Calculus Problem From OpenStax

 I was trying to explain to someone how I got a solution to OpenStax Calculus Vol 1 section 4.1 (related rates) exercise 43, so first I did a google search to see if there was a posted answer. I found two answers posted, and I disagreed with both of them, and they also disagreed with the OpenStax solutions (which only had a numerical answer and not an explanation).

So I worked it again, and my work agrees numerically with the OpenStax answer and I'm satisfied. If anyone else is searching for the supporting work, this post is an attempt to document that.

First, the problem: A batter hits a ball toward second base at 80 ft/sec and runs toward first base at a rate of 30 ft/sec. At what rate does the distance between the ball and the batter change when the runner has covered one-third of the distance to first base? (Hint: Recall the law of cosines.)

Whatever triangle we construct to visualize this set up has sides along the line from home to 1st base, and from home to 2nd base. The third side of the triangle is from the runner's position to the ball's position. Since a baseball "diamond" is actually a square, we can say the angle between the 1st baseline and the line passing through home and 2nd base is 45 degrees (or Pi/4).

If r is the runner's distance from home and b is the ball's distance from home then Law of Cosines tells us the relationship to d, the distance between runner and ball, is d²=r²+b²-2r*b*Cos(π/4). At the instant the runner is 1/3 the distance to 1st base r=30 (1/3 of 90 feet). This implies time t=1. This tells us b=80. We can calculate d=62.497

If we differentiate that equation relating d, r, and b (with respect to time, as r, b, and d are functions of time) we get d*d' = r*r' + b*b' - Cos(π/4)*(r*b' +b*r'). The problem statement tells us r'=30 and b'=80. Substituting in known values and solving for d' we get d'=62.497.

The solutions manual gives 62.50, which is the same answer to two decimal places precision.

Berlin Food Memories April 2022

Notes on most of our restaurant experiences during a one week trip to Berlin. 

Sunday 10 April

Mogg, Auguststr. 11-13, is a limited-menu sandwich shop in the Old Jewish Quarter. They're located in a building which was formerly the Jewish Girls School (Ehemalige Jüdische Mädchenschule). The building dates to 1927, That, of course, was closed down as Jews were deported to be murdered under National Socialist policies. After the war the building saw a number of uses. More recently it was refurbed and now houses a number of small restaurants and art galleries. 

We went back a few times over six days. We had both Rubens and Pastrami. Both are on a German rye bread, square loaf, sliced thickly (about 2 cm). Generous portion of meat, nice balance of cheese, dressing, and sauerkraut. Side of cole slaw and half of a small pickle. Seats were in demand, and we were lucky to get in when we went. If you can, I recommend you make a reservation.

On the way back to our hotel we noticed Wiener Brot Holzofenbäckerei at Tucholskystraße 31. The shop window is only a couple meters wide, but we liked the look of what we saw and went in. Good choice! The pastries are, not as is common in US bakeries, overly sweet. This allows the flavors of the fillings (fruit, cinnamon, chocolate, poppyseed) to come through clearly. Over several visits we tried strudels, struesels, and rolls. Hard to say a favorite, but I'll give my top three as the sour cherry streusel, and the cinnamon and chocolate rolls. This place is well worth going out of your way for a treat. They have a cute logo of a fox (with long eyelashes) carrying a baguette in its mouth. I would have bought one, but they don't sell t-shirts.

Monday 11 April

Jewish Museum cafeteria inside the Berlin Jewish Museum (Jüdische Museum Berlin) there is a well-reviewed cafeteria. The offerings seems to change, and the day we visited the items on the printed menu were not available. They did have daily specials, and we tried the cauliflower curry soup, and tandoori chicken. The soup was good, the chicken not so much. Although I would go back for the soup, I don't know how to be certain what offerings they have on any given day. Hit or miss.

Tuesday 12 April

There is a big deal in Berlin popular culture (maybe in German popular culture) for a dish called curry wurst. The legend is that it was invented in Berlin, and now there is a lot of buzz about various curry wurst shops. We went to one of the famous ones Konnopke's Diner. They are convenient by public transit, located underneath the U-Bahn tracks at Schönhauser Allee.

Okay, so now I've eaten official (East) German currywurst. And I don't see what the big deal is. It's a boiled bratwurst, cut into chunks, smothered in ketchup, and sprinkled with what seems to be canned curry powder. Some people claim, quite passionately, that this one was the best, but I can't see why any boiled bratwurst smothered in ketchup wouldn't serve just as well.

Rausch Schocoladehause is a fine chocolatier located at Charlottenstraße 60. They have three stories open, with the first devoted to sale of chocolate goods, the second where you can custom order a particular blend of ingredients at a particular "darkness" level. The third level (in European numbering, the second floor) has a cafe. We ordered coffee and cake. Quite delicious. Quite pricey. Nice for a splurge, but I wouldn't eat here on a regular basis. The store is decorated with a number of sculptures made of milk chocolate. Prominently displayed in the cafe is a chocolate Titanic. Wow! I hope their air conditioning system never breaks down.

Down the street from the New Synagogue we noticed a wall decoration characteristic of Thailand. On investigation it turned out to be the front of a Thai restaurant, Kamala at Oranienburger Str. 69. Their menu looked authentic enough (many of the Asian restaurant menus we had been seeing had odd combinations of offerings that left us unsure of what cuisine the chef actually knows how to make) so we gave it a try. From the street you take a few steps down to the restaurant door. Inside the restaurant is in three or four levels, with a few tables on each. Still concerned about COVID we liked that we could be mostly isolated in our own room (it was not crowded on our visit). Although the menu has many offerings, it seems that for the most part they have a few sauces which you can order with a few different proteins. We both ordered a Phanäng curry, mine with chicken and my wife's with duck. We were both satisfied. It had just enough spiciness to leave a little tingle in the mouth, and good balance of flavor. 

Wednesday 13 April

After touring the Pergamon we felt we needed a snack, so we went to their cafeteria. It was a warm day and they were only seating outside on the terrace. This was nice because it overlooks the Spree River, and was quite pleasant surroundings. We ordered a Flamkuchen, a thin-crust German version of pizza. It was good enough. This was the only place where the waiter solicited a tip, and I gave them none.

For dinner I went to Pho at Linienstraße 134, a moderate walk from our hotel. I chose outdoor seating, and soon realized that many other people in outdoor seating are smokers. Oh well. Their menu adopts the conceit of an airline ticket. Cute. I ordered the summer rolls and a bowl of pho. The summer rolls were mostly okay, but in making them there were too many layers of the rice wrapper, making an unchewable chunk at one end. The pho was okay. I would have liked more vegetables in the broth. It had a good level of spiciness for my taste.

Thursday 14 April

After touring the Memorial to Murdered European Jews, we went to Noodles Plus at  Hannah-Arendt-Straße 1. The menu had a mix of Chinese-looking items, and some with atypical ingredients, like peanut sauce. We ordered a dim sum assortment (four each of 3 different dumplings) and I tried their mapo dofu. Both were good.

Friday 15 April

Encouraged by favorable online reviews, we tried Bentokado at Eberswalder Str. 5. When we got there, we saw the sushi fish did not look particularly fresh, so opted against any of the sushi offerings. I ordered the miso ramen, which was okay. My wife ordered the shiitake vegan udon, which she said was very strange. She was very unhappy with her choice. She had been looking forward to ordering a Japanese cake for dessert based on the photos on the website, but on seeing the actual options changed her mind. Essentially, the name (which could translate from Japanese as "lunch box corner") is not at all what their menu is about.

Another Wasted PD Day

 I just survived another wasted two days of what passes for "Professional Development" at my school.

Thursday session 1: Toxic positivity 101. This is not to warn us of the perils of toxic positivity, but to encourage us to gaslight each other. Session started with that classic red-flag warning, "Studies show that ..." without bothering to cite any study.

Session 2: Visioning exercise. I am sorry, but I literally have absolutely no recall of what we did during this session. Sessions are named in ways which are not helpful in remembering content. This may have been the session where our principal described Hattie's work as "doing a bunch of math" on research to show that an "effect size" of 0.4 is the minimum for one year's growth. There was no mention that the "bunch of math" which underlies much of Hattie's work is statistically invalid. I often mentally check out when material depends on belief in Hattie.

Session 3: Dictate to teachers how school trips (towards the end of the school year) will be handled.

Session 4: How to write letters of recommendation. Geared exclusively  to Gr 11 and 12 teachers, but all teachers must attend.

Session 5: For advisory teachers, and I was excused, as I do not teach an advisory.

Session 6: Read through the IB evaluation received, note the areas that were highlighted for improvement, and explain to us that the school won't really address those areas despite giving lip service to the changes.

Friday session 1: Calisthenics.

Session 2: Read and summarize an article of entry-level undergrad educational psychology. I would have expected anyone who is a certified teacher had already read this material.

Session 3: Share "Best Practices." Require everybody to write up a "best practice" and then walk around reading what everybody has written. This session used the vocabulary word "plenary" in a way I had never heard before. That is, in this session "plenary" means a summary of what happened during the previous period.

Session 4: Follow up on school trips. Staff are to decide what activities students will engage in on school trip, based on student survey results that are unrelated to school trip activities. Note that during this session the facilitator sat reading their email.

Session 5: So-called Pub Quiz about social studies topics and movie titles.

COVID-19 in Frankfurt am Main

 Some people have expressed interest in the numbers I've been copying down, from Robert Koch Institute, since April 2021.

I think this link will continue to update while the document is alive.

Vietnamese Food in München

Today we tried Oanh 65 for lunch. One order goi cuon, two orders hoanh thanh soup, one banh canh xao ga, and one curry. All were excellent.

Lindwurmstraße 65, about a block from the Goetheplatz U-bahn stop. Highly recommended.

Hohenschwangau

About two hours by train from Munich, the town of Füssen is the gateway to Hohenschwangau, home of two delightfully accessible Bavarian castles.

To

We got the Bahn Card 25 which gives 25% discount on train tickets, so we opted for first class. It turns out the first class sections of the RB68 trains amount to about 20 seats in a section of two cars. The consist seems to be three cars, with one married pair coupled with a single. As fate would have it we opted for the first class section in the single.

Also as fate would have it, there was some mechanical problem with that car about halfway to Füssen. They needed to uncouple the car and remove it from the train. That meant all passengers on that car had to move to the remaining part of the train.

Fortunately there were few first class passengers and we had no issue finding a new seat. There was a delay of about 15 minutes as they completed the work. The train made up some of the time during the rest of the journey.

From Füssen we took a taxi to our hotel in Hohenschwangau, the Hotel Müller.

Stay

Hotel Müller gives a good first impression. Friendly check in. Got our room ready early, as we arrived before check in time. Generous with advice without being pushy.

Our room had a balcony with view of Hohenschwangau Castle. Nicely appointed. But then we noticed odd things. The TV would not turn on. The phone did not work so we couldn't call the front desk. There was a safe in the room, but it was locked open with no means to unlock it. There were no racks to put suitcases on. Not horrible, but slightly off-putting.

Breakfast included, ordered from a wide variety of options and served by friendly staff dressed in dirndl. Recommend the fresh-squeezed orange juice.

Hohenschwangau Castle

A footpath from the left of the hotel leads up the mountain to the castle.

There are no guided tours, and the only option is audio guide. They are very strict enforcing COVID regulations. Must have "3G" proof. Small groups. Make at all times.

Ticket order came with harsh instructions demanding were arrive well in advance if the appointed time. If more than five minutes late no admittance and no refund. But in reality we did not need to arrive at all before the tour time, as all we could do was stand around and wait.

When our time arrived we scan or QR code at the entry gate, and the turnstile allowed us to proceed. If we don't move fast enough the turnstile whaps us on the backside.

Inside they check out vaccine status and hand out audio guides, burning 10 or do minutes. Guide walks with us from room to room, activating that room's content on the little audio device. Very robotic. This is a Castle that was actually used by Bavarian royals, and in that respect is interesting. The rooms are decorated not with things the people may have used, but with portraits and statues of former residents. I wonder if that was the actual decor at any point in time, or just to make it a museum.

A big deal is made of Ludwig II.

Overall impression is that it's a nice Castle and all, but I wish I knew more about German and Bavarian history.

Neuschwanstein Castle

This is a very pretty fantasy castle situated higher above the hamlet of Hohenschwangau. 

My hiking app, Gaia GPS, says the walk up from hamlet level was one mile with an elevation increase of 450 feet (2840 feet to 3290 feel above sea level). It took us just under 30 minutes.

Visitors are permitted into the outer courtyard from 15 minutes before their tour time. Again, why did they give instructions to arrive over an hour in advance? In the courtyard people are killing about waiting for the entry time. We were advised by the Steves book of a staircase over to the left which leads to an upper courtyard. This is worth the additional climb.

There are signboards at the upper level explaining that Ludwig intended this courtyard be blocked by a chapel, and the footprint of that intended chapel is visible in the paving stones.

Eventually the tour time comes. And you begin climbing stairs! OMG, so many stairs. Only a dozen or so rooms are completed, and those are the only rooms on the tour. As I saw rooms and heard more Ludwig II anecdotes I got the impression that Ludwig was not so much mad as her was eccentric. The castle send to have been a short of private themed park, where the theme was Richard Wagner.

So anyhow of you're going to the area definitely see both castles. But if I were forced to choose just one I would probably choose Hohenschwangau.

Other

We allowed waaaaaay too much time, guided by pre-COVID advice. We likely could have seen both castles in a single day. The guided portion of each tour was not more than 40 minutes.

But because we had so much time we had a very relaxed pace. We even made an unplanned side trip to a nearby thermal bath, Königlich Krystal Therme. Wow! 22€ well spent!

The region offer a transit pass for guests in the local hotels, so we were able to move around freely.  The hotel did not offer this on their own, but as soon as I asked it was quickly generated.

Füssen, the nearest train station town, is also worth spending some time walking in.

Palindromic Depth

 I tried a problem today in three classes, two 9th grade and on 7th grade, which I consider successful. I found the problem from Alicia Burdess (aliciaburdess.com) in a collection called Teaching Through Problem Solving. Also credited are Daniel Student, Geri Ann Lafleur, Dawn Morris Blackburn, Doris Duret, and Jonathan Scott.

The basic setup involves talking about number palindromes. I get into that with sentence palindromes, and my current favorite (which I first heard from my son) is, "No sir! Away! A papaya war is on!" I stretched this part more with the 7th graders, but eventually get into number palindromes, and then introduce an idea called palindrome depth.

If you take a number like 84, which clearly is not a palindrome and reverse the digits you get 48.
Add the two, 84 + 48 = 132, which is still not a palindrome. Reverse the digits and add again.
132 + 231 = 363, a palindrome.
Because it took us two operations to get a palindrome, we say that 84 has a palindrome depth of 2. Some numbers, like multiples of 11, have a depth of 0.

The task: find the palindrome depth of all two-digit numbers.

It was great that all students are able to access the problem. I definitely consider it low threshold. Students of all levels were excitedly exploring. I use visibly random groupings of three, and almost all groups were dividing labor and sharing results, and both making and testing conjectures.

One of the most interesting extensions a student with colored pencils came up with was to make a 9 x 10 table with each two-digit number, and they color-coded each number palindrome depth. Wow!

I will definitely use this one again.