How many lines of symmetry does a regular hexagon have?
In the U of Chicago text, regular polygons are introduced in Lesson 7-6, but that lesson has little to say on its lines of symmetry.
Fortunately, the modern Third Edition of the U of Chicago text discusses the symmetries of a regular polygon in much more detail. The relevant theorem, which appears in Lesson 6-8, is:
Regular Polygon Reflection Symmetry Theorem:
Every regular polygon has reflection symmetry about:
- each line containing its center and a vertex.
- each perpendicular bisector of its sides.
And that text adds:
"You may have noticed...that when a regular n-gon has an odd number of sides, the perpendicular bisector of one side contains a vertex on the opposite side of the polygon. When n is even, each symmetry line either bisects two angles of the n-gon or is the perpendicular bisector of two opposite sides. Thus, every n-gon has n lines of symmetry."
Today's Pappas question asks about the hexagon (n = 6). Therefore the regular hexagon has six lines of symmetry -- and of course, today's date is the sixth.
It's a shame that the symmetries of the regular polygon don't appear in the old version of the text, but I'm glad they appear in the new edition. We notice the following Common Core Standard:
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.
And yes, the modern text writes about the rotations that map the regular polygon to itself as well. We notice that the n lines of symmetry form 2n rays emanating from the center of the polygon, and so the angle between two consecutive rays is 360/(2n). Then the magnitude of the smallest rotation mapping the polygon to itself must be exactly twice this value, or 360/n (according to the Two Reflection Theorem for Rotations).
But actually, the new U of Chicago text proves that a regular polygon has rotation symmetry first, and then we can prove that it has reflection symmetry. That a regular polygon has rotation symmetry follows from the existence of its center -- the Center of a Regular Polygon Theorem. And this is the one theorem that is actually proved in the old Second Edition of the text. Perhaps Lesson 6-8 of the Third Edition is a good lesson to emphasize for the SBAC -- but then again, we didn't see any symmetry problem during our recent look of the practice SBAC questions.
Today is the last day of school in my old district. It isn't the last day of school in my new district, where it is only Day 174. But the blog is following the old calendar.
Usually, today is when I post a preview of the upcoming school year, but of course, summer school is on my mind right now. In my new district, today is the last summer school training meeting.
As of now, I still don't know whether enough students will sign up for summer school. Remember that even if my summer class is canceled, there won't be a Great Post Purge of 2018 -- that is, I won't go back and delete every post that mentions summer school. Even when I first found out about summer school, I knew that it was dependent on there being enough students to sign up for the class.
Assuming that the summer school class happens, then what are my plans? Of course, the class doesn't start next week (since this is only Day 174), but the week after. The classes are three weeks, four days per week, and I'm scheduled to teach two classes, each a little less than two hours.
When I see my fellow summer Algebra I teacher today (yes, she's the current student teacher at one of the district high schools), we confirm the following plans. She continues to think in terms of the Glencoe text that she uses during the school year, even though we're actually using Edgenuity, an online curriculum:
First Week: Chapters 1-2
Second Week: Chapters 3-4
Third Week: Chapter 5 and District Final
In the name of purity, I should use the names of the actual units in Edgenuity:
- Solving Linear Equations
- Introduction to Functions
- Analyzing Functions
- Linear Functions
- Point-Slope Form and Linear Equations
- Solving Equations and Inequalities
The first unit, Solving Linear Equations, actually corresponds to Chapter 2 of Glencoe. Again, Chapter 1 of Glencoe doesn't actually appear in Edgenuity (since Chapter 1 is based on middle school standards in Common Core), and so we'll teach them Chapter 1 material without a computer -- mostly Order of Operations and the Distributive Property.
In fact, it's possible that my supplemental lessons could come the U of Chicago Algebra I text. The Order of Operations is taught well in Lesson 1-4 of the text. Unfortunately, the Distributive Property is spread out among several lessons in Chapter 6. The distributive property is first introduced in Lesson 6-3, but this is mostly about combining like terms -- as in 2x + 2x = (2 + 2)x = 4x -- as well as discount and markup questions -- as in x - 0.25x = (1 - 0.25)x = 0.75x for 25% off.
More general examples of the Distributive Property appear in Lesson 6-8, whose title is "Why the Distributive Property Is So Named." Examples of distributing a negative value appear in the next lesson, "Subtracting Quantities." All three of these Chapter 6 lessons contain equations to solve, since the U of Chicago text introduces solving equations before the Distributive Property.
Along with us Algebra 1A teachers, the Algebra 1B and Algebra II teachers continue to be concerned with how Edgenuity presents some of the lessons. In fact, when I explored some Edgenuity lessons, I observe that in the videos, fraction and decimal equations are solved directly without clearing the fractions or decimals first. But then the ensuing quiz asks students to identify the number by which they must multiply both sides to clear the fractions or decimals -- and there are two or three such questions on a ten-question quiz! The Algebra II teachers notice similar problems on their respective quizzes -- to the extent that they're seriously considering foregoing the Edgenuity quizzes and just creating and printing their own tests.
This just goes to show us that we teachers shouldn't blindly assume that Edgenuity is teaching all of the students properly. Throughout the entire process, we must continue to monitor the students to see whether they are actually learning -- or are the videos too confusing.
Especially since I still don't know whether I'll have an actual class this summer, I do want to look ahead to the upcoming school year. Of course, it's always possible that I'll be hired as a regular teacher in the fall. But assuming that my employment situation doesn't change, I'll be preparing the fifth year of this blog and the fourth year of posting U of Chicago Geometry.
This year, the digit pattern for pacing served me well. I liked how the digit pattern allows me to schedule lessons -- Lesson 2-1 on Day 21, Lesson 3-2 on Day 32, Lesson 3-4 on Day 34, Lesson 13-3 on Day 133, and so on -- without needing to think about it. If I were teaching a real class with real students, then I might want to be more careful with my pacing. And as my partner summer teacher informs me, the district encourages her to cover the chapters in a different order anyway. But on my blog, it's easiest for me just to post the lessons in order. So I'll continue the digit pattern next year.
Meanwhile, I'll continue to follow the calendar for my old district where I do still get occasional subbing calls, even though the lion's share of calls will be for my new district. I prefer letting the blog follow an Early Start Calendar (that is, where first semester finals are before winter break) rather than a near-Labor Day Calendar (that is, where the first day of school is within a week of that holiday).
On the Early Start Calendar, the first semester is usually slightly less than 90 days -- so that only seven full chapters are covered (with Chapter 7 on Days 71-80) rather than eight chapters. I find that the end of Chapter 7 is a natural semester dividing point. On the other hand, the end of the semester on near-Labor Day Calendars are unconstrained by winter break, and so that first semester ends on the actual Day 90 (which would force eight chapters into the first semester).
But there are a few differences between this year's district calendar and next year's. This year, the first semester contained 83 days, so we finished Chapter 7 (Lesson 7-8) on Day 78, reviewed two days for the final, and then we had the three official finals days.
This year, the first semester contains 85 days. This means that according to the digit pattern, Lessons 8-1 and 8-2 are before the final, Lessons 8-3 through 8-5 are blocked by the final (just as Lessons 8-1 through 8-3 were blocked this year), and the new semester begins with Lesson 8-6. I don't mind squeezing in Lessons 8-1 and 8-2 into the first semester, as these are easy lessons (which are on perimeter and tessellations). But Lesson 8-5, on the areas of triangles, is important. I definitely don't want to skip this lesson and force the students to begin with Lesson 8-6, on the areas of trapezoids, right after returning from winter break. An interesting way to combine Lessons 8-5 and 8-6 is to think of a triangle as a trapezoid with one of its bases having length zero -- then the trapezoid area formula reduces to the triangle area formula.
In this district, students always return from winter break on a Tuesday, since Monday is a PD day for teachers only. So Days 86-89 are on Tuesday through Friday, and then Day 90, the Chapter 8 Test, lands on a Monday. I know that Monday tests are tough since students forget stuff over the weekend, but it beats the alternative of squeezing Chapter 8 even shorter. I have plenty of time to decide how exactly I'll teach Chapter 8 this year. Once again, I could just switch to the other district calendar, where there's plenty of time to teach Chapter 8 in the first semester. (After seeing my partner teacher delay Glencoe Chapter 6 to the second semester of Algebra I, this reminds me that I don't wish to end the semester with a tough chapter like 8).
Let's look at how other tests fit the pacing plan, besides the first semester final. This year, the PSAT occurred near the end of Chapter 3, which fits since Chapter 3 has only six sections. Also, I like the idea of not having so much algebra at the start of the Geometry course -- but Chapter 3 is a great time to teaching equations of lines (Lesson 3-4 on parallel lines) since it's right before the PSAT.
We see that in this district, the SBAC is given around Chapter 14 -- which isn't good because Chapter 14 material and Lesson 15-3 appear on the SBAC. There's not much I can do about this -- but I will move the Chapter 15 Test up to Day 160 and start the SBAC review on Day 161. This will allow me to cover all 34 questions on the practice SBAC (two a day on Days 161-177) before finals week begins with Day 178. Unfortunately, Day 160 is on a Monday next year, so this will be yet another Monday test. (Both Chapters 8 and 15 contain nine sections, so neither test can be moved up to the previous Friday.)
Oh, and there's one more problem to watch out for. This year, Pi Day is on Day 130, so that it's the date of the Chapter 12 Test. Giving a test interferes with the celebratory nature of Pi Day, but there's no alternative since Chapter 12 has ten sections, and then we're running up against the day that third quarter grades are due. (Then again, last year in my new district, some eighth graders had to take a test on Pi Day -- but that matters less since eighth grade isn't the year students learn about pi.)
It might be better to use the new Third Edition of the U of Chicago text next year rather than my trusty rusty old Second Edition. It fits the SBAC better, since there are only 14 chapters, and no Chapter 14 material appears on the state standardized test. Chapter 12 is shorter so we can avoid a test on Pi Day, but Chapters 3 (PSAT) and 8 (first semester finals) are worse in the new edition than in the old one. I'll probably stick to the Second Edition, since that's the text I've been using to blog lessons for years now.
I've been continuing to reflect upon yesterday's subbing, especially the sixth period incident. No, I really didn't break the third resolution on bring up past incidents, but now I worry that I might have broken the second resolution on arguing/yelling in class.
You see, before the boys started throwing rulers, I asked one girl to answer some short questions about the Quadratic Formula (such as how to find -b when b = -3). But she didn't want to answer -- she claimed that she already answered all the problems on the worksheet in her head. (The Quadratic Formula in your head -- really?)
I eventually write her name on the Bad List of names to give the regular teacher -- but I continued to talk to her about the incident anyway. Even though I don't think I yelled at her, I must admit that anything I said to her after writing her name down counts as arguing. As soon as I write down the name, the incident is over.
I suspect that by continuing the argument, it made me appear to lose control of the class -- at least in the eyes of the students. And since I was no longer in control, the kids felt that they could do whatever they wanted, including throwing rulers.
Also, I could have given the girl an incentive to do the work -- singing the song. I remember that she in particular wanted me to sing "Pop Goes the Weasel." I could have told her that I'd sing the song if she completed the work. Again, this naturally fit the lesson -- each time we started a new problem, I must set up the Quadratic Formula, and so I could sing the song. So if the class gets through more problems, then I could sing more renditions of the song. I embarrass myself in the eyes of blog readers -- I devote so many posts to music, specifically music in the classroom. But then when I have a chance to sing something, I don't use the songs to the fullest extent. A powerful use of my songs is as incentives for the kids to work.
The week between the end of regular school (in my old district) and the start of summer school (in my new district) is similar to spring break. My plan is to post twice during this break. One of these posts will probably be on Friday, June 15th -- the day we have access our summer classrooms. This assumes that the class isn't cancelled -- in which case I'll use those posts to announce the cancellation.
In fact, it's possible that my supplemental lessons could come the U of Chicago Algebra I text. The Order of Operations is taught well in Lesson 1-4 of the text. Unfortunately, the Distributive Property is spread out among several lessons in Chapter 6. The distributive property is first introduced in Lesson 6-3, but this is mostly about combining like terms -- as in 2x + 2x = (2 + 2)x = 4x -- as well as discount and markup questions -- as in x - 0.25x = (1 - 0.25)x = 0.75x for 25% off.
More general examples of the Distributive Property appear in Lesson 6-8, whose title is "Why the Distributive Property Is So Named." Examples of distributing a negative value appear in the next lesson, "Subtracting Quantities." All three of these Chapter 6 lessons contain equations to solve, since the U of Chicago text introduces solving equations before the Distributive Property.
Along with us Algebra 1A teachers, the Algebra 1B and Algebra II teachers continue to be concerned with how Edgenuity presents some of the lessons. In fact, when I explored some Edgenuity lessons, I observe that in the videos, fraction and decimal equations are solved directly without clearing the fractions or decimals first. But then the ensuing quiz asks students to identify the number by which they must multiply both sides to clear the fractions or decimals -- and there are two or three such questions on a ten-question quiz! The Algebra II teachers notice similar problems on their respective quizzes -- to the extent that they're seriously considering foregoing the Edgenuity quizzes and just creating and printing their own tests.
This just goes to show us that we teachers shouldn't blindly assume that Edgenuity is teaching all of the students properly. Throughout the entire process, we must continue to monitor the students to see whether they are actually learning -- or are the videos too confusing.
Especially since I still don't know whether I'll have an actual class this summer, I do want to look ahead to the upcoming school year. Of course, it's always possible that I'll be hired as a regular teacher in the fall. But assuming that my employment situation doesn't change, I'll be preparing the fifth year of this blog and the fourth year of posting U of Chicago Geometry.
This year, the digit pattern for pacing served me well. I liked how the digit pattern allows me to schedule lessons -- Lesson 2-1 on Day 21, Lesson 3-2 on Day 32, Lesson 3-4 on Day 34, Lesson 13-3 on Day 133, and so on -- without needing to think about it. If I were teaching a real class with real students, then I might want to be more careful with my pacing. And as my partner summer teacher informs me, the district encourages her to cover the chapters in a different order anyway. But on my blog, it's easiest for me just to post the lessons in order. So I'll continue the digit pattern next year.
Meanwhile, I'll continue to follow the calendar for my old district where I do still get occasional subbing calls, even though the lion's share of calls will be for my new district. I prefer letting the blog follow an Early Start Calendar (that is, where first semester finals are before winter break) rather than a near-Labor Day Calendar (that is, where the first day of school is within a week of that holiday).
On the Early Start Calendar, the first semester is usually slightly less than 90 days -- so that only seven full chapters are covered (with Chapter 7 on Days 71-80) rather than eight chapters. I find that the end of Chapter 7 is a natural semester dividing point. On the other hand, the end of the semester on near-Labor Day Calendars are unconstrained by winter break, and so that first semester ends on the actual Day 90 (which would force eight chapters into the first semester).
But there are a few differences between this year's district calendar and next year's. This year, the first semester contained 83 days, so we finished Chapter 7 (Lesson 7-8) on Day 78, reviewed two days for the final, and then we had the three official finals days.
This year, the first semester contains 85 days. This means that according to the digit pattern, Lessons 8-1 and 8-2 are before the final, Lessons 8-3 through 8-5 are blocked by the final (just as Lessons 8-1 through 8-3 were blocked this year), and the new semester begins with Lesson 8-6. I don't mind squeezing in Lessons 8-1 and 8-2 into the first semester, as these are easy lessons (which are on perimeter and tessellations). But Lesson 8-5, on the areas of triangles, is important. I definitely don't want to skip this lesson and force the students to begin with Lesson 8-6, on the areas of trapezoids, right after returning from winter break. An interesting way to combine Lessons 8-5 and 8-6 is to think of a triangle as a trapezoid with one of its bases having length zero -- then the trapezoid area formula reduces to the triangle area formula.
In this district, students always return from winter break on a Tuesday, since Monday is a PD day for teachers only. So Days 86-89 are on Tuesday through Friday, and then Day 90, the Chapter 8 Test, lands on a Monday. I know that Monday tests are tough since students forget stuff over the weekend, but it beats the alternative of squeezing Chapter 8 even shorter. I have plenty of time to decide how exactly I'll teach Chapter 8 this year. Once again, I could just switch to the other district calendar, where there's plenty of time to teach Chapter 8 in the first semester. (After seeing my partner teacher delay Glencoe Chapter 6 to the second semester of Algebra I, this reminds me that I don't wish to end the semester with a tough chapter like 8).
Let's look at how other tests fit the pacing plan, besides the first semester final. This year, the PSAT occurred near the end of Chapter 3, which fits since Chapter 3 has only six sections. Also, I like the idea of not having so much algebra at the start of the Geometry course -- but Chapter 3 is a great time to teaching equations of lines (Lesson 3-4 on parallel lines) since it's right before the PSAT.
We see that in this district, the SBAC is given around Chapter 14 -- which isn't good because Chapter 14 material and Lesson 15-3 appear on the SBAC. There's not much I can do about this -- but I will move the Chapter 15 Test up to Day 160 and start the SBAC review on Day 161. This will allow me to cover all 34 questions on the practice SBAC (two a day on Days 161-177) before finals week begins with Day 178. Unfortunately, Day 160 is on a Monday next year, so this will be yet another Monday test. (Both Chapters 8 and 15 contain nine sections, so neither test can be moved up to the previous Friday.)
Oh, and there's one more problem to watch out for. This year, Pi Day is on Day 130, so that it's the date of the Chapter 12 Test. Giving a test interferes with the celebratory nature of Pi Day, but there's no alternative since Chapter 12 has ten sections, and then we're running up against the day that third quarter grades are due. (Then again, last year in my new district, some eighth graders had to take a test on Pi Day -- but that matters less since eighth grade isn't the year students learn about pi.)
It might be better to use the new Third Edition of the U of Chicago text next year rather than my trusty rusty old Second Edition. It fits the SBAC better, since there are only 14 chapters, and no Chapter 14 material appears on the state standardized test. Chapter 12 is shorter so we can avoid a test on Pi Day, but Chapters 3 (PSAT) and 8 (first semester finals) are worse in the new edition than in the old one. I'll probably stick to the Second Edition, since that's the text I've been using to blog lessons for years now.
I've been continuing to reflect upon yesterday's subbing, especially the sixth period incident. No, I really didn't break the third resolution on bring up past incidents, but now I worry that I might have broken the second resolution on arguing/yelling in class.
You see, before the boys started throwing rulers, I asked one girl to answer some short questions about the Quadratic Formula (such as how to find -b when b = -3). But she didn't want to answer -- she claimed that she already answered all the problems on the worksheet in her head. (The Quadratic Formula in your head -- really?)
I eventually write her name on the Bad List of names to give the regular teacher -- but I continued to talk to her about the incident anyway. Even though I don't think I yelled at her, I must admit that anything I said to her after writing her name down counts as arguing. As soon as I write down the name, the incident is over.
I suspect that by continuing the argument, it made me appear to lose control of the class -- at least in the eyes of the students. And since I was no longer in control, the kids felt that they could do whatever they wanted, including throwing rulers.
Also, I could have given the girl an incentive to do the work -- singing the song. I remember that she in particular wanted me to sing "Pop Goes the Weasel." I could have told her that I'd sing the song if she completed the work. Again, this naturally fit the lesson -- each time we started a new problem, I must set up the Quadratic Formula, and so I could sing the song. So if the class gets through more problems, then I could sing more renditions of the song. I embarrass myself in the eyes of blog readers -- I devote so many posts to music, specifically music in the classroom. But then when I have a chance to sing something, I don't use the songs to the fullest extent. A powerful use of my songs is as incentives for the kids to work.
The week between the end of regular school (in my old district) and the start of summer school (in my new district) is similar to spring break. My plan is to post twice during this break. One of these posts will probably be on Friday, June 15th -- the day we have access our summer classrooms. This assumes that the class isn't cancelled -- in which case I'll use those posts to announce the cancellation.
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