Discovering Pi in Squeak

Hi,

I am trying to create an experiment to help children my kids discover numbers
like Pi. I don't want Pi to be introduced to kids as an "irrational" number.
It is a real number that exists in curved shapes. While countables can be
understood with beads or pebbles and fractions with slices, numbers like Pi
will need continuous things like sticks and strings[1].

The kids start the play by placing two sticks in a V-shape and use a string to
span the other ends. Add another stick to the mix and spread the sticks out
radially. Extend the string to the tip of the new stick and back again to the
starting point to form a triangle. Keep increasing the number of sticks and
use the string to form squares, pentagons and so on. Soon a circle takes
shape and the string converges to its perimeter. Now get the child to mark
this length and express it in terms of stick units (fractions allowed).
Repeat with different lengths of sticks. Let the child discover that some
measures are not countable or even expressible easily as a fraction. Now the
name 'Pi' can be introduced and the perimeter could be expressed as 2*Pi.  
Pictures of village blacksmith trying to cut a strip of iron to rim a bullock
cart wheel set the tone for the exercise.

As a parent of two young kids, I worry about kids hurting themselves with the
sticks. Squeak is a lot safer for such experiments. The nearest object that I
could use in Squeak is the Star. But the number of sticks (vertices count) or
stick length (distance between center and vertex) or the string segment
length (distance between adjacent vertices) are not computable from the
properties visible in the viewer.

Did I miss something or is there a better way to do this in Squeak?

Thanks in advance .. Subbu
[1] Sutra in Sanskrit. The humble string is so useful in conveying complex
concepts that the term Sutra also gets applied for formulae (e.g. E=mc^2) and
theory, theses etc.
_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

Using "forward by, turn by" you can draw a circle of known  
circumference, similar to how kids can measure the circumference of a  
large circle in the sand by counting footsteps. Measuring the  
diameter is easy - take the ratio and you're done.

So "turn by 120" three times makes your "three stick" version. You  
can go up to 360 sticks by using 1 degree turns ...

There also is a project for discovering pi (using real-world  
materials) in the "Powerful Ideas" book.

- Bert -


On May 30, 2007, at 7:41 , subbukk wrote:


_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

On Wednesday 30 May 2007 1:54 pm, Bert Freudenberg wrote:
> Using "forward by, turn by" you can draw a circle of known
> circumference, similar to how kids can measure the circumference of a
> large circle in the sand by counting footsteps. Measuring the
> diameter is easy - take the ratio and you're done.
> So "turn by 120" three times makes your "three stick" version. You
> can go up to 360 sticks by using 1 degree turns ...

I ruled this out because the 'turn' method uses Pi. To discover Pi, we cannot
use Pi. That would be a circular argument.

Subbu
_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

Hi --

Bert points out that it is easy to use forward by, turn by to make
pologons whose diameters can be measured.

For example, you can make a big circle with a turn by 1 and sum the
forwards, and also remember max y and min y to get the diameter. This
will give you a pretty good value for Pi.

You didn't mention the ages of your children.

But it is always good to get them to do some reasoning about measures
of various kinds and areas. I think that the manipulation of the
strings, etc.,  might be too awkward (but see the discussion in the
"Powerful Ideas" book about measurement).

I would just give them squares of different sizes and see if they can
work out how a side might relate to the perimeter, and if so, why
something like that would also work for a diagonal. The idea that the
relationship is the same regardless of scale is a biggie for
children. Discovering the relation for the area is even bigger.

Before that I would use rulers with different scales to make similar
figures (starting with triangles), and get them to see that nature
doesn't care how our rulers are laid out (a measurement taken with
one ruler can be used to make a similar figure of a different scale
using a different ruler). This is a very good way to show how and why
proportions work (and many studies have shown that proportions and
the normalizations associated with them are not learned well by most children).

Cheers,

Alan

At 10:41 PM 5/29/2007, subbukk wrote:
_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

On May 30, 2007, at 12:04 , subbukk wrote:

Not quite. If you would do this with an actual turtle robot, you  
certainly would accept the approach, wouldn't you? Why would you  
dismiss the simulation of that robot?

- Bert -


_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

On Wednesday 30 May 2007 5:35 pm, Alan Kay wrote:
> Hi --
> For example, you can make a big circle with a turn by 1 and sum the
> forwards, and also remember max y and min y to get the diameter. This
> will give you a pretty good value for Pi.
Unfortunately, what they will discover is the number programmed into the Float
class initialize method :-). I wanted to explore a method without built in
bias.
> You didn't mention the ages of your children.
10 and 6. The exercise is for the older one (and her
friends/classmates/cousins). The younger one went to a Montessori and watches
everything the elder one does :-).
> But it is always good to get them to do some reasoning about measures
> of various kinds and areas. I think that the manipulation of the
> strings, etc.,  might be too awkward (but see the discussion in the
> "Powerful Ideas" book about measurement).
I used 'string' in the generic sense. Actual stuff could be anything that we
can find around the house - strip of paper, cloth or ribbon, buntings etc
(e.g. cutting a wrap for a round pencil holder). There is a long tradition in
India of learning by immersion and many lessons are woven into daily
activities, so there is no awkwardness in using 'strings'.

"Powerful ideas..." is not available in bookstores here in Bangalore :-( and
Amazon doesn't ship direct to India. Shipping it in is quite unreliable (a
great incentive to go online :-)).

> I would just give them squares of different sizes and see if they can
> work out how a side might relate to the perimeter, and if so, why
> something like that would also work for a diagonal. The idea that the
> relationship is the same regardless of scale is a biggie for
> children. Discovering the relation for the area is even bigger.
Perimeter to side ratio is grasped quickly because 'addition' is readily
apparent. Diagonals and Circles require deeper thinking. I will try out your
suggestions.

> This is a very good way to show how and why
> proportions work (and many studies have shown that proportions and
> the normalizations associated with them are not learned well by most
> children).
Very true. The casual way proportions are treated in math text books is
disturbing. I always wondered how a child looks at things like shadows or
dolls before discovering proportions. This is something my daughter could
teach me.

Thank you very much for your suggestions .. Subbu
_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

On May 30, 2007, at 19:02 , subbukk wrote:

I bet if they are able to trace from the "turn by" tile into the  
method where pi is actually used directly, they are able to  
understand why this is not cheating. Besides, you could use radians  
directly for turning and then pi would not be involved directly.

It's like your "blacksmith" argument - he doesn't actually need pi  
but could simply roll the wheel on the strip of metal to find the  
length. But that would be cheating because pi is built into the  
wheel, yes?

- Bert -


_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

Hi Subbu --

And of course Pi is not at all required for "turn by" or "forward".
It is one of many constants in the init method for float but is not
needed for these behaviors. "foo turn by ang" is easily written in
Etoys as "foo heading increase by ang" where "heading" is a simple
property, so there is no Pi involved here. "foo forward by 5" is more
complicated (because it involves a vector addition to foo's location)
but still requires no Pi.

The other thing to think about that if it was "vectors all the way
down to the display" (as indeed it was in the old days of
calligraphic displays) then one would have a direct analog to a
coordinate free system in the hardware. The artifact today of having
to supply the display machinery with a 2D array of pixels forces a
kind of transformation and a kind of coordinate system, but still
doesn't require Pi.

Cheers,

Alan

At 12:21 PM 5/30/2007, Bert Freudenberg wrote:

_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

On Thursday 31 May 2007 6:18 pm, Alan Kay wrote:
> Hi Subbu --
>
> And of course Pi is not at all required for "turn by" or "forward".
> It is one of many constants in the init method for float but is not
> needed for these behaviors. "foo turn by ang" is easily written in
> Etoys as "foo heading increase by ang" where "heading" is a simple
> property, so there is no Pi involved here. "foo forward by 5" is more
> complicated (because it involves a vector addition to foo's location)
> but still requires no Pi.
It is true that turn: just sets the heading and doesn't use Pi. But the
heading gets used in forward operation which uses degreesToRadians from Float
class which ends up using Pi. The x and y offsets are computed using sin/cos
primitive functions whose computations, I suspect, would have used Pi.
Therefore, the diameter computed as (xmax - xmin) would have used Pi. I just
assumed the worst case :-).

The forward/turn method is same as triangular approximation when the turn
angle is a factor of 360. A child who happens to chose 7 (3.176) will wonder
why those who chose 8 (3.143) or 6 (3.141) have closer results. Triangular
approximation is self-correcting. The error isnt significant for angles below
5, so this aspect is of academic interest only.

Anyway, the idea behind triangular approximation is to let kids learn that
there are named numbers (pi, e) whose values are approximated unlike named
numbers like half, quarter etc.

I noticed something strange about the numbers calculated by Squeak. The ratio
of perimeter of inscribed polygon to diameter should be less than Pi. But for
turns of 10 degrees (3.150) and 8 degrees (3.143) I get overshoots.

Regards .. Subbu
_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland

At 12:21 PM 5/30/2007, Bert Freudenberg wrote:
>I bet if they are able to trace from the "turn by" tile into the
>method where pi is actually used directly, they are able to
>understand why this is not cheating. Besides, you could use radians
>directly for turning and then pi would not be involved directly.
The issue not about 'cheating', but in the equation:
 pi = perimeter/diameter where diameter = xmax - xmin
the rhs should not have used pi. degrees to radians conversion, sin and cos
functions used in offset calculations all use pi.

>It's like your "blacksmith" argument - he doesn't actually need pi
>but could simply roll the wheel on the strip of metal to find the
>length. But that would be cheating because pi is built into the
>wheel, yes?
The purpose of the exercise is not to discover the length but that the ratio
of perimeter to width is same for all circles and that this ratio is unique
enough to deserve a separate name. The blacksmith who knows this ratio does
not have to take the wheel off the cart to rim it :-).

Regards .. Subbu
_______________________________________________
Squeakland mailing list
[hidden email]
http://squeakland.org/mailman/listinfo/squeakland