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Topic: Real Numbers in the Primary Grades
Replies: 39   Last Post: Jan 2, 2010 11:36 PM

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kirby urner

Posts: 3,198
Registered: 11/29/05
Re: Real Numbers in the Primary Grades
Posted: Dec 26, 2009 8:00 PM
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On Thu, Dec 24, 2009 at 1:15 PM, Anna Roys <roys.anna@gmail.com> wrote:

> RE: On Thu, Dec 24, 2009 at 10:29 AM, Hecman Gun <sirgodel@live.com>wrote:
>

>> For students of all ages, definitions of basic
>> mathematical concepts have to be framed with care: not too formal, not too
>> informal...
>>
>>
>> ...What should be argued for a class (learning) the real numbers,
>> fractions, etc., is to learn by intuition.
>>

>
>


Hi Anna --

Let me try with the topic of real numbers again. How might we spark the
intuition in some curriculum?

REALS AS EDGE LENGTHS:

First, I agree with Bill and Dr. Wu that the number line is important, but
why exactly? Because it's graphical, not lexical, is part of the answer,
and because it relates number to length, a distance. The relation of
numbers to length is primal and should drive many of our early lesson plans.

However, instead of starting with just any old lengths, it'd make sense to
look at some specific distances in a geometric context. We'll want to build
some specific objects, will need specific lengths to do so.

FRACTIONS AS VOLUMES:

What's a common context for fractions, one with practical applications and
associations? Cooking skills, following recipes, in terms of whatever
units. Half cups, three quarter teaspoons etc. Nothing new here (so far).

But what might our measuring bowls actually look like? Here's one version:

http://www.wikieducator.org/Image:Still_life_sm.jpg ("mixing bowls", world
class).

http://www.wikieducator.org/Image:Syn_volumes_sm.jpg

http://worldgame.blogspot.com/2009/12/district-standard.html

COMBINING EDGES AND VOLUMES:

There's much to recommend this bevy of inter-connecting concepts, in that
we've got spatial geometry going in tandem with both fractional and real
number relationships. In addition to lengths, we have angles. We could add
rotation matrices, spherical coordinates.... looking down the road, yet
keeping something already familiar.

SUMMARY:

We have some specific fractions relating to volumes, measuring discrete
amounts, motivating computations and detection of equivalancies.

We have irrationals relating to lengths, motivating a discussion of Real
Number, their appearance in the history of ideas. Bill's use of intervals
might feature here.

MEETING A STANDARD

With polyhedra in view, we've got a state standard to hit: understand how
area and volume change as a 2nd and 3rd power of changes to linear scale.
Double edges, four fold area, eight fold volume. Have edges, reduce area
4x, volume 8x.

SOURCES

Published sources for using polyhedra in this way include a 1965 paper in
Math Teacher by MIT crystallographer Arthur Loeb. Robert Williams in his
The Geometrical Foundation of Natural Structure (Dover), and of course Bucky
Fuller himself (the dome guy). Lotsa street cred here, websites, books,
teaching supplies... everything one might need (including free computer
source code if you're taking a more digital approach -- on tap at my site,
with expository video).

I go into more biblographic detail here. Bill, this post mentions our
discussion and links back to this thread FYI:

http://worldgame.blogspot.com/2009/12/real-and-rational-numbers.html

Kirby

PS: I didn't learn all this in ed school, which should be a point in my
favor with Haim.


Date Subject Author
12/23/09
Read Real Numbers in the Primary Grades
Bill Marsh
12/24/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/24/09
Read Re: Real Numbers in the Primary Grades
Kirby Urner
12/24/09
Read Re: Real Numbers in the Primary Grades
Hecman Gun
12/24/09
Read Re: Real Numbers in the Primary Grades
Anna Roys
12/25/09
Read Re: Real Numbers in the Primary Grades
Bishop, Wayne
12/26/09
Read Re: Real Numbers in the Primary Grades
kirby urner
12/24/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/24/09
Read Re: Real Numbers in the Primary Grades
Bill Marsh
12/25/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/25/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/25/09
Read Re: Real Numbers in the Primary Grades
Hecman Gun
12/26/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/26/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/27/09
Read Re: Real Numbers in the Primary Grades
Haim
12/27/09
Read Re: Real Numbers in the Primary Grades
kirby urner
12/28/09
Read Re: Real Numbers in the Primary Grades
Bill Marsh
12/28/09
Read Re: Real Numbers in the Primary Grades
Haim
12/29/09
Read Re: Real Numbers in the Primary Grades
Bill Marsh
12/29/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/29/09
Read Re: Real Numbers in the Primary Grades
Michael Paul Goldenberg
12/29/09
Read Re: Real Numbers in the Primary Grades
Bill Marsh
12/29/09
Read Re: Real Numbers in the Primary Grades
Bill Marsh
12/30/09
Read Re: Real Numbers in the Primary Grades
GS Chandy
12/30/09
Read Re: Real Numbers in the Primary Grades
Michael Paul Goldenberg
12/30/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/30/09
Read Re: Real Numbers in the Primary Grades
Michael Paul Goldenberg
12/30/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/30/09
Read Re: Real Numbers in the Primary Grades
Michael Paul Goldenberg
12/30/09
Read Re: Real Numbers in the Primary Grades
Dave L. Renfro
12/30/09
Read Re: Real Numbers in the Primary Grades
Michael Paul Goldenberg
12/30/09
Read Re: Real Numbers in the Primary Grades
Bill Marsh
12/30/09
Read Re: Real Numbers in the Primary Grades
Louis Talman
12/30/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/30/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
12/31/09
Read Re: Real Numbers in the Primary Grades
Michael Paul Goldenberg
12/30/09
Read Re: Real Numbers in the Primary Grades
Dave L. Renfro
12/30/09
Read Re: Real Numbers in the Primary Grades
Bill Marsh
12/31/09
Read Re: Real Numbers in the Primary Grades
Robert Hansen
1/2/10
Read Re: Real Numbers in the Primary Grades
Bill Marsh

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