Today I subbed at the same continuation school I was at last week, except that this time it's a science class, not English. Thus I won't do "A Day in the Life" today.
This is my second visit to this class. My first visit was back in December, and so I wrote more about this class in my December 17th post.
They are to watch a David Attenborough video on predators and prey. It's possible that I could have shown a video like this to my seventh graders at the old charter school for life science. It might also have fit the sixth grade curriculum -- as that was the year of transition from "earth science" (old CA standards) to "integrated science" (new NGSS standards).
(By the way, I didn't realize that Attenborough is still alive at 92 years old. And indeed he's still producing nature videos -- his series One Planet will be released on Netflix in three weeks.)
In December, I had several problems with the fifth period class, and so I anticipate today's fifth period class with worry. Then I find out that fifth period is listed as a "first semester" class -- in other words, the troublemakers from December have already completed the class. The only student still enrolled in fifth period is the TA -- and he obviously wasn't one of the December troublemakers.
(Recall that the continuation school uses trimesters, so here "first semester" means "equivalent to the first semester at a comprehensive school." As I wrote in yesterday's post, the trimester in my new district has just started.)
And so I realize that I can have a better experience today than I did three months ago provided that I avoid the mistakes I made earlier. First and foremost, I actually take attendance rather than rely on taking the names off of the worksheets, since many students won't complete them. Also, unlike in December, I actually turn off all the lights during the video.
There are two main rules that I enforce today. One is the "no cell phone" rule, which the regular teacher insists that I enforce. In order to avoid an argument from December, I tell the students in each class "Yes, I'm the only sub in the whole world who enforces the no cell phone rule as strictly as a regular teacher or principal would." The other rule is that the students can't sit in the far lab station during the video. This isn't a rule from the regular teacher, but I recall from three months ago that most of the troublemakers sat in the far lab station that day.
I wrote in December that after fifth period, first period was the worst class of the day. Today, one group of guys enters the classroom, and of course they try to sit in the far lab station. After they move to other locations, one of them keeps his cell phone out. Per the teacher's instructions, I call security to send the student away, and as I do so, the student says, "You're wasting your time!" Fortunately, by the time security arrives he's put the phone away, and so I don't write any referral.
The only other student who refuses to put a phone away is a third period girl. She then plays the old trick, "I'm texting my mom," which I can neither confirm nor deny. My basic assumption is that almost any student who says this is lying and actually texting another teenager. In the end, I don't call security on her, but I inform the regular teacher of everything that she says and does. On her video sheet of 20 questions, she leaves more than half blank, including all of the first nine. This I present to the teacher as evidence of when and how often she's texting "her mom."
Fourth period is the best class of the day -- all the students do the work in this class. The next best class is third period. One student tells me that he doesn't have a pencil -- and I believe him, since he says this at the start of class. (If he'd waited until the end of class to say it, I assume that he's lying.) I don't have any pencils to give him.
But this raises another question -- on holidays I often have pencils to give away. Since St. Patrick's Day is on a Sunday, today is St. Paddy's Day Observed in schools. It's possible that I could have had some holiday pencils to give away, but I don't. The store I bought the St. Patrick's Day pencils from last year didn't have any such pencils this year. But yesterday, I did mention that I could have bought some green pencils and pass them off as holiday pencils.
Of all the pencil holidays, I always have the most trouble with St. Patrick. In my eyes, the holiday is overshadowed by Pi Day, especially in years when March 17th falls on the weekend and so the holiday is observed on the 15th or 16th.
And this also brings up a problem that I had back at the old charter school -- I considered pencils to be a holiday reward, but then students ask for pencils all year because they don't have any. It would water down the holiday reward to give the students pencils all the time. But if I don't always have pencils to give them, the students assume that I don't care about them, and so they disrespect me.
The best thing to do in my own classroom (where I'm the regular teacher) is to give the holiday pencil rewards on a day when the regular assignment is on Chromebooks (or other laptops). Then their lack of pencils doesn't hinder them from doing the assignment -- and by completing it, they can earn the pencils that they can use in their other classes the rest of the day.
As a day-to-day sub, if I don't have enough holiday pencils for every student, then I should just give the pencils to anyone who needs one. If I never call the pencils a "reward," then there's nothing wrong with giving them away by need rather than by merit. So for today, I could have purchased some green pencils (since explicit St. Patrick pencils are unavailable) and then given them away to anyone who asks for them today.
Speaking of the old charter school, I was thinking about Pi Day. Two years ago, I delivered my pizza to the sixth grade class on Pi Day. This was around noon, not 1:59 PM, since it was the week of Parent Conferences and school was dismissed well before 1:59. Last year, the math holiday fell on a Wednesday, which meant that school was out before 1:59 again. This year, Pi Day was yesterday, a Thursday, and I believe Parent Conferences week is two weeks later this year. Therefore, this year would have been the first year when a true Pi Day celebration would have been possible -- if only I were still teaching there.
Six months ago, back in my September 14th post, I described what that week would have looked like if only I were still teaching there. Well, today I found out what my old school did to celebrate Pi Week, and so let me do the same for this special week. Yes, this is now a "spilled milk" post.
This Week at the Old Charter School:
Monday, March 11th: Coding Monday???
Two years ago, the coding teacher came in on Mondays to teach. I don't know whether he still teaches at all at our school, much less on Mondays. For the sake of this post, let's assume that Mondays are still for coding.
Tuesday, March 12th: STEM Project and WSU Visit
Recall that our school now has a middle school science teacher, so if I were there, I'd no longer have to worry about squeezing in science lessons. In my September 14th post, I followed the Illinois State weekly plan, and so I do the same here. In that plan, we begin with a STEM project.
I would have tried to time it so that the 7.G Common Core lessons on pi occur this week. I do believe that there was a pi project in the seventh grade STEM text.
Meanwhile, I do know that a few male students from Washington State University visited the school on Tuesday, during their spring break. (At colleges, spring break occurs midway through the semester, and at WSU, the spring semester begins just after January 1st and the summer classes begin very soon after May 1st.) The whole point is to inspire our middle school students to start thinking about college, especially our male students. (I wrote about Bart/Lisa Simpson syndrome in several earlier blog posts.)
Wednesday, March 13th: Traditional Lesson
The seventh graders would receive a traditional lesson on pi -- provided, of course, that seventh graders actually have math on Wednesdays. Two years ago they didn't -- but I don't know exactly what the middle school schedule is on Wednesdays. Otherwise, I'd probably be forced to give the pi lesson on Tuesday and drop the STEM project. (Sixth and eighth graders would still get the STEM project, of course.)
In yesterday's post, I alluded to "Concrete/Pictorial/Abstract," which is another way to organize modern math lessons. I still haven't explained what that is in full detail, and so this has nothing to do with my plans for the week. (The Concrete/Pictorial/Abstract post is still forthcoming.)
Thursday, March 14th: Pi Day Project
Normally this would be Learning Centers, but I found out that my old school actually had a special project on Pi Day. The middle school students made a paper chain to represent the digits of pi -- each digit 0-9 was one of ten colors. All three grades added to it and it was a total of 40.5 meters long.
I don't know how many digits that would be! If each ring had a diameter of 10 cm, then the chain would contain 405 digits. I'm hoping that the rings were much smaller -- around 5 cm each -- so that the students would reach the Feynman point -- six 9's in a row, starting with the 762nd digit (763rd counting the initial 3).
Apparently all three grades participated in the chain, even though the sixth graders haven't learned about pi yet and the eighth graders only use pi for one unit (volume of cylinders/cones/spheres). If we assume that the three middle grades were about the same size now as two years ago (about 64 students), then each student would have needed to make 12 rings in order to reach the Feynman point.
Of course, two years ago I purchased the Pi Day pizza for the sixth graders, but that was an artifact of the Parent Conferences week schedule. Those sixth graders are now eighth graders participating in our school's first true Pi Day celebration.
And if I were still a teacher at the school this week, I'd want to buy either another pizza or some pie, depending on the schedule. Once again, I don't know which grade has math at 1:59 on Thursdays. It's possible that all afternoon classes were suspended and the students were busy assembling the Pi Day chain at 1:59.
With all middle school students participating in the chain, it's likely that I would have wanted to buy food for all three grades. Perhaps I would have done something similar to what I actually did in yesterday's post -- cookies for the morning class and pie after lunch (approaching 1:59). And like two years ago, I could order a pizza for whichever grade met just before lunch.
Friday, March 15th: LA County Fair Trip
And now you're thinking, isn't the LA County Fair in September? Well apparently, there was a special STEM fair for students this week, and my old school visited it as a field trip:
https://fairplex.com/events/fairplex-presents-extreme-steam-fair
Apparently, this event is called the "STEA^2 M Fair." I've seen the term "STEAM" before where the "A" stands for arts. As it turns out, the second "A" stands for agriculture -- which makes sense to include at a fair located at the Fairplex in Pomona.
According to the link, tomorrow popular scientist Bill Nye will be a guest at the fair. The students at my old school went there today, so unfortunately they didn't meet him.
Once again, no Pi Day is complete without music, so what songs would I have performed for my students during Pi Week? With today's trip to the fair, I probably would have wanted to sing "Meet Me in Pomona, Mona" -- perhaps even add an extra verse to represent what the students might see in particular at a STEA^2 M fair as opposed to the regular fair. Since the field trip is today, I would have needed to sing the Pomona song yesterday -- Pi Day itself.
And so I would have added Pomona to the Pi Day songs. Some of these might have been the YouTube links from yesterday's post. But in my classroom, I would've had my guitar, and so it's more important for me to perform the songs. The entire time that students are eating refreshments could be considered music break, and so I have time to play both the Pi Day and Pomona songs.
It's also possible for me to have played Pomona on Tuesday -- after all, I first played Pomona in my classroom on the Tuesday before our Friday field trip that year. Then again, on Tuesday I might have played a new parody of the WSU fight song instead, in honor of the visiting college students. (Recall "U-N-I-T Rate! Rate! Rate!" -- the UCLA fight song parody that I created that year.)
With all of this going on, would I have had time to hand out St. Patrick's Day pencils? Well, I explain above that the best day for me to hand out St. Paddy's Day pencils (or green pencils, since again true holiday pencils were unavailable) would be on computer day -- Monday in the schedule above.
Friday would ordinarily be assessment day, according to the schedule I wrote on September 14th. But with the field trip, there would be no quiz or test today. Well, if the week after next is Parent Conferences Week, then next week might be second trimester Benchmark Tests -- and so I may not give a quiz/test just before the Benchmarks anyway.
Aren't you glad I don't do as many "spilled milk" posts anymore? But Pi Day is an exciting time to be a regular teacher in any math class, and so I can't help but wonder what would have happened if I were still teaching at the old charter school this week.
Speaking of the old charter school, I was thinking about Pi Day. Two years ago, I delivered my pizza to the sixth grade class on Pi Day. This was around noon, not 1:59 PM, since it was the week of Parent Conferences and school was dismissed well before 1:59. Last year, the math holiday fell on a Wednesday, which meant that school was out before 1:59 again. This year, Pi Day was yesterday, a Thursday, and I believe Parent Conferences week is two weeks later this year. Therefore, this year would have been the first year when a true Pi Day celebration would have been possible -- if only I were still teaching there.
Six months ago, back in my September 14th post, I described what that week would have looked like if only I were still teaching there. Well, today I found out what my old school did to celebrate Pi Week, and so let me do the same for this special week. Yes, this is now a "spilled milk" post.
This Week at the Old Charter School:
Monday, March 11th: Coding Monday???
Two years ago, the coding teacher came in on Mondays to teach. I don't know whether he still teaches at all at our school, much less on Mondays. For the sake of this post, let's assume that Mondays are still for coding.
Tuesday, March 12th: STEM Project and WSU Visit
Recall that our school now has a middle school science teacher, so if I were there, I'd no longer have to worry about squeezing in science lessons. In my September 14th post, I followed the Illinois State weekly plan, and so I do the same here. In that plan, we begin with a STEM project.
I would have tried to time it so that the 7.G Common Core lessons on pi occur this week. I do believe that there was a pi project in the seventh grade STEM text.
Meanwhile, I do know that a few male students from Washington State University visited the school on Tuesday, during their spring break. (At colleges, spring break occurs midway through the semester, and at WSU, the spring semester begins just after January 1st and the summer classes begin very soon after May 1st.) The whole point is to inspire our middle school students to start thinking about college, especially our male students. (I wrote about Bart/Lisa Simpson syndrome in several earlier blog posts.)
Wednesday, March 13th: Traditional Lesson
The seventh graders would receive a traditional lesson on pi -- provided, of course, that seventh graders actually have math on Wednesdays. Two years ago they didn't -- but I don't know exactly what the middle school schedule is on Wednesdays. Otherwise, I'd probably be forced to give the pi lesson on Tuesday and drop the STEM project. (Sixth and eighth graders would still get the STEM project, of course.)
In yesterday's post, I alluded to "Concrete/Pictorial/Abstract," which is another way to organize modern math lessons. I still haven't explained what that is in full detail, and so this has nothing to do with my plans for the week. (The Concrete/Pictorial/Abstract post is still forthcoming.)
Thursday, March 14th: Pi Day Project
Normally this would be Learning Centers, but I found out that my old school actually had a special project on Pi Day. The middle school students made a paper chain to represent the digits of pi -- each digit 0-9 was one of ten colors. All three grades added to it and it was a total of 40.5 meters long.
I don't know how many digits that would be! If each ring had a diameter of 10 cm, then the chain would contain 405 digits. I'm hoping that the rings were much smaller -- around 5 cm each -- so that the students would reach the Feynman point -- six 9's in a row, starting with the 762nd digit (763rd counting the initial 3).
Apparently all three grades participated in the chain, even though the sixth graders haven't learned about pi yet and the eighth graders only use pi for one unit (volume of cylinders/cones/spheres). If we assume that the three middle grades were about the same size now as two years ago (about 64 students), then each student would have needed to make 12 rings in order to reach the Feynman point.
Of course, two years ago I purchased the Pi Day pizza for the sixth graders, but that was an artifact of the Parent Conferences week schedule. Those sixth graders are now eighth graders participating in our school's first true Pi Day celebration.
And if I were still a teacher at the school this week, I'd want to buy either another pizza or some pie, depending on the schedule. Once again, I don't know which grade has math at 1:59 on Thursdays. It's possible that all afternoon classes were suspended and the students were busy assembling the Pi Day chain at 1:59.
With all middle school students participating in the chain, it's likely that I would have wanted to buy food for all three grades. Perhaps I would have done something similar to what I actually did in yesterday's post -- cookies for the morning class and pie after lunch (approaching 1:59). And like two years ago, I could order a pizza for whichever grade met just before lunch.
Friday, March 15th: LA County Fair Trip
And now you're thinking, isn't the LA County Fair in September? Well apparently, there was a special STEM fair for students this week, and my old school visited it as a field trip:
https://fairplex.com/events/fairplex-presents-extreme-steam-fair
Apparently, this event is called the "STEA^2 M Fair." I've seen the term "STEAM" before where the "A" stands for arts. As it turns out, the second "A" stands for agriculture -- which makes sense to include at a fair located at the Fairplex in Pomona.
According to the link, tomorrow popular scientist Bill Nye will be a guest at the fair. The students at my old school went there today, so unfortunately they didn't meet him.
Once again, no Pi Day is complete without music, so what songs would I have performed for my students during Pi Week? With today's trip to the fair, I probably would have wanted to sing "Meet Me in Pomona, Mona" -- perhaps even add an extra verse to represent what the students might see in particular at a STEA^2 M fair as opposed to the regular fair. Since the field trip is today, I would have needed to sing the Pomona song yesterday -- Pi Day itself.
And so I would have added Pomona to the Pi Day songs. Some of these might have been the YouTube links from yesterday's post. But in my classroom, I would've had my guitar, and so it's more important for me to perform the songs. The entire time that students are eating refreshments could be considered music break, and so I have time to play both the Pi Day and Pomona songs.
It's also possible for me to have played Pomona on Tuesday -- after all, I first played Pomona in my classroom on the Tuesday before our Friday field trip that year. Then again, on Tuesday I might have played a new parody of the WSU fight song instead, in honor of the visiting college students. (Recall "U-N-I-T Rate! Rate! Rate!" -- the UCLA fight song parody that I created that year.)
With all of this going on, would I have had time to hand out St. Patrick's Day pencils? Well, I explain above that the best day for me to hand out St. Paddy's Day pencils (or green pencils, since again true holiday pencils were unavailable) would be on computer day -- Monday in the schedule above.
Friday would ordinarily be assessment day, according to the schedule I wrote on September 14th. But with the field trip, there would be no quiz or test today. Well, if the week after next is Parent Conferences Week, then next week might be second trimester Benchmark Tests -- and so I may not give a quiz/test just before the Benchmarks anyway.
Aren't you glad I don't do as many "spilled milk" posts anymore? But Pi Day is an exciting time to be a regular teacher in any math class, and so I can't help but wonder what would have happened if I were still teaching at the old charter school this week.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
M is the midpoint of segmentAB. Find A's x-coordinate.
[Here's the additional given info: A(x, 9), B(-2, -7), M(6.5, 1).]
We use the Midpoint Formula of Lesson 11-4, focusing only on the x-coordinates:
(-2 + x)/2 = 6.5
-2 + x = 13
x = 15
Therefore the desired x-coordinate of A is 15 -- and of course, today's date is the fifteenth.
I did say that there would be Geometry questions on Pappas everyday this week except one. As it turns out, the day without a Geometry question was yesterday -- Pi Day, the most geometrical day of the year. Tuesday's question was the only one in which the constant pi was mentioned.
Chapter 13 of the U of Chicago text is called "Logic and Indirect Reasoning." It is a strange chapter, for two reasons.
First, in past years on the blog, I didn't like the fact that some lessons on logic that appear early in most texts don't appear until Chapter 13 of the U of Chicago text. For example, "Reasoning and Proof" is only Chapter 2 of the Glencoe text. And so I broke up Chapter 13 and scattered its lessons among other chapters.
And second, it turns out that the modern Third Edition of the U of Chicago text does the same thing! I see that Chapters 13 and 14 of the new version correspond to Chapters 14 and 15, respectively, of the old version. The content of the old Chapter 13 is now included in other chapters.
On this blog I continue to follow my old Second Edition of the text. Therefore, we will continue to cover the logic of Chapter 13 as a separate chapter.
This is what I wrote last year about today's lesson:
The other is that I've been meaning to move the first two lessons of Chapter 13 -- namely 13-1 on the Logic of Making Conclusions and 13-2 on Negations -- up to Chapter 2. Dropping Lessons 2-3 leaves a hole right in the middle of Chapter 2, and conveniently, 13-1 and 13-2 fit here. Indeed, 13-1 on Making Conclusions makes perfect sense right after Lesson 2-2 on If-then Statements.
Not only does the modern Third Edition of the text include the old Lesson 13-1 in another chapter, namely Chapter 11, but just as I did in past year, the new edition also combines Lessons 13-1 and 13-2 in its new lesson, which is Lesson 11-2.
This chapter focuses on mathematical logic, which ultimately helps the students write proofs. I mentioned earlier that the Law of Detachment is often known by its Latin name, modus ponens. In fact, I pointed out that on the Metamath website -- a website full of mathematical proofs -- modus ponens is one of the most used justifications:
http://us.metamath.org/mpegif/ax-mp.html
Notice that I only mention the Metamath website for general information. This website is definitely not suitable for use in a high school math classroom. At Metamath, even a simple proof like that of 2+2=4 is very complex:
http://us.metamath.org/mpegif/2p2e4.html
In fact, believe it or not the proof was once even more complicated because it tried to use pure set theory to prove that 2+2=4, and then later on more axioms (postulates) were added to make the proofs easier -- similar to the postulates for real numbers mentioned in Lesson 1-7. To repeat, the basic idea is that one makes a proof simpler by adding more axioms/postulates.
This is when students often ask, "Why do we have to learn proofs?" Of course, they ask because proofs are perhaps the most difficult part of a geometry course. The answer is that even though mathematical proofs may not be important per se -- but proofs are. Many fields, from law to medicine, depend on proving things. We don't want to guess that a certain person is guilty or that taking a certain medicine is effective -- we want to prove it. For centuries, the dominant way to learn how logical arguments work was to read Euclid. Let's learn about how Honest Abe learned about logical arguments from Euclid:
http://the-american-catholic.com/2012/08/16/lincoln-and-euclid/
Unfortunately, the above link is a political and religious website. Well, I suppose it's impossible to avoid politics when discussing Lincoln, but the webpage is also a Catholic site.
By the way, this is our first study of Chapter 13 since we read Eugenia Cheng's third book, which is all about logic. Thus some ideas from Cheng's book are explored in this chapter.
Today is an activity day. The Exploration questions for Lesson 13-1 are based on a famous puzzle called "Who Owns the Zebra?" I decided to include a description of the puzzle -- but instead of typing it in by hand, I cut-and-paste from the following link:
https://sites.lsa.umich.edu/inclusive-teaching/2017/08/16/who-owns-the-zebra/
As the U of Chicago text notes, "This puzzle dates from the 1950's, before the dangers of smoking were widely recognized." By now, of course, we understand the perils of smoking. And so the U of Michigan link above replaces "the Old Gold smoker" with "the aerobic exerciser."
M is the midpoint of segment
[Here's the additional given info: A(x, 9), B(-2, -7), M(6.5, 1).]
We use the Midpoint Formula of Lesson 11-4, focusing only on the x-coordinates:
(-2 + x)/2 = 6.5
-2 + x = 13
x = 15
Therefore the desired x-coordinate of A is 15 -- and of course, today's date is the fifteenth.
I did say that there would be Geometry questions on Pappas everyday this week except one. As it turns out, the day without a Geometry question was yesterday -- Pi Day, the most geometrical day of the year. Tuesday's question was the only one in which the constant pi was mentioned.
Chapter 13 of the U of Chicago text is called "Logic and Indirect Reasoning." It is a strange chapter, for two reasons.
First, in past years on the blog, I didn't like the fact that some lessons on logic that appear early in most texts don't appear until Chapter 13 of the U of Chicago text. For example, "Reasoning and Proof" is only Chapter 2 of the Glencoe text. And so I broke up Chapter 13 and scattered its lessons among other chapters.
And second, it turns out that the modern Third Edition of the U of Chicago text does the same thing! I see that Chapters 13 and 14 of the new version correspond to Chapters 14 and 15, respectively, of the old version. The content of the old Chapter 13 is now included in other chapters.
On this blog I continue to follow my old Second Edition of the text. Therefore, we will continue to cover the logic of Chapter 13 as a separate chapter.
This is what I wrote last year about today's lesson:
The other is that I've been meaning to move the first two lessons of Chapter 13 -- namely 13-1 on the Logic of Making Conclusions and 13-2 on Negations -- up to Chapter 2. Dropping Lessons 2-3 leaves a hole right in the middle of Chapter 2, and conveniently, 13-1 and 13-2 fit here. Indeed, 13-1 on Making Conclusions makes perfect sense right after Lesson 2-2 on If-then Statements.
Not only does the modern Third Edition of the text include the old Lesson 13-1 in another chapter, namely Chapter 11, but just as I did in past year, the new edition also combines Lessons 13-1 and 13-2 in its new lesson, which is Lesson 11-2.
This chapter focuses on mathematical logic, which ultimately helps the students write proofs. I mentioned earlier that the Law of Detachment is often known by its Latin name, modus ponens. In fact, I pointed out that on the Metamath website -- a website full of mathematical proofs -- modus ponens is one of the most used justifications:
http://us.metamath.org/mpegif/ax-mp.html
Notice that I only mention the Metamath website for general information. This website is definitely not suitable for use in a high school math classroom. At Metamath, even a simple proof like that of 2+2=4 is very complex:
http://us.metamath.org/mpegif/2p2e4.html
In fact, believe it or not the proof was once even more complicated because it tried to use pure set theory to prove that 2+2=4, and then later on more axioms (postulates) were added to make the proofs easier -- similar to the postulates for real numbers mentioned in Lesson 1-7. To repeat, the basic idea is that one makes a proof simpler by adding more axioms/postulates.
This is when students often ask, "Why do we have to learn proofs?" Of course, they ask because proofs are perhaps the most difficult part of a geometry course. The answer is that even though mathematical proofs may not be important per se -- but proofs are. Many fields, from law to medicine, depend on proving things. We don't want to guess that a certain person is guilty or that taking a certain medicine is effective -- we want to prove it. For centuries, the dominant way to learn how logical arguments work was to read Euclid. Let's learn about how Honest Abe learned about logical arguments from Euclid:
http://the-american-catholic.com/2012/08/16/lincoln-and-euclid/
Unfortunately, the above link is a political and religious website. Well, I suppose it's impossible to avoid politics when discussing Lincoln, but the webpage is also a Catholic site.
By the way, this is our first study of Chapter 13 since we read Eugenia Cheng's third book, which is all about logic. Thus some ideas from Cheng's book are explored in this chapter.
Today is an activity day. The Exploration questions for Lesson 13-1 are based on a famous puzzle called "Who Owns the Zebra?" I decided to include a description of the puzzle -- but instead of typing it in by hand, I cut-and-paste from the following link:
https://sites.lsa.umich.edu/inclusive-teaching/2017/08/16/who-owns-the-zebra/
As the U of Chicago text notes, "This puzzle dates from the 1950's, before the dangers of smoking were widely recognized." By now, of course, we understand the perils of smoking. And so the U of Michigan link above replaces "the Old Gold smoker" with "the aerobic exerciser."
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