You said you wanted to see my notes on their calendar, didn't you? But my notes are boring.

By coincidence, that Monday morning in Dan and Kristie's overview of physical science education topics, they were discussing teaching about skip years. We'll pick up a little after Professor Billings's lesson began.

The professor posed this question: "Many students will find the subject of skip years interesting on the last day of January or the first of February in a skip year. But when it's a regular day of a regular year, how can we get students interested in the length of the year?"

Bess thought out loud. "Models of the solar system are generally good for raising interest."

"Like this one?" The professor took a solar system model out from under his desk and demonstrated rotating the planets a bit. "What does it tell the students about skip years?" he asked, and set it on the lab table beside his lectern.

Kristie responded, "I guess that many students won't draw a connection without help?"

"Well?"

Merill suggested, "If someone has a birthday that falls on January 30th, that could also motivate interest. You could raise the question of whether or when to count January 30th birthdays on skip years."

Some of the students nodded.

Dan commented, "But I'd guess it's going to have to be something related to either astronomy or the calendar."

"How about the math?" Professor Billings suggested. "Do you think young students would find that interesting?"

Merill's response was fast. "No way!"

Ted's response was a little more positive. "Maybe. Sometimes math that seems to break rules is interesting to children. Skip years being exceptions in a cyclic series, they might be presented as breaking rules."

"Okay, who agrees with Merill," and he paused as some students raised their hands, "and who agrees with Ted?" and he paused again as other students raised theirs. Dan and Kristie raised their hands both ways.

"Kristie, can't you and Dan make up your minds?"

"It's not a binary question." Kristie said, and the professor nodded.

Dan said, "My buddy Karel used to entertain the football team with oddball math, but he had a sense for it that's a little unusual."

"So what do you think?"

"I learned some of his tricks, but the math sure doesn't come natural to me. I think I might use it in class sometimes, but not as much as Karel could."

The professor chuckled. "I guess I need to try harder to get you guys to really dig into the math. The more you guys are interested, the more your students will be."

"Would asking why skip years happen also help?" Kristie asked.

"Kids are often interested in reasons," the professor encouraged.

Merill laughed. "My little sister once asked me why someone couldn't have just passed a law to make all the years the same."

"There's a possibly good approach, asking why just making a law wouldn't work."

"Why not?" asked Karen. "I mean, isn't it just like π, numbers somebody made up?"

Yes, I'm translating that. The character was not π, and it was not pronounced as in "pie". Strangely enough, however, one of their words for "pie" did have the same pronunciation as the name of the constant. More than coincidence, maybe?

Against the general chorus of groans and cat-calls, Dan said, "Take it easy on Karen. I think more than half of us have at sometime in our college lives had the same question."

"Thank you, Karen and Dan. You're both welcome to a maple cookie for your contributions today."

Professor Billings liked to encourage various kinds of independent thinking with lab-baked sweetbreads.

And now you're wondering that they have maple in their world. Well, the tree and its leaves look a lot like the maple, and its sap serves pretty much the same as the sap of maple serves us. The flavor is slightly different -- we might think someone had mixed in a little chocolate or cinnamon, depending on the variety.

You ask how either you or I could taste their maple, considering the distances between their planet and ours?

Good question. Back to the lesson.

There were still complaints. Jack was among those complaining: "But we're college students. Surely we all should know by now that the ratio of the radius of a circle to its circumference is a physical property of the real world, not a product of the whimsy of lawyers and legislators."

The classroom became quiet. The professor waited a moment and then asked, "Okay, who is willing to admit that you really aren't quite sure of either of these questions?"

A few hands hesitantly drifted up, almost as if without the will of their owners.

"Thanks for being honest." He continued, "If you couldn't look it up, how many of you would be unable to calculate an accurate value of π?" He paused. More students started raising their hands. Then he asked, "And how many of you would not be confident of being able to design a proper experiment to measure the length of the year?"

More hands were raised. He continued, "... especially since outer space has no reasonable way to set down a signpost that says, 'This is where we were this time last year.'?"

Jack was looking rebellious, but everyone around him had raised their hands.

"And now, how many of you think this discussion is interesting? Are we all awake enough to review the math?"

Many in the class nodded in partial understanding.

Kristie commented, "Asking questions in a way that asks children to think is generally pretty good for keeping the students awake, right?"

"You've heard me say that before, too. Good. Now we've talked a little about some lead-ins, how about the explanations? Who thinks they have an approach to the math?"

"Start with the shifting seasons?" Mark suggested.

"Who has ever seen a season shift?" the professor asked.

No one had.

Jack said, "Well, I really think we should start with demonstrating how to tell when a year has passed."

"Do tell." The professor motioned Jack forward.

Jack stood up at the chalkboard and drew a diagram showing their planet in orbit around their sun, showing the four positions of maximum and minimum tilt with the seasons labeled. For simplicity's sake, he avoided using the terms that would translate as equinox and solstice:

Chalkboard. Yes, they had calcium-rich minerals in abundance, and they made large boards to write on with the chalk. And, yes Jack used colored chalk.

"There you go," he said. "Standard stuff."

"And the kids really understand it?" the professor asked.

"Sure, well enough."

"How do you explain the part about marking the start position in outer space?"

"Oh. Yeah. We tell them we are counting the minutes from noon on the longest day of the year to noon on the longest day of the next year."

"And if the students understand how hard that is?"

"Most of the kids who will understand what the marker question means will have dads or moms who have already explained the precession of the equinoxes. Those who haven't will probably take the question for granted, and that should be okay for this level of lesson. Any oddballs, we can invite to help us make a class experiment."

"Good enough. Especially that last part about making experiments. Thank you. Do you want an oatmeal cookie?"

He grinned. "Thanks, but I take so many home, my wife has started baking cookies for me to bring. I actually have some oatmeal raisin cookies to share today." Jack sat down.

Oats. Wheat. Not quite the same, but closer to wheat than to rice. Oh. And they have a grain quite similar to rice, as well. And the raisins were a dried fruit that would be hard to say was not a grape, or maybe a cross between a grape and a currant.

"Okay, let's compare recipes after class," the professor laughed. "So where do we go from here?"

Mirium had a suggestion. "Some students won't really be familiar with the calendar, and reviewing a little will probably keep them listening. We should explain about how many weeks there are in a year, with how many left-over days, and how there is one less left over in skip years."

"Why not explain the varying number of days in the months, and the reason January may be twenty-nine or thirty days? We could talk about that." Merill countered.

"Wouldn't days in the week be simpler as a starting point?" Mirium responded.

"True, the weeks look simpler, but is simpler always simpler?" the professor asked.

The students all thought for a moment, then Ted spoke up. "Sometimes I've found it harder to get kids to talk about the simpler approach. Especially if they think they've heard it before. Simple sometimes means boring, uhm, because simple doesn't necessarily offer many features for them to get a grip on, to manipulate the ideas. I think we need to prepare multiple approaches."

"Thank you, Ted. I think there's an almond-flake doughnut in the box today."

And lot's of students said, "Yum!"

The almond seems to be an important form factor for nuts in many worlds.

Doughnuts? ... Uhm, yeah. Someone in their country had dropped a ring of sweet pastry batter into a pot of hot oil and discovered that it made an interesting way to cook sweetbreads.

"Okay, let's walk through both. Mirium, could you go first on the number of weeks?"

She nodded and stood.

"And, Merill, either you or Ted could do the days in the months?"

"I'll let Ted. I think he's brought something interesting."

Ted agreed to do the months, and Mirium went to the chalkboard.

"How many days this year?" Mirium asked.

Several students responded. "Three hundred fifty-three."

Mirium wrote the number on the chalkboard as she asked, "How many days in a week?"

Everyone answered seven, and Mirium wrote:

353 days ÷ 7 days per week =and paused.

"Seven times five is thirty-five," Georgia said.

"Fifty, remainder three."

353 days ÷ 7 says per week = 50 rem 3,"Fifty and three sevenths, or ..." Jack recalled his sevenths. "50.428571, repeating from the four."

or 50 weeks and 3 days

"Too many decimals," Georgia suggested.

"Maybe," replied Professor Billings. "But you want to be sensitive to the class. Some classes, some days, details can keep the students awake instead of put them to sleep. Some days, your plan may even require details."

Mirium continued, "Is next year a skip year?" General noises of agreement. "How many days?" And she continued the exchange while she wrote.

352 ÷ 7 = 50 rem 2, or 50 2/7, or 50."What does two sevenths mean?" she asked.285714

"Two days out of a week," Georgia volunteered.

"Okay," Mirium continued, "in seven years, how many skip years do we have?"

"Two," in chorus.

"How many non-skip years?"

"Five."

"How many days in seven years?"

"Don't forget to give the slow students a little time to see the answers if they can." the professor reminded them. "Even if you have to wait a minute, try to keep them answering in their own heads."

And the exchange continued while Mirium wrote on the board:

352 days/yr × 2 yrs + 353 days/yr × 5 yrs"Okay, how many weeks?"

704 + 1765 = 2469 days in seven years, usually

Georgia suggested, "Divide by seven."

"Seven what?" Mirium asked. "Seven years?"

"Seven days in a week," Georgia replied again.

And Mirium wrote on the board,

2469 days in 7 years ÷ 7 days/week"Again, how many weeks in a regular year?"

= 352 weeks 5 days in 7 years

Karen answered, "Three fifty-three divided by seven is fifty with three days left over. Fifty weeks and three days."

"How many in a skip year?"

Georgia answered, "Three fifty-two divided by seven is fifty with two days left over. But, just one day less, so, of course fifty weeks with just two days left over."

Bess spoke up. "I've always wondered, is there a reason that the number of weeks in seven years is the same as the number of days in a skip year?"

"Good question. Anyone?"

Ted raised his hand.

"Ted?"

Ted said, "If we had, for example, 366 days in a regular year, and 365 in skip years, the numbers would be similar."

Merill worked the math on some scratch paper, reading it out loud. "Three sixty-six times five plus three sixty-five times two is twenty-five sixty. Divide that by seven and we get three hundred sixty-five and five days. Wait. Of course. There are five years that have one more day than the short year, so of course it's the shorter number plus the extra days. It's because there are seven years in the cycle and seven days in the week."

Professor Billings laughed. "Okay, Merill, if you'll put that on the board, you can have the apple fritter in the box."

Apples, yes, they had several fruit trees that produced fruit with flesh and skin like apples. Some varieties were naturally tinged with a cinnamon-like flavor. Some tasted slightly like persimmon.

(Oh. They had a persimmon-like fruit, too. And you are asking again how we would know about the tastes of these. X^)

"Can I skip Ted's theoretical years from some science fiction planet and just do real years?"

"Sure."

So Merill put his work on the chalkboard, carefully organized:

Whole weeks per year: 50"Which is just to show that dividing by seven is dividing by seven, I guess. Since we have seven years in the small cycle of years, and we have seven days in a week, there are numbers that will appear the same."

( 50 wks/yr × 7 days/wk × 7 years ) ÷ 7 years

== 350 days/yr

( 50 wks/yr × 7 days/wk × 7 years ) ÷ 7 days/wk

== 350 wks in 7 years

leftover days --

2 days in 2 years, and 3 days in five years:

2 days/yr × 2 yrs/cycle + 3 days/yr × 5 yrs/cycle

4 + 15 or 19 days in seven years

or, leftover days --

2 days in 7 years, and 1 more in five years:

2 days/yr × 7 yrs/cycle + 1 day/yr × 5 yrs/cycle

14 + 5 or 19 days in seven years

19 days/cycle ÷ 7 days/wk == 2 wks 5 days,

or 2 5/7 weeks/cycle

19 days/cycle ÷ 7 yrs/cycle == 2 5/7 days/yr

The professor asked, "Do you think you could explain that to elementary school children?"

"Maybe. It'd take a lot more thought than this, I think."

"Good enough for now?"

No one could think of questions, so Merill sat down and Mirium continued. "Now, is it always going to be like this?" she asked. "Is every seven year cycle going to be the same?"

"This is where you are going to start to lose students." the professor warned.

Turning to the professor, she asked, "Then, should I just finish the math out myself?"

Mark suggested, "Would that also depend on the class?"

"What do you all think?"

Ted suggested, "Would it be a good idea to work on the months now?"

No one had a complaint, so Ted stood up and Mirium sat down.

Patience. We're getting there.

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*[Backup and edit history will be here: http://joel-rees-economics.blogspot.com/2017/04/backup-soc500-03-08-calendar-math.html.]*

*[Chapter 3 part 8 is original to the second draft, and is not found in the first draft. Chronologically, it would come in chapter five of the first draft: http://free-is-not-free.blogspot.com/2016/05/economics-101-novel-ch05-first-semester.html.]*