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.
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:
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 Squeakland@squeakland.org 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
On May 30, 2007, at 12:04 , subbukk wrote:
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.
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 -
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:
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 Squeakland@squeakland.org 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
On May 30, 2007, at 19:02 , subbukk wrote:
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.
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 -
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:
On May 30, 2007, at 19:02 , subbukk wrote:
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.
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 Squeakland@squeakland.org 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
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@lists.squeakfoundation.org
-
Alan Kay -
Bert Freudenberg -
subbukk