11:25 -- That's right, my day doesn't start until 11:25. There are several reasons. First of all, Mondays in this district are late days. Second, this teacher has only four classes -- periods 4-7 -- because her fifth class is an online computer science course. As a sub, I'm responsible only for the classes that are actually face-to-face.
Fourth period is an AP Computer Science Principles class. This is one of the newest AP tests, with the first administration of this test being just two years ago. There are 34 students in this class -- half of them sophomores, about a third juniors, and the rest seniors.
The AP exam was given three days ago, on Friday. Since this is the first day they're attending this class since taking the exam, I decide to ask them how they felt about the test. Many of them believe that they think they've passed the exam. For many of the sophomores, this is the first AP test they've ever taken, since other tests commonly taken in tenth grade (such as World History) aren't tested until this week, the second week of AP's. I warn the students that it's time to play the waiting game -- they won't get their scores until July or later.
So what are the students doing today now that they've completed the AP exam? Actually, they must take another test now -- the "mock AP" given as the final exam. Many AP teachers give their finals close to the AP rather than wait until finals week, since by then they may be starting to forget what they've learned. Today is Part 2 of the final -- presumably Part 1 was on Thursday (or earlier), as I doubt the teacher would give any part of the final on the same day as the AP itself.
12:05 -- Fourth period leaves and fifth period arrives. This is Exploring Computer Science. Many of these students are freshmen, perhaps preparing to take AP Comp Sci Principles next year.
These students have a choice to make -- either an Hour of Code or an hour of detention. This class actually has a co-teacher, and she helps to make sure that they are on task.
Both computer classes remind me of coding Mondays at the old charter school. Indeed, I can't help but notice that the very first year of the AP Comp Sci Principles exam was the same as the year I spent at the old charter. Thus I wonder whether the coding teacher's intention was to prepare our students to take the AP Comp Sci Principles exam in high school. My eighth graders that year would now be sophomores who might have attempted the AP exam last Friday.
And in fact, as I glance at the final exams and Hour of Code assignments, I see the connections to what the coding teacher tried to teach my eighth graders. Some of the fifth period students appear to be creating video games during their Hour of Code. And even though Java is the language of choice on the main AP Comp Sci exam, in Principles they work in several computer languages. I notice a test question that might have been written in a real language, but it could have been pseudocode (or even Hofstadter's BlooP).
But unfortunately, the coding class two years ago was not a success. And I suspect that part of the reason has to do with my classroom management problems -- even though there was a separate coding teacher, the classes were in my room. Thus many students misbehaved for the coding teacher more than they would have if I'd been a stronger teacher.
12:50 -- Fifth period leaves for lunch.
1:35 -- Sixth period arrives. This is the first of two freshman Algebra I classes.
These students are preparing for a test that's coming up this week. They are working in Chapter 9 of the Glencoe Algebra I text, which is on quadratic equations.
2:15 -- Sixth period leaves and seventh period arrives. This is the second of two Algebra I classes.
I'm ashamed to admit that I don't sing the Quadratic Weasel song today. Quadratic Weasel should be automatic in any algebra class that's studying second-degree equations. What throws me off is that the students can use any method to solve the equations since this is review, so I can never be sure when a student will choose the Quadratic Formula.
Perhaps I should have chosen one question in advance and tell them to use the Quadratic Formula to solve it. Then this would give me a reason to sing the song. It's possible that the song might help them remember the formula for this week's test.
3:00 -- Seventh period leaves, thus ending my day.
The main classroom management issue today is in sixth period, as several students keep talking loudly during the review. But once again, this is where Quadratic Weasel comes in handy -- I could have sung the song once for each problem that we use the Quadratic Formula. More problems completed means more renditions of the song. This would have been an incentive for the students to remain quiet so we can get through more problems.
Sometimes I wonder whether singing songs might become a management crutch -- can I get students to behave without promising them a song? But when a lesson has a song naturally built in to the lesson, such as Quadratic Weasel, then why not sing it?
You might notice that today's post has been labeled "traditionalists." Yes, our main traditionalists were active over the weekend. The topic is state testing -- ubiquitous at this time of year. Since this blog is already in SBAC Prep mode anyway, I might as well cover the traditionalists today.
https://traditionalmath.wordpress.com/2019/05/11/principal-gladhand-dept-coexistence-in-the-land-of-oz-and-kansas/
Barry Garelick:
With the Common Core annual testing coming up in California, Principal Gladhand’s weekly missive to parents brings good tidings about how his school is dealing with it.
He starts with the age-old premise that tests don’t matter.
First of all, I'm having trouble figuring out who this "Principal Gladhand" is. Is this (a pseudonym for) Garelick's own principal? On one hand, Garelick is in California while this Gladhand appears to be from Kansas. But judging from the rest of Garelick's post, "Kansas" may be an allusion to The Wizard of Oz, since he mentions "Oz" later in the post.And unfortunately, Garelick suddenly jumps from The Wizard of Oz to 1984. I might have understood his post better if he had stuck to either novel rather than switching allusions.
Here Garelick quotes "Gladhand":
We work hard to ensure our students are learning not because we want them to do well on any given test, but because the learning is important. We don’t
work to “teach to the test” or take excessive practice tests to “get our students used to” taking standardized tests. This isn’t teaching for mastery, it’s just teaching about testing.”
It appears that Garelick is accusing Gladhand of contradicting himself. First, Gladhand says that standardized tests aren't important (presumably from September to April) and then, once May comes, suddenly the Common Core tests matter to him. From Garelick's perspective, the only thing that Gladhand consistently accepts is that memorization is bad.
One major traditionalist, Ze'ev Wurman, leaves a short comment about Gladhand:
Ze'ev Wurman:
I have but my standard comment: Idiots will be idiots.
But as usual, the main commenter is SteveH:
SteveH:
“Please encourage your child to try hard each day, eat a good breakfast, and get plenty of sleep this week!”
… and ensure the mastery we do not enforce, even at the Common Core non-STEM level …. but don’t tell us because we like our dream world. If we wait long enough, then even the students will blame themselves.
Oh, and since when do they need to wait for a yearly state test to get feedback on their methods?
Here, SteveH presumably means that "they" (Gladhand and other anti-traditionalists) make several assumptions about the best way to teach students without proof (such as state test scores). The underlying contention by SteveH is that traditional methods -- including memorization -- lead to higher test scores by far.Once again, the traditionalists overlook the students' attitude towards memorization. For example, when Garelick writes:
... in this digital age, we can just Google information ...
many students believe that, and many students will refuse to memorize, and many students will give Google as a reason not to memorize. All Principal Gladhand wants to do is find a way to get such students to do something rather than nothing in their classes.
But instead, as usual, the traditionalists assume that students would gladly memorize anything we ask them to, and once they memorize they will start earning higher test scores. To them, the only thing blocking the students from their beloved memorization is the attitude of this Principal Gladhand.
Notice that I do agree with traditionalists that students would be better off if they at least memorize their times tables. Given this, what should Gladhand say to third graders who don't want to memorize them -- and would be traditionalist-approved? Perhaps something like this:
Students, I know that you don't want to memorize your times tables. But if you memorize them, you will do better in your math classes this year, next year, and through middle school. You are smart enough to learn them and not have to rely on Google or a calculator. You might have to work a little, but you WILL know them all. You'll be able to join the mathematical majority, a world where most people are smart enough to KNOW their times tables. With so many smart people in the world, we will be able to build great things such as new video games, and even colonies on the moon!
Speaking of standardized testing, one thing of concern is the Islamic holiday of Ramadan. This is a purely lunar holiday that doesn't coincide with any season of the year. Thus it falls slightly earlier each year on the solar Gregorian Calendar. For most of this decade Ramadan fell in the summer when schools are closed, but now it's moving into spring -- during state testing season. Many teachers are concerned that their Muslim students won't be able to concentrate on getting correct answers on the SBAC when they are so hungry from the Ramadan fast.
The current controversy is in Seattle:
https://thetakeout.com/seattle-school-muslim-ramadan-fast-standardized-test-1833295368
I've added the "Calendar" label to this post, since it's a good time to discuss solar vs. lunar calendars and how religious observances affect the school calendar. (On this blog, I try to limit discussion of religion to the "Calendar"-labeled posts.)
Before we begin, keep in mind that I am not a Muslim. Therefore it's not my place to tell adherents of Islam what their calendar should be like. Here I only wish to discuss calendars and whether there can be a solution to this calendar controversy.
In the link above, we see the following lines:
“This is 100 percent the same thing as making children take their test on Christmas or Easter,” someone commented on Facebook, according to Crosscut.
But here's the problem -- Christmas is always on December 25th. It's easy to avoid making children take a standardized test on Christmas -- just don't give the SBAC at all in December. Meanwhile, Easter admittedly is closer to testing season, and like Ramadan, Easter depends on the moon. But Easter is a lunisolar holiday, so at least it's always predictably in March and April. We see that middle schools wait until May to test. And even at districts such as my old district where spring break has nothing to do with Easter, there will never be an SBAC on Easter because it's a Sunday.
But Ramadan lasts a full month, and so it contains every day of the week. And because it's purely lunar, it can fall during any season. In some years it falls in the spring during testing season, while in other years it's in the fall when there's no testing, and in other years it's in the summer when there's no school, much less any standardized testing.
The problem is that it's inconvenient for users of a solar calendar such as the Gregorian Calendar to accommodate a purely lunar holiday such as Ramadan. Lunisolar holidays are also inconvenient (which is my many districts no longer tie spring break to Easter), but clearly not as much as purely lunar holidays.
Why is the Islamic Calendar purely lunar, unlike the lunisolar Hebrew or Chinese Calendars? It's all because of an interpretation of a verse of the Koran:
(Disclaimer: The author of the link below is a former Muslim.)
http://abdullahsameer.com/blog/the-flawed-islamic-calendar/
Time has completed a cycle and assumed the form of the day when Allah created the heavens and the earth. The year contains twelve months of which four are sacred, three of them consecutive, viz. Dhul-Qa’dah, Dhul-Hijjah and Muharram and also Rajab of Mudar which comes between Jumadah and Sha’ban.
The three consecutive months referred to are the eleventh, twelfth, and first months of the lunar Islamic Calendar. If a Leap Month were to be inserted at the end of the year (to make it lunisolar), then the three holy months would no longer be consecutive. Thus there must be no Leap Months.
It would be convenient for solar calendar users if the Islamic Calendar were lunisolar. But the Islamic Calendar is for Muslims, not for non-Muslim solar calendar users. Once again, it's not up to those who aren't Muslim to dictate what the Islamic Calendar should be like.
But sometimes I wonder, what if there could be a World Peace Calendar that could incorporate all of the major calendars of the world? Surely it would be a lunisolar calendar, since it incorporates both solar and lunar calendars that exist in the world.
Theoretically, every group would have to make some concession to use this calendar. Christians, for example, already have a lunisolar holiday in Easter, but now Christmas would have to become lunisolar as well, as well as all other holidays not currently tied to Easter (saints' days, and so on).
The main accommodation Muslims would need to make is the addition of Leap Months. And it's still possible to do so without violating the Koran verse quoted above. It only states that the 11th, 12th, and first months be consecutive. If we were to add the Leap Month after, say, the tenth month instead, then the three holy months would still be consecutive.
One calendar I once considered using as the World Peace Calendar is the Meyer-Palmen Calendar:
https://www.hermetic.ch/cal_stud/nlsc/nlsc.htm
Many world holidays already fit readily into this calendar. Since the first month (Aristarchus) falls near the spring equinox, this is the month to place both Easter and the Jewish Passover. Then Rosh Hashanah would end up on the first of Galileo. It turns out that Chinese New Year should be on the first of Lilius, since this date always falls between late January and mid-February.
For Muslims, I like the idea of simply declaring the ninth month to be Ramadan (as it already is in the Islamic Calendar). Notice that this month already has an Arabic name -- Ibrahim. Peter Meyer tells us he named the month for an Arabic astronomer -- but it can also represent "Abraham," as in the three Abrahamic religions. This month falls near the winter solstice -- which would be convenient for Muslims, since they'd have a shorter fasting period (in the hemisphere where most Muslims live).
In fact, all three Abrahamic faiths would celebrate a holiday in Ibrahim. For Jews this would be Hanukkah, while Christians can place Christmas in Ibrahim (maybe even on the 25th, so as to agree with the current calendar). Then the twelve days of Christmas and eight days of Hanukkah would extend towards the first of Julius, which would replace both the Gregorian (or Julian!) New Year as well as the end of Ramadan (Eid-al-Fitr). And there would be no standardized testing at this time, since all three Abrahamic faiths are celebrating a major holiday.
But this solution isn't perfect. The Leap Month, Meton, is the thirteenth month. This would violate the Koran's three consecutive holy months. We could instead declare Khayyam to be Leap Month so that Lilius, Meton, and Aristarchus are the three holy months. Notice that Jews would probably declare Lilius to be the Leap Month instead -- this reflects the fact that the Jewish Leap Month is considered to be Adar I, with Purim in Adar II. (Oh, and so as not to leave the calendar of India out, Holi can coincide with Purim, as it often already does.)
Also, we made Ibrahim the fasting month so that Muslims can enjoy a shorter fast. But there are Muslims in the Southern Hemisphere. (Unfortunately, we've heard about the recent tragedy that struck Muslims in Christchurch, New Zealand.) They might not appreciate a calendar where Ramadan is stuck near their summer solstice!
We might try placing the first month near the September equinox instead, so that Ramadan falls near the northern solstice instead. This gives a shorter fast in the south. Some say that a northern summer Ramadan more authentically reflects a pre-Islamic Calendar (Arabic ramada = "to heat"). But then the twelfth month of the hajj would fall in late summer, when it's hot in Mecca. (Since hajj is the pilgrimage to Mecca, only the hemisphere of Mecca matters here.) Our original calendar with the first month at Aristarchus places the pilgrimage in Mecca's late winter.
Of course, we can avoid hemisphere debates by placing Ramadan near an equinox instead. At any rate, I like the idea of the first, second, etc., months in the World Peace Calendar corresponding to the Muslim months -- since they must already concede something (the nonexistence of Leap Months), let's at least accommodate their month numbering. This includes the need for the 11th, 12th, and first months to be consecutive holy months.
Consider David's Lunar Calendar from my January 6th post. This calendar starts near the winter solstice, and so it places its ninth month (Ramadan) near the September equinox. The twelfth month (hajj) becomes late fall and Leap Month is between the seventh and eighth months, so that the 11th, 12th, and first months are still three consecutive holy months. Since Ramadan is near the first day of school, there would be no standardized testing during Ramadan.
If we were to make David's Lunar Calendar the World Peace Calendar though, we should probably leave out the 12-day week and just keep the current seven-day week. The intent of this calendar was to accommodate religions for which the seven-day week matters. (Sorry base 12 advocates, but dozenalism isn't considered to be a religion.)
Finally, note that there are objections to any lunar calendar at all:
(x) the solar year cannot be evenly divided into lunar monthsI once read that there may be a very good reason to adopt a lunar calendar -- what if, in the future, we were to build colonies on the moon? If and when that ever happens, lunar months would suddenly become significant. Future lunar colonists could then say:
(x) lunar months are real and the calendar year needs to sync with themsince on the moon, lunar months are very real.
That takes us full back to traditionalists. Before we can build lunar colonies, we need to graduate more STEM students to build them. Of the classes I subbed for today, the Algebra I freshmen aren't bound for STEM, at least not according to SteveH (who wants Algebra I in eighth grade). The only class today which the traditionalists would respect is the AP computer class.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
Shown are a white and gray square and a black circle. The 4 right triangles are congruent. The black circle's area is 12.25pi square units. Find the length of the side of the white square.
[Once again, much of the information is given in an unlabeled diagram, so let me label it. The white square is ABCD, the gray square is EFGH, and the circle is Circle O. Square EFGH circumscribes O, and ABCD is also centered at O. Also, A-E-F (E is between A and F), B-F-G, C-G-H, and D-H-E, and finally, we have the length AE = 5.]
Since the area of the black circle of 12.25pi, we can find its radius and diameter:
12.25pi = pi r^2
12.25 = r^2
r = 3.5
2r = 7
Notice that the diameter of the circle equals the side of the gray square, and so EF = 7. Since AE = 5 and A-E-F, we conclude AF = 12. Also, Triangles ABF and DAE are congruent and AE = 5, and so we conclude that BF = 5.
Thus the desired length of the white square, AB, is the hypotenuse of a right triangle ABF whose legs are BF = 5 and AF = 12. So we can use the Pythagorean Theorem:
a^2 + b^2 = c^2
5^2 + 12^2 = c^2
25 + 144 = c^2
169 = c^2
c = 13
Therefore the desired side length is 13 -- and of course, today's date is the thirteenth.
Speaking of dates, lately there have been some medicine-related Google Doodles. I usually highlight all STEM-related Google Doodles, so once again, it all depends on whether medicine is considered to be part of STEM. The two medicine-related Doodles are for Lucy Wills and George Papanikolaou.
This is what I wrote last year about today's lesson:
Question 9 of the SBAC Practice Exam is on irrational numbers:
Cheryl claims that any irrational number squared will result in a rational number.
Part A
Drag an irrational number into the first response box that when squared will result in a rational number.
Part B
Drag an irrational number into the second response box that when squared will result in an irrational number.
Here are irrationals that can be dragged: cbrt(2)/sqrt(3), sqrt(3)/sqrt(2), cbrt(2), sqrt(2), pi, sqrt(pi).
This is a tricky one to place. In the Common Core Standards, rational and irrational numbers appear in the eighth grade, and so it's arguably not a high school topic at all. If they do appear in an Algebra I text, it's likely to be in the context of quadratic equations, thus it's a second semester topic.
It's also the first question in the calculator section. Then again, calculators won't really help students with rational and irrational numbers.
Cheryl's idea that the square of an irrational number must be rational is an attractive one. After all, the first irrational numbers we encounter are numbers like sqrt(3), which when squared produces the rational number 3.
But, as Georg Cantor shows us (discussed in old posts), most numbers are irrational. Therefore, the square of most irrational numbers is still irrational. The list of choices includes cube roots, so their squares are still irrational. In fact, Cheryl's conjecture that the square of an irrational must be rational is just like a claim that doubling any fraction produces a whole number (presumably because the most commonly used fractions like 1/2 and 1 1/2 indeed have that property).
The choices involving pi are tricky. We know that the square of sqrt(pi) is pi, which is clearly irrational, and so sqrt(pi) would be dragged into the second box. But to which box should the number pi itself be dragged. I doubt that any high school text explains that pi^2 is irrational. We know that pi is transcendental, and so no integer power of pi can be rational -- but high school students wouldn't be expected to know this. Of course, students can forget about pi and just drag sqrt(pi) or one of the cube roots into the second box, since only one number needs to be dragged there.
So here is a complete answer: only sqrt(3)/sqrt(2) or sqrt(2) can be dragged into the first box. All other numbers are possibilities for the second box.
Both the girl and the guy from the Pre-Calc class correctly answer this question, even though they choose different answers. The girl uses sqrt(3)/sqrt(2) for the first box and cbrt(2) for the second, while the guy uses sqrt(2) for the first box and pi for the second.
Question 10 of the SBAC Practice Exam is on building equations:
A train travels 250 miles at a constant speed (x), in miles per hour.
Enter an equation that can be used to find the speed of the train, if the time to travel 250 miles is 5 hours.
The guiding equation is d = rt, rate times time equals distance. The rate of speed is x, the time is 5, and the distance is 250. Therefore the equation is 5x = 250.
I consider this to be a first-semester Algebra I problem. While we might avoid the formula in middle school (or perhaps even mention dimensional analysis), by the time the students reach Algebra I, I should teach them the formula, the guiding equation.
Both the girl and the guy from the Pre-Calc class correctly answer this question, but I wonder whether the SBAC computer would mark them as correct. The guy writes 250 = x * 5 (and gives the solution as 50 mph). This is correct, but mathematicians usually write 5x not x * 5. The girl writes her equation as s = 250/5. Technically this equation can be used to find the speed, but I suspect SBAC is looking for a multiplication equation, not division with the variable isolated. (Actually, now I definitely think SBAC will mark it as wrong because she uses the variable s when the directions plainly state to use the variable x.)
SBAC Practice Exam Question 9
Common Core Standard:
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
SBAC Practice Exam Question 10
Common Core Standard:
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
Commentary: Lesson 12-6 of the U of Chicago Algebra I text is on Rational and Irrational Numbers, and the Exploration Question there is on rational solutions of quadratics. Lesson 4-4 of the U of Chicago text is on Solving ax = b, with d = rt mentioned as an example. Students should have no problem with this question if they know the guiding equation.
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