Ted picked up the model of the solar system and demonstrated the orbit of their planet again. "We should focus the children's attention on the tilt of the earth, as Jack diagrammed. It would probably be good to review the reasons for the seasons."
Georgia asked, "When the north pole is tilted towards the sun, that's summer for us in the northern hemisphere, right?"
Professor Billings asked, "Anyone still not clear on that point? You don't want to find yourself in class, wondering which is which, with students looking at you and expecting you to know all the answers."
Jack asked, "Isn't the physics of the seasons obvious? I mean, we would expect to not even need that model in high school, right?"
Karen complained. "It's obvious to you. I'm still trying to figure it out. Is the sun that much closer to us in summer?"
"Yeah," Jack replied sarcastically. "A fraction of a percent makes that much difference."
"Jack, you'll confuse people." said Merill. "Karen, think of the sun against your car roof. At an angle, the light tends to reflect more."
Jack grumbled. "Maybe, maybe not."
"Okay," Karen said hesitantly. "I can see the angle being important. There was a teacher who explained it to us as distance when I was in elementary school, and I've been confused since then."
Ted said, "Well, it is partly distance, but the distance that is relevant is the distance through the atmosphere due to the angle, not the distance from the sun. But, more basically, like Merill says, the angle reduces the amount of sunlight that reaches the surface. And that's more important." And he drew a diagram on the chalkboard, of sunlight hitting their earth:
Karen still looked puzzled. "I'm not seeing it."
So Kristie stood up and said, "There's another way to draw that diagram that helped me." And she drew a second version of the diagram, using a polygon instead of a circle:
"It helped me see that the deeper angle reduces the amount of sunlight for the same amount of surface area." Then she picked up a textbook and held it in front of a window, rotating it to show the area being struck by sunlight. "Like this."
Karen thought a few moments and said, "Thanks, I think I can see that."
Kristie gave the chalk back to Ted and he started started writing out a chart on the chalkboard.
Dan grumbled. "You and Karel."
Kristie smiled as she sat down, and members of the class who knew our four protagonists laughed.
Karen continued after some thought, "I think I can probably draw diagrams like that in a class."
While Ted wrote, Professor Billings pointed out, "This kind of discussion often does much more than a lecture." He paused. "You may feel a little out of control, during the discussion, but I strongly urge you to learn to use this kind of discussion, in every grade you teach."
Ted turned around. "And if we can get the students to draw the diagrams and write the math on the board themselves, that works even better, right?"
"You've heard it before."
General laughter.
"So," Ted returned to the board, to the left of the chart he had been working on, and said, "if we can satisfy the students about the measured length of the solar year, we can tell them that the length of the year is three hundred fifty-two whole days and four hundred eighty-five six eighty-sixths of a day. But we probably want to show them decimal fractions, too:
1 year: 352 485/686 days (about 352.7070)Carl asked, "How do we connect that with sevens?"
Dan said, "Write the factors on the board, two times seven cubed."
Ted stopped, and he and many of the students turned to look at Dan.
"What? I didn't say I couldn't do any math at all, just not comfortable with some of it." Dan was grinning sheepishly.
More laughter, which Ted and Dan joined as Ted wrote out the factors on the board:
2 × 7 = 14, 14 × 7 = 98Dan continued, "And we could work out the decimal fraction, to show how close it is to five sevenths. Should I work that out?"
98 × 7 = 686, 686 ÷ 2 = 343
Professor Billing smiled and held up a piece of chalk, which Dan took to a chalkboard at the side of the room, to show the long division for 5/7 and 485/686:
0.714287 0.70699And Dan sat down.
----------- -------------
7| 5.0000000 686| 485.0000000
4.9 480.2
10 4.800
07 4.116
30 6840
28 6174
20 6660
14 6174
60 4860
56
40
35
50
Carl pressed the question. "Is that the closest small fraction?"
Ted said, "Actually, I've got a chart of fractions that I printed up on the computer, where we can get a look at that question."
There were some complaints.
"Braggart."
And, "How come he's so lucky?"
Etc.
To which Professor Billings said, "Ted is being kind enough to loan it to the department." The department had had to scrounge for the budget for the electricity, and we'll find out more about that later. The professor asked, "Do you have the printout with you?"
"Yeah."
"I'll go get the overhead projector."
"I'll help you," Mark volunteered, and Mark followed the professor into the prep room as Ted went back to writing on the board.
Do you remember those heavy, hot, smelly, fragile overhead projectors we used to use, back in the 1950s and '60s? And big. That was what Mark and the professor emerged carrying. They set it up on a desk in front of a screen and plugged it into the wall to get the lamp warming up while Ted continued his explanation.
"We probably want to ask how many months in a year and show the division:"
1 ideal month: 352.7070 ÷ 12 = 29.39225 days"Ideal month?" queried the professor.
"We can say it's some crazy professor's idea of a better month, but it isn't a real month."
"That explanation of ideal might work."
Ted continued, "Here's a question I think young kids will find interesting: If we used this exact length for the month, at what hour of the day on January 30th would it become February 1st?" And he showed them where he had written out the math for that, to the side of the still incomplete chart:
0.39225 * 24 = 9.414 hoursWhile the other students absorbed that, he continued, "Then we can ask whether they would think it interesting for January thirtieth to suddenly turn into February first at about nine twenty five in the morning."
0.414 * 60 = 24.84 minutes
Before 9:25 A.M., January 30th.
After 9:25, February 1st.
Jack suggested, "That's going to get some students excited, maybe too excited."
Dan chuckled. "It's the kind of thing Karel would say just as we split up for our specialty practice. Make sure we had plenty of time to think about it."
Kristie suggested, "Maybe you could save it until just before lunchtime? then the students would have lots of time to talk about it and forget their excitement."
Jack scratched his head. "Maybe, if you could get the timing right."
The professor was chuckling. He suggested, "Ted, let's look at your chart of fractions."
Ted dug a computer printout out of his books and handed it to the professor, who put it on the overhead projector's light stage.
: PRCH EMIT ;
: COMMA 44 PRCH ;
: COLON 58 PRCH ;
( No trailing space. )
: PSNUM ( number -- )
0 .R ;
: NUMERATORS ( count -- )
DUP 1+ 0 DO
I PSNUM COLON SPACE
I 1000 * OVER / PSNUM COMMA ( 1000 times I divided by count )
SPACE LOOP
DROP ;
: FRACTIONS ( count -- )
CR
1+ 1 DO
I PSNUM SPACE I NUMERATORS CR
LOOP ;
"What's this?"
"That's the program."
Okay, it isn't exactly the program code he showed. Although the colon and semicolon were similar, most of the characters did not look much like our Latin-based alphabet. And I've translated it into symbols derived from English.
But the postfix grammar was surprisingly similar to the language Forth.
Maybe not so surprising, if we consider what a progressive definition grammar would naturally look like.
I made sure the translation would run in most Forth interpreters, if you are interested in trying it out. It definitely works in gforth, which is relatively easy to get for most PCs and for Android phones. Probably the easiest way to get it on a MSWindows PC is to go to the Cygwin site and install that, then use Cygwin's installer to get gforth. Android's Play Store has it, and, if you are using a Linux OS, it is probably in the packages.
(It also runs in my bif-c interpreter, but that is not easy to get if you aren't into compiling things yourself.)
In our world, it would have been a few more years before Forth became somewhat available, but we'll hear more about that later.
Professor Billings moved the printout down on the staging table to show the chart:
9 fractions
1 0: 0, 1: 1000,
2 0: 0, 1: 500, 2: 1000,
3 0: 0, 1: 333, 2: 666, 3: 1000,
4 0: 0, 1: 250, 2: 500, 3: 750, 4: 1000,
5 0: 0, 1: 200, 2: 400, 3: 600, 4: 800, 5: 1000,
6 0: 0, 1: 166, 2: 333, 3: 500, 4: 666, 5: 833, 6: 1000,
7 0: 0, 1: 142, 2: 285, 3: 428, 4: 571, 5: 714, 6: 857, 7: 1000,
8 0: 0, 1: 125, 2: 250, 3: 375, 4: 500, 5: 625, 6: 750, 7: 875, 8: 1000,
9 0: 0, 1: 111, 2: 222, 3: 333, 4: 444, 5: 555, 6: 666, 7: 777, 8: 888, 9: 1000,
He used a pen to trace through the listing. "Okay, these are the first three digits to the right of the decimal point, right?"
"Right. So we can see that 0.714 in the fives column of the sevens row is about the closest fraction less than ten."
Jack was still doubtful. "But seven tenths is closer than five sevenths."
"Well," Carl spoke up again, "When somebody way back there in our history first set up our modern calendar, he wouldn't have even had mechanical calculators, and the numbering system wasn't always base ten. And it took several centuries to get the length of the year accurately measured, didn't it?"
While the rest of the class talked about this, Ted finished his chart on the chalkboard.
"What's this?" Professor Billings asked, as he slid the listing across the stage table so the class could see it:
7 showmonths
0 0: 30 (30, 29.392 )
0 1: 29 (59, 58.784 )
0 2: 30 (89, 88.177 )
0 3: 29 (118, 117.569 )
0 4: 29 (147, 146.961 )
0 5: 30 (177, 176.353 )
0 6: 29 (206, 205.746 )
0 7: 30 (236, 235.138 )
0 8: 29 (265, 264.530 )
0 9: 29 (294, 293.922 )
0 10: 30 (324, 323.315 )
0 11: 29 (353, 352.707 )
1 0: 30 (383, 382.099 )
1 1: 29 (412, 411.491 )
1 2: 29 (441, 440.884 )
1 3: 30 (471, 470.276 )
1 4: 29 (500, 499.668 )
1 5: 30 (530, 529.060 )
1 6: 29 (559, 558.453 )
1 7: 29 (588, 587.845 )
1 8: 30 (618, 617.237 )
1 9: 29 (647, 646.629 )
1 10: 30 (677, 676.022 )
1 11: 29 (706, 705.414 )
2 0: 29 (735, 734.806 )
2 1: 30 (765, 764.198 )
2 2: 29 (794, 793.591 )
2 3: 29 (823, 822.983 )
2 4: 30 (853, 852.375 )
2 5: 29 (882, 881.767 )
2 6: 30 (912, 911.160 )
2 7: 29 (941, 940.552 )
2 8: 29 (970, 969.944 )
2 9: 30 (1000, 999.336 )
2 10: 29 (1029, 1028.729 )
2 11: 30 (1059, 1058.121 )
3 0: 29 (1088, 1087.513 )
3 1: 29 (1117, 1116.905 )
3 2: 30 (1147, 1146.298 )
3 3: 29 (1176, 1175.690 )
3 4: 30 (1206, 1205.082 )
3 5: 29 (1235, 1234.474 )
3 6: 29 (1264, 1263.867 )
3 7: 30 (1294, 1293.259 )
3 8: 29 (1323, 1322.651 )
3 9: 30 (1353, 1352.043 )
3 10: 29 (1382, 1381.436 )
3 11: 29 (1411, 1410.828 )
4 0: 30 (1441, 1440.220 )
4 1: 29 (1470, 1469.612 )
4 2: 30 (1500, 1499.005 )
4 3: 29 (1529, 1528.397 )
4 4: 29 (1558, 1557.789 )
4 5: 30 (1588, 1587.181 )
4 6: 29 (1617, 1616.574 )
4 7: 29 (1646, 1645.966 )
4 8: 30 (1676, 1675.358 )
4 9: 29 (1705, 1704.750 )
4 10: 30 (1735, 1734.143 )
4 11: 29 (1764, 1763.535 )
5 0: 29 (1793, 1792.927 )
5 1: 30 (1823, 1822.319 )
5 2: 29 (1852, 1851.712 )
5 3: 30 (1882, 1881.104 )
5 4: 29 (1911, 1910.496 )
5 5: 29 (1940, 1939.888 )
5 6: 30 (1970, 1969.281 )
5 7: 29 (1999, 1998.673 )
5 8: 30 (2029, 2028.065 )
5 9: 29 (2058, 2057.457 )
5 10: 29 (2087, 2086.850 )
5 11: 30 (2117, 2116.242 )
6 0: 29 (2146, 2145.634 )
6 1: 30 (2176, 2175.026 )
6 2: 29 (2205, 2204.419 )
6 3: 29 (2234, 2233.811 )
6 4: 30 (2264, 2263.203 )
6 5: 29 (2293, 2292.595 )
6 6: 29 (2322, 2321.988 )
6 7: 30 (2352, 2351.380 )
6 8: 29 (2381, 2380.772 )
6 9: 30 (2411, 2410.164 )
6 10: 29 (2440, 2439.557 )
6 11: 29 (2469, 2468.949 )
ok
"I calculated out a chart to show how the lengths of months would mesh with the lengths of days cumulatively across seven years. It might be useful for demonstrating how variable a calendar would be if we didn't use something like skip years. Typing it up for a duplicator would be a lot of work. But you can see how the first year in that chart looks a lot like the calendar we use."
The professor moved the chart back up to year zero and let the students look it over. Then he asked, "How hard would it be to go to 700 years, so we could get a look at the entire cycle?"
"Waste of paper and ink, I'd say. Also, I'd have to fix part of the program that can only calculate to a bit more than thirty thousand days."
"Oh. So even a hundred years is going to be too many, then?"
"If I don't fix it."
"Well, it would be nice to pass this table out to everyone. Anyone want to type it up for the duplicator?"
Xerography was still too expensive to substitute for stencil or spirits duplicators, so copying out material like this still required a lot of manual labor. Georgia and Karen volunteered to type it up.
And this chapter is getting a bit long, so I'm going to arbitrarily end it here so you and I can take a break.
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[Backup and edit history are here: http://joel-rees-economics.blogspot.com/2017/04/backup-soc500-03-09-calculating-months-skip-years.html.]
[Chapter 3 part 9 is original to the second draft, and is not found in the first draft. Chronologically, it would be placed in chapter five of the first draft: http://free-is-not-free.blogspot.com/2016/05/economics-101-novel-ch05-first-semester.html.]
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