In the district whose calendar I'm following, Wednesday, Thursday, and Friday are the official finals days next week. This means that today and tomorrow are excellent days for review.
But therein lies the big difference between last year and past years. In past years the first semester finals were held Mon.-Wed., and this year they're Wed.-Fri., so you may ask, why the change? Well, the answer is that it all depends on what day of the week Christmas happens to fall in a given year.
[2019 update: This is the second straight year with the current winter break schedule, but I'll retain some of this discussion anyway.]
We know that in New York, school often lasts all the way until December 23rd (provided, of course, that it's a weekday). But in the district whose calendar we're following on the blog, there can never be school on the 23rd, or even the 22nd.
Meanwhile, the last day before winter break must be either Wednesday or Friday. Friday allows for a full week at the end of the semester, while Wednesday allows three days for finals.
So the rule, presumably, is that the first semester ends either the last Wednesday or Friday that is no later than December 21st. Since 21 = 3 * 7, an equivalent definition is that it ends either the third Wednesday or third Friday in December, whichever is later.
Two years ago, December 22nd fell on a Friday. Since there couldn't be school on the 22nd, this Friday was part of winter break. Then there couldn't be school on the 21st either, because the last day could never be Thursday. Thus the semester ended the 20th. This year, the 20th falls on a Friday, and so that is the last day of the semester.
This also affects the day that the second semester begins in January. As in many districts, there must always be one PD for teachers between the semesters. If the first semester ends on Wednesday, then Thursday is the PD day before teachers leave on winter break. But if the first semester ends on Friday, then the PD day is in January, on a Monday. The second semester then starts on Tuesday -- and this is the case this year. Therefore school resumes on Tuesday, January 7th.
The Friday before first semester finals is always Day 80 on this calendar. So the first semester contains either 83 days (if it ends on Wednesday) or 85 days (if Friday). The latter situation applies this year -- and in fact, the 85-day semester ending on Friday is more common.
In past years, we reviewed for the final on Days 79-80 and gave the final on Days 81-83. But this year, the final is on Days 83-85, so what do we do on Days 81-82?
Well, our rule is that we follow the digit pattern. That means that Days 81 and 82 must be reserved for Lessons 8-1 and 8-2, which are now pushed up into the first semester! But don't worry -- these are two fairly easy lessons. Lesson 8-1 is on perimeter (hint: just add up all the sides), while Lesson 8-2 is on tessellations (how fun).
The lessons corresponding to final days are always skipped. On this calendar, Lesson 8-3 is never covered, since this is always part of finals week. The tricky part is that the second semester now begins with Lesson 8-6, with 8-4 and 8-5 also skipped. Lesson 8-5 is very important (areas of triangles), and so this is a problem. We'll wait to worry about it until after winter break.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
a_1, a_2, a_3, ..., a_n, ... is an arithmetic sequence with constant difference 7 and a_1 = 50. Which term of this sequence has value 127?
In Lesson 6-4 of the U of Chicago Algebra I text, we learn that the nth term of the sequence is:
43 + 7n
where n is the phantom "zeroth term" of the sequence. (In the text, these are often found using charts with a "row 0".) This we must set to 127 and solve:
43 + 7n = 127
7n = 84
n = 12
Therefore 127 is the twelfth term of the sequence -- and of course, today's date is the twelfth. (Notice that some modern texts teach the formula a_n = a_1 + (n - 1)d, with constant difference d. On the other hand, arithmetic sequences don't appear in Dolciani at all. This shows that sequences in Algebra I have only been emphasized in the past two decades.)
Speaking of math calendars, today I also received my new Rebecca Rapoport mathematics calendar. I hope I'll enjoy it, but I know that I'll still miss the Theoni Pappas calendar. I hope Pappas will return to publishing her calendar next year.
Why am I posting this non-Geometry Pappas problem on the blog? It's because it leads to discussion of another Putnam problem. I haven't talked about the B session of the exam yet. Problem B2 does contain a sequence -- of course, this sequence a_1, a_2, a_3, ..., a_n, ... is anything but arithmetic:
https://artofproblemsolving.com/community/c7h1966299_putnam_2019_b2
The goal is to find the sum of the terms of this sequence (that is, an infinite series) and then use it to find a certain limit. Even though finding this limit is technically Calculus, most of the work in the proof is trig. In the thread, the solvers tell us that they use obscure trig identities to rewrite the series as a "telescoping sum" -- that is, one where all the terms cancel except the first. Then that remaining first term is, in fact, the sum of the original series.
Here's an example of telescoping to find the sum of a simple geometric series:
1 + 1/2 + 1/4 + 1/8 + ...
= (2 - 1) + (1 - 1/2) + (1/2 - 1/4) + (1/4 - 1/8) + ..
= 2 (since all the other terms cancel)
Once again, you can read the link to learn more about Problem B2 in full.
This is what I wrote last year about today's final review:
Now I don't presume to know what your district is giving for a final. Most likely, your district is already giving some sort of common final, so I wouldn't need to post my own final. But I've decided that I will post my own final this week, as well as a review for that final. You, as a teacher, can still give my final review, if it can help your students review for your final.
All of these questions are based on the SPUR objectives from the U of Chicago text. I chose to leave out Chapter 1, since we only covered parts of that chapter. For Chapters 2-7, the breakdown by chapter will be as follows:
Chapter 2: 8 questions
Chapter 3: 10 questions
Chapter 4: 6 questions
Chapter 5: 8 questions
Chapter 6: 9 questions
Chapter 7: 9 questions
Like most high school finals, the exam will be multiple choice -- since most teachers don't have time to grade free-response finals before grades are due. But the review worksheets for this final will not be multiple choice, since it's just the review.
A few notes about the questions included on this review worksheet, which covers mostly Chapters 2 through 4 of the U of Chicago text (with two questions that require parallels from Chapter 5):
Question 8 requires a TI calculator, since it's a TI-BASIC question from Chapter 2. On the actual final, if the students don't have a calculator, they can simply use the formula to compute the number of diagonals in some polygon without needing a calculator at all.
Questions 11 and 17 require the students to draw angles of a certain measure. Hopefully, the students at least have protractors. The multiple choice questions on the final will be written such that students won't need the protractors.
Questions 14 and 18 require constructions -- that is, straightedge and compass. If these are not available to the students, then they can just freehand the necessary lines.
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