Monday, November 2, 2015

Lesson 5-1: Isosceles Triangles (Day 47)

This weekend marked the end of Daylight Saving Time, with the clocks set back one hour. Each time we set the clocks, I post about the reasons for the time change here on the blog:

This is what I wrote last fall about DST:

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.

OK, so that's what I wrote last year. Now each year, I write about various states that are considering making changes to DST. This time, it's Michigan:

In March of this year, Rep. Jeff Irwin, D-Ann Arbor, introduced House Bill 4342, to get rid of DST and place the state on Eastern Standard Time year-round. Last week Rep. Peter Lucido, R-Shelby Township, introduced House Bill 4986 to do the same thing. Lucido says he is open to tweaking his bill to give some Upper Peninsula counties more flexibility. (Four U.P. counties that border Wisconsin are on Central Standard Time.)

The fact that Michigan lies in both the Central and Eastern time zones adds to the complexity of the situation considering whether to keep or eliminate DST. We see in the comments listed at the above link that some posters favoring keeping the biannual clock shift the way it is now, some prefer to keep year-round standard time as proposed in the bill, while others prefer year-round DST. But still other posters propose putting all of Michigan on Central Time without mentioning whether should be any DST or not. (Presumably, those who say that Michigan should be in the same time zone as Chicago must favor the biannual clock shift since that city shifts its clocks as well.)

To determine which is best, we remind ourselves that the goal of DST is to avoid extreme sunrise times such as 4 AM or 9 AM. We look up the current sunrise times (Eastern time, biannual clock shift) for the summer and winter solstices for the state of Michigan -- we might as well look at the state capital of Lansing. (Detroit would be a few minutes earlier than Lansing.)

Sunrise on June 21st: 6:00 AM
Sunrise on December 21st: 8:05 AM

Now the proposal would use year-round Eastern Standard Time. This would set the sunrise time in June to 5 AM, which is still acceptable, and keep the December sunrise at 8:05. Year-round DST would make the sun rise at 9:05 in the winter, which is unacceptable as the tardy bells at most schools would ring well before sunrise. And year-round Central Standard Time would make the sun rise at 4 AM in the summer, which is unacceptable as hardly anyone is awake at that time. (But using Chicago time would have the sun rise at 5 AM and 7:05 AM, which is acceptable for both times.)

So if I were a Michigander who desired a single year-round time, then year-round Eastern Standard Time would be my preference. Michigan's clocks would be where Indiana was over a decade ago. It would work towards what I call the Danzig plan, named for Sheila Danzig, who proposes dividing the contiguous 48 states into two year-round time zones:

Michigan is well-positioned to lead a conversion to Danzig time, as it's one of the rare northern states that can keep its sunrises mostly between 5 and 8 AM using Danzig time. As for my own state of California, I'd favor keeping the biannual clock shift unless Nevada changes as well. Then both states should convert to year-round Arizona time, which is also the Danzig recommendation for all three of our states.

As I mentioned last week, today I subbed in a middle school math class -- an eighth grade class. The students were working in Lesson 2-5 of the Glencoe 8th grade text, which is on solving multi-step equations -- a tricky problem at this level. Students really had trouble with this type of equation:

(7t - 2) - (-3t + 1) = -3(1 - 3t)

especially since the answer turned out to be t = 0, which many students tend to confuse with "no solution" or "all real numbers." The students will finish Chapter 2 by the end of this week -- which make sense, as there are only nine chapters in the text.

What makes this class interesting is that one of the periods was actually a combined math/computer programming class. The students in that class were well into Lesson 3-5, Graphing Using Intercepts. I see that this class is further along, and they even had time to write their first C program -- by tradition, it's the "Hello World" program -- on the side. Most likely, these students are stronger math students than those still on Lesson 2-5 -- which is why they were placed in a computer class.

I saw posted on the wall that for the previous lesson on slope, the teacher used another one of those Sarah Hagan mnemonics -- Slope Dude:

As I said earlier, I am scheduled to sub in this class for two days while all of the eighth grade teachers have meetings. So we'll see how much the students have learned for tomorrow's class.

Over the weekend -- besides turning back all of my clocks -- I had a discussion about Common Core Geometry with the author of this blog:

The author is James, a Maryland high school math teacher. He noticed that on my Day 28 post, I asked about how to prove the Converse of the Perpendicular Bisector Theorem, and he ended up providing a proof that's similar to what I eventually posted later on Day 38.

I noticed that on that I Heart Geo blog linked above, there is a question about how to prove the Vertical Angles Theorem. We can prove it either by Approach #1, using algebra (the traditionalist method), or by Approach #2, using a rotation (the Common Core method). On this blog back on Day 20, I used the traditionalist approach to prove that vertical angles have equal measure -- but it's not until Lesson 6-5 of the U of Chicago text (which we've skipped over so far) when we finally find out that angles of equal measure are actually congruent. Then again, I also sneaking in a question about using 180-degree rotations to generate vertical angles on Day 38.

And now that takes us into today's lesson. Lesson 5-1 of the U of Chicago text is on the Isosceles Triangle Theorem. The text uses the reflection about the angle bisector to prove the theorem. But we instead are using SAS to prove the theorem -- instead the reflection proof is actually what both James and I came up with to prove the Perpendicular Bisector Converse instead!

But we know that symmetry -- the fact that the reflection maps the isosceles triangle to itself -- is a key Common Core Standard. There's a trick to incorporate both SAS and symmetry into our Isosceles Triangle proof -- and that's to use the Pappas proof:

Given: AB = AC
Prove: Angle B = Angle C

Statements                    Reasons
1. AB = AC                   1. Given
2. Angle A = Angle A     2. Reflexive Property of Congruence
3. AC = AB                   3. Symmetric Property of Congruence
4. Triangle ABC = ACB 4. SAS Congruence Theorem
5. Angle B = Angle C     5. CPCTC

So here we prove that the isosceles triangle ABC is congruent to itself, but with the vertices given in a different order. This is exactly what symmetry is -- there exists an isometry mapping the isosceles triangle to itself, which is what the Common Core asks for. We see that there exists an isometry mapping A to itself, and B and C to each other. This tells us that most likely, the reflection over the perpendicular bisector of BC is exactly that isometry, and it maps A to itself, so that A lies on that perpendicular bisector.

The Converse to the Isosceles Triangle Theorem has a Pappas proof similar to that of forward Isosceles Triangle, except that it uses ASA instead of SAS.

For today's worksheet, I didn't get these questions from the U of Chicago text. Instead, I posted that 1970 Weeks and Adkins page on isosceles triangles (that I first mentioned on Day 32). I promised that I'd post it in order to show that students can still do traditionalist proofs, even in the Common Core classroom.

Of course, all of today's discussion underlies the questions that all of us Geometry teachers face -- should we continue to use the traditionalist proofs with which we are all familiar, or should we try to use Common Core transformations as often as possible? The U of Chicago text uses transformations until Chapter 7, when it finally proves the SAS and other congruence theorems. But we've seen that the Common Core tests, such as PARCC actually do expect students to use SAS on proofs for which the U of Chicago uses transformations.

And so I've made the decision to prove as little as possible using transformations -- that is, to prove just SAS and the other congruence theorems, and any lemmas required to prove those theorems (such as Perpendicular Bisector and its converse). Then once we have SAS, we use it to prove the other theorems such as Isosceles Triangle the traditionalist way. And now we can use both SAS and Isosceles Triangle to prove the Weeks and Adkins Theorems just as that text expects.

In other words, we can use transformations to prove as theorems many statements that traditionalists take as postulates, such as SAS. But once we have SAS, it is now what many computer scientists call a black box -- it doesn't matter whether SAS was a postulate or a theorem when it's time to use SAS in proofs.

And so I hope that today's worksheet will illuminate the true relationship between the Common Core and the traditionalist building blocks of proof. Of course, a traditionalist may still point out that even though I'm posting this today, Day 47, in a traditionalist course, it could have been posted back on the day I first mentioned it, Day 32 -- by skipping transformations and making SAS a postulate, we can reach Chapter 5 much earlier.

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