Find x.
Once again, all the given information is in an unlabeled diagram, so let me label it:
In Circle O, Arc AB = 152, Angle OAB = x.
Actually, it might be helpful to draw in and label one more point -- C, so that
The Inscribed Angle Theorem appears in Lesson 15-3 of the U of Chicago text. It's the last Geometry theorem that regularly appears on the SBAC as well as the Pappas problem. But today's post, of course, is all about the beginning of our Geometry course, not the end.
Today is the first day of school according to the blog calendar. The blog calendar is based on one of the districts from which I hope to receive subbing calls. This district is not LAUSD, but in fact is the district whose calendar I used last year. I first followed this calendar five years ago -- the first year of this blog.
I could have used another calendar -- the other district where I currently sub. This calendar has a near Labor Day start. Actually, I propose another name for this calendar -- the Labor Thanksgiving Calendar. In other words, years ago school started after Labor Day, but in order to allow for school to be closed the entire week of Thanksgiving, the start was moved up in recent years to the week before Labor Day. The reason I favor the other district calendar (the Early Start or Pre-Christmas Finals Calendar) is that it's more convenient for my purposes.
Recall that I'm following the digit pattern for days in the U of Chicago text. Chapter 7 is the last chapter of the first semester, and it contains eight sections. Following the district calendar puts Lesson 7-8 ("The SAS Inequality" or Hinge Theorem) on Day 78. Then Days 79-80 can be review days for the final. But this year, the finals will be given on Days 83-85. This means that Lessons 8-1 and 8-2 are taught before the final.
Notice that the district calendar starts the second semester with Day 86, which would be the day for Lesson 8-6. Lesson 8-1, "Perimeter Formulas," can be summed up in a single sentence ("Just add up all the sides!") and Lesson 8-2, "Tiling the Plane," is also easy since it's on tessellations, so I don't mind squeezing these in before the final. Lesson 8-3, "Fundamental Properties of Area," does give the formula for the area of a rectangle, but then this can easily be sneaked into some of the later lessons. Lesson 8-4, "Areas of Irregular Regions," can safely be skipped, but Lesson 8-5 ("Areas of Triangles") is important. Instead, this year I'll do something special for Lesson 8-6 ("Areas of Trapezoids") on the day after winter break.
On the other hand, the Labor Thanksgiving Calendar has a true 90 days in the first semester, which would force all of Chapter 8 to be in the first semester (and on the first final). I'd rather keep most of Chapter 8 in the second semester, and so I choose the Early Start Calendar to base the blog on (even though I'm much more likely to get subbing calls from the Labor Thanksgiving Calendar district).
For future reference, here is a pacing guide for the entire year:
Chapter 0: August 14th-27th
Chapter 1: August 28th-September 11th
Chapter 2: September 12th-25th
Chapter 3: September 26th-October 9th
Chapter 4: October 10th-24th
Chapter 5: October 25th-November 7th
Chapter 6: November 8th-22nd
Chapter 7: December 2nd-13th
Lessons 8-1 to 8-2: December 16th-17th
Semester 1 Finals: December 18th-20th
Lessons 8-6 to 8-9: January 7th-13th
Chapter 9: January 14th-28th
Chapter 10: January 29th-February 12th
Chapter 11: February 13th-27th
Chapter 12: February 28th-March 12th
Chapter 13: March 13th-April 2nd
Chapter 14: April 3rd-20th
Chapter 15: April 21st-May 4th
State Testing Window: May 5th-29th
Semester 2 Finals: June 1st-3rd
Notice that Day 14 is the day after Labor Day. This means that teachers at schools that have a Labor Day Start, but wish to follow my pacing guide, can simply pick it up at Lesson 1-4, "Points in Networks" -- which was my traditional first day of school lesson for the first three years of this blog.
But at the district whose calendar I'm following on the blog, today is the first day of school. Actually, there was a "Freshman First Day" yesterday, so ninth graders attend 181 days of school. We could thus count yesterday as "Day 0" -- but if I were a regular Geometry teacher in this district, I wouldn't bother to teach any math on Day 0 anyway. Some of the Geometry students might be freshmen, but many would be sophomores, and so the first real lesson would be today, Day 1.
Even though there is no Day 0 on the blog, there is a Chapter 0, covering Days 1-10. Since the U of Chicago text doesn't have a Chapter 0, we use Michael Serra's Discovering Geometry instead. Don't forget that my copy of Discovering Geometry is an old text, dated 1997 (Second Edition). My old version goes up to Chapter 16. The modern (3rd-5th) editions only go up to Chapter 13. But Chapter 0 is essentially the same in all editions -- the only difference is that Lessons 0.5 and 0.8 no longer exist in any edition later than my Second Edition.
This is what I wrote last year about today's lesson:
Lesson 0.1 of the Discovering Geometry text is called "Geometry in Nature and Art." In this lesson, students learn that Geometry is all around them.
The main theme of this lesson is symmetry. Serra writes about two types of symmetry -- reflectional symmetry and rotational symmetry. And so we are introduced to two of the main Common Core transformations, reflections and rotations, in the very first lesson.
By the way, I've noticed that other teachers use the Serra text as well. For example, here is a link to Lucy Logsdon -- a Michigan Geometry teacher and Blaugust participant:
https://lsquared76.wordpress.com/2018/08/14/honors-geometry-unit-5-polygon-properties/
[2019 update: Logsdon was a Blaugust participant last year. She's isn't blogging this year -- in fact, she's made only one post since the one linked to above.]
Even though Logsdon never mentions the name Serra or title Discovering Geometry, she's clearly referring to this text. Notice that she begins by discussing "Polygon Angle Conjectures" -- and calling these "conjectures" instead of "theorems" is a giveaway that this is Serra's text. I see that the lessons follow the same order as in Serra -- polygon angle conjectures are in Lessons 6.1 and 6.2, kite and trapezoid properties are in Lesson 6.3, midsegment properties are in Lesson 6.4, and parallelogram properties are in Lessons 6.5 and 6.6. The only reason she calls this "Unit 5" instead of 6 is that in his modern editions, Serra combines his old Chapters 1 and 2. Thus the modern chapter numbers tend to be one less than the Second Edition chapter numbers.
The reason that Serra calls these "conjectures" instead of "theorems" is that officially, nothing is proved until the final chapter. But he does sneak a few coordinate proofs and flow proofs in earlier chapters, which is why these proofs appear on Logsdon's blog as well.
Oops -- I'm not an official Blaugust participant, yet I can't help myself from looking at Shelli's list of Blaugust prompts and participants. I still won't actually add this blog to her list (since only real teachers belong there, and I'm not a real teacher). But I still want to look at her prompts anyway. As today is the fourteenth, let's look at the 14th prompt on Shelli's list:
- My favorite go-to ____(Online resource, book, blog). Share an idea of how you have utilized this source.
In the past the first online resource I'd automatically list here would be Fawn Nguyen's blog. But unfortunately, she hasn't posted since Janu --
Hold on a minute. Yesterday, Nguyen finally posted on her website!
I just accepted a math TOSA position (grades 5-8) with the Rio School District.
I’d spent the last 30 years in the classroom – my school as my second home, my colleagues as my family, my work as my life and identity, but most of all, my students as my children, my babies, my heart.
Then, I thought about not having to grade papers and not having to write sub plans when I’m deathly ill. Hell, yeah, I wanna be a TOSA.
So that's where she's been all this time. She's now a "TOSA" (teacher on special assignment), which is sort of like a math coach. Since she's no longer a classroom teacher, I wonder whether I should continue to count her as one of my favorite go-to resources. Of course, I can always refer to her old posts from back when she was a classroom teacher.
Let me cut-and-paste some old posts of mine where I linked to other online resources. In January 2017, back when I was still at the old charter middle school, I linked to some of my fellow middle school teachers. I'll double-check that post to see whether any of these teachers are still active bloggers -- and are still classroom teachers, not TOSA's:
13. Kit Golan's blog: https://teachdomore.wordpress.com/
His MLK Day post: https://teachdomore.wordpress.com/2019/01/22/honoring-mlk-with-action-from-anger/
Golan teaches sixth and seventh grades in New York. Here's how he describes his blog:
After my fifth year of teaching 8th grade math, I’m transferring schools and I’m going to be teaching 6th and 7th graders. I’m excited for the change in content/curriculum, and I’m hoping the switch will provide me an opportunity to innovate and revisit some of the things I’ve done in the past and consider how to do them better in the future.
This most recent post is from right around the same time as Nguyen's last post before yesterday's. So I still count it as an active blog, but who knows -- Golan might post tomorrow that like Nguyen, he's no longer in the classroom.
22. Jonathan Newman's blog: https://hilbertshotel.wordpress.com/
His Tau Day post: https://hilbertshotel.wordpress.com/2019/06/28/vanguard-program-day-2/
Newman is still in Maryland -- but he's now a high school teacher. This post is about a teacher PD that he attended over the summer.
And that's it! All of the others haven't posted in at least a year, and one of the middle school teachers is now in high school (but at least he's still teaching). I ought to get some new links, but I have no incentive to until I'm a full-time teacher again -- when I'd link to those teaching the same grade or class as I'd be teaching.
One blog I used to link to all the time is Dan Meyer. It's still active. I used to call Dan Meyer "the King of the MTBoS," but he's since rejected the label MTBoS in favor of "I teach math":
https://blog.mrmeyer.com/
https://blog.mrmeyer.com/2019/the-limits-of-just-teaching-math/
At the time, Fawn Nguyen took up the MTBoS mantle, but now I wonder whether she should still be considered Queen of the MTBoS now that she's a TOSA.
Meanwhile, I've also linked to some Geometry teacher blogs since this is a Geometry blog. One of my most linked-filled posts was in January 2018. That day I was covering Lesson 9-7 of the U of Chicago text, on Making (Nets for) Surfaces. Let's see whether any links are active:
Question 26-30, the nets themselves, come from the following link:
https://www.math-drills.com/geometry/net_platonic_solids.pdf
The Math Drills link provides two nets for the dodecahedron. I chose the second one, since it more closely resembles the net in the U of Chicago text. On the other hand, their icosahedron net is very different from ours in the U of Chicago text.
I first found this link via another teacher website -- that's also inactive. Well, at list this page still exists -- in her prompt, Shelli did ask for online resources, which this is.
Later in that post, I linked to David Joyce's website. David Joyce is a Clark U (Massachusetts) professor who advocates returning to Euclid's Elements to teach Geometry. He laments that there isn't much 3D geometry in most high school classes these days outside of measurement. So I linked to Joyce's Euclid website -- particularly his Book XI, on solid geometry:
https://mathcs.clarku.edu/~djoyce/java/elements/toc.html
https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html
Finally, that day I linked to Bizzie Lizzie/Elizabeth Landau. She now works for JPL, but as a young high school student, she wrote some songs about pi. The following link, while quite old, is to a version of her "American Pi," a parody of Don McLean's "American Pie." It's actually not my favorite version -- my favorite version is on her website that's -- you guessed it -- no longer active.
Let's continue Shelli's prompt by looking at my go-to books -- unlike websites, books aren't just going to disappear. On the last day of 2017, I blogged about several books I've read during that year:
Ogilvy, Stanley. Excursions in Number Theory
Pappas, Theoni. Magic of Mathematics
And I've doing the Theoni Pappas Mathematics Calendar in most other years. (Note that 2017 was the year that Pappas didn't publish her calendar, which is why I read her book instead.)
I also wrote that the same day that I purchased the Pappas book at a library book sale (in March 2017, just after I left the old charter school), I purchased two Integrated Math I texts -- these were published by McDougall Littell and Pearson. Oh, and there's one more graphic novel I can't forget:
Ottaviani, Jim. Hawking
After all, we just finished it in my last post. I'm considering reading another one of his other graphic novels -- the one about the Apollo 11 moon landings, before this 50th anniversary year is over.
13. Kit Golan's blog: https://teachdomore.wordpress.com/
His MLK Day post: https://teachdomore.wordpress.com/2019/01/22/honoring-mlk-with-action-from-anger/
Golan teaches sixth and seventh grades in New York. Here's how he describes his blog:
After my fifth year of teaching 8th grade math, I’m transferring schools and I’m going to be teaching 6th and 7th graders. I’m excited for the change in content/curriculum, and I’m hoping the switch will provide me an opportunity to innovate and revisit some of the things I’ve done in the past and consider how to do them better in the future.
This most recent post is from right around the same time as Nguyen's last post before yesterday's. So I still count it as an active blog, but who knows -- Golan might post tomorrow that like Nguyen, he's no longer in the classroom.
22. Jonathan Newman's blog: https://hilbertshotel.wordpress.com/
His Tau Day post: https://hilbertshotel.wordpress.com/2019/06/28/vanguard-program-day-2/
Newman is still in Maryland -- but he's now a high school teacher. This post is about a teacher PD that he attended over the summer.
And that's it! All of the others haven't posted in at least a year, and one of the middle school teachers is now in high school (but at least he's still teaching). I ought to get some new links, but I have no incentive to until I'm a full-time teacher again -- when I'd link to those teaching the same grade or class as I'd be teaching.
One blog I used to link to all the time is Dan Meyer. It's still active. I used to call Dan Meyer "the King of the MTBoS," but he's since rejected the label MTBoS in favor of "I teach math":
https://blog.mrmeyer.com/
https://blog.mrmeyer.com/2019/the-limits-of-just-teaching-math/
At the time, Fawn Nguyen took up the MTBoS mantle, but now I wonder whether she should still be considered Queen of the MTBoS now that she's a TOSA.
Meanwhile, I've also linked to some Geometry teacher blogs since this is a Geometry blog. One of my most linked-filled posts was in January 2018. That day I was covering Lesson 9-7 of the U of Chicago text, on Making (Nets for) Surfaces. Let's see whether any links are active:
Question 26-30, the nets themselves, come from the following link:
https://www.math-drills.com/geometry/net_platonic_solids.pdf
The Math Drills link provides two nets for the dodecahedron. I chose the second one, since it more closely resembles the net in the U of Chicago text. On the other hand, their icosahedron net is very different from ours in the U of Chicago text.
I first found this link via another teacher website -- that's also inactive. Well, at list this page still exists -- in her prompt, Shelli did ask for online resources, which this is.
Later in that post, I linked to David Joyce's website. David Joyce is a Clark U (Massachusetts) professor who advocates returning to Euclid's Elements to teach Geometry. He laments that there isn't much 3D geometry in most high school classes these days outside of measurement. So I linked to Joyce's Euclid website -- particularly his Book XI, on solid geometry:
https://mathcs.clarku.edu/~djoyce/java/elements/toc.html
https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html
Finally, that day I linked to Bizzie Lizzie/Elizabeth Landau. She now works for JPL, but as a young high school student, she wrote some songs about pi. The following link, while quite old, is to a version of her "American Pi," a parody of Don McLean's "American Pie." It's actually not my favorite version -- my favorite version is on her website that's -- you guessed it -- no longer active.
Let's continue Shelli's prompt by looking at my go-to books -- unlike websites, books aren't just going to disappear. On the last day of 2017, I blogged about several books I've read during that year:
Ogilvy, Stanley. Excursions in Number Theory
Pappas, Theoni. Magic of Mathematics
And I've doing the Theoni Pappas Mathematics Calendar in most other years. (Note that 2017 was the year that Pappas didn't publish her calendar, which is why I read her book instead.)
I also wrote that the same day that I purchased the Pappas book at a library book sale (in March 2017, just after I left the old charter school), I purchased two Integrated Math I texts -- these were published by McDougall Littell and Pearson. Oh, and there's one more graphic novel I can't forget:
Ottaviani, Jim. Hawking
After all, we just finished it in my last post. I'm considering reading another one of his other graphic novels -- the one about the Apollo 11 moon landings, before this 50th anniversary year is over.
Notice that these books probably aren't the type of book Shelli had in mind when she first came up with this prompt. Instead, she likely means a teacher book -- speaking of Fawn Nguyen, but she wrote the foreword of a teacher book -- Geoff Krall's Necessary Conditions. Indeed, Shelli herself devoted one of her Blaugust posts to this book.
But as you have seen, I'm not good at acquiring teacher books, either for sale or from the library. One of the few teacher books I've ever read was Harry Wong's The First Days of School -- and that was years ago, before I started this blog. Then again, I might as well mention this book today, since it is the first day of school today.
As usual during Blaugust, I like to highlight one of the other participants. Well, Shelli, the leader of the Blaugust challenge, made her own post today:
http://statteacher.blogspot.com/2019/08/mtbosblaugust-rocketbook-hack.html
I LOVE a-ha moments!!! :)
This post might not make a ton of sense, but I wanted to share it with everyone since it's back to school time and this might be a hack that helps you out!
If you've not heard of Rocketbooks, it's a text to digital notebook where you can write on the notebook, color in a little circle at the bottom to choose a destination, snap a photo with the RB App and your notebook page gets sent to the folder automatically!
No, I've never heard of Rocketbooks, but it sounds interesting and useful. She writes:
I just put my little laminated frame right ON TOP OF my INB and it will scan it!
Recall that INB is an interactive notebook -- and those are getting more popular lately. Once again, this makes me upset that I didn't try using INB's at the old charter school (with out without this Rocketbooks app).
Returning to Serra:
"Nature displays an infinite array of geometric shapes, from the small atom to the greatest of the spiral galaxies. Crystalline solids..."
Atoms -- but then again, Serra focuses more on macroscopic shapes rather than microscopic shapes. His other examples include honeycombs, snowflakes, and pine cones.
I created the first worksheet of the year from questions from Serra's text. Some of these are labeled as Exercises, while others come from his first "Project." And some of these questions ask students to bring objects in to class. This seems awkward for the first day of school -- but then again, this is the very first lesson in Serra (so it's intended for early in the year). It might be good for teachers to find photos on the internet and show them to the class. Or better yet, the students might be able to come up with pictures of their own to draw -- especially the questions about art from various cultures (which include the students' own cultures).
Yet there are two questions that students might enjoy as an opening day activity. One of them asks students to find the line of symmetry in a work by British artist Andy Goldsworthy, who indeed is still alive (hint: H2O). The other asks students to name playing cards with point symmetry. They might want to draw these -- the three of diamonds has point symmetry, but not the three of clubs. Yes, they might want to try drawing the three of clubs with point symmetry to see why it is impossible.
But as you have seen, I'm not good at acquiring teacher books, either for sale or from the library. One of the few teacher books I've ever read was Harry Wong's The First Days of School -- and that was years ago, before I started this blog. Then again, I might as well mention this book today, since it is the first day of school today.
As usual during Blaugust, I like to highlight one of the other participants. Well, Shelli, the leader of the Blaugust challenge, made her own post today:
http://statteacher.blogspot.com/2019/08/mtbosblaugust-rocketbook-hack.html
I LOVE a-ha moments!!! :)
This post might not make a ton of sense, but I wanted to share it with everyone since it's back to school time and this might be a hack that helps you out!
If you've not heard of Rocketbooks, it's a text to digital notebook where you can write on the notebook, color in a little circle at the bottom to choose a destination, snap a photo with the RB App and your notebook page gets sent to the folder automatically!
No, I've never heard of Rocketbooks, but it sounds interesting and useful. She writes:
I just put my little laminated frame right ON TOP OF my INB and it will scan it!
Recall that INB is an interactive notebook -- and those are getting more popular lately. Once again, this makes me upset that I didn't try using INB's at the old charter school (with out without this Rocketbooks app).
Returning to Serra:
"Nature displays an infinite array of geometric shapes, from the small atom to the greatest of the spiral galaxies. Crystalline solids..."
Atoms -- but then again, Serra focuses more on macroscopic shapes rather than microscopic shapes. His other examples include honeycombs, snowflakes, and pine cones.
I created the first worksheet of the year from questions from Serra's text. Some of these are labeled as Exercises, while others come from his first "Project." And some of these questions ask students to bring objects in to class. This seems awkward for the first day of school -- but then again, this is the very first lesson in Serra (so it's intended for early in the year). It might be good for teachers to find photos on the internet and show them to the class. Or better yet, the students might be able to come up with pictures of their own to draw -- especially the questions about art from various cultures (which include the students' own cultures).
Yet there are two questions that students might enjoy as an opening day activity. One of them asks students to find the line of symmetry in a work by British artist Andy Goldsworthy, who indeed is still alive (hint: H2O). The other asks students to name playing cards with point symmetry. They might want to draw these -- the three of diamonds has point symmetry, but not the three of clubs. Yes, they might want to try drawing the three of clubs with point symmetry to see why it is impossible.
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