Monday, November 3, 2014

Section 6-1: Transformations (Day 57)

Did you remember to set your clocks back an hour this weekend?

This is a question that people commonly ask -- why do we have Daylight Saving Time? Many people oppose DST and prefer that a single time be used the entire year. So, they wonder, why do we bother with DST?

Well, one of the first people to propose the biannual clock change was a British builder named William Willett, just over a hundred years ago. In Great Britain, the sun rose at around 4 AM on the day of the summer solstice, and around 8 AM on the day of the winter solstice. The purpose of DST is to make the sun rise as close to the time that people wake up as possible. This way, the hours that the sun is up more closely match the hours that people are awake. With DST, the sun rises at around 5 AM on the day of the summer solstice. But with year-round DST, the sun wouldn't rise until around 9 AM on the day of the winter solstice. The only way to avoid both undesirable sunrise times, 4 and 9 AM, is with a biannual clock change.

If one wants to do away with the biannual clock change, then one must decide whether one wants year-round standard time or year-round DST. Those who have young children tend to favor standard time, since wake-up and bedtime for young school-age children tend to match standard time's sunrise and sunset times better. Adults without young children, on the other hand, prefer year-round DST since their wake-up and bedtimes fit DST better. Students in the grade levels that I focus on -- middle and high school -- are in the middle of a bodily transformation. They need as much sleep as a young child, yet can stay awake as long as an adult. The later wake-up and bedtimes of secondary students end up suiting year-round DST better. Notice that if modern adults were to go to bed closer to sunset, so that noon would be halfway during the waking hours and midnight halfway during the sleeping hours, then DST would be completely unnecessary.

Many people state that the purpose of a biannual clock change is to save energy. The best argument in favor of a clock change would be to show that DST saves energy in the summer and wastes energy in the winter. But among the many studies on the relationship between DST and energy, none of them demonstrate this result.

Other alternatives to DST that regularly appear in debates about this subject are:

  • A year-round half-hour compromise between standard time and DST.
  • Reversing it so that one sets the clocks forward in the fall (in an attempt to keep sunset at around the same time all year rather than sunrise)
  • Keeping it, but changing the dates of the clock shifts closer to the equinoxes (rather than have nearly two-thirds of the year in DST)
  • Abolishing both DST and time zones and having a single timezone worldwide.
  • And my favorite, the one that appears at, where there are two time zones in the continental U.S. separated by a two-hour time difference.
If I were voting in a poll where the choices are to abolish the biannual clock change, keep it, and it doesn't matter either way, I'd choose the third option. Here in California, which is closer to the equator than Willett's Great Britain (there's that spherical geometry again), both year-round standard time and year-round DST keep sunrise near the 5 to 8 AM range without any clock shifts. But since I have no children of my own, I would lean towards year-round DST.

But I also like the plan proposed by Sheila Danzig (at the above link). The Danzig plan can easily be extended around the world, so that there are only twelve time zones worldwide. (Russia recently changed its time zones -- much of the current time zones now resemble Danzig time zones.) Also, the two-hour time difference of Danzig also solves a problem faced by sports networks -- if a Monday Night Football game were to start at 8 PM Eastern, West Coasters are still at work at the start of the game, but if it started an hour later, East Coasters are already in bed at the end of the game. In the Danzig plan, a game can start at 8 Eastern and 6 Western, satisfying both coasts. (Actually, Danzig doesn't name her two time zones -- I'd name them "Arizona time" and "Indiana time," after the last two major holdouts of DST. And yes, I know that Indiana observes DST now.) California would observe Arizona time, the equivalent of year-round DST, which is the time I prefer anyway.

Finally, here's an article from the Atlantic:

This article proposes a plan similar to Danzig. The East Coast would have the same time as Danzig but the West Coast would be one hour ahead of Danzig. Although this would give us only an hour's time difference between the coasts, this would give the equivalent of year-round Double DST here on the West Coast. The sun wouldn't rise until almost 9 AM at the winter solstice. And so I prefer the Danzig plan to the Abad-Santos plan.

Here's one more DST plan, proposed only as a joke:

Meanwhile, today I finally subbed in a math class, a sixth grade pre-algebra class. The class was studying division -- the traditional long division method, with three- and four-digit dividends (and one- and two- digit divisors.

Today we begin Chapter 6, which is on transformations -- the heart of Common Core Geometry. We see that the first lesson, Section 6-1, is simply a general introduction to transformations. Of course, on this blog we've already covered reflections and rotations, since we jumped around a little.

This lesson begins with a definition of transformation. Once again, I omit the function notation T(P) for transformations since I fear that they'll confuse students. But I do use prime notation. The best way to demonstrate transformations is on the coordinate plane, so I do use them.

The book uses N(S) to denote the number of elements of a set S, an example of function notation. I include this question in the review, since the number of elements in a set (cardinality) is such a basic concept for students to understand.

Interestingly enough, in college one learns about these special types of transformations:

  • A transformation preserving only betweenness is called a homeomorphism. (Actually, this is based on continuity, not betweenness, but these are closely related.)
  • Add in collinearity, and it becomes an affine transformation.
  • Add in angle measure, and it becomes a similarity transformation.
  • And finally, add in distance, and now we have an isometry.
The final question on my worksheet brings back shades of Jen Silverman -- the distance between lines is constant if and only if they are parallel in Euclidean geometry.

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