And so I'd like to add one more data point about school start times, since this is the school that doesn't start until 8:50. In order to shorten last Monday's post (which was one of my longest ever), I have edited that post by cutting out the DST/school start time parts and moving it to this post.
This is now officially my fall DST discussion post.
Last year, Californians voted for Proposition 7, which would establish Year-Round DST. But of course, last weekend, Californians still set the clocks back to Standard Time. This, of course, means that Congress has yet to approve the change, which is necessary to establish Year-Round DST.
Instead, Congress had been focused on something else that's huge in the news. You know -- it starts with "imp-" and ends with "-ment." I wish to avoid politics in this post, but I assume most of you know exactly what "imp****ment" I'm talking about.
Even though Congress has yet to act on Year-Round DST in California, I do wish to discuss the progress that DST-related bills have made in various states. Last year I mentioned that while I personally have no problem with the biannual clock change, I do prefer DST to Standard Time, and that if Year-Round DST would appear on a ballot, I would vote for it. It did, and I did.
I did say, though, that Year-Round DST in California would make no sense unless at least Nevada supports it as well. Well, not only has Nevada approved Year-Round DST, but so has the entire Pacific Time Zone -- this includes Oregon, Washington, and even British Columbia in Canada! (Note that some of these states, implementation of Year-Round DST is contingent on the rest of the time zone adopting it as well.)
OK, so Year-Round DST is on its way in the West. Various other states throughout the country have also passed DST-related bills. At first, I was hoping that the states passing Year-Round DST would be mainly in the Pacific and Central time zones, which would follow the Sheila Danzig plan:
http://standardtime.com/proposal.html
But I'm sure that Danzig herself would approve of Year-Round DST even if her dream of there being only two time zones, two hours apart, fails.
By the way, here are two other pro-Year-Round DST sites that I've found:
https://www.sco.tt/time/
https://enddaylightsavingtime.org/
There's also a site that is officially pro-Year-Round Standard Time, but also supports Year-Round DST, which it calls "Forward Time":
http://timezonereport.com/?p=192
I will refer to Year-Round DST as "Forward Time" or "Prop 7 Time" for the rest of this post.
By counting the number of states that support Forward Time, we can estimate how much support a possible bill may have in Congress. For example, if 26 states support Forward Time, and every Senator from a Forward Time state supports it, then that's a 52-48 lead in the Senate. With thirty states, that gives Forward Time a filibuster-proof majority. (Notice that Forward Time is bipartisan in that support for it is independent of party affiliation -- a rarity in this political climate.)
Bills needs to pass both the Senate and the House. Of course, it's likely that if thirty states support Forward Time, then it would have a majority in the House as well. Finally, President Trump has already tweeted his support of Year-Round DST.
There's one bill, though, that already passed the California Legislature, received Governor Newsom's signature, and doesn't require Congressional approval. It declares that secondary schools should not start before 8:30:
https://edsource.org/2019/california-mandates-later-start-times-for-middle-and-high-school-students/618546
As we see in the comments, not everyone agrees with later school start times -- and to me, those arguments have a point. But later start times are definitely compatible with Prop 7 -- after all, with Year-Round DST, sunrise is much later in the winter. Later start times make it less likely that school will start before sunrise in the winter, reducing the need for students to attend school in the dark. In fact, we see that this is why parents often prefer Year-Round Standard Time to Forward Time -- earlier winter sunrises provide light to students on the way to school. We address this concern by making school start later. (Also, Forward Time results in later sunsets, meaning that the sun is still lit after the later dismissal times for after-school activities.)
The high school that I subbed in today is unusual in that it doesn't start until 8:50. Thus it would be unaffected by the new law. Some people might wonder, does a later start time actually reduce tardies at the high school?
Well, last Friday -- the day after Halloween -- was hardly a representative day. I wrote that there were plenty of tardies that day -- but most of those were for zero period. According to the link, zero period would be unaffected by the new law.
I also subbed at that same school on October 15th, the day after Columbus Day (and the first day of school since Newsom signed the law). That day, I didn't have zero period. In order to make a true apples-to-apples comparison, we should look at the number of tardies in first period both days. First period begins at 8:50, which respects the new law.
Actually the tardy situation wasn't that bad last Friday in first period. There were only three tardies -- about the same as third and fifth periods.
On the other hand, October 15th was one of the worst days I've ever had for first period tardies in terms of minutes tardy (as opposed to a bunch of students coming in seconds after the bell). The regular teacher warned me that one student claimed he simply can never wake up easily. As it turned out, he and another student arrived at the same time, 9:35, with a third student coming just before the couple at 9:30. All three tardies were considered severe (more than 30 minutes late), while two more tardies were just shy of severe (around 9:15).
Meanwhile, today in first period P.E., I don't take attendance until after students dress. But the other P.E. teachers give the students 15 minutes or even more to dress, since they don't take attendance until the locker rooms are empty. I suspect that it takes so long to dress because many students are arriving to school a few minutes late. Then again, there are no severe tardies today as compared to October 15th.
If the stated reason for starting school later is to avoid tardies, then perhaps this means that instead of 8:30 or 8:50, we should wait until 9:40 to start school. Then again, we must understand that some students arrive late because they simply don't like school, and want to find any excuse to make class time a little shorter. We could start school at 9:40, noon, or even 8:30 PM, and these same students will arrive late. On the other hand, if the same guy who claims he can't wake up early had tickets to a highly-anticipated movie at 8:50 AM (because, say, tickets to more reasonable showtimes are sold out), he'll probably make it there with bells on. That is, the only way to get certain students to show up on time to school is to make entertaining -- that is, no longer school.
But if this is the case, then why do I support later school start times? Well, I already told you why -- because I voted for Prop 7, the Year-Round DST bill, and later school start times are more compatible with Year-Round DST, which I voted for!
As a case study, we should look at school start times in Saskatchewan. This is a Canadian province with Year-Round Standard Time -- but since that's Central Standard Time (with longitude 105W, the reference meridian for Mountain Time, bisecting the province), it essentially observes the equivalent of Forward Time.
I decided to check times for Saskatoon (the largest city) and Regina (the capital), as well as several more rural areas. In Saskatoon, both elementary and high schools started school within a few minutes of 9:00 either way (the range is around 8:50-9:10), and the same is true for rural schools. In the capital, some high schools start at 8:30 while elementary schools start later -- I found a few schools starting at 9:08. This is right around sunrise on the winter solstice.
Thus we see what schools should do if Forward Time becomes official -- start at 8:30 or later, just as the new law demands, for secondary schools, and perhaps even later for elementary schools if this is necessary (say, for bus schedules). This should keep school start time after winter sunrise for all states and Canadian provinces that implement Forward Time.
Notice that the new California law doesn't take effect until 2022. It would be nice if Congress approves Forward Time to start that same year (by then, the imp****ment mess should be over, so they can focus on DST-related issues). California clocks can be set forward to Prop 7 time in March 2022, and then remain there permanently. Then the new law will fix school start times so that they are compatible with Forward Time starting August 2022.
In my new district, the high schools have both (a zero-like) "first period" as well as "second period" currently both starting before 8:30. I suspect that under the new law, "first period" at those schools will be renumbered "zero period" in order to comply with the new law (since "zero periods" may start before 8:30, but not "first periods") and then delayed so that the new "first period" doesn't start until 8:30 or later. To me, the early "first periods" should have been numbered "zero" all along.
This completes the DST portion of this post.
Meanwhile, today is an odd block day, as Wednesdays usually are at this school. (There's no funny business affecting the schedule this week, as opposed to my last two visits here.) But this teacher has a strange schedule -- first period is his only P.E. class! Third period appears to be some sort of study hall, with eight student TA's (or "mentors") available to help the students study in groups.
Fifth period is the teacher's conference period, but it's taken away because the teacher must attend a one-hour meeting. This is the most interesting class of all -- Engineering Graphic Design. The students have a drawing assignment -- they must take the picture of a three-dimensional object and draw what it looks like using the oblique technique. Then they have a quiz where they must draw a similar-looking object obliquely. The students must be careful not to draw the part isometrically instead of obliquely.
If all of this sounds familiar, it should. Three years ago at the old charter school, the fourth Illinois State project of the year (the last one common to all three grades) was all about drawing cubes using oblique and isometric graph paper. And I remember that the students struggled on it. Most of today's students fare much better with their drawings.
The purpose of that Illinois State project is to introduce the students to engineering design. The kids in today's class are mostly freshmen. It would be nice if my eighth graders had succeeded on that Illinois State project and as a result become inspired to take a class like this the following year at their high school. But based on what that project looked like, none of them would be inspired to continue on with engineering.
I look around the classroom and spot something that looks like an interactive notebook. Inside are various drawings (oblique, isometric, and even perspective). There is also a stats lesson -- the lengths of various copies of the same part are analyzed. Students must calculate the mean, median, and modal lengths, and even the standard deviation.
I've discussed ad nauseam whether I should have even given that fourth project or replaced it with, say, a true science project instead. Indeed, the three October days that I assigned that project fit neatly between Metric Day and Mole Day. But assuming that I keep this lesson, it's easy to see how I could have taught it differently.
I should have created a worksheet that gave more examples. (Since all three grades received this same project, having "six preps" or lacking prep time isn't an excuse.) If I had decided to use interactive notebooks, they could have glue these worksheets there. Assuming that we keep this as a three-day project, the first day could be for drawing a cube obliquely, and then the second for drawing it isometrically. The third day could be for designing something like a mousetrap car using one of the two drawing methods.
(By the way, I sing the Mousetrap Car song in class today before the quiz, since it fits this class.)
On today's quiz, the students must produce the oblique version of the object. So how is the object originally presented on the quiz for them to draw? That's easy -- they're given front, side, and top views of the object. These views are mentioned in Lesson 9-6 of the U of Chicago text.
Why are three views enough to specify what the object looks like? That's easy -- it's because the object is three-dimensional. A fourth view isn't needed unless we wanted to draw an object that was actually four-dimensional. Speaking of which...
Chapter 16 of Ian Stewart's The Story of Mathematics is called "The Fourth Dimension: Geometry out of This World." Here's how it begins:
"In his science fiction novel The Time Machine, Herbert George Wells described the underlying nature of space and time in a way that we now find familiar, but which must have raised some eyebrows among his Victorian readers: 'There are really four dimensions, three which we call the three planes of Space, and a fourth, Time.'"
This chapter is, of course, all about fourth dimension. We've had another side-along reading book about the fourth dimension a few years ago, by Rudy Rucker. So once again, we'll revisit some of those ideas from Stewart's perspective. The author writes:
"The fourth dimension was championed by charlatans, exploited by novelists, speculated upon by scientists and formalized by mathematicians."
And the mathematicians' ideas are still used today:
"Higher-dimensional spaces remain almost unknown outside the scientific community, but very few areas of human thought could now function effectively without these techniques, remote though they may seem from ordinary human affairs."
Stewart tells us that Euclid originally distinguished between two and three dimensions:
"The first part of the book is about the geometry of the plane, a space of two dimensions. The second part is about solid geometry -- the geometry of three-dimensional space."
One of the first 19th century mathematicians to consider the fourth dimension was William Hamilton, who did so from an algebraic perspective:
"The idea is simple, but appreciating requires a sophisticated concept of mathematical existence. Hamilton then set his sights on something more ambitious."
This was a full algebra of 4D numbers -- the quaternions. Unfortunately, multiplication of these quaternions is not commutative. Meanwhile, Hermann Grassmann wanted to generalize the quaternions to yet higher dimensions:
"He published his idea in 1844 as Lectures on Lineal Extension. His presentation was mystical and rather abstract, so the work attracted little attention."
Instead, the author tells us that the lead of making sense of 4D was taken by scientists:
"Meanwhile, physicists were developing their own notions of higher-dimensional spaces, motivated not by geometry, but by Maxwell's equations for electromagnetism."
Indeed, Maxwell himself used quaternions before simplifying these to vectors:
"Mathematicians wondered just how many hypercomplex number systems there might be. To them, the question was not 'are they useful?' but 'are they interesting?'"
The author now returns to Riemann and his geometrical perspective. He was invited by Gauss to lecture about his research:
"Riemann was terrified -- he disliked public speaking and he hadn't fully worked out his ideas. But what he had in mind was explosive: a geometry of n dimensions, by which he meant a system of n coordinates (x_1, x_2, ..., x_n), equipped with a notion of distance between nearby points."
Stewart informs us that his ideas were a major triumph:
"Soon scientists were giving popular lectures on the new geometry. Among them was Hermann von Helmholtz, who gave talks about beings that lived on a sphere or some other curved surface."
We now move on to matrix algebra, first developed by Arthur Cayley. Stewart quotes George Salmon when he writes about how matrices can be used to solve problems:
"'The question now before us may be stated as the corresponding problem in space of p dimensions. But we consider it as a purely algebraic question, apart from any geometrical considerations.'"
We ask, does the fourth dimension exist? The author mentions Shakespearean scholar Clement Ingleby, who insisted that 3D is an essential part of real space. But Stewart counters:
"The nature of real space is irrelevant to the mathematical issues. Nevertheless, for a time most British mathematicians agreed with Ingleby."
But instead, it was the ideas of mathematician James Sylvester that prevailed:
"In this sense, multidimensional spaces are just as real as the familiar space of three dimensions, because it is just as easy to provide a formal definition."
And the idea that matrices can be used to solve problems lives on today:
"Nowadays, this point of view is called linear algebra. It is used throughout mathematics and science, especially in engineering and statistics."
Even Cayley himself doubted that his matrices would ever have any practical use, but:
"By 1900 Sylvester's predictions were coming true, with an explosion of mathematical and physical areas where the concept of multidimensional space was having a serious impact."
And of course, one of these areas was Einstein's relativity, with time as the fourth dimension.
Meanwhile, Stewart points out that we can consider any extra variable in a real-world problem to be an extra dimension. For example, the position of a bicycle can be considered six-dimensional -- one for the handlebars, two for the wheels, one for the axle, and two for the pedals:
"Your brain has to construct an internal representation of how those six variables interact -- you have to learn to navigate in the six-dimensional geometry of the bicycle-space."
These six dimensions, of course, aren't all spatial (or even temporal) dimensions:
"However, physicists working in the theory of superstrings currently think that our universe may actually have ten dimensions, not four."
Nowadays, high-dimensional spaces are used all the time:
"They are standard in economics, biology, physics, engineering, astronomy ... the list is endless. The advantage of high-dimensional geometry is that it brings human visual abilities to bear on problems that are not initially visual at all."
On that note, Stewart ends the chapter as follows:
"Mathematical concepts that have no direct connection with the real world often have deeper, indirect connections. It is those hidden links that make mathematics so useful."
The sidebars in this chapter include a biography of William Rowan Hamilton, what high-dimensional geometry did for them, and what high-dimensional geometry does for us.
This is our second review day, which means it's time to look around the web for an activity. The following worksheet comes from Elissa Miller -- yes, I know I haven't linked to Miller's website since August. This link is nearly four years old:
http://misscalculate.blogspot.com/2016/03/geometry-unit-6-quadrilaterals.html
As usual, these don't print well on my computer, so you may wish to go directly to the source. One thing about Miller's quadrilateral worksheets is that some of them mention parallelogram properties that don't appear in the U of Chicago text until Chapter 7. And so instead, I decided to post her trapezoid worksheet instead, even though the other quadrilaterals will appear on tomorrow's test.
Notice that in addition to "isosceles trapezoid," Miller uses the terms "scalene trapezoid" and "right angled trapezoid," neither of which appear in the U of Chicago text. Of course, we can derive their definitions from the analogous terms for triangles.
It also appears that Miller is using the exclusive definition of a trapezoid. For example, Trapezoid Problem #5 asks:
5.
The intended answer is
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