LOGO-L> Re: DIST and Math
Author Message
LOGO-L> Re: DIST and Math

Quote:
Yehuda Katz writes:
> Hello Turtlers,

> Recently I proposed a class-activity (see: 4-sided polygons). I want to
> propose here a family of activities, which suits students of various
> math (mainly geometry) levels.

> Here are 2 sample problems:

> ********************************************************************
> 1. In a square, how many times is the diagonal longer than its side?
> ********************************************************************

> Clear the screen, it's very important that the turle begins from its
> home. Draw a square with side(s) of 100:
>         CS REPEAT 4[FD 100 RT 90]
> Now we know that the coordinates of the point opposite the HOME is [100
> 100], so you say:
>         PR DIST[100 100]
> and Logo replies with 141.421... . That's 1.414 times longer than the
> square's side.
> (actually you don't need that drawing, it's done only for visualization
> reasons).

> Try it again with other squares, you always come to the same conclusion.

> Result: The diagonal of a squre is 1.414 (more precisely - the square
> root of 2) times the square;s side.
> No need to use the Pythagorean Theorem nor the Geometric Supposer...

BTW, if you measure the distance to (0,0) after half a square -- REPEAT
2[FD 100 RT 90], you can show that it's always the same even if the square
is rotated.

Anyway, I do not mean to be provocative -- I am asking the following
question out of genuine interest.  What do the Logo-L readers think about
the value of teaching math (or supplementing conventional math teaching)
via experimenting with Logo?  Having the students actually measure the
diagonals of several squares with a ruler is surely a great way to
introduce them to subject, but with Logo they rely on some mysterious DIST.
We all know that DIST computes its value using the Pythagorean formula,
and it does not actually measure distance.  So it is not replacing the
ruler.  It shows that when one applies the Pythagorean formula to the
diagonal of a square, one gets a number which is 1.41... (and not sqrt(2))
times the side.  How is that number related to the actual distance?

Thanks,
Chuck Shavit

---------------------------------------------------------------

Sun, 05 Mar 2000 03:00:00 GMT
LOGO-L> Re: DIST and Math

Quote:
> Yehuda Katz writes:
> > Hello Turtlers,

> > Recently I proposed a class-activity (see: 4-sided polygons). I want to
> > propose here a family of activities, which suits students of various
> > math (mainly geometry) levels.

> > Here are 2 sample problems:

[...]

Quote:
>Chuck Shavit writes:
> Anyway, I do not mean to be provocative -- I am asking the following
> question out of genuine interest.  What do the Logo-L readers think about
> the value of teaching math (or supplementing conventional math teaching)
> via experimenting with Logo?  Having the students actually measure the
> diagonals of several squares with a ruler is surely a great way to
> introduce them to subject, but with Logo they rely on some mysterious DIST.
>  We all know that DIST computes its value using the Pythagorean formula,
> and it does not actually measure distance.  So it is not replacing the
> ruler.  It shows that when one applies the Pythagorean formula to the
> diagonal of a square, one gets a number which is 1.41... (and not sqrt(2))
> times the side.  How is that number related to the actual distance?

> Thanks,
> Chuck Shavit

I have a little  experiences with Math and Logo, but I'd like to
share one episode that surprised me. Some time ago I proposed my 8th
graders (boys and girls of 13) to compare nearly the same way the
length of a circle and it's diameter. It turned out, only few of them
heard about PI. Even after they have got something about 3.14 for
very different circles, they couldn't formulate the general
conclusion. Some of them even didn't notice it.
I see in Yehuda's post the idea of making systematical investigation
of the geometrical shapes, so that children could observe and
generalize the results of their experiments.

Regards,
Olga.
---------------------------------------------------------------

Sun, 05 Mar 2000 03:00:00 GMT
LOGO-L> Re: DIST and Math

Quote:

> > Yehuda Katz writes:
> > > Hello Turtlers,

> > > Recently I proposed a class-activity (see: 4-sided polygons). I want to
> > > propose here a family of activities, which suits students of various
> > > math (mainly geometry) levels.

> > > Here are 2 sample problems:
> [...]

> >Chuck Shavit writes:

> > Anyway, I do not mean to be provocative -- I am asking the following
> > question out of genuine interest.  What do the Logo-L readers think about
> > the value of teaching math (or supplementing conventional math teaching)
> > via experimenting with Logo?  Having the students actually measure the
> > diagonals of several squares with a ruler is surely a great way to
> > introduce them to subject, but with Logo they rely on some mysterious DIST.
> >  We all know that DIST computes its value using the Pythagorean formula,
> > and it does not actually measure distance.  So it is not replacing the
> > ruler.  It shows that when one applies the Pythagorean formula to the
> > diagonal of a square, one gets a number which is 1.41... (and not sqrt(2))
> > times the side.  How is that number related to the actual distance?

> > Thanks,
> > Chuck Shavit

> I have a little  experiences with Math and Logo, but I'd like to
> share one episode that surprised me. Some time ago I proposed my 8th
> graders (boys and girls of 13) to compare nearly the same way the
> length of a circle and it's diameter. It turned out, only few of them
> heard about PI. Even after they have got something about 3.14 for
> very different circles, they couldn't formulate the general
> conclusion. Some of them even didn't notice it.
> I see in Yehuda's post the idea of making systematical investigation
> of the geometrical shapes, so that children could observe and
> generalize the results of their experiments.

My teacher did exactly this experiment in 6th grade.
With String, Hoops, Rulers and guidance.
First everyone said about 3. Then she kept on saying
are sure it's exactly 3. Then later everyone concluded
it was 3.1. Then she said are you sure it's exactly 3.1
and so on.

I'll never forget it.

--
===============================================================
George Mills

http://www.softronix.com
The www page contains some very powerful educational software.
Our single most important investment is our kids.
---------------------------------------------------------------

Sun, 05 Mar 2000 03:00:00 GMT
LOGO-L> Re: DIST and Math

Quote:
Brian Harvey writes:

> >  Having the students actually measure the
> >diagonals of several squares with a ruler is surely a great way to
> >introduce them to subject, but with Logo they rely on some mysterious DIST.

> I think that what comes up more often in kids' spontaneous work with Logo
> is the opposite problem:  Instead of drawing a picture that you know how
> to draw, and then ask Logo for the distance, kids want to draw a picture
> and don't know how far to ask the turtle to move.  For example, you want
> to draw an isosceles right triangle, so you say

>         fd 100
>         rt 90
>         fd 100
>         rt ???
>         fd ???

> and you find out what to do by trial and error.

Excellent point.  Thanks, Brian.

So applying that to Yehuda's exercise, the students could draw a square,
then turn right 45 degrees and move xxx pixels in a straight line.  After
trial and error, they will find the diagonal's length and hopefully the
mathematical rule.  This is a great variation on the theme of using a ruler.

Chuck

---------------------------------------------------------------

Sun, 05 Mar 2000 03:00:00 GMT
LOGO-L> Re: DIST and Math

Quote:

>  Having the students actually measure the
>diagonals of several squares with a ruler is surely a great way to
>introduce them to subject, but with Logo they rely on some mysterious DIST.

I think that what comes up more often in kids' spontaneous work with Logo
is the opposite problem:  Instead of drawing a picture that you know how
to draw, and then ask Logo for the distance, kids want to draw a picture
and don't know how far to ask the turtle to move.  For example, you want
to draw an isosceles right triangle, so you say

fd 100
rt 90
fd 100
rt ???
fd ???

and you find out what to do by trial and error.

Sun, 05 Mar 2000 03:00:00 GMT

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