Dear Roger,
I enjoyed your response to my playground thoughts
and I enjoyed your puzzle even more...
> Calculate the time for the sun to be at azimuth
> of 45º for the summer and winter solstices for
> a given latitude, say 50º.
Alas, I can add nothing to Fred Sawyer's reply.
It is easy to express azimuth given latitude,
hour-angle and declination but any expression for
hour-angle given latitude, declination and azimuth
is messy. I have tried my full repertoire of tricks
but cannot come up with anything better than the
version Fred offered.
> The difference is remarkable.
Yes it is. Fred correctly shows the difference
to be about 25 degrees or, say, 1 hour 40 minutes.
You likewise are correct to say that...
> if the class repeats the experiment with a vertical
> gnomon six months later they will find that the
> hour lines are very much different.
Yes, indeed they will, but this seems to me to be an
advantage rather than a disadvantage! The goal (at
this stage) is to get the class to ask awkward
questions not to make a well-engineered sundial!
> Will this destroy their trust in science?
My view of science (perhaps idiosyncratic) is that
it is about formulating and refining models using the
"scientific method" which I think of as:
1. Design of experiment
2. Observation
3. Inference
4. Analysis of error
So, you take the children outside at the start of the
school day. [Let's suppose this is close to the summer
solstice and 9am on the clock but 8am sun time.]
You show them a vertical post. You point out its shadow
and you propose various models of what is going to happen
to this shadow:
A. It will stay pointing in the same direction
indefinitely (as long as the sun stays out).
B. It will move round the stick in an hour.
C. It will move round the stick in 12 hours.
D. It will move round the stick in 24 hours.
If the children have a short attention span, you could
argue for A quite convincingly. They stare at the shadow
and it doesn't seem to shift.
About now, you get a child to make the first mark. You
then ask them all to guess where the shadow will be when
they go out at break time. It is important to record
the guesses in some way so you can build up a little
bit of anticipation for later on. Also, with luck, this
guessing process will take long enough for model A to be
rejected!
When break time comes you congratulate the winner and
get that winner to make the second mark. This will be
about two hours after the first mark. [Let's say 10am
by the sun.] We can probably reject model B here.
By now, we have exercised quite a bit of the scientific
method without the teacher having to be any kind of
expert [see later]! We have designed an experiment,
we have made some observations and made a few inferences.
We can now look at the difference between the first
time mark and the second time mark and ask where the
shadow will be at lunch time another two hours hence.
We can also ask how long it will take to go right
round the post.
[By chance, at 50 degrees north, the azimuth at the
summer solstice, changes about 30 degrees between
8am and 10am which is almost too good to be true!!]
My target age group (6 to 8 years) probably don't
know about angular measure yet but they will know
about slices of cake. How many of these 30-degree
slices will fit round the circle? With luck we
can now reject model C.
When asked to guess where the shadow will be at
lunch time, two hours later, they will probably
fall into the trap of adding one slice of cake.
This is fine. Let them all make guesses and
keep the excitement going.
When they come out at lunch (1pm on the clock
but 12 noon by the sun) they will find the
shadow has gone much further than they expected.
They will be much more cautious about guessing
where the shadow will be two hours later still.
This is about as far as I would take the story
with 6- to 8-year olds. Maybe they could have
another look next day but I would guess they are
just not up to retaining all the inferences for
six months.
I do take your point ...
> The ancients building megaliths etc learned a lot
> about their world by observing the sun.
Indeed so, and 2000+ years before the polar-oriented
gnomon was devised.
I just might be persuaded to ornament my post with
a nodus of sorts about 1m above the ground and
plot some hyperbolic arcs. This would be for an
older age group I suspect.
> Unfortunately a lot of fires were burned and virgins
> sacrificed to get the sun to come back.
Hmmm. Your school curriculum sounds much more interesting
than mine!! Perhaps I should put in for a research grant
to see how many virgins it takes to make the sun come back?
> The important thing for teachers today is to get the kids
> to keep asking why...
Yes, yes, yes.
Alas, in Britain at least, science teaching in schools is
becoming increasingly dire. Physics teachers are no longer
merely an endangered species, they are almost extinct.
If the sundial community can get people to look at shadows
and think a little about what they see then we shall be
doing our bit!
Frank
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