The measure of arc ABC = 154. Find x.
(Here is the given info from the diagram: O is the center of the circle. Line OB and the tangent lines through A and C are concurrent, intersecting at P. Angle BPC = x.)
For starters, we know that Angle AOC is a central angle of the same measure as arc ABC = 154.
Now we consider quadrilateral AOCP. Two of its angles OAP and OCP are right angles because of the Radius-Tangent Theorem. This means that AOC = 154 and APC are supplementary, so that APC is 26 degrees.
Now Triangles AOP and COP are congruent by HL, since they are both right (OAP = OCP = 90) with congruent legs (OA = OC both radii) and hypotenuses (OP = OP). Thus BPA = BPC, each being half of Angle APC = 26. So BPC = x = 13.
Therefore the desired angle is 13 degrees -- and of course, today's date is the thirteenth. And yes, that's Friday the thirteenth. And yes, this is a Chapter 13 problem as that's where Radius-Tangent occurs (though it can be argued that arc measure isn't defined until Lesson 15-1). But look at all those thirteens today -- triskaidekaphobia running rampant!
Meanwhile, today, in the district whose calendar the blog observes, is Day 22. We are approximately midway through the first quarter -- that is, it's the end of the first quaver.
Lesson 2-2 of the U of Chicago text is called "If-then Statements." This is what I wrote two years ago about today's lesson:
Lesson 2-2 of the U of Chicago text continues the study of logic by focusing on "if-then" statements. I certainly agree with the text when it writes:
"The small word 'if' is among the most important words in the language of logic and reasoning."
There are a few changes that I will make to the text. First of all, the text refers to the two parts of a conditional statement as the antecedent and the consequent -- although it does mention hypothesis and conclusion as acceptable alternatives. I'm going to follow what the majority of texts do and just use the words hypothesis and conclusion. Actually, Dr. Franklin Mason doesn't even use the word hypothesis -- he simply uses the word given -- since after all, the hypothesis of a theorem corresponds to the "given" statement in a two-column proof.
When I teach or tutor students in geometry, one of my favorite examples is "if a pencil is in my right hand, then it is yellow." So I pick up three yellow pencils, and we observe that the conditional is true. But let's suppose that I pick up a blue pencil in addition to the three yellow pencils. Now the conditional is false, since we can find a counterexample -- the blue pencil, since that's a pencil in my right hand yet isn't yellow.
Notice that I decided to replace the word instance with the word example -- so that the connection between examples and counterexamples becomes evident.
The text has to go back to an example from that dreaded algebra again. Of course, it's an important example, since students often forget that 9 has two square roots, 3 and -3. But I decided to include it anyway since it's simple -- it's not as if I'm making students use the quadratic formula or anything like that.
Then the book moves on to a famous mathematical statement: Goldbach's conjecture, named after the German mathematician Christian Goldbach who lived 300 years ago:
If n is an even number greater than 2, then there are always two primes whose sum is n.
At the time the book was written, the conjecture had been verified up to 100 million, but the conjecture had yet to be proved. But what about now -- has anyone proved Goldbach's conjecture yet? As it turns out, the answer is still no -- but now the conjecture has been verified up to four quintillion -- that is, the number 4 followed by 18 zeros.
But there has been work on a similar statement, called Goldbach's weak conjecture:
If n is an odd number greater than 5, then there are always three primes whose sum is n.
This is called weak because if the better-known (or strong) conjecture is true, the weak is automatically true because we can always let the third prime just be 3. Ironically, when Goldbach himself actually stated his conjecture, he stated the weak version of the conjecture. It was a letter from Euler -- you know, the same Euler who solved the bridge problem that we discussed as an Opening Activity -- that convinced Goldbach to state the strong conjecture instead.
Now as it turns out, someone has claimed a proof of Goldbach's weak conjecture -- namely the Peruvian mathematician Harald Helfgott. Recently, Helfgott's proof was still being peer-reviewed -- that is, checked by other mathematicians to find out whether the proof is correct. By now, Helfgott's proof has finally been verified. Yes, mathematicians are still proving new theorems everyday.
Dr. M also mentions Goldbach's conjecture, on a worksheet for his Lesson 2-1. Often students are fascinated when they hear about conjectures that take centuries to prove, such as Goldbach's conjecture or Fermat's Last Theorem. I often use these examples to motivate students to be persistent when trying to come up with proofs in geometry -- if mathematicians Helfgott and Wiles didn't give up even after centuries of trying to prove these conjectures, then why should they give up after minutes?
The final example in this section has students rewrite statements into if-then form. With the newly released Common Core scores still fresh on my mind [...]
[2019 Update: Notice that the 2019 SBAC scores still haven't been released yet. Apparently, at this time in other years the scores had been released, but for some reason, this year the release of the scores is even slower. This is one reason for opposition to Common Core and its associated computer-based standardized tests -- they take too long to score. To me, this defeats the main purpose of taking tests on the computer. Once again, I believe that (at least for math) computer scores should be given instantaneously. Any math question that can't be scored immediately isn't worth asking. So instead, let's skip to the next relevant part of last year's post.]
If you thought that there would be no "spilled milk" in this post, guess again. That's because it's become a tradition to write about the LA County Fair on Day 22. Back at the old charter school, the field trip was usually on Day 22.
[2019 Update: Of course, that school is now closed, so I can't say "Today is the day I would have gone to the fair had I not left the school" anymore. No one from my old school is at the fair today, because my old school no longer exists. I will still cut-and-paste from my old LA County Fair posts today. Lately, I've been singing the LA County Fair song in class -- except that it's an old version. Last year I blogged a new version of the song, but the old version is still in my notebook. I should change my notebook so that I can sing the new version in class instead -- and I'm now repeating that part of the post to remind me of the new version. On the other hand, I'm cutting out the part "Here's where I imagine what the first month of school would have looked like had I not left." The school is now closed, so there's no imagining what I'd be doing if only I were still working there.]
Anyway, this is what I wrote three years ago about the field trip:
1) Teachers make a lot of decisions throughout the day. Sometimes we make so many it feels overwhelming. When you think about today, what is a decision/teacher move you made that you are proud of? What is one you are worried wasn’t ideal?
I think that the best decision I made during the first 22 days of school was to include a music break as part of my daily lesson. As I wrote in my First Day of School (August 16th) and August monthly posts, I try to sing a math related song three times a week. This motivates the students to want to sing along -- and by learning the words, they are learning math without realizing it. One of my most popular songs is the one I mentioned in my August monthly post, Count on It. Music break is ten minutes out of an 80-minute block -- but as an incentive, I extend the break to 15 minutes if the students are singing along.
As for the worst decision I made -- well, the field trip to the LA County Fair was two days ago, and so it's still fresh on my mind. There were a number of poor decisions I made on that trip. I know that this isn't supposed to be a Day in the Life post, but here is a brief overview of my field trip:
10:00 -- We arrived at the fair. All groups -- including mine of half a dozen sixth graders, five boys, one girl -- walked through the Jurassic Planet exhibit. My students were hungry and wanted to eat their lunch, but I tell them that all groups would eat near Mojo's Wild and Crazy Island.
12:00 -- The students eventually spent all of their money on the Extreme Thrills tickets. Since all of the other rides were now open, we walked towards the Carnival section -- only to find out that all of the rides require purchasing tickets. The kids kept walking hoping to find a free ride, but we didn't.
2:00 -- As we get ready to board the bus to leave, I met my Support Staff aide, who had a small group of sixth graders of her own. She told me that her group had taken a tram to the farm area, rode a few extreme rides, and still had money left over for the carnival rides!
At that point, one of my group members proceeded to blame me for giving them such a miserable day at the fair -- even though I wasn't the one who wouldn't let them ride. (That would be the carnies who told them that they needed tickets to ride.) On the other hand, he had a point, as there actually were a few things that I could have done to improve my group's experience at the fair.
Until I arrived, I didn't even know that there was a tram. That was something I should have looked into ahead of time -- when I was doing research for my song "Meet Me in Pomona, Mona." Finally, I should have found out that all of the rides require tickets -- perhaps if I'd told my students this, they would have saved money for the Carnival section.
First of all, I wrote that when I was writing the music break song, "Meet Me in Pomona, Mona" (as in Pomona, the city where the fair is located), I should have looked up info about the tram and how to get to the farm area.
And so let me fix that error today. Today I'll post a new, better version of the song. Three years ago, I was so obsessed with trying to match the lyrics of the song I was parodying ("Meet Me in St. Louis, Louis") that I kept writing about a "janitor" instead of the field trip itself. In this new version, I keep the first verse but change the second verse to reflect what the kids would see at the fair. Some of that info was included in my original refrain, and so I must change the refrain as well:
MEET ME IN POMONA, MONA
First Verse:
When Mona came up to the school, as she sat,
She hung up her coat and her hat.
She gazed around, but no teacher she found,
So she said "Where can the class be at?"
She remembered the noted, she flipped,
She saw it was a permission slip.
It said, "Hear, hear, it's too slow to learn here,
So let's go on this crazy field trip."
Refrain:
Meet me in Pomona, Mona,
Meet me at the fair.
Don't tell me that I'll learn science,
Any place but there.
The bus will leave for the fair soon,
We can stay there all afternoon.
Meet me in Pomona, Mona,
Meet me at the fair!
Second Verse:
At the fair Mona said, "Here I am!
So first I will get on this tram."
She went to the farm and she saw at the barn,
Cows, pigs, chickens, and even a lamb.
And then Mona wanted to go,
To see the monkey named Mojo,.
Peacocks and giraffes and a whole lot of laughs,
And when to leave there she didn't know.
(Repeat Refrain)
In addition to today's worksheet, I'm restoring my old pattern of posting a weekly activity. In a way, nearly all of Chapters 0 and 1 are activities, so I resume this tradition here in Chapter 2. Our first activity from last year is a list of logic puzzles, to go along with the logic that we learn here in this chapter. Yes, I know I just wrote a few lines above that for Illinois State, I do projects on Tuesdays (or maybe Thursdays) and assessment on Fridays, but for now I'm posting tests on Wednesdays and activities on Fridays.
Here's a little of what I wrote last year about the logic puzzles:
As it turns out, I've seen a version of this puzzle before last spring. It is a similar brain teaser known as the "Sum and Product Puzzle." The next link contains a statement and solution of the puzzle:
http://www.qbyte.org/puzzles/p003s.html
Notice that in describing the solution, the author actually uses Goldbach's conjecture -- the unproved conjecture that I mentioned earlier in this post. Of course, the numbers involved in this problem are much too small to be counterexamples to Goldbach.
I'll repeat the same activity worksheet from last year, although it might be interesting to replace the old Puzzle #10 with Cheryl's birthday problem. The sum and product version of this puzzle might be suitable in an algebra class, especially near the lesson on factoring quadratic polynomials.
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