 2013, it's no longer tenable to avoid all this
development even at the high school level -- IF, that is, you're using one
of the better, more enlightened curricula (might be from Russia who knows
-- not saying competition with Common Core can't come from outside the US).

Today, when people say "cubed" unconsciously, because that's what they
learned in school, they don't have any sense of a branch, an alternative.
This will not be so true of people coming to adulthood today. More and
more of them are getting a better STEM education than their parents did, or
at least they have that potential.

Kirby

Date Subject Author
7/13/13 Peter Duveen
7/13/13 Robert Hansen
7/13/13 kirby urner
7/14/13 Robert Hansen
7/14/13 kirby urner
7/14/13 Robert Hansen
7/14/13 kirby urner
7/14/13 Robert Hansen
7/14/13 Robert Hansen
7/14/13 kirby urner
7/14/13 Robert Hansen
7/14/13 kirby urner
7/14/13 Robert Hansen
7/15/13 kirby urner
7/14/13 Wayne Bishop
7/14/13 kirby urner
7/14/13 Wayne Bishop
7/15/13 kirby urner
7/15/13 Robert Hansen
7/15/13 Christian Baune
7/15/13 kirby urner
7/16/13 Wayne Bishop
7/16/13 kirby urner
7/14/13 Wayne Bishop
7/14/13 Joe Niederberger
7/15/13 Joe Niederberger
7/15/13 Joe Niederberger
7/15/13 Robert Hansen
7/15/13 kirby urner
7/15/13 Robert Hansen
7/16/13 kirby urner
7/16/13 Robert Hansen
7/17/13 kirby urner
7/17/13 Robert Hansen
7/17/13 kirby urner
7/17/13 Robert Hansen
7/17/13 kirby urner
7/15/13 CCSSIMath
7/16/13 Joe Niederberger
7/16/13 Robert Hansen
7/16/13 kirby urner
7/17/13 Joe Niederberger
7/17/13 Robert Hansen
7/17/13 kirby urner
7/17/13 Robert Hansen
7/17/13 kirby urner
7/17/13 kirby urner
7/17/13 Robert Hansen
7/18/13 Wayne Bishop
7/17/13 Joe Niederberger
7/17/13 Joe Niederberger
7/17/13 Robert Hansen
7/18/13 Wayne Bishop
7/18/13 Joe Niederberger
7/18/13 kirby urner
7/18/13 Joe Niederberger
7/20/13 kirby urner
7/19/13 Joe Niederberger
7/19/13 kirby urner
7/19/13 Joe Niederberger
7/19/13 Joe Niederberger
7/20/13 kirby urner
7/21/13 Joe Niederberger
7/23/13 kirby urner
7/24/13 frank zubek
7/24/13 frank zubek
7/24/13 kirby urner
7/25/13 frank zubek
7/25/13 kirby urner
7/25/13 frank zubek
7/25/13 frank zubek
7/25/13 frank zubek
7/26/13 frank zubek
7/26/13 frank zubek