Wednesday, October 5, 2016

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:

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:


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:


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:

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:

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:

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.

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