8:15 -- Homeroom leads directly into first period. This is the only class that isn't English, but AVID.
Just after first period proper begins, there is a lockdown drill -- the second such drill that I've had to participate in this month and the third this school year. (The earlier lockdown drill occurred in my old district during nutrition.) Once again, this is a sad commentary on what public schools are like now that so many lockdown drills have become necessary.
Disclaimer: Today, there is a shooting at a high school in Southern California. Clearly, I'm subbing at a middle school today -- and its nowhere near Saugus High School. Note that the Saugus shooting occurs an hour before our drill. It's remotely possible that today's lockdown is a reaction to the shooting, but it's more likely that the lockdown drill had already long been scheduled for today. Even if it's indeed a coincidence, it bears repeating -- this is a sad commentary on what public schools are like now.
After the drill, AVID tutorials begin, as they do every Thursday. A special adult tutor comes in every Thursday to help.
And of course, I assist the students who are working on math. Some eighth grade Algebra I students are working on systems of linear equations. Notice that they are currently ahead of the freshmen Algebra I students who were graphing by intercepts earlier this week. This is because the two grades have different Algebra I texts (Big Ideas for eighth grade, Glencoe for ninth). Besides, eighth grade must move faster since these students must also learn the non-algebraic Common Core 8 standards necessary to pass the eighth grade SBAC.
I've mentioned before that many math classes are now implementing VNPS -- in other words, vertical non-permanent surfaces. Actually, VNPS is a staple in the AVID classroom. At tutorials, students always set up the problems to be solved on the large whiteboards placed around the room.
In her lesson plan, the regular teacher specifies a strict restroom policy -- there are to be no restroom passes unless it's an emergency. This is always a tough one -- the teacher gives a strict policy because she knows students want to take advantage of subs. But then the students complain and tell the subs that they are wrong to enforce the policy, because the regular teacher usually isn't as strict.
Two students ask for restroom passes. I generally assume that if two students in one period ask to go, they aren't both genuine emergencies. This, of course, leads to argument -- and I have to tell a third student that she can't go to the restroom.
9:20 -- First period leaves. This is the school where all periods after homeroom/first rotate. The rotation for Thursday is 1-5-6-2-3-4. And this teacher doesn't have a conference period (probably because she teaches AVID), and so it's eighth grade English the rest of the day.
The students have just finished reading Edgar Allen Poe's The Telltale Heart. They must complete a "closing argument" writing assignment, then study for tomorrow's test on Poe's short story.
I anticipate that there will be a restroom argument again. My recent go-to for ending arguments is simply to -- sing a song. Once again, I choose Square One TV's "One Billion Is Big." Normally, I would base my music incentive on how many students do the work. But since the main assignment today is once again on Chromebooks (where it's often difficult to determine how many students are doing the work), I take advantage and switch the music incentive to restroom passes instead. In order for me to rap the entire song, the number of passes for the period must be zero.
Ultimately, the incentive works. One student changes his mind when he tries asking for a pass, and in the end, I do sing the entire song.
10:15 -- Fifth period leaves for snack.
10:30 -- Sixth period arrives. This class ends up being the best class of the day.
11:25 -- Sixth period leaves and second period arrives. This is anything but the best class of the day.
Students continually make noise and become distracted from work. I ultimately write one student on the bad list -- the guy who appears to be in the center of it all. The fact that he later receives a lunch detention from another teacher only corroborates my decision to write his name.
Meanwhile, one girl does claim an emergency and asks for the restroom pass. One thing I don't tell the students about the music incentive -- until it's necessary, that is -- is that if only one student goes during a period, I sing one verse of the song anyway. This is so that there isn't a flood of restroom passes right after "zero" is no longer achievable -- and besides, it's plausible that one pass in a period may be a genuine emergency. (I sing them the second verse, since I'd already sung the first verse to whet their appetites at the start of class. Classes with "zero" get all three verses.)
12:20 -- Second period leaves for lunch.
1:05 -- Third period arrives. This class ends up being the second best class of the day.
2:00 -- Third period leaves and fourth period arrives.
A major issue in this class is the presence of a few special ed students. The regular teacher often gives them special accommodations -- for example, they wouldn't have to complete today's paragraph on the Chromebook (because they might not be capable of writing a paragraph). Also, they might be assigned special seats. While I'm trying to get the students to return to their correct seats, some of them call each other highly disparaging (and possibly racially charged) names.
But there is no mention of any accommodations in the lesson plan. And so of course this leads to more arguments. It doesn't help indeed that I don't know who the special ed students are at first -- so they look just like (gen ed) students just trying to sit wherever they want and avoid doing work. Oh, and the seating chart argument also leads to my making an error on the attendance.
Is there any way I could have avoided this argument? I'm not sure. Perhaps I should have recognized one of the students as special ed -- he was the new student back when I subbed at this same school on October 31st-November 4th. Then again, I never knew whether he was capable of writing paragraphs until today. Also, the seating chart -- while not mentioning the special accommodated seats -- mentions something about "vocabulary." There is a vocab worksheet for tomorrow's test, but the special ed students refuse to work on any vocab. (Then again, the note about vocab is at least several days old, and it might refer to different vocabulary.) One special ed girl insists that I email the regular teacher to find out whether she needs to take tomorrow's test or not.
But I do know how to end most arguments. My recent go-to for ending arguments is simply to -- sing a song. So I sing the billion rap again.
There is one more problem here. Earlier this week, I had a problem with collecting Chromebooks to be charged at the end of the day. So this time, I change up the song incentive so that it's for cleaning up at the end of the day. I tell the students that I will call them up one table at a time (since there's nothing like "Student A," "Student B" in this class) to place their laptops in the correct slot, charge them, and then return to their seats.
Then one special ed student asks, "What's a seat?" It's obvious that he's saying this just because he doesn't want to sit down anywhere -- and I tell him this. A better response might have been simply to say nothing at all -- including no rap, if he's the only one not sitting down. He wouldn't want to be the one to cost the class the incentive.
As it turns out, many students still fail to charge the laptops, and one student is left trying to charge many Chromebooks. He doesn't finish until around the time the bell rings -- but some students stay to hear me finish the rap they earned anyway.
2:55 -- Fourth period leaves.
Another teacher arrives in the classroom. She tells me that she's a laid-off teacher who's been assigned to sub for this class tomorrow. (So it's not a true multi-day assignment, since she'd already picked up tomorrow's job before today's became available.) Thus fortunately she has the regular teacher's number -- so at my request, she texts the regular teacher to ask about what to do about the special ed students and tomorrow's test.
I still wish to work on avoiding arguments completely -- especially with special ed students. They are already having a tough time in school, and I don't need to make it tougher by arguing. I can't help but think back to this time of year at the old charter school -- three years ago in mid-November -- when I argued with "the special scholar" about the bell schedule during Parent Conferences week. Incidents like today demonstrate that I still need to work on communication with special ed students who may be mainstreamed in a gen ed classroom.
Yesterday, we finished Ian Stewart's The Story of Mathematics. But Stewart is one of the most prolific authors when it comes to writing popular math and science books. (I even saw another one of his books, dated this year, at the local bookstore.)
Anyway, I'd already purchased a Stewart book -- Calculating the Cosmos: How Mathematics Unveils the Universe -- a few years ago. I already blogged a chapter or two here. Let's round out the week with a few randomly chosen chapters from this book under the Stewart label. Perhaps this summer, I'll finally read this book from cover to cover, and blog about it.
So here's my randomly chosen chapter:
Chapter 11 of Ian Stewart's Calculating the Cosmos: How Mathematics Unveils the Universe is called "Great Balls of Fire." Chapters in this book begin with a quote:
"We may determine the forms of planets, their distances, and their motions -- but we can never know anything of their chemical composition." -- Auguste Comte, The Positive Philosophy.
This chapter is all about, well, the chemical composition of planets and stars. (I've heard of Comte before -- he is the originator of the 13-Month Calendar Reform. He named his calendar after the philosophy mentioned above -- the Positivist Calendar.) Stewart begins:
"With twenty-twenty hindsight it's easy to poke fun at poor Comte, but in 1835 it was inconceivable that we could ever find out what a planet is made of, let alone a star."
And the author tells us how we did finally figure out what celestial bodies are made of:
"By combining observations with mathematical models, scientists have inferred detailed answers to all of these questions, even though visiting a star with today's technology is virtually impossible. Let along tunneling inside one. The discovery that rubbished Comte's example was an accident."
Stewart begins with the story of Joseph Fraunhofer, a glassmaker's apprentice. He inadvertently discovered a spectroscope which can measure wavelengths. Using his spectroscope, he could compare the wavelengths in a star's spectrum to that produced by a chemical element:
"Fraunhofer applied this idea to Sirius, thereby observing the first stellar spectrum. Looking at other stars, he noticed that they had different spectra."
The astronomer Jules Janssen then sought the chemistry of the sun's chromosphere -- as opposed to the part of the sun that we see, the photosphere:
"The chromosphere is so faint that it can be observed only during a total eclipse, when it has a reddish hue."
He and Norman Lockyer eventually discovered a new element in the periodic table -- helium, named after the Greek word for "sun":
"We see it in the Sun because the Sun isn't just made of it, along with a lot of hydrogen and lots of other elements in lesser amounts: it makes it...from hydrogen."
At this point, Stewart discusses other stars and their classification. Each class is assigned a letter based on its temperature:
"Stars are also given a luminosity class, mostly written as a Roman numeral, so this scheme has two parameters, corresponding roughly to temperature and luminosity. Class O stars, for instance, have a surface temperature in excess of 30,000 K, appear blue to the eye, have mass at least 16 times that of the Sun, show weak hydrogen lines, and are very rare."
The author tells us that our own sun is classified as a G2 star. The vast majority of all stars are in Class M -- the coolest and smallest. He displays a scatter plot of known stars with their temperatures and magnitude (or brightness) known as the Hertzsprung-Russell diagram:
"The most prominent features are a cluster of bright, coolish giants and supergiants at top right, a curvy diagonal 'main sequence' from hot and bright to cool and dim, and a sparse cluster of hot, dim white dwarfs at bottom left. The study of stellar spectra went beyond butterfly collecting when scientists started using them to work out how stars produce their light and other radiation."
At this point Stewart discusses how stars make their energy. He begins by discussing "deuterium," which is a type (isotope) of hydrogen that contains a proton and a neutron in its nucleus:
"After about four seconds the deuterium nucleus fuses with another proton to make an isotope of helium, helium-3: quite a lot more energy is released."
But eventually, the energy will run out:
"This takes hundreds of billions of years for slow-burning red dwarfs, 10 billion years or so for stars like the Sun, and a few million years for hot, massive O-type stars."
Our sun will become a red giant in about five billion years. Other larger stars have a different fate:
"The core of the star can end up as a white dwarf or a black dwarf, which is a white dwarf that has lost so much energy that it stops shining."
Another possibility is a black hole -- but Stewart saves that for a later chapter. Instead, he discusses the origin of the heavier elements which we are made of. In short, we're stardust -- it was discovered by Fred Hoyle:
"He published a lengthy analysis of reaction routes leading to all elements up to iron. The older the galaxy, the richer its brew of elements."
Some elements also formed by supernovas, or exploding stars. The author tells us that this theory doesn't account for how much of the element lithium there is in the universe:
"Some scientists think this is a minor error that can probably be fixed up by finding new pathways or new scenarios for lithium formation."
This error appears with the heavier elements as well, leading some scientists to fudge the numbers:
"I'm not suggesting anyone deliberately does this kind of thing, but selective reporting like this is natural, and it's happened elsewhere in science."
One element of particular concern is carbon, since we're carbon-based lifeforms. According to Hoyle, there shouldn't be enough carbon in the universe to make us:
"Unless...the energy of carbon is very close to the combined energies of beryllium-8 and helium. This is a nuclear resonance, and it led Hoyle to predict a then unknown state of carbon at an energy 7.6 MeV above the lowest energy state."
Stewart warns the reader against concluding that the universe has been "fine-tuned" for life:
"But we shouldn't confuse outcomes with causes, and imagine that the purpose of the universe is to make humans. One reason I've mentioned this (aside from a distaste for exaggerated fine-tuning claims) is that the whole story has been made irrelevant by the discovery of a new way for stars to make carbon."
In this theory, the carbon comes from young stars, not supernovas. At this point, Stewart now switches to a discussion of sunspots, and whether sunspots cause climate change. In particular, minimal sunspot activity sometimes coincides with lower temperatures on earth:
"So did a previous period of low sunspot activity, the Dalton minimum (1790-1830), which includes the famous 'year without a summer' of 1816, but the low temperatures that year resulted from a huge volcanic explosion, Mount Tambora in Sumbawa, Indonesia."
Later scientists continued to study sunspots and their effects:
"Horace Babcock modelled the dynamics of the Sun's magnetic field in its outermost layers, relating the sunspot cycle to periodic reversals of the solar dynamo."
The author tells us how the sun's magnetic field indeed causes sunspots:
"The result is a long-period oscillation in the average size of the field during a cycle, and when it dies down few sunspots appear anywhere."
Now Stewart moves on to finding the distance to the stars. The history of calculating long distances -- just like many chapters we just read in his Story of Mathematics -- goes back a few millennia:
"In the sixth century BC the ancient Greek philosopher and mathematician Thales estimated the height of an Egyptian pyramid using geometry, by measuring its shadow and his own."
And of course, this type of question shows up in the similarity chapters of Geometry texts. We learn that from the distance from the earth to the sun, we can calculate the distance to the stars by observing them six months apart -- a technique known as parallax:
"The star's distance is approximately proportional to the parallax, and a parallax of one second of arc corresponds to about 3.26 light years."
This unit is now known as the parsec, or parallax arcsecond. At this point, Stewart tells us the story of an American astronomer, Henrietta Leavitt:
"In the 1920's Pickering hired her as a human 'computer,' carrying out the repetitive task of measuring and cataloguing the luminosities of stars in the Harvard College Observatory's collection of photographic plates. Most stars have the same apparent brightness all the time, but some, which naturally arouse special interest among astronomers, are variable: their apparent brightness increases and decreases in a regular periodic pattern."
Some such variable stars are known as Cepheids. She discovered a formula to calculate their brightness, thus giving a new meaning to "Twinkle, Twinkle Little Star":
"And those results then extended to all Cepheids, using the formula relating the period to the intrinsic brightness. Cepheids were the long-sought standard candle."
Stewart concludes the chapter by relating brightness of stars to their distance:
"Each step involved a mixture of observations, theory, and mathematical inference: numbers, geometry, statistics, optics, astrophysics. But the final step -- a truly giant one -- was yet to come."
Lesson 6-4 of the U of Chicago text is called "Miniature Golf and Billiards." In the modern Third Edition, we must backtrack to Lesson 4-3 to play miniature golf.
This is what I wrote last year about today's lesson:
Today we proceed with the next lesson in the text. Lesson 6-4 of the U of Chicago text is all about applying reflections to games such as miniature golf and billiards. I don't need to make any changes to the lesson, so I can just keep what I wrote last year for this lesson almost intact:
One of my favorite TV programs is The Simpsons -- I've been watching it for decades. One of its earliest episodes, having aired exactly 28 years ago today (November 15th, 1990), was called "Dead Putting Society." In this episode, Bart Simpson is preparing for a miniature golf competition. His sister Lisa shows him how he can use geometry to help him make a difficult shot. After saying this, Bart proclaims, "You've actually found a practical use for geometry."
[2019 update: The story my students read today, "The Telltale Heart," is actually mentioned twice on The Simpsons. First, an allusion appears in the first season episode title "The Telltale Head," when Bart cuts down a buries the head of a statue of the town's founder. The other is "Lisa's Rival," where Lisa and a new, smart girl in her class each create a diorama of Poe's short story.]
One of my favorite TV programs is The Simpsons -- I've been watching it for decades. One of its earliest episodes, having aired exactly 28 years ago today (November 15th, 1990), was called "Dead Putting Society." In this episode, Bart Simpson is preparing for a miniature golf competition. His sister Lisa shows him how he can use geometry to help him make a difficult shot. After saying this, Bart proclaims, "You've actually found a practical use for geometry."
[2019 update: The story my students read today, "The Telltale Heart," is actually mentioned twice on The Simpsons. First, an allusion appears in the first season episode title "The Telltale Head," when Bart cuts down a buries the head of a statue of the town's founder. The other is "Lisa's Rival," where Lisa and a new, smart girl in her class each create a diorama of Poe's short story.]
Lesson 6-4 of the U of Chicago text discusses miniature golf and billiards. Just as Bart learns in this above video, one can use geometry to determine where to aim.
The key is reflections -- one of the important transformations in Common Core Geometry. It is often said that when a ball bounces off a wall, the angle of incidence equals the angle of reflection. The text describes where to aim a golf ball G so that it bounces off of a wall and reaches the hole H:
"In this situation, a good strategy is to bounce (carom) the ball off a board, as shown [in the text]. To find where to aim the ball, reflect the hole H over line AB. If you shoot for image H', the ball will bounce offAB at P and go toward the hole."
We can write a two-column proof to show why the angle of incidence -- the angle at which the ball approaches the board, which is BPG -- equals the angle of reflection APH:
Given: H' is the reflection of H over line AB.
Prove: Angle APH = Angle BPG
Statements Reasons
1. H refl. over line AB is H' 1. Given
2. Angle APH = Angle APH' 2. Reflections preserve angle measure
3. Angle APH' = Angle BPG 3. Vertical Angle Theorem
4. Angle APH = Angle BPG 4. Transitive Property of Equality
Notice that for this proof, I've skipped a few steps. Technically, we should write that the reflection images of both A and P are the points themselves, since they lie on the mirror (Definition of Reflection), and so angle APH' is the image of APH (Figure Reflection Theorem). But I'm tired of writing that over and over again -- how much less, then, will the students want to write that.
Of course, this only works if the ball caroms only once. The U of Chicago text describes the game of billiards, where the player is required to bounce the ball off of three cushions. The text writes:
"Pictured [in text] is a table with cushions w, x, y, and z, the cue ball C, and another ball B. Suppose you want to shoot C off x, then y, then z, and finally hit B. Reflect the target B successfully over the sides in reverse order: first z, then y, then x. Shoot in the direction of B"' [...]
"Notice what happens with the shot. [...] On the way toward B"', it bounces off side x in the direction of B". On the way toward B", it bounces off y in the direction of B. Finally it hits z, and is reflected to B. [End of quote]"
In the video clip above, four caroms are required for the golf ball shot by Bart to find the hole. But unfortunately, the path of the ball as drawn on the show is impossible. To see why, let's label the direction from the starting triangle to the hole "North," and all the walls appear to meet at right angles, so they are all oriented in the north-south or east-west directions. Bart begins with the ball slightly to the right side of the starting triangle, so the initial direction of the ball is northwest. After hitting the first east-west wall, the ball is now traveling southwest. But then, after hitting the second east-west wall, the ball should be traveling northwest again. Indeed, we can use the Alternate Interior Angles Consequence and Test Theorems to prove that the path of the ball after hitting two walls should be parallel to the original direction of the ball. Yet the show depicts the ball as travelling due north after hitting two walls.
In fact, we can show that the only correct path to make the ball arrive in the hole involves hitting a north-south wall -- most likely the wall to the far west (in front of "Do not sit on statuary"). Only by hitting a north-south wall can the direction change from anything-west to anything-east, which is necessary for the ball to approach the hole. Lisa's advice to her brother is sound, but the way that it is animated is geometrically impossible.
For this worksheet, I decided to reproduce Bart's golf course, but tilt two of the walls in order for the path of the ball to be geometrically correct. I used equilateral triangle paper to create this page, so that the students will be able to figure out the paths without needing a ruler or protractor, for classrooms in which these are not available. Just as for Bart, four caroms are needed to get from G to H.
Working backwards on this worksheet, we can determine the path from A to H by reflecting H to the point H', then aiming from A to H'. But to determine a path from B to H using two caroms, we can't reflect H to first H' and then H", because H" would be well off the page. It may be better to aim from B to A', the reflection of A in the necessary wall.
Notice that if there are two walls and the paper is large enough, it may be actually possible to perform both of the necessary reflections. If the two walls meet at right angles, it is fairly easy to perform both reflections -- because the composite of those two reflections is a rotation. So to find the direction to aim at, we take the target point and rotate it twice 90, or 180, degrees around the point where the two walls meet. And if the ball is bouncing off of two parallel walls, then the composite of the two reflections is a translation, so we can just translate the target twice the distance between the two walls.
Officially, this lesson is the Guided Notes for Lesson 6-4 of the U of Chicago text. But this lesson naturally lends itself into a group activity -- the teacher can provide additional golf courses for the students to solve, or even allow the students to make up their own via the Bonus Question. It is a nice activity to give right before the week-long Thanksgiving break.
The key is reflections -- one of the important transformations in Common Core Geometry. It is often said that when a ball bounces off a wall, the angle of incidence equals the angle of reflection. The text describes where to aim a golf ball G so that it bounces off of a wall and reaches the hole H:
"In this situation, a good strategy is to bounce (carom) the ball off a board, as shown [in the text]. To find where to aim the ball, reflect the hole H over line AB. If you shoot for image H', the ball will bounce off
We can write a two-column proof to show why the angle of incidence -- the angle at which the ball approaches the board, which is BPG -- equals the angle of reflection APH:
Given: H' is the reflection of H over line AB.
Prove: Angle APH = Angle BPG
Statements Reasons
1. H refl. over line AB is H' 1. Given
2. Angle APH = Angle APH' 2. Reflections preserve angle measure
3. Angle APH' = Angle BPG 3. Vertical Angle Theorem
4. Angle APH = Angle BPG 4. Transitive Property of Equality
Notice that for this proof, I've skipped a few steps. Technically, we should write that the reflection images of both A and P are the points themselves, since they lie on the mirror (Definition of Reflection), and so angle APH' is the image of APH (Figure Reflection Theorem). But I'm tired of writing that over and over again -- how much less, then, will the students want to write that.
Of course, this only works if the ball caroms only once. The U of Chicago text describes the game of billiards, where the player is required to bounce the ball off of three cushions. The text writes:
"Pictured [in text] is a table with cushions w, x, y, and z, the cue ball C, and another ball B. Suppose you want to shoot C off x, then y, then z, and finally hit B. Reflect the target B successfully over the sides in reverse order: first z, then y, then x. Shoot in the direction of B"' [...]
"Notice what happens with the shot. [...] On the way toward B"', it bounces off side x in the direction of B". On the way toward B", it bounces off y in the direction of B. Finally it hits z, and is reflected to B. [End of quote]"
In the video clip above, four caroms are required for the golf ball shot by Bart to find the hole. But unfortunately, the path of the ball as drawn on the show is impossible. To see why, let's label the direction from the starting triangle to the hole "North," and all the walls appear to meet at right angles, so they are all oriented in the north-south or east-west directions. Bart begins with the ball slightly to the right side of the starting triangle, so the initial direction of the ball is northwest. After hitting the first east-west wall, the ball is now traveling southwest. But then, after hitting the second east-west wall, the ball should be traveling northwest again. Indeed, we can use the Alternate Interior Angles Consequence and Test Theorems to prove that the path of the ball after hitting two walls should be parallel to the original direction of the ball. Yet the show depicts the ball as travelling due north after hitting two walls.
In fact, we can show that the only correct path to make the ball arrive in the hole involves hitting a north-south wall -- most likely the wall to the far west (in front of "Do not sit on statuary"). Only by hitting a north-south wall can the direction change from anything-west to anything-east, which is necessary for the ball to approach the hole. Lisa's advice to her brother is sound, but the way that it is animated is geometrically impossible.
For this worksheet, I decided to reproduce Bart's golf course, but tilt two of the walls in order for the path of the ball to be geometrically correct. I used equilateral triangle paper to create this page, so that the students will be able to figure out the paths without needing a ruler or protractor, for classrooms in which these are not available. Just as for Bart, four caroms are needed to get from G to H.
Working backwards on this worksheet, we can determine the path from A to H by reflecting H to the point H', then aiming from A to H'. But to determine a path from B to H using two caroms, we can't reflect H to first H' and then H", because H" would be well off the page. It may be better to aim from B to A', the reflection of A in the necessary wall.
Notice that if there are two walls and the paper is large enough, it may be actually possible to perform both of the necessary reflections. If the two walls meet at right angles, it is fairly easy to perform both reflections -- because the composite of those two reflections is a rotation. So to find the direction to aim at, we take the target point and rotate it twice 90, or 180, degrees around the point where the two walls meet. And if the ball is bouncing off of two parallel walls, then the composite of the two reflections is a translation, so we can just translate the target twice the distance between the two walls.
Officially, this lesson is the Guided Notes for Lesson 6-4 of the U of Chicago text. But this lesson naturally lends itself into a group activity -- the teacher can provide additional golf courses for the students to solve, or even allow the students to make up their own via the Bonus Question. It is a nice activity to give right before the week-long Thanksgiving break.
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