> Den 13. okt. 2024 kl. 03.13 skrev Cliff Landesman via Origami
> <[email protected]>:
>
> I have created a video to show anyone (including, say, a fifth grader) how to
> fold what I call a "delta sailboat", an easily folded origami model I
> designed.
> The folder is able to deploy differentials usefully without knowing what the
> word "differential" means. The folder need not know the jargon of calculus
> (such as being able to say what "dy/dx" means) or have the related
> mathematical skills.
Hi Cliff,
That is a nifty idea, and I really like the delta boat fleet.
Video 1: The goal needs to be stated clearer: You have a curve and a point, and
you want to estimate the slope of the tangent in that point. This shiould be
drawn. Also, you assume the student knows what a tangent is - I'm not sure how
old you are when the school introduces that.
It is a fine idea to start showing the delta fleet. Consider showing the fleet
against the curve, so that we visually can see the approximation improves with
the smaller boats. I know, as the slope grows for bigger x in x^2, the fleet
will hide the curve. Consider drawing the curve on a transparent slide and put
it on top of the fleet.
You might also consider showing a curve with decreasing slope, say ln(x). Here
you can see the curve behind. Though using the same slide idea throughout might
be more consistent.
Video 2: All the numbers seem to come in the way of the idea of difference
equations. There is a lot of effort that is merely an artefact of how to find
specific coordinates through folding. The last part where you just select two
points, is much clearer. Maybe you should just start there, but with a large
boat first, and then coming increasingly close, e.g. halving distances from the
point to the right edge 1/1, 1/2, 1/4, etc. This way the halving notches you
used to find coordinates, instead become an integral part of the approximation
method - and might lead to a discussion of other approximation sequences like
linear, maybe in pedagogical teacher material outside the videos.
The two videos should cross-reference.
You are right in using a simple, monotone, increasing curve with a not too
small slope. x^2 is fine.
Obviously, you want to continue investigating other kinds of curves. At least I
do :-) It could be interesting to play around with other curves. Not as part
of the video, but letting the students try them out. Some examples (some of
which are definitely for students beyond the fifth form, even pending the
curriculum in the specific country):
ln(x): What happens to curves with a negative slope change?
1/x: What happens to curves with a negative slope? Do we fold capsized boats?
Or with the same hull, but broken masts pointing downwards?
sin(x):
Scope: How much of the period do you include? A quarter? A half period? More
than one period? This can also be seen as scaling or zooming in/out: Do you let
the paper width represent a small or large neighbourhood?
Negative slopes: What if your point of approximation has negative slope?
Non-monotone functions: What happens to the delta fleet if the your first
approximation is on the other side of a minimum or maximum?
Arbitrary curve: where you don't know the answer. That is, you cannot just
enter the formula into a calculator and get the result. You *have* to measure
somehow, here through delta approximations.
x + 1/(2-x): Discontinuity, and scale. If you include the interval 0-100 and
solve for x=0, all your delta boats would have close to 45 degree sails. Use
0-4 on the x-axis, and weird things happen. Solve for x = 0, 1, 2, and 3.
Practical curve drawing: How do you mark the same curve on several pieces of
paper? Scoring through a template? See-through the folding paper (requires a
light colour)? Just eye-balling it is likely too imprecise for most schoolers
to be meaningful.
You could print the curve on several pieces of paper, but I believe there is a
solid point in letting students construct their own materials.
The limit boat: In theory, it would have length 0.
Battling fleets: Let the point to solve be at the centre of the x-axis. Make
boats sailing from the left and the right. Which fleet approximates the
fastest? Of course, you need a suitable function. Try e.g. (x-3)^3+8(x-2)^2+x-1
for the interval 0-3, and solve for x=1.5.
A final remark: It is interesting to consider how origami can be used to teach
principles in math and physics. Some of them have a practical role in origami,
such as Fujimoto's iterative method to finding thirds by guesstimating a third
initially - only problem when I teach that is that my own eyeballing often is
close to exact, thus voiding the iteration ...
Best regards,
Hans
Hans Dybkjær
http://papirfoldning.dk
Society: http://foldning.dk
