Quiz #2 (Day 34)

There are several things that I wish to discuss in this post. First of all, the first Saturday in October marks the biannual book sale at my local library. Recall that last year, I purchased Mandelbrot's book on Fractal Geometry from the October book sale.

So what did I purchase at the book sale this year? Well, I always like to purchase at least one textbook on a subject or grade level that I want to teach. Since I'm now working at a middle school, of course I wanted a middle school text. And the book I got was Saxon's Algebra 1/2, third edition.

This marks the second time I've bought a Saxon text -- the first was Math 65, second edition. The Saxon texts lie right at the heart of the battle between traditionalists and progressives. Now that I am a full-time teacher, I don't wish to spend as much time discussing the traditionalist debates. But in today's post, I will bring up a little bit of the debate since I just purchased another Saxon text.

The Saxon texts are popular with traditionalists. Integrated math texts, on the other hand, are unpopular with traditionalists. The irony here is that the Saxon texts for high school math are -- you guessed it -- integrated, long before Common Core! So far, I've yet to see a traditionalist satisfactorily explain why he or she likes Saxon integrated math but not Common Core integrated math. Notice that the fourth editions of Saxon are not integrated, but the third editions are -- and it was a third edition that I purchased at the book sale.

One major concern of traditionalists is that students make it to AP Calculus by senior year, which usually means eighth grade Algebra I. But notice that the Saxon Algebra 1/2 text -- which is basically a Pre-Algebra text -- can nonetheless be given in eighth grade. Then freshmen can take Saxon Algebra 1, sophomores Saxon Algebra 2, and juniors Saxon Advanced Mathematics, as these are integrated texts that include Geometry. This does prepare students for senior-year Calculus.

I won't type out the entire contents of Saxon Algebra 1/2, but let's at least look at the lessons that are multiples of five:

5. Addition and Subtraction Word Problems
10. Divisibility
15. Fractions and Decimals
20. Multiples
25. Area as a Difference
30. Adding and Subtracting Fractions
35. Subtracting Mixed Numbers
40. Reciprocals
45. Volume
50. Scientific Notation
55. Fractions, Decimals, and Percents
60. Circles
65. Proportions with Mixed Numbers
70. Rules for Addition of Signed Numbers
75. Implied Ratios
80. Increases in Percent
85. Equation of a Line
90. Algebraic Sentences
95. Variables on Both Sides
100. Advanced Ratio Problems
105. Evaluating Powers of Negative Bases
110. Markup and Markdown
115. Polygons, Congruence, and Transformation
120. Volume of Pyramids, Cones, and Spheres

Notice that Saxon Algebra 1/2 is approximately equivalent to Common Core 8. Indeed, there are many topics in Common Core 8 missing in Saxon Algebra 1/2. We see that Lesson 115 mentions transformations, but of course Saxon, like most pre-Core texts, doesn't cover transformations as extensively as Common Core. But more surprisingly, slope doesn't appear in Saxon Algebra 1/2 -- actually, there is an appendix covering both transformational geometry and slope as well as several other additional topics. But the main text omits slope. Systems of equations is yet another topic that Common Core 8 expects students to learn, but not Saxon Algebra 1/2.

We see that eighth graders can study from Saxon 1/2 -- a text less rigorous than Common Core 8 -- and yet make it to senior-year Calculus. This is part of why I think that students can proceed directly from Common Core 8 to Integrated Math II as freshmen, en route to AP Calculus.

I actually purchased one more math text, but I must admit that my reason for purchasing this book is pure nostalgia. I bought Mathematics Around the Clock, published in 1970. This text is considered part of the "California State Series," even though this was well before California State Standards.

The subject of this text is "clock arithmetic," more formally called "modular arithmetic." Here is a link discussing modular arithmetic:

https://nrich.maths.org/4350

There is no grade level associated with this text. The above link labels this as a "Stage 4" topic, which corresponds roughly to junior and senior years in high school, but I suspect that this text can be used with much younger students (though probably not younger than fourth grade, as these kids are still learning ordinary arithmetic). In fact, I was given a copy of this text when I was younger -- I no longer own that copy, which is why I said that I bought the book for nostalgic reasons.

Even though the above link describes what clock/modular arithmetic is, there's actually a song from Square One TV that discusses clock arithmetic:

The text begins with the modulo twelve system, just like the clock. It then moves on to modulo 7, 5, 4, 6, and others. Modular systems are examples of rings (as I mentioned in spring 2015 when reading another book), and prime modular systems are examples of fields. The words "ring" and "field" don't appear in the text, although the properties these satisfy (closure, commutative, etc.) do appear.

On page 39 of the text, I see the following section header:

"SOLVING NUMBER SENTENCES"

What the -- I heard from traditionalists that "number sentences" is a phrase invented by the Common Core, so how can it appear in a text written 40 years before the Core? Actually, this text usually uses the phrases "addition sentences" or "multiplication sentences" -- "number sentences" is an umbrella term covering both. The Common Core "number sentences" cover both equations and inequalities. I point out that inequalities don't make sense in modular arithmetic. Despite this, the clock arithmetic text doesn't use the word "equation" -- perhaps since it's awkward to call "9 + 8 = 5" an equation, even in mod 12 (or even more to the point, 9 + 8 = N).

Even though modular arithmetic is associated with the clock, on page 71, the modulo seven system is connected to the calendar. Notice that many of the algorithms I mentioned last week for calculating Rosh Hashanah and other holidays are examples of modular arithmetic -- for example, Conway Doomsday is done in mod 7, while Golden Numbers are an example of a modulo 19 system.

On page 86, there is a discussion of exponents in modular systems. There is a series of exercises that leads to the following theorem:

If N is any number in a prime modular system and N is not equal to zero, then N^(P - 1) = 1, where P is the prime number that is the modulus of the system.

This is actually a famous theorem -- even though the text doesn't call it this, it is actually known as Fermat's Little Theorem, named for the 17th century mathematician Pierre de Fermat. (I mentioned Fermat's Last Theorem in another post last year.)

When I first read about Fermat's Little Theorem in this book so many years ago, I knew how hard it is to determine whether or not a number is prime. I wondered to myself whether this theorem could be used to determine primality -- until I told myself, "Don't be silly -- of course we can't use this theorem, since raising numbers to powers is more difficult than trial division."

As it turns out, the first instinct of my young self was correct. The dominant method of determining whether a large number is prime is the Fermat method! As it turns out, exponentiation in a modular system is quite simple, especially if one uses repeated squaring -- for example, squaring a number four times is equivalent to raising it to the 16th power. With trial division, we can't prove a number is composite unless we stumble upon a factor and divide by it, but with the Fermat method, we could just plug in N = 2 into the formula, get a value other than 1, and instantly know that the number we're testing is composite.

The real problem with the Fermat method is the existence of Carmichael numbers -- numbers which are composite, yet the formula produces 1 for almost every value of N anyway. Modern primality tests usually begin with trial division for a few small factors, then the Fermat method, and then finally a more powerful method such as Miller-Rabin to test against Carmichael numbers.

Yesterday there was a Google Doodle to celebrate the anniversary of the Gregorian Calendar -- the calendar with which we are the most familiar. Actually, I'd argue that October 4th, 1582, was the last day of the old style Julian Calendar, as the first day of the new style calendar was October 15th. I would have posted about the Google Doodle yesterday, except that yesterday was my scheduled day off from blogging (as it was Day 33, and 33 is in the 0-class mod 3).

Oh, and by the way, not only was it Jewish New Year, but it was Islamic New Year as well. Notice that the Islamic Calendar is a pure lunar calendar -- each year is twelve lunar months, or about 354 days, with no attempt to harmonize with the solar calendar. For three consecutive years, the Jewish and Islamic New Years will be at the same time, until 2019 when the Jewish Calendar adds a leap month, but not the Islamic Calendar.

By the way, New York school system not only takes off the Jewish holidays also but also two major Islamic holidays. The Islamic New Year isn't one of them, and so it's not a day off in New York unless it coincides with the Jewish, or even (in over a decade, since there's no leap months) the Chinese New Year.

With all of this talk about calendars, yesterday I told my students about the Google Doodle and mentioned calendars as part of the review for today's quiz. It was a science quiz about the movements of the earth, moon, and sun.

Today I give today's quiz to all three grades. As I mentioned last week, the Wednesday schedule has been modified slightly. Now I see each of the three grades for one hour until nutrition, and then the seventh graders return after break for music.

Actually, the results of today's quiz are mixed. There are perfect scores in sixth and seventh, but not eighth grade. But eighth grade had relatively more students actually pass the quiz. The student with the lowest math grade ended up passing the quiz, while one of my harder working math students got only one question correct. Oh well -- at least I see my students' relative strengths in math and science.

Here is the song I sang yesterday while reviewing for the quiz. No, it wasn't Square One TV's "Time Keeper," since that has to do with a book I bought, not anything I taught in class:

EARTH, MOON, AND SUN

I know of how the earth goes,
It revolves around the sun every year.
I know of how the earth goes,
It revolves around the sun every year.
The earth's tilt is the reason,
That we have four seasons.
Winter, spring, summer, and fall,
That is all.
In the north, remember,
Winter's in December.
Tilts toward the sun in June,
It'll be summer soon.
I know of how the earth goes,
I know of how the earth goes,
I know of how the earth goes,
It revolves around the sun every year.

I know of how the moon goes,
It revolves around the earth every month.
I know of how the earth goes,
It revolves around the sun every year.
That's why every 30 days,
We can see every phase.
New moon, waxing crescent,
Half moon, waxing gibbous.
That's why every 30 days,
We can see every phase,
Full moon, waning gibbous,
Half moon, waning crescent.
I know of how the moon goes,
I know of how the moon goes,
I know of how the moon goes,
It revolves around the earth every month.

Note: In honor of Rosh Hashanah, this song is sung to the tune of the Hebrew song Hava Nagila.

Meanwhile, let's get back to the "Day in the Life" project. Today is the 5th, and so the monthly poster for the fifth is James Cleveland:

https://rootsoftheequation.wordpress.com/

James Cleveland is a New York high school -- teacher? Unfortunately, I can't tell from his post whether or not he's teaching this year. He hasn't made his October 5th post yet, and his September 5th post was on Labor Day. He did write a post for the next day, which he labeled as the "First Day" (of school), but it's apparently more like a PD day, since he spent the entire day programming students for the new school year. Notice that New York is one of the few regions in the country where the first day of school is still after Labor Day.

So Cleveland might be a teacher who had to do office work before students arrived, but I can't be sure until he posts again. I was hoping that he might have caught up blogging on Monday, since after all, New York had a three-day weekend for Rosh Hashanah as well. All I can do is wait and see.

Despite the lack of "Day in the Life" teaching posts on Cleveland's blog, I do enjoy one of his recent blog posts. At the following link, he writes about the AP Calculus curriculum:

https://rootsoftheequation.wordpress.com/2016/09/05/integration-first/

Readers of my blog know that I devote many posts describing the order in which math -- especially Geometry -- should be taught. Well, here Cleveland writes about why AP Calculus students should be taught integration before differentiation:

In general, I feel like area is a much more approachable subject than slope. My years of teaching Algebra I to 9th graders certainly seems to support that claim. But I also think it’s easier to understand the linearity of integration than the linearity of slope. “If you add together two functions, the area under the new function is the sum of the areas under the old functions” seems much more evidently true than “If you add together two functions, the slope of the tangent line for each point of the new function is equal to the sum of the slopes of the tangent lines at the same points on the old functions.”

I am not a Calculus teacher, of course. Still, I've discussed some of these same ideas in thinking about the Geometry curriculum. Which should we teach first: area or similarity (which leads to slope as per the Common Core)? The Geometry text we've been following for the first two years of this blog gives area first, but I rearranged it to similarity first so that we can get to slope faster. Nonetheless, I mentioned how a few proofs that use similarity (leading to slope) can be rewritten using area instead, which might be easier for students. After all, as Cleveland writes, "area is a much more approachable subject than slope." (Again, notice that Saxon Algebra 1/2 contains lessons on area, but not slope!)

Of course, I'm not really a Geometry teacher either. But, as we know, the middle school curriculum (especially Common Core 8) has a strong geometry component. So this is something I should be thinking about before my eighth grade class reaches the geometry unit.

Oh, and speaking of Geometry, let me conclude this post with a link to Sarah Carter's blog:

http://mathequalslove.blogspot.com/2016/10/guest-post-composition-of.html

I didn't actually teach a science lesson from Carter's website today -- in fact, if this new Wednesday schedule holds, I won't have an extra period for eighth grade science (as a trade-off, I get to see my seventh graders now).

Then again, this post isn't actually Carter's. It's a "guest post" from an anonymous middle school teacher in Washington state -- and I'm always looking for middle school bloggers! But this is actually an advanced eighth grade class taking high school Geometry. In this post, the teacher explains how she teaches transformations to her class, using Foldable notes.

So I have plenty to think about from these links.

Science: The Earth-Moon-Sun System (Days 32-33)

This upcoming Monday is Rosh Hashanah, the Jewish new year. For decades, the LAUSD has observed both Rosh Hashanah and Yom Kippur as holidays. Thus my charter school, which mostly follows the LAUSD calendar, is closed both Monday, October 3rd and Wednesday, October 12th.

I've decided that this is a great time to teach a science lesson about the earth, moon, and sun. I told my students about the earth's revolution around the sun, which leads to the seasons of winter, spring, summer, and fall. Then I moved on to the moon's revolution around the earth, which leads to the phases of new moon, waxing crescent, first quarter, waxing gibbous, full moon, waning gibbous, last quarter, and waning crescent.

Then I told my students about the calendar used by the ancient Hebrews. On this calendar, which is still used today by the Jews, the new month begins at the new moon and the new year begins near the fall equinox. I said that calculating the date of Rosh Hashanah is quite complicated, but I did show them the following link:

http://quasar.as.utexas.edu/BillInfo/ReligiousCalendars.html

(I've mentioned this link in previous blog spots when trying to explain the Easter date.)

A simplified formula for the date of Rosh Hashanah on the Gregorian calendar for 1900-2099 is gotten by calculating

N + fraction = 6.057778996 + 1.554241797*Remainder(12G|19) + 0.25*Remainder(y|4) - 0.003177794*y,

where y=Y-1900. Use the same postponement rules (note that 23269/25920=0.898, and 1367/2160=0.633). This method is easier to calculate using a pocket calculator.

I showed them the calculation for this year. Here y = 116 and the Golden Number G is 3, since 114 is a multiple of 19. The first division is 36 (12 times 3) divided by 19, which has remainder 17, and the second is 116 divided by 4, which has remainder 0. So the calculation is:

6.057778996 + 1.554241797*17 + 0.25*0 - 0.003177794*116 = 32.57204557

This gives the date as September 32nd, which really means October 2nd. But October 2nd is a Sunday, and the rules given above state that Rosh Hashanah can't fall on Sunday. (The reason for this is indirectly related to the fact that Saturday is the Jewish Sabbath. Neither Rosh Hashanah nor Yom Kippur can fall on the day before or after the Sabbath.) Thus  the holiday is on Monday, October 3rd.

Then I showed the students the calculation for next year. Some students guessed that Rosh Hashanah will fall on a Tuesday next year, since October 3rd next year will be a Tuesday (that is, in analogy with their birthdays). But let's see -- the Golden Number will be 4, and the first division is 48 (12 times 4) divided by 19, which has remainder 10, and the second is 117 divided by 4, which has remainder 1. So the calculation is:

6.057778996 + 1.554241797*10 + 0.25*0 - 0.003177794*117 = 21.69314744

September 21st, 2017 is a Thursday, and so no postponement is required. Next year, Rosh Hashanah will indeed be on that Thursday, not Tuesday. (Notice that Yom Kippur will fall on a Saturday. When either high holiday falls on the Sabbath, the LAUSD does not take an extra day off.)

But how do we know that September 21st next year will be a Thursday? According to the link, the simplified Rosh Hashanah formula shown above is attributable to the British mathematician John Conway, whom I've mentioned on the blog before. And so that means Conway Doomsday.

I told my students the Conway Doomsday formula. I figured they'd get a kick out of finding out what day of the week they were born. Interestingly enough, the first birthdays I randomly selected ended up being in either June or October, where 6/6 and 10/10 are easily identified as Doomsday. But one sixth grader gave me a January birthday, which is the hardest month in the Doomsday formula.

The earth-moon-sun system is included in the Next Generation standards for eighth grade. And so after this lesson, I give my eighth graders the online science assignment for this system.

This is a Calendar-labeled blog post due to the mention of a religious holiday. I didn't tell my students the following, but it may be instructive to blog readers to figure out where all the magic numbers in the Conway formula come from:

N + fraction = 6.057778996 + 1.554241797*Remainder(12G|19) + 0.25*Remainder(y|4) - 0.003177794*y

Here are my best guesses:

-- 6.05777... means that the earliest Rosh Hashanah is September 6th. Actually, it could be on the 5th due to the minus term later in the formula (and it did fall on September 5th, 2013 -- the early Rosh Hashanah that led to "Thanksgivukkah").

-- 19 refers to the 19 years of the Metonic cycle. The significance of the 19-year cycle is that new moons link up with solar years approximately 19 years apart. For example, I was born under a new moon, and there was a new moon on my 19th birthday as well.

-- 1.55424... doesn't mean anything on its own, but 1.55424...*19 = 29.53059..., which is about the length of a lunar month.

-- Likewise, 12 doesn't mean anything on its own, but 1.55424... * 12 = 18.65090... If there is a new moon one year on September 1st, there will be a new moon the following year about 18.65 days later, which is September 19th or 20th.

-- The numbers 0.25 and 4 obviously refer to Leap Days.

-- 0.003177794 is a slight adjustment to the Metonic cycle. The author at the above link writes that the 19-year Metonic cycle isn't exact, and so this term gives a slight adjustment (though it's still not as accurate as the Gregorian calendar).

Originally, I wanted to give this lesson today, since it's the last day before the three-day weekend caused by the Rosh Hashanah holiday. But there were continued problems with the mousetrap cars, and in fact I actually gave this lesson yesterday (even though I waited until today to post it here on the blog). This gave me an extra chance to figure out how to make the cars work before having my students complete the cars today.

The problem is that the launching string is so hard to wrap around the axle (a bit like wrapping spaghetti around a fork). It is just like setting up dominoes, where placing them too close risks making them fall before you're ready, and placing them too far apart risks having them remain standing when you are ready to let them fall. If the string is loaded improperly, the mousetrap is released too early or too late for the car to move.

I don't like how difficult it is to launch the cars. The reason for the project title "What's the Best Advantage?" is that students are supposed to make changes such as adjusting the size of the wheels and seeing what effect this has on the distance the car travels. But it is so hard to make the car go even once that neither the students nor I really want to perform that experiment. If Illinois State, who provided us with the cars, really wanted students to make this investigation, then the effort required to load the string should have been trivial.

But I am able to salvage some science out of this project. I have the students consider whether the cars will travel farther on the indoor carpet or on a smoother surface, such as the tiles located near the sink or the corridor outside the room. The students figure out that the car travels a shorter distance on the carpet due to greater friction.

The seventh graders are the most successful in launching the cars. The sixth graders are too confused, and some of the eighth graders are just lazy. We take videos of some of the seventh and eighth graders who race their mousetrap cars, but again I don't post the videos on the blog in order to maintain student privacy.

Thinking about the "Day in the Life" project, there technically is a participant whose monthly posting day is the 30th -- Kevin Cormier, who is apparently a Massachusetts middle school teacher. But unfortunately, the list of participating teachers provides only Cormier's email address, not his blog. It is a shame, though, since I enjoy reading the blogs of fellow middle school teachers. (Naturally, there is no participant for the 31st, since not every month has a 31st day.)

I wish everyone who celebrates it a happy Rosh Hashanah! There will be no blog post on Monday due to the holiday, and there is no post on Tuesday as this will be my scheduled day off (since the remainder of 33 divided by 3 is zero). So the next post will be Wednesday, October 5th.

What's the Best Advantage? Continued (Day 31)

Well, here are a few things I want to say about the new mousetrap project thus far:

First of all, it turns out that the music teacher was absent on Wednesday. Therefore I was able to perform the project with all three grades as well as have an extra period with my eighth graders for the online assignment on forces and motion. This still, of course, doesn't clear up the confusion regarding the new music schedule.

Students were confused with how to attach the string to launch the mousetraps. And so I'm definitely extending the project an extra day.

The eighth graders were also having trouble trying to figure out the online assignment on Newton's Laws. I think part of the problem is that physical science -- the main topic of the upcoming NGSS test -- is always more obscure than life science. Some students in all grades wonder whether I will give any life science projects this year. The problem is that most of my projects go back to either the NGSS physical science test or the Illinois State text -- which also appears to focus on physical science, even in the sixth and seventh grade texts.

Meanwhile, at the Common Planning meeting on Wednesday, the administrators introduced us to the new online software that we will be using to take attendance and grades -- yes, we're just now being introduced to it fully a hexter into the year. And, sure enough, the software has a built-in grading scale that we're required to use.

Remember how I once mentioned a quick-and-dirty weighting for the grades:

40% Tests

30% Quizzes

20% Classwork

10% Homework

Then I figured that with so many Illinois State projects -- such as the current mousetrap project -- the classwork section needs to be elevated:

30% Tests

30% Quizzes

30% Classwork

10% Homework

Well, here is the actual weighting that the new software will be using to calculate grades:

40% Tests & Projects

20% Quizzes

15% Homework

15% Participation

10% Classwork

Recall what I've said about previous online grading software -- it doesn't matter how many tests or homework assignments I give or how many points each it worth, because they will automatically be weighted so that tests are part of the 40% and homework part of the 15%. I've said before that this is deceptive from the students' perspective -- one homework point isn't the same as one test point.

I've already promised to circumvent this weighting simply by making the trimester 1000 points. Then I make sure that of these 1000 points, the tests add up to 400, the quizzes add up to 200, the homework up to 150, and so on. Then a point is a point no matter what, since the computer will calculate that the number of points I'm giving already fits the weighting percentages.

Earlier I wanted to include the projects as Classwork, but now I see that the projects now belong under the Tests category. This already fits my plans -- I want to give three tests per trimester (and if there's a fourth, the lowest score is dropped) for 300 points, and there will probably be about five projects per trimester worth 20 points each, for a total of 400 points.

But I will have to tone done the quizzes a little. The quizzes I've given so far are already worth 50 points each. My plan was to give three Dren Quizzes and three general quizzes for a total of 300 points (with a fourth quiz of either type to allow dropping the lowest grade). Now instead, I'll give only three quizzes of each type per trimester. This means that there will be only nine Dren Quizzes (10's, and then 2's through 9's). Scoring below an A will still result in a score of 1/50, but now it can only be made up by taking the next Dren Quiz -- for example, if a student fails the 7's, he/she will still have to take the 8's the next time.

I'm also having to differentiate between Participation and Classwork. Warm-Ups, Exit Passes, and of course Participation Points are now considered Participation, leaving Classwork (the smallest category) for worksheets only.

Here is the song I sang today:

What's the Best Advantage?

Life is full of patterns.
They show us the way.
School starts at the same
Time everyday.
What's the best advantage?
To tell us every time
We can go how far
When we build a better mousetrap...
Mousetrap car!

Life is full of patterns.
The sun and moon follow patterns.
We can use math
To learn about patterns.
What's the best advantage?
To tell us just how fast
We can be a star
When we build a better mousetrap...
Mousetrap car!

Yeah, that's right -- this is mostly the same song that I sang for "The Need for Speed," the first project where students worked on the mousetrap cars.

I did take pictures during the project, but most of these include students' faces, and so in order to protect their privacy, I will not be posting them here on the blog.

Test #2 and What's the Best Advantage? (Days 29-30)

Today the students take a test. This is for all three grade levels, now that I've changed my original assessment schedule. The eighth grade test is on rational approximations. So far, many students fare well on the test, since much of it involves approximating a square root on the calculator and rounding it off from zero to three decimal places.

Day 30 marks the midpoint of the trimester. In the past I've referred to half of a trimester by a special name -- the "hexter." This is what I wrote in the past about the name "hexter":

The name hexter is interesting indeed. But to discover the origin of this name, we must first consider the origins of the words semester and trimester.

Where does the word semester come from? Some people might recognize a prefix semi- meaning "half" -- for example, in geometry a semicircle is half of a circle. Since a semester is half of an academic year, this seems logical -- but it's wrong. As it turns out, the word semester actually means "six months" -- it comes from Latin sex-, "six," plus mes- or mens-, "month." (Notice that in Spanish, the word mes still means "month.") But a semester can't possibly last six months, since then two semesters would be twelve months, the entire year, with no time for summer vacation. As it turns out, the word semester doesn't come directly from Latin, but passed through German. In German universities, the two semesters actually are six months long -- the winter semester lasting from October to March, and the summer semester lasting from April to September. There actually are breaks corresponding to our summer break, but they're actually included as part of the semesters! So semester means "six months," sex- plus mes-, but Latin speakers often drop the letter x when it appears right before the letter m, just as emigrate is really ex- (out of) plus migrate.

Therefore, a trimester actually means "three months" -- since it comes from tri-, "three," plus mes-, which we already identified as "month." It does not mean "one-third of a year." But since the school year is approximately nine, or three-squared, months long, one-third of the year just happens to be around three months. The term of a woman's pregnancy is also around three-squared months, and so some might believe that trimester means one-third of a pregnancy, but it still means "three months."

And so what about hexter? Now hex- is Greek for six (think hexagon), but is a hexter six of something, or one-sixth of something else? This word doesn't contain mes-, so it has nothing to do with six months or one-sixth of a month. On one hand, there are six hexters in a year, so this word,hexter, appears to be one-sixth of an academic year. But a hexter is also six of something -- it is close to six weeks in length, since there are approximately 36 or six-squared weeks in a school year! The answer is that we can't be sure, since the academic term hexter, while used at some schools, doesn't appear in a dictionary where we can discover its etymology.

Finally, notice that hex- is Greek while all the other numerical prefixes for academic terms are derived from Latin. Recall what I wrote about this lack of linguistic purity in geometry, where we have both hexagons (hex-, Greek) and nonagons (non-, Latin). To be linguistically consistent, we ought to use the Latin prefix sex- and call it a "sexter." The problem is that most schools using hexters are middle schools, and students at that age will assume that this has something to do with sexuality, even though the Latin sex, "six," has nothing to do with the Latin sexus, "sex." In order to avoid trying to explain to middle schoolers how "sexter" and "sexual" come from two completely unrelated Latin roots, the schools just throw linguistic purity out the window and use Greek-based "hexter" instead.

Notice that trimesters, and therefore hexters, appear mainly at the middle school level. High schools almost always use semesters instead, as this is what the colleges expect on the transcripts. But I have seen a few high schools give report cards three times per semester -- in other words, the progress report occurs at the end of every hexter.

By the way, since I wrote the above, I found a link to an actual high school using the term "hexter":

http://www.springpointschools.org/blog/2016/08/iterating-college-and-career-readiness/

In 10th grade, beginning at the end of the first hexter (six week period), students who have demonstrated mastery of Habits of [...]

This is at a high school in New York. I've never seen any California school use the term "hexter" -- and that includes my own middle school. But still, I will use the term "hexter" both in the classroom and on the blog as a convenient word to refer to the progress reporting period.

I've written about my plans to give four tests this trimester. Therefore, I ought to have two tests during each hexter. But as it turns out, I ended up giving only one test the first hexter, since I printed up the progress reports before grading the tests. Furthermore, the last major grade before printing the first hexter progress reports was a Dren Quiz, which was easy.

As it turns out, all of my eighth graders are earning a C or better. But there are a few students who were failing until the Dren Quiz raised their grades to a C. This might make the progress reports misleading, since the grades were artificially inflated by a Dren Quiz -- oops! As it turns out, most of my failing students are seventh graders. The first test was difficult, and no Dren Quiz can erase all the 10% and 20% scores received on the test.

Here is the song for today:

UNIT RATES

If you want to find unit rates,
There's one thing you must know.
To find a unit rate,
All you do is divide!
To see if it's proportional,
All you do is divide!
Write it as a fraction,
Reduce it then you're fine.
Graph it at (0, 0),
Then just draw a line.

If you want to find square roots,
There's one thing you must know.
To find an estimate,
4 and below, round down!
To find an estimate,
5 and above, round up!
1 place for tenths, 2 for hundredths,
3 for thousandths, you're fine.
Graph it between two values,
Right on the number line.

Tomorrow is also the beginning of a new module. Learning Module 3 of the Illinois State text is called "What's the Best Advantage?" In this module, students will finish the mousetrap cars that they started back in Module 1.

For my eighth graders, this will be an excellent opportunity to integrate science in the lesson. As I wrote earlier, the next NGSS science lesson on the computer is on motion and force. I've been delaying it until the students can learn about force and Newton's Laws. Well, as it just so happens, the students are supposed to measure the force used to launch the mousetrap cars -- in Newtons! So the idea is to have the eighth graders use the first hour to use the mousetrap cars and then the second hour to complete the online assignment.

But the problem is that our mixed-up Wednesday schedule might finally be changing. Here's how the old schedule worked: first period I would have sixth grade, then second period I'd have the eighth graders for "science" (the online assignment), and then third period I'd keep the eighth graders for STEM (which I'd use for either math or an Illinois State project). The problem with the old schedule is trying to fit music into the schedule. According to the music teacher's schedule, eighth grade music started near the end of first period and was intended to last into second period. So the eighth graders began the day in the history classroom and switched to music when the music teacher arrived -- only to have it end 15 minutes later when the sixth graders arrived to the history classroom.

The new, more logical schedule has eighth grade music line up with second period. This means, among other things, that the eighth graders won't be in my room for both an online science lesson and a STEM project, since they'll still be in music. I assume that I will begin the day with the sixth graders in my room -- but I can't send them to the history classroom after first period, since the eighth graders will still be in there for the music lesson. So I'd either keep the sixth graders an extra period or have them go to English and have the seventh graders come to my room.

Under the old schedule, the seventh graders came to my room for fourth period -- but then their music lesson took place in my room, and I didn't see them for math or science at all! Frankly, I wouldn't mind seeing the seventh graders tomorrow, and I bet they'll enjoy beginning the project. If they're in my room only for music as usual, then I'll just have them do the project on Thursday -- indeed, I suspect the project will bleed into Thursday for all the grades no matter what.

And if I do lose an hour with my eighth graders, then I'll just do the project today and have them do the online science assignment tomorrow after lunch -- that time on Thursday is usually for online math assignments, but I'll just have them do science instead.

But this is a two-day post, and I won't know what happens until tomorrow. That's right -- we received an email informing us of the new music schedule, but we were never told what to do with the students outside of music time! The English teacher has given up trying to figure out the schedule and says that she'll just give an online English assignment to whatever kids show up in her classroom! So this is what Wednesdays are like at our middle school, even one hexter into the year!

My next post will be Thursday.

Coding: Editing in Word (Day 28)

Today is a coding Monday. Obviously, I don't have much to say about math in today's post, since I spend most of today watching the coding teacher do his work.

In today's computer lesson, the teacher provides each eighth grader with a Google account, then shows the kids how to edit in Word. (As usual, this blog's focus is on the eighth graders, thus this post is titled "Editing in Word.") Some of the students already know what they are doing and attack the teacher-provided sample document with ease. Other students struggle, and the teacher warns them that they'll be in trouble if they make it to high school and don't know how to use a word processor.

I think back to the first time I used Word -- it was in a summer coding class I took in high school nearly 20 years ago. I took an exploratory computer class in middle school as well, but I don't remember whether or not the word processor we used was Word -- I suspect it wasn't.

The sixth and seventh graders had a lesson on cyberbullies and safety. Last week, the lessons were reversed -- the eighth graders learned about safety and the younger kids used Word.

Since there's no math for me to discuss today, this is a great time to catch you readers up with my year-long goal to become an ideal classroom manager. I assume that the incident that I'm about to describe are quite common in classrooms.

It begins as my eighth graders arrive. Usually, the coding teacher arrives 20-25 minutes after the block begins, so I use that time for a Warm-Up and passing out the homework for the week, and then the students can use the extra time to start on the homework until the coding teacher gets here. I pass out calculators for the Warm-Up, since it is still on square roots and irrational numbers.

But then the students continue to use the calculators on the homework. The questions on the homework (that come from a practice workbook) aren't quite dren-level questions like single-digit multiplication -- indeed, the first question is long division. But still, this is the type of question that some people (like traditionalists, for example), say should be done without a calculator.

Of course, you readers may notice that this is homework, and so there's actually nothing stopping them from using calculators at home to do the homework. Nonetheless, I'll do whatever it takes to stop them from using calculators in front of me on the homework.

So I take the calculators away. And this causes one student -- and she happens to be the lowest student in the class -- to say, "I need the calculator because I'm not smart like you are!"

I know that from a traditionalist perspective, the people who can do long division by hand are actually the normal ones and those who can't divide are on the outs! I don't bring up traditionalists in class anymore, but I do say, "I'm not smart -- I'm normal."

The girl's response is, "I need it because I'm not normal like you are!"

Keep this in mind as I discuss what happens when the eighth graders return after lunch -- a Math Intervention block that also used for an online math curriculum. The girl enters the classroom upset because someone has taken her cellphone, and so she disrupts the classroom by telling the other students to empty out their backpacks in search of the phone.

At this point I begin to yell at the students to stop the search immediately and get back to work. The problem, I tell them, is that phones are forbidden in the classroom. In theory, all phones are supposed to be confiscated at the start of the day, to be returned at the end of the day. But you can probably figure out why this is doomed to failure -- a student who turns it in is guaranteed to be without a phone for a full seven hours, while someone who keeps it can probably sneak in even just a few minutes on the phone at some point without being caught.

I tell the students that I can't make them turn in their phones, but I can enforce the rules by at least not granting any class time to search for missing phones. The student begins to cry, thinking about what her mother will say when she finds out that the phone is missing.

Then she tries to ask for a restroom pass -- but the problem is that last week, the principal told me that I can't allow restroom passes anymore either. Formerly I've been allowing students to leave during Music Break, but lately they've been taking advantage and using that time to go places other than the restroom. At this point the girl complains, "I hate this class because you care about the rules so much!"

At this point I tell her that if she leaves, I'll have to give her a detention for using the restroom during class time. (Of course she has no intention of going to the restroom -- she just wants to go out and search for the phone.) She accepts my detention and leaves the room. But then the principal shows up to the classroom, having been called in by the English teacher due to the riot in my class. The principal asks for the girl who first caused the disruption, and upon her return from the "restroom," she is forced to go to the office.

Let's step back now and think about what's happening here. The reason that I enforce the rules is that students who use the restroom only during breaks and puts their education about the entertainment of a cell phone find themselves with lots and lots of A's on their report cards. Given a choice between being bored with an A and entertained with an F, I'd always choose boredom with the A.

But let's think about this from the girl's perspective. She is at the bottom of my class -- so she's most likely failed math her entire time as a scholar. Think back to what she said earlier about calculators -- only special "smart" people like me can do math without a calculator, not those like herself. She believes that no matter how hard she works, she'll get a bad grade in math. So to her, the choice is between being bored with an F and entertained with an F -- and in that situation, she might as well choose entertainment. And in fact, she immediately puts her head down when I start talking about getting A's as a reason for following the rules -- since she believes that she'll never get A's.

I've mentioned it this here on the blog before -- this student thinks that only special "smart" people can be good at math, not ordinary people like herself. Now imagine if the Cleveland Cavaliers believed that only "special" teams like the Lakers, Celtics, or even the Warriors win championships, not ordinary teams like themselves. Or if the Chicago Cubs now assume that only special teams like the Yankees and Cardinals win World Series, not ordinary teams like themselves. Sure -- the girl has struggled in math since kindergarten, but the Cubs have struggled in baseball for a century before her troubles began.

This, of course, is all related to the "growth mindset." But unfortunately, "growth mindset" has turned into a buzzword that can mean anything the writer wants it to mean. Still, the important thing I should ask is, how can I get this girl to realize that she should work hard so her grades will rise?

So far, I've seen that endless stories about traditionalists, A's, and their futures doesn't work. And of course, yelling at her as I did today doesn't work at all. Of course, there is no simple answer -- otherwise we'd all be excellent math teachers.

Since today is the 26th, let's look at the blog of Tara Daas, whose monthly posting day is today:

https://hazeleyedmathnut.blogspot.com/

Here is a link to her September 26th post:

https://hazeleyedmathnut.blogspot.com/2016/09/a-day-in-thelife-92616-615am-husband.html

Daas is a Georgia high school teacher who teaches mostly Algebra I. Her regular classes are learning about compound inequalities, but she also has three Accelerated Algebra I classes, which are learning about even and odd functions. When I took Algebra I as a student so many years ago, it was a regular class, so I didn't learn about even and odd functions.

Daas writes about the struggles her third period class is having. This is technically an Accelerated class, but they aren't as successful as her other classes. She points out how she tries to connect even and odd functions to the transformations (y-axis reflections and 180-degree rotations about the origin) that they learned the previous year in Common Core 8, but this doesn't help them.

Well, hopefully we'll both improve with time.

Whiteboard Lesson: Estimating Square Roots (Days 26-27)

Tonight is Back to School Night. With a long evening spent at school, it's a good thing that this is a two-day post, with no scheduled post tomorrow.

In all classes, the kids continue work in the so-called "Student Journals." Then tomorrow, the students will practice on the whiteboards. This should provide adequate preparation for the major test that all students will take on Tuesday. For eighth graders, the test will be on rational approximations to irrational numbers, particularly square roots.

During the music break, I choose a song from Square One TV: "Nine Nine Nine." The reason I chose this song is that this is one of the oldest videos on YouTube -- in fact, Monday, September 26th marks the tenth anniversary of its first posting on the web. I might have sung this song on the anniversary itself (or maybe not, since it's a coding Monday), but I decided that this would be a great song to sing for Back to School Night. (Actually, it would have made more sense to play it on the ninth anniversary since it's a song about the number 9. Well, today is the 9.99th anniversary, so there are your nines!)

Here is that old YouTube video:

Here are the lyrics to this song, courtesy Barry Carter:

http://wordpress.barrycarter.org/index.php/2011/06/07/square-one-tv-more-lyrics/#.V-SYfCgrKUk

Nine Nine Nine

Lead vocals by Reg E. Cathey

Backup vocals by Cynthia Darlow

Nine nine nine
Fantastic number nine
It’s perfectly consistent
It works out every time
Nine nine nine
That crazy number nine
Times any number you can find
It all comes back to nine

Two times nine is eighteen
Eight and one is nine
Three times nine is twenty-seven
Seven and two is nine
Four times nine is thirty-six
Six and three is nine
Five times nine is forty-five
Five and four is nine

Six times nine is fifty-four
Five and four is nine
Seven times nine is sixty-three
Six and three is nine
Eight times nine is seventy-two
Seven and two is nine
Nine times nine is eighty-one
Eight and one is nine

Nine nine nine
Fantastic number nine
It’s perfectly consistent
It works out every time
Nine nine nine
That crazy number nine
Times any number you can find
It all comes back to nine

Nine times ten is ninety
Just drop the zero sign
Nine times eleven is ninety-nine
Makes me rhyme with another line
Then nine and nine is eighteen
And eight and one is nine
Times any number you can find
It all comes back to nine

This work for bigger numbers, too?
Yup.
Let’s try this’un: three thousand four hundred’n eighty seven
Hmm … nine times three thousand four hundred eighty seven is
Thirty-one thousand three hundred eighty-three
Three plus one plus three plus eight plus three is eighteen;
One plus eight is nine.
It always works!

Nine nine nine
Fantastic number nine
It’s perfectly consistent
And it works out every time
Nine nine nine
That crazy number nine
Times any number you can find
It all comes back to nine
It all comes back to nine
It all comes back to nine

As it turns out, a second YouTube video was posted on September 26th, 2006 -- "One Billion Is Big", which I mentioned on the blog around the time of the billion-dollar Powerball jackpot. The reason I favored Nine Nine Nine over One Billion Is Big is that Barry Carter doesn't have the lyrics to the latter song available. I'll have to play and pause the video myself to get the lyrics. I might do so within the next month or so, since there's a lesson coming up in the eighth grade Illinois State text soon about large numbers.

Meanwhile, today's song would be a great song to play around the time of the Dren Quiz for 9's, which will be during the third trimester. In many ways, the hardest digit by which to multiply in base ten is not 9, but 7. The 8's have the advantage of being all even, while the 9's follow the pattern mentioned in the song. (At the Dozenal Forum, it's mentioned that in base 12, it's the 11's times tables that follow the pattern mentioned in the song -- a pattern known as the omega rule.)

At Back to School Night I met several of the parents and sang the song for them. One parent was concerned about our school's lack of a science teacher, but many enjoyed the song. I ended up giving away my old "Meet Me in Pomona, Mona" song to the one parent who volunteered at the fair.

Here is a link to the "A Day in the Life" poster for the 22nd, Jonathan Newman:

hilbertshotel.wordpress.com

Newman is a Maryland middle school teacher -- that's right, many of the participants in the "A Day in the Life" project are middle school teachers! So I definitely want to discuss Newman's blog in much more detail. Here is a link to his September 22nd post:

https://hilbertshotel.wordpress.com/2016/09/22/ditl-september-22nd/

Most of Newman's classes are Common Core Math 8, but he does have one section of Algebra I. He writes that today is the end of the first quaver at his school. Even though he has only one Algebra I class, almost all of the photos in his post are from that class.

Newman writes that he gives his Algebra I students a Three-Act Lesson -- which means that it goes back to the King of the MTBoS, Dan Meyer. This activity is all about taxi fares and ultimately leads to graphing linear functions.

I admit that from my perspective, this is a bit early in Algebra I to be teaching linear functions -- that topic usually appears later in the first quarter or early in the second. Of course, my "Show Me the Numbers" activity is all about graphing what should be a linear function -- but it's more about recording and graphing data and less about the structure of a linear function.

Notice the name of Newman's blog -- Hilbert's Hotel. Back in January, I mentioned the idea of an infinite hotel, dating back to the mathematicians Georg Cantor and David Hilbert.

My next post will be Monday.

Student Journal: Rational Approximations (Day 25)

There are several things I wish to discuss in today's post. First of all, my "Day in the Life" links are now visible at the Tina Cardone's official website:

http://ditlife.tumblr.com/

And let's celebrate my official joining of Cardone's project with a link to another participant, Wendy Menard, whose monthly posting date is today, the 21st:

https://hermathness.wordpress.com/

Menard is a New York high school teacher. She begins her September 21st post by announcing that it's also her daughter's 25th birthday -- maybe that's why she chose the 21st:

https://hermathness.wordpress.com/2016/09/21/day-in-the-life-september-21-2016/

Here are a few interesting things I found in Menard's post. First of all, she writes that she always begins the day with a math problem whose answer is -- you guessed it, the date. As it turns out, I'm not doing it that much anymore. The problems I gave were too easy when it's already known that the answer is the date, plus I'm supposed to be giving warm-up problems from the Illinois State text.

Menard teaches Algebra II, which is learning about linear systems, as well as "Discrete Math." She writes that "Discrete Math" (which is learning about "matrix logic") is considered to be a downgrade from Algebra II -- and indeed, some students are disappointed that they can't take the higher course.

The next thing I want to discuss is the school calendar. As it turns out, the LAUSD school board has just approved a new school calendar for next year. There has been a strong enough opposition to the Early Start Calendar that the board voted to start school gradually later. The first day of school will be August 22nd, 2017 and then August 28th, 2018 -- just one week before Labor Day. Recall that I work at a charter school, not LAUSD -- but since my school is co-located, there's a good chance that it will follow suit and start later as well.

I'm actually curious what will happen to Admissions Day on the new calendar. It seems awkward to have three days of school followed by a four-day weekend. Perhaps "Admissions Day" will be mysteriously moved to make Veteran's Day weekend or spring break longer, as in other districts.

Normally I'd call a calendar that starts the fourth week of August a "Middle Start" Calendar. But somehow, it's stated that first semester finals will still take place before Christmas. Notice that this year, the fall "semester" is only 79 days long -- so this means that in two years, the first semester will only be about 70 days long, unless there some other changes (such as the shortening of Thanksgiving or winter break).

No, 70 out of 180 days is not half of the year, but it is much closer to two-fifths of the year. This is why I still say that the best way to deal with the problem with trying to squeeze a full semester between Labor Day (or just before it) and Christmas is to divide the year into five "quinters" rather than four quarters. Then the second quinter can take place before Christmas, while the fourth quinter can finish before AP and SBAC testing. The fifth quinter can be reserved for testing, credit recovery, and other end-of-year activities.

I'm still a little upset that my best eighth grader is no longer a student at my school. Our English teacher saw the girl walking to school one day, but she never arrived at our school, and the next thing we hear about her is that she attends another school. I was looking forward to teaching her Algebra I and recommending her to take Geometry next year, but now she's gone. Of course, it's still possible that I might help another student with Algebra I this year -- and even before this year began, I wrote on the blog that I wanted to try teaching some Algebra I to my eighth graders -- especially during the statistics unit. But still, the obvious student to give such lessons to is no longer here.

In class today, I begin having the students work out of the so-called "Student Journals," which are really just alternate textbooks. The traditional lesson for eighth graders is on rational approximations to irrational numbers, following the standard:

CCSS.MATH.CONTENT.8.NS.A.2
Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

I continue to use the extra Wednesday block I have with eighth graders for science. But this time, I don't give a lesson from Sarah Carter, who is still working on things like scientific notation and significant figures. The goal is to get to the computer lesson on forces and motion.

I decide to give my students guided notes as found in the following link:

http://systems-of-the-earth-0421.ellensburg.ehs.schoolfusion.us/modules/locker/files/get_group_file.phtml?gid=4819558&fid=27293008

Let's see whether this will help them do well on the computer lesson next week. But as it turns out, our mixed-up Wednesday schedule may finally be getting fixed. I might get to teach my seventh graders math rather than just watch them do music, but it could be at the cost of not seeing my eighth graders that extra block.

Still, I'm going to give my eighth graders the science lesson no matter what, since they'll need it for the NGSS test in May.