(Recall that in this district, "Integrated Science" is for special ed students only -- even though I myself took Integrated Sci as a young high school student. There are students of all grade levels in these classes, but the majority are juniors.)
All four of the regular teacher's classes have the same assignment on Mondays -- a current event, which they look up on Chromebooks. This is after the students watch "CNN 10," a daily ten-minute news video intended for school students. (From my experience, special ed teachers are apparently more likely to show "CNN 10" than gen ed teachers.) For the two senior classes, a representative from Jostens gives a presentation on caps, gowns, and class rings. (He even shows the students an actual Super Bowl ring, produced by Jostens.)
There will be no "Day in the Life" since most of the classes have a special aide. This includes the class I must cover during my conference period -- which turns out to be yet another special ed class (this time a self-contained class). During the period I'm in there, three students in this class are giving a presentation on "How I Spent My Weekend."
But seventh period, the last class of the day, has no aide. It's an Integrated Science class with mostly juniors (so there's no Jostens this period either).
The students are supposed to complete the current event assignments on forms. Earlier in the day, the aide notices that there aren't any forms, so she makes more copies -- but unfortunately, she neglects to make enough copies to last all the way through seventh period. I have one of the students pass out the CE forms, and he realizes that there's only enough for about half of the 18 students in this class. (By contrast, the other Integrated Science class has only four students -- and two of them are absent!)
I also point out that sometimes, the students complete the CE forms in Google Classroom rather than on paper forms -- but there's no place for this week's CE forms on Google. (Keep in mind that the regular teacher probably went home Friday thinking he'd be here today, so he had no reason to set up Monday's CE assignment in advance, either on paper or online.)
This sets up a domino effect. One guy refuses to work on his current event -- he just starts playing nonacademic videos on YouTube instead of searching for an article. I tell him that he's required to answer the first three questions on the CE, and he says that he will -- just to get me off his back. He has no intention of stopping the video. Of course, if I try to force him to do the current event, all he has to do is point to the side of the classroom without CE forms and insist on not working, just as those students aren't.
And then some students start packing up with about six or seven minutes to go! Part of it is because a student with a walker is allowed to leave early with her assistant. But it's more because I check to see whether the other guy has answered his three questions, and so many students associate checking work with time to go. Students are now lined up at the door wondering what's taking so long for the bell to ring (answer: they lined up too soon). Three students leave early -- I catch one girl. She insists that she should be allowed to wait outside for the bell -- after all, two others are already gone.
In the past, I've asked, is it a good idea to ask students to answer a certain number of questions to avoid being written on the bad list. We've seen before that, especially in special ed classes, I might set the bar too high and hard-working students are unable to reach it. But in this case, I only require the students to answer the following three questions:
This sets up a domino effect. One guy refuses to work on his current event -- he just starts playing nonacademic videos on YouTube instead of searching for an article. I tell him that he's required to answer the first three questions on the CE, and he says that he will -- just to get me off his back. He has no intention of stopping the video. Of course, if I try to force him to do the current event, all he has to do is point to the side of the classroom without CE forms and insist on not working, just as those students aren't.
And then some students start packing up with about six or seven minutes to go! Part of it is because a student with a walker is allowed to leave early with her assistant. But it's more because I check to see whether the other guy has answered his three questions, and so many students associate checking work with time to go. Students are now lined up at the door wondering what's taking so long for the bell to ring (answer: they lined up too soon). Three students leave early -- I catch one girl. She insists that she should be allowed to wait outside for the bell -- after all, two others are already gone.
In the past, I've asked, is it a good idea to ask students to answer a certain number of questions to avoid being written on the bad list. We've seen before that, especially in special ed classes, I might set the bar too high and hard-working students are unable to reach it. But in this case, I only require the students to answer the following three questions:
- What is the title of the article?
- What is the source of the article?
- What is the date of the article?
Students should be able to answer these trivial questions in under ten minutes, if not five. Any student who fails is most likely doing something nonacademic.
But the main issue here is, what should I do if I'm passing out papers and run out of copies. It's possible that had I not lost my conference period, I might have noticed the lack of papers during the break -- enough time to find a Xerox machine. (This has happened recently to me while subbing at another school.)
A simple solution might have been to drop current events completely to play the "Conjectures"/"Who Am I?" game. (I don't want to wait until my next birthday to play it -- next year, my birthday will fall on Saturday, and I'll have to wait two years, until my 40th in 2020, for it to fall on Monday.) Of course I'd inform the regular teacher of the reason to deviate from his plan (the lack of copies). It might have been tricky to come up with Integrated Science questions -- I see a General Science text in the classroom but I don't know what page they're on. I can always stall for time with the "What's my age/weight?" questions with "Let me give you time to guess," while I search the teacher's desk for evidence of a science lesson.
I've made many changes to this lesson over the past four years. (The new Third Edition also blows up the old Chapter 13 by moving all of its topics to other chapters. Ironically, Chapter 7 isn't one of them -- Lesson 7-6 of the new edition, on tessellations, appears in Chapter 8 of the old edition, and the other two extra lessons in the new edition don't appear in the old edition at all.)
Anyway, this is what I wrote last year about today's lesson:
The unit test will not be given until just before winter break, leaving us with an extra week to fill. So I asked myself, what else can I fit into this unit?
So here is the proof of the Center of a Regular Polygon Theorem as given by the U of Chicago -- in paragraph form, just as printed in the text (rather than converted to two columns as I usually do):
We have seen how powerful a proof by induction can be. We have proved the Center of a Regular Polygon Theorem. But now we wish to prove another related theorem -- one that is specifically mentioned in the Common Core Standards:
The modern Third Edition of the text actually mentions rotation symmetry. This section tells us that a parallelogram has 2-fold rotation symmetry, and the statements about regular polygons actually appear in Lesson 6-8 of the new edition.
Before we end this post, let me add the "traditionalists" label. Our main traditionalists have posted over the weekend:
https://traditionalmath.wordpress.com/2018/12/08/3145/
I wrote this 4 years ago regarding a column in USA Today. I commented and got into an argument with Linda Gojak, former president of NCTM. She presents the usual obfuscation and claims as evidence that students lack ‘understanding” because they cannot apply procedures in a variety of different problem solving situations. Well, if you ignore the novice-expert spectrum and put an expectation of expert thinking on novices, then yes, there’s your evidence I guess.
Now here Garelick quotes himself from four years ago. But who am I to complain, considering that I just quoted myself from four years ago in our Lesson 7-6 discussion.
Lesson 7-6 of the U of Chicago text is called "Properties of Special Figures." (In the new Third Edition of the U of Chicago text, this is Lesson 7-7. Oh, and the title of the new Lesson 7-7 makes it clear that the "special figures" referred to here are parallelograms.)
I've made many changes to this lesson over the past four years. (The new Third Edition also blows up the old Chapter 13 by moving all of its topics to other chapters. Ironically, Chapter 7 isn't one of them -- Lesson 7-6 of the new edition, on tessellations, appears in Chapter 8 of the old edition, and the other two extra lessons in the new edition don't appear in the old edition at all.)
Anyway, this is what I wrote last year about today's lesson:
The unit test will not be given until just before winter break, leaving us with an extra week to fill. So I asked myself, what else can I fit into this unit?
And so I decided just to post the parallelogram properties after all. This week we cover Lesson 7-6, on the Parallelogram Consequences, and Lesson 7-7, on the Parallelogram Tests.
We've already seen how useful the Parallelogram Consequences really are. The reason that they are delayed until 7-6 in the U of Chicago text is that they are best proved using triangle congruence, but triangle congruence doesn't appear until Chapter 7.
It would probably make more sense to cover parallelograms along with the other quadrilaterals in Chapter 5, but it's too late now. Fortunately, I have this opening in the schedule now to cover both Lessons 7-6 and 7-7.
And today's Lesson 7-6 also includes the Center of a Regular Polygon Theorem, which the U of Chicago text proves using induction. This fits with the Putnam-based lesson that I posted last week.
Oh, and by the way, I found the following page about the Fibonacci sequence and generalizations:
http://mrob.com/pub/seq/linrec2.html
(That's right -- this is my second straight week on the same Putnam problem.) It shows why numbers of the Fibonacci sequence have other such numbers as factors -- it's because the Fibonacci numbers can be written as polynomials that can be factored to get other Fibonacci polynomials.
So how does Dr. Merryfield solve Putnam problems like 2014 B1 or 2015 A2? As it turns out, he uses a proof technique called mathematical induction. Most proofs in a high school geometry course aren't proved used mathematical induction. Indeed, only one proof in the U of Chicago text is proved this way -- and it just happens to be the Center of a Regular Polygon Theorem! Furthermore, Dr. Wu uses induction in his proofs on similar triangles, so this is a powerful proof technique indeed. (On the other hand, Wu simply defines a regular polygon as an equilateral polygon with a circle through its vertices, so he would have no need to prove the following theorem at all.)
We've already seen how useful the Parallelogram Consequences really are. The reason that they are delayed until 7-6 in the U of Chicago text is that they are best proved using triangle congruence, but triangle congruence doesn't appear until Chapter 7.
It would probably make more sense to cover parallelograms along with the other quadrilaterals in Chapter 5, but it's too late now. Fortunately, I have this opening in the schedule now to cover both Lessons 7-6 and 7-7.
And today's Lesson 7-6 also includes the Center of a Regular Polygon Theorem, which the U of Chicago text proves using induction. This fits with the Putnam-based lesson that I posted last week.
Oh, and by the way, I found the following page about the Fibonacci sequence and generalizations:
http://mrob.com/pub/seq/linrec2.html
(That's right -- this is my second straight week on the same Putnam problem.) It shows why numbers of the Fibonacci sequence have other such numbers as factors -- it's because the Fibonacci numbers can be written as polynomials that can be factored to get other Fibonacci polynomials.
So how does Dr. Merryfield solve Putnam problems like 2014 B1 or 2015 A2? As it turns out, he uses a proof technique called mathematical induction. Most proofs in a high school geometry course aren't proved used mathematical induction. Indeed, only one proof in the U of Chicago text is proved this way -- and it just happens to be the Center of a Regular Polygon Theorem! Furthermore, Dr. Wu uses induction in his proofs on similar triangles, so this is a powerful proof technique indeed. (On the other hand, Wu simply defines a regular polygon as an equilateral polygon with a circle through its vertices, so he would have no need to prove the following theorem at all.)
[2018 update: I retain this discussion from 3-4 years ago about old Putnam problems in order to demonstrate mathematical induction. In a way, this year's 2018 A6 is sort of like induction in the way we prove certain numbers are rational, but not quite. Last year, I wrote about 2017 A1 where we proved that certain numbers are "sexy," and we can do the same with 2018 and replace "rational" with "sexy." So the sum of two sexy numbers is sexy, as is their difference, product, and so on. Oh, and RIP Dr. Merryfield yet again.]
So here is the proof of the Center of a Regular Polygon Theorem as given by the U of Chicago -- in paragraph form, just as printed in the text (rather than converted to two columns as I usually do):
Proof:
Analyze: Since the theorem is known to be true for regular polygons of 3 and 4 sides, the cases that need to be dealt with have 5 or more sides. What is done is to show that the circle through three consecutive vertices of the regular polygon contains the next vertex. Then that fourth vertex can be used with two others to obtain the fifth, and so on, as many times as needed.
Given: regular polygon ABCD...
Prove: There is a point O equidistant from A, B, C, D, ...
Draw: ABCD...
Write: Let O be the center of the circle containing A, B, and C. Then OA = OB = OC. Since AB = BC by the definition of regular polygon, OABC is a kite with symmetry diagonal OB. Thus ray BO bisects angle ABC. Let x be the common measure of angles ABO and OBC. Since triangle OBC is isosceles, angle OCB must have the same measure as angle OBC, namely x. Now the measure of the angles of the regular polygon are equal to 2x, so angle OCD has measure x also. Then triangles OCB and OCD are congruent by the SAS Congruence Theorem, and so by CPCTC, OC = OD. QED
Now the "Analyze" part of this proof contains the induction. If the first three vertices lie on the circle, then so does the fourth. If the fourth vertex lies on the circle, then so does the fifth. If the fifth vertex lies on the circle, then so does the sixth. If the nth vertex lies on the circle, then so does the (n+1)st. I point out that this is induction -- from n to n+1.
Every induction proof begins with an initial step, or "base case." In this proof, the base case is that the first three points lie on a circle. This is true because any three noncollinear points lie on a circle -- mentioned in Section 4-5 of the U of Chicago. The induction step allows us to prove that one more point at a time is on the circle, until all of the vertices of the regular polygon are on the circle.
We have seen how powerful a proof by induction can be. We have proved the Center of a Regular Polygon Theorem. But now we wish to prove another related theorem -- one that is specifically mentioned in the Common Core Standards:
CCSS.Math.Content.HSG.CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
What we wish to derive from the Center of a Regular Polygon Theorem is that we can rotate this polygon a certain number of degrees -- about that aforementioned center, of course -- or reflect it over any angle bisector or perpendicular bisector. But the U of Chicago text, unfortunately, doesn't give us a Regular Polygon Symmetry Theorem or anything like that.
I'm of two minds on this issue. One way would be to take this theorem and use it to prove that when rotating about O, the image of one of those isosceles triangles with vertex O and base one side of the polygon is another such triangle. The other way is to do Dr. Wu's trick -- he defines regular polygon so that it's vertices are already on the circle. Then we can perform rotations on the entire circle. (Rotations are easier to see, but it's preferable to do reflections because a rotation is the composite of two reflections.) Notice that the number of degrees of the rotation depends on the number of sides. In particular, for a regular n-gon we must rotate it 360/n degrees, or any multiple thereof.
The modern Third Edition of the text actually mentions rotation symmetry. This section tells us that a parallelogram has 2-fold rotation symmetry, and the statements about regular polygons actually appear in Lesson 6-8 of the new edition.
https://traditionalmath.wordpress.com/2018/12/08/3145/
I wrote this 4 years ago regarding a column in USA Today. I commented and got into an argument with Linda Gojak, former president of NCTM. She presents the usual obfuscation and claims as evidence that students lack ‘understanding” because they cannot apply procedures in a variety of different problem solving situations. Well, if you ignore the novice-expert spectrum and put an expectation of expert thinking on novices, then yes, there’s your evidence I guess.
Now here Garelick quotes himself from four years ago. But who am I to complain, considering that I just quoted myself from four years ago in our Lesson 7-6 discussion.
Linda Gojak, former president of NCTM, decides to answer my comment on a comment she made in response to someone else and … Where was I? Well, it was a USA Today article proclaiming that Common Core math is not fuzzy.
Here’s what I said: “Linda Gojak Some understanding is critical, but not all. Sometimes procedural fluency leads to that understanding. It works in tandem. “
Recall that NCTM is strongly Pro-Core -- in fact, this is why the U of Chicago text resembles the Common Core. Both were influenced by NCTM.
We might as well skip to SteveH's comment, since you know that's where we're going:
SteveH:
Linda Gojak is a past president of NCTM with a, BSEd, Elementary Education and Teaching, from Miami University.
Kids struggle in math because schools don’t enforce mastery of basic skills in K-6, no matter how much talk of understanding and in-class group work they offer – which has now been going on in K-6 for 2+ decades! All proper skills require and ensure some understanding. After that, students still might have understanding/flexibility issues. The solution is not top down, but bottom up from mastery of scaffolded skills, and Ms. Gojak has no clue what we mathematicians, scientists, and engineers went through and really know about the many levels of understanding. We don’t expect kids to be able to add and subtract in binary or octal, even though they already know something about dozens and 60 minutes in an hour. Do we expect them to know why months have different numbers of days?
I assume SteveH means "dozenal or sexagesimal," since those bases fit better with his later references to dozens and 60's.
No, SteveH, we don't need to know other number bases or why months are different lengths to use a calendar. Nor do they need to know how arithmetic works in order to use arithmetic. But if they view arithmetic as a set of unrelated procedures, then they're likely to get confused at to which procedure at to use and at what time.
SteveH:
They need to first ensure individual mastery of homework P-sets in K-6 (which they supposedly believe in) and then we can talk about higher levels of understanding that might be missing.
P-sets that are left blank lead to neither learning nor understanding. Just ask that student whose class I subbed in today. How much does he learn about current events by leaving his CE sheet blank today?
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