Monday, July 31, 2017

Pappas Computer Post: Number Bases

Table of Contents

1. A Minor Announcement
2. Pappas Page of the Day
3. Pappas and the Binary System
4. Divisibility Rules in Decimal
5. Divisibility by 7 in Decimal
6. Finding the Perfect Base
7. Divisibility by 5 in Dozenal
8. SPD in Other Bases
9. Other Uses of Dozenal
a. Lower Bases
b. Higher Bases
10. Conclusion

A Minor Announcement

Today I'm close to securing a substitute position for the fall. It's actually located at one of the charter schools to which I applied -- it turns out that a teacher who was expected to leave will be staying.

If I become a sub there, then I'll adapt its calendar as the official blog calendar and return to posting from the U of Chicago text. The calendar at this charter is similar to the LAUSD calendar, except that the holidays unique to the LAUSD (Admissions Day, Rosh Hashanah, etc.) are not observed -- instead there are more PD days. Also, the spring break at the charter may be different from LAUSD.

Pappas Page of the Day

This is what Theoni Pappas writes on page 212 of her Magic of Mathematics:

"Speculations on the fourth dimension surfaced in the 19th century when August Mobius noticed that the shadow of a right hand could be made into a left handed shadow simply by passing one's hand through the third dimension. Little did one realize that the term hypercube would lead to a generation of such terms as hyperspace, hyperbeing, hypercard, and now hypertext."

I've written about the fourth dimension several times on the blog before -- first when discussing Rudy Rucker's book The Fourth Dimension, and then in passing with Eugenia Cheng's Beyond Infinity, when she writes about infinitely many dimensions. Pappas writes about the fourth dimension as well, but we're already well past that chapter, as this was back on page 43. She only mentions 4D again here because she's trying to compare a hypercube (a 4D cube) to "Hypertext," which is the actual title of this new section.

Notice that Mobius, the 19th century German mathematician mentioned on this page, is better known as the creator of the Mobius strip. But he also worked in the fourth dimension -- his name appears in title of a Futurama episode about a four-dimensional whale, "Mobius Dick."

This is another section in Pappas which predicts the power of the internet, which was still in its infancy when Pappas wrote her 1994 book. Every time we use the internet, we use hypertext:

-- The web address of the very page that you're reading right now begins with https:// -- which stands for "hypertext transfer protocol (secure)."
-- The web address of the very page that you're reading right now ends with .html -- which stands for hypertext markup language.

Therefore you're reading hypertext this very moment! Pappas proceeds:

"While reading the story on the computer, you can choose the direction in which you want the story line to move."

This sounded very exciting back in 1994, but it seems quite ordinary now.

I've decided to use these pages in the Pappas computer chapter to discuss number bases. She writes a little about the binary system in this computer chapter. I'll borrow information about other bases from the Dozens Online Forum (written in hypertext, of course). As I promised earlier, this will become a traditionalist post where I compare the addition and multiplication tables in other bases to those in our current base ten system.

Pappas and the Binary System

Pappas writes about base two, or binary, due to its relationship with computers. In an earlier post, I wrote about the Scottish mathematician John Napier, who moved markers on a chessboard in order to add two numbers in binary.

But now suppose that we wish to multiply two numbers in binary instead. Pappas tells us how Napier would have performed the multiplication on the same chessboard:

"Suppose we want to multiply 15 * 11. One number is expressed with markers along the bottom row and the other along the vertical column at the right. Then a new marker is placed at the intersecting square where a row with a marker meets a column with a marker. After this is done, the multiplication process is simply done by sliding the markers from the bottom row diagonally to the vertical row. As with addition, any place two markers occupy the same square, they are removed and one marker is placed in the square above them."

It might seem amazing that this works, but there's a reason that it does work. As it turns out, this is identical to the lattice method for multiplication -- except in base two rather than ten!

Think about it -- the lattice method begins by placing the factors on the bottom (or top) and the side of the lattice, and Napier does the same on the chessboard. The statement:

"Then a new marker is placed at the intersecting square where a row with a marker meets a column with a marker."

takes advantage of the fact that 1 * 1 = 1 while 0 * 0, 0 * 1, and 1 * 0 are all 0. This multiplication step is more complicated in decimal, since the product of two digits could be as large as 81. And of course, the diagonal addition is the same in both methods.

As it turns out, Pappas mentions the lattice method on the page previous to the Napier page. She displays a page from a 15th century Italian book which includes four methods of multiplication. One of them is the lattice method, and the second is the lattice method again but with the numbers placed slightly differently. The third looks similar to the standard algorithm, and the last has multiplication similar to the standard algorithm but with the addition step diagonally as with the lattice methods.

The whole point of this is that many traditionalists hate the lattice method. They act as if the lattice method is something invented by the Common Core authors, who are more interested in controlling students than educating them. But as we see here, the lattice method existed more than 500 years before the Common Core.

I already mentioned the simple multiplication table in binary, and the addition table is also easy, since 0 + 0 = 0, 0 + 1 and 1 + 0 are both 1, and 1 + 1 = 10 in binary. I suspect that many students struggling with the tables wish that they only had to learn the binary tables instead of the decimal tables!

The main reason that binary isn't suitable as a general purpose base is that it's so small, so numbers in binary are longer than their decimal equivalents. Consider the binary multiplication:

111 * 111

By the time a binary user has found the answer as 110001, a decimal user -- provided he has learned the decimal times tables -- would have multiplied 7 * 7 = 49 in seconds (since after all, 111 in binary is just 7 in decimal). And two-digit numbers in decimal can contain anywhere from four to seven digits in binary. It's easier to multiply two-digit numbers than seven-digit numbers, even though the decimal tables are much larger than the binary tables.

Suppose our goal is to find the perfect base. We see that there is a trade-off -- smaller bases with fewer digits have easier tables, but with fewer digits, the numbers must be longer. Computers can easily handle longer binary numbers, since having a simple table is more important for computers (where 1 represents "on" and 0 represents "off"). But for humans, we'd prefer to have a slightly larger table if it shortens our numbers. The question is, what size base is suitable for human use?

Divisibility Rules in Decimal

I've mentioned the Dozens Online Forum earlier in this post. As its name implies, that forum draws users who believe that the perfect base is dozenal, or base 12.

But before we try to discover the perfect base ourselves, note that there are other considerations when determining the perfect base between the size of the base. For example, since our current base is 10 and the dozenalists advocate base 12, why don't we compromise and just use base 11. After all, if 10 and 12 are both human-scale bases, then surely base 11 must be human-scale as well.

At Dozens Online, there are many threads devoted to finding the perfect base. The dozenalists have developed a special jargon for describing number bases, which include:

-- alpha and omega
-- log wheel
-- opaque
-- sevenite
-- SPD
-- Stevinian algorithm
-- tuning
-- twelfty
-- uncial and unqual

Don't worry if you don't understand any of these words now -- by the end of this post, I'll have described all of them. The concepts represented by these terms are used to determine which base is the best.

Let's begin with "alpha" and "omega." These are used for the various divisibility rules. The following is a link to a typical list of divisibility rules:

We notice that many of these divisibility rules are different depending on the factor. For example, divisibility by two only requires looking at the last digit:

A number is divisible by 5 if its last digit is a 0 or 5.

But divisibility by four requires looking at the last two digits:

A number is divisible by 4 if the number's  last two digits are divisible by 4.

And divisibility by three requires looking at all the digits:

A number is divisible by 3 if the sum of its digits  is divisible by 3.

Why do some factors require one digit, others two digits, and others all digits? Well, let's look at the most straightforward divisibility rule:

A number passes the test for 10 if its final digit is 0

If we think about this for a moment, it should be obvious why this works -- a multiple of ten should have a whole number of tens, with no ones left over, so the digit in the ones place should be 0. Of course, this is all because we're working in base 10. In ternary (base 3), a number ending in 0 is a multiple of three, in quaternary a number ending in 0 is a multiple of four, and so on.

This strongly suggests that divisibility rules are base-dependent. The divisibility rules for a base like dozenal are different from the corresponding rules in decimal.

We notice that the divisibility rules requiring only the last digit are 2, 5, and 10 -- and it doesn't escape our notice that 2 * 5 = 10. In general, all factors or divisors of the base have this simplest possible divisibility rule.

The divisibility rule for 9 should sound familiar:

A number is divisible by 9 if the sum of the digits  are evenly divisible by 9.

This rule is mentioned in the Square One TV song "Nine, Nine, Nine." At the Dozens Online Forum, the number one less than the base is called the omega of the base. All factors of the omega have the same divisibility rule, so the digit-sum works for 3 as well as for 9.

On the other hand, 11 is one more than the base:

A number passes the test for 11 if the difference of the sums of alternating digits is divisible by 11.

At the Dozens Online Form, the number one more than the base is called the alpha of the base. All factors of the alpha have the same divisibility rule, but unfortunately 11 is prime, so no factors inherit the alpha rule.

There are other divisibility rules mentioned at the link above. Let's look at six:

Since 6 is a multiple of 2 and 3, the rules for divisibility by 6 are a combination of the rule for 2 and the rule for 3.

This is called a compound test. In this case, it combines the divisor rule for the factor 2 with the omega rule for the factor 3.

Earlier, we mentioned that for 4, we must look at the last two digits. This works because square of the base, 100, is divisible by 4. For 8, we must look at the last three digits. This works because the cube of the base, 1000, is divisible by 8. At Dozens Online, these are called regular tests.

(By the way, another word on the list is "log wheel." Apparently, "log wheel" refers to writing the regular numbers in a base on a wheel. Here "log" means logarithmic -- the regular numbers are spaced logarithmically. In a base like hexadecimal, the regulars are 1, 2, 4, and 8 spaced equally around the wheel, but in bases that aren't powers this is a little more complex.)

Divisibility by Seven

We notice that the factor 7 is missing. This is because seven is a divisor of neither the base, nor the omega, nor the alpha, nor the square of the base, nor the cube of the base. At Dozens Online, the factor seven is considered to be opaque in base 10.

When I was young, I once saw the following divisibility rule for 7:

We wish to test the number 742 (N1) for divisibility by 7. We get to the smaller number (N2) by chopping off the units digit, multiplying it by 5 and adding it to the number of tens in the orginal number (N1):
742 -> 74 + (2 x 5) = 84, which is clearly a multiple of 7
Therefore 742 is also a multiple of 7.
Notice that seven is opaque in our base 10, but this process involves multiplying by 5. If we multiply 10 * 5 we get 50. Now let's look at base 50 -- suddenly seven is no longer opaque. Instead, we see that the omega, 49, is a multiple of 7.

The above link also gives an alternate rule for 7:

Take the last digit, double it, and subtract it from the rest of the number; 
if the answer is divisible by 7 (including 0), then the number is also.

If we double the base 10, we obtain base 20. Again, while 7 is opaque in base 10, it is not opaque in base 20, since the alpha, 21, is a multiple of 7. Since this is an alpha, we must subtract the last digit rather than add it, just as we're required to subtract for alpha divisibility (11) in the original base.

And so we see that these rules are just omega and alpha rules in higher bases -- specifically those that are multiples of the original base. The divisibility rule for 13 given at the above link is merely the omega rule for base 40. And the divisibility rule for 17 given at the above link is merely the alpha rule for base 50. And the divisibility rule for 19 given at the above link is merely the omega rule for base 20, and so on.

I also saw the following divisibility rule for 7 in the book Quick Arithmetic (the same book in which I found the "sexy six"):

-- Divide the number in question by 50. Add the quotient and remainder. If the result is divisible by seven, then so is the original number.

Notice that this directly converts the number to base 50 (the remainders are actually the base 50 digits written in reverse). But I don't like having to divide in a divisibility rule. Otherwise, if we have to divide, we might as well have written the rule as:

-- Divide the number in question by 7. If the remainder is 0, then the number is divisible by seven.

The multiply by 5 rule takes advantage of base 50 without having to divide by 50. To see why, let's look at the number 742 again -- this number has 74 tens and 2 ones. But the omega rule in base 50 means that we must add the number of fifties and ones, not the tens and ones. Unfortunately, 74 tens is the same as 74/5 fifties, and 74/5 isn't a whole number.

Now if 742 is divisible by seven, then so is 742 * 5. This new number has 74 fifties and 2 * 5 ones -- and now we can add the number of fifties and ones, to obtain 84, a multiple of seven. So this explains why the multiply by 5 rule works, and it allows us to avoid division in a divisibility rule.

Then again, we might want to avoid multiplication in our divisibility rules as well. In this case, seven is truly opaque in base 10. If we include only divisor, omega, and alpha tests, then base 10 provides six divisibility rules -- 2, 3, 5, 9, 10, and 11.

Finding the Perfect Base

It's now time to look at the divisibility rules in other bases. The following link gives a list of bases that have many divisibility rules relative to the size of the base:

The list of good bases according to this link is:

2, 3, 4, 6, 9, 12, 16, 21, 25, 36, 60, 81, 85, 120, 225, 240

Notice that the link provides the following disclaimer:

21 is the lowest base with 'easy' divisibility tests for 8 different numbers, assuming that the casting out 11's method is not considered 'easy'.

The "casting out 11's" method is what Dozens Online calls the "alpha" method. Surely we want to count the "alpha" method, since Dozens Online counts it and even gives it a special name.

But as it turns out, even though alpha counts, the list of "best bases" including alpha doesn't provide as good a list as the previous list:

2, 3, 4, 5, 7, 9, 11, 15, 19, 25, 29, 35, 41, 49, 55, 71, 119

We notice that most of these bases are odd -- and in fact, many of them are exactly one less than the bases from the first list. For example, we see that base 11 appears on the second list, while base 12 (dozenal) appears on the first list.

At Dozens Online, there is a strong preference for even bases over odd bases. We can look at the multiplication table for decimal to see why -- half of our base is 5. And the 5's times tables are very easy -- 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. In even bases, the times tables for half of the base are just as easy. So the 4's in octal, the 6's in dozenal, and the 7's in base 14 are just as easy as the 5's are in decimal.

Notice that 9 makes both lists -- now let's look at the times tables in base 9. This base has one fewer digit than decimal, and since it's odd, it doesn't have an easy row in the times table similar to the 5's in decimal. So base 9 is like taking base 10 and eliminating the 5's row -- that is, we're "simplifying" the table by removing one of its easiest rows!

At Dozens Online, odd bases cause a "tuning problem." The user who created this term (icarus) compares this to an old-fashioned radio. We solve the tuning problem by tuning an odd base either up to the next even base or down to the previous even base.

Let's return to bases 11 and 12. Base 11 has divisibility rules for 2, 3, 4, 5, 6, 10, 11, and 12. When we tune this base up to the next even base of 12, the old divisor (11) becomes an omega, while the factors of the old alpha (2, 3, 4, 6, 12) become divisors.

Thus in determining which base has more divisibility rules, we must compare the omega of the odd base to the alpha of the even base. And base 11 has divisibility rules for both 5 and 10, while dozenal has a divisibility rule for 13. Base 11 has one more divisibility rule than 12, and therefore 11 makes the list while 12 doesn't.

This pattern occurs with many of the odd bases on the second list -- the odd base has an even omega while the even base has an odd alpha, and so tuning up from odd base to even base renders both the old omega and its half opaque, while rendering the new alpha (often a useless prime) transparent. So tuning up from base 9 to decimal gives us 11 at the expense of 4 and 8, and tuning up from base 15 to hexadecimal gives us 17 at the expense of 7 and 14. Some cases are even worse -- tuning up from base 29 to base 30 (trigesimal) gives us 31 at the expense of 7, 14, and 28, and so on. This is why mainly odd bases appear on the second list.

As it turns out, the second list overvalues the alpha. Given a choice among the divisor, omega, and alpha rules, we prefer the divisor rule as it's the simplest. We avoid overvaluing the alpha by choosing the first list, where only the divisor and omega rules are considered. Even though this list contains odd bases, there are a healthy number of even bases as well.

Let's look at this first list again:

2, 3, 4, 6, 9, 12, 16, 21, 25, 36, 60, 81, 85, 120, 225, 240

We assume that our ideal base appears somewhere on this list -- and now, finally, we can consider the size of the base. We see that of the smaller bases, of course binary makes the list as it's the smallest possible base, ternary makes the list (as it adds a divisibility rule for 3 to binary), and quaternary makes the list (as it adds a divisibility rule for 4 to ternary). It doesn't take much for small bases to make the list -- and so we consider the smallest nontrivial base on the list to be six. Senary is the first base that makes the list after a base that misses it (base five, or quinary). And so we consider all smaller bases to be below human scale. Like binary, bases 3 and 4 are more suitable for computers (which can handle numbers with many digits) than for humans.

Now we need an upper limit to the human scale. We notice that since we are skipping odd bases, there is a huge gap between hexadecimal and the next even base, 36. This increases the size of the table greatly, and so we consider base 36 to be above the human scale. Base 36 is interesting in its own right, if only because its 36 digits could be the decimal digits 0-9 plus 26 letters a-z.

The larger even bases on the list, 60, 120, and 240, are definitely on the superhuman scale. Of these, 60 and 240 have primes for both alpha and omega, while 120 has these as composites. Because of this, one Dozens Online user, Wendy Krieger, actually advocates 120 as the perfect base. She often refers to this base as "twelfty."

And indeed, base 120 almost made the second list despite it being even. Tuning from base 119 to twelfty gives us 11 and 121 at the expense of 59 and 118. Yet 120 fails to make the list because it merely ties the record set by 119, but only outright record breakers make the list.

Anyway, our human scale runs from base 6 to base 16. In this range, there are four bases that make the list, namely 6, 9, 12, and 16. But base 6 is borderline human scale -- many consider it to be too low of a base (like bases 2-4). And base 16 is also borderline -- many consider it to be too high of a base (like base 36). With base 9 already eliminated as an odd base, this leaves only one base -- the perfect base.

And this is why the Dozens Online Forum was created -- its users view it as the perfect base.

For the rest of this post, I actually want to focus not as much on dozenal as on the borderline cases, and whether we should adjust the borders up or down. This is when we must consider how easy it is to learn arithmetic in the various bases.

By the way, before we leave the list of bases, I point out that the odd bases in the first list can be tuned down to the previous even base -- then the old base's omegas become divisors and the old base's divisors become alphas (thereby restoring the importance of alpha). This means that we could add bases 8, 20, 24, 80, 84, and 224 to the list. (Notice that our usual base, decimal, doesn't make any version of list at all.)

The official name of base 12 is duodecimal, just as base 16 is hexadecimal. Yet advocates of base 12 like to refer to their chosen base as dozenal. This is because "duo" means two, so "duodecimal" actually means two beyond "decimal" (base 10). But base 12 advocates prefer complete independence from decimal, so they don't want a base with "decimal" in its name. Since many people are already familiar with groups of 12 via the concept of a "dozen," the preferred name of the base is "dozenal."

Some base 12 advocates want to go a step further. Even "dozenal" is unacceptable because "dozen" comes from the French word douze, which is the same as the Latin duodecim. So even "dozen" indirectly refers to "decimal" in its name.

There is another Latin word, uncia, that actually means "twelfth." It appears in the English words "inch" (1/12 of a foot) and "ounce" (1/2 of a troy pound). So some Dozens Onlune users prefer using "uncial" as the name of base 12. The name "unqual" is a modification of "uncial" -- the thought being that "uncial" refers to the fraction 1/12 while "unqual" refers to the whole number 12.

We normally write numbers in other bases by writing the name of the base as a subscript (which doesn't show up well in ASCII). So, for example, 43_7 means 43 (base 7). Sometimes Pappas even writes the base out in letters, as in 43_seven. The problem with writing 27_12 to mean 27 (dozenal) is that the base 12 is itself written in decimal -- so again we're favoring decimal.

On the Dozens Online Forum, a special notation is used to name bases:

{a} (decimal)
{c} (dozenal)
{e} (14-mal)
{g} (hexadecimal)

and so on. The idea is that in hexadecimal, the digits are 0-9 and a-f, and so the first letter that isn't used in the base is g, and so {g} is the name of the base. Dozenal requires 0-9 and a-b, and so the next letter {c} is the name of the base. Decimal uses only 0-9, and so the first letter of the alphabet, {a}, can name the base.

To me, it might have been interesting to use {a} for base 9 rather than 10. Then {d} works out to be base 12 (as in dozenal), {f} would be base 14 (as in fourteen-mal), and {h} would be base 16 (as in hexadecimal), and so on. But actually, no one on Dozens Online ever uses "d" for "dozenal." If they do use the subscript"d" to stand for the first letter of a base, it instead means decimal. Instead, the letter "z" is used for dozenal ("dozen" is often shortened to "zen"), "o" is used for octal, and so on. Again, these are subscripts and are separate from the braces, where {a} is decimal and {c} is dozenal.

Here's a link to a Dozens Online post, by the user Double Sharp, which uses the symbols {a} and {c} in order to switch from decimal to dozenal in a single post:

{a} (default decimal)

Indeed some numbers don't have ten or eleven in them, but most do. The reason why separate-identity is a flawed idea is because it presupposes that ten-ness is inherent in the 1 of "10". Indeed ten-ness is inherent somewhere in that numeral, but it is not to be found in that one. It is to be found in our mental assumption that any number with multiple digits is decimal. If we just overturn that and say that any number on a particular page or post is dozenal, then "10" would instantly switch to having inherent twelve-ness, and it would once again not be found in the "1" but in our assumption.

{c} (default dozenal)

And to me, it is as simple as flicking a mental switch, like I just did, upon which I can say that 7*5=2b or 9*8=60 without batting an eyelid. Yes, in fact, I am fluent in dozenal and know its multiplication tables. I also know them in senary, octal, tetradecimal, and hexadecimal, and hope to soon add octodecimal and vigesimal to that list. I just find it a little uncomfortably constraining in having only two transparent important primes rather than three or four.

By the way, notice that not every dozenalist uses "a" and "b" for digits ten and eleven. Many prefer some version of "X" for ten and "E" for eleven. These many have been influenced by Schoolhouse Rock, which uses them in its multiplication video "Little Twelvetoes." That's right -- in teaching students how to multiply by twelve, the video actually introduces dozenal numerals. A version of "X" is "dek," a version of "E" is "el," and 10, the dozen, is "doh."

In this post, I will stick to "a" for ten and "b" for eleven, especially if I use the marker {c} to denote the dozenal base.

In this above post, Double Sharp is writing about the divisibility rules for dozenal. We've seen that dozenal made the earlier list due to having so many divisibility rules. In this base, the factors which are transparent in dozenal are:

-- 2 (divisor)
-- 3 (divisor)
-- 4 (divisor)
-- 6 (divisor)
-- 8 (regular)
-- 9 (regular)
-- b (omega)
-- 10 (divisor)
-- 11 (alpha)

But Double Sharp laments that only the first two primes are transparent (2 and 3), while the next two primes (5 and 7) are opaque. Double Sharp is willing to skip 5 in favor of 7 (as in octal, where 7 is the omega), but skipping both 5 and 7 in favor of the large primes b and 11 is undesirable.

Divisibility by 5 in Dozenal

It's possible to find rules for 5 and 7 similar to the rule for 7 in decimal. We see that 5 * 7 = 2b, and so 30 is a dozenal multiple with both 5 and 7 in the omega. Therefore both 5 and 7 can be resolved by tripling the last digit and adding to the rest of the number. Five also has another rule based on the alpha of 20 -- double the last digit and subtract. But I'd prefer tripling and adding, since a single rule tests for both five and seven.

But this isn't the usual rule for 5 on Dozens Online. Instead, another rule that avoids multiplication is used for 5 -- and this rule is known as SPD.

We notice that not only does 20 have 5 in its alpha, but so does 100, the square of the base (also known as "gross"). So according to the pattern, we can take the last digit, multiply it by 10 (dozen), and then subtract it from the remaining digits. In other words, we take the last digit and subtract it from the third-to-last digit (rather than the second-to-last digit for ordinary alpha). We can actually perform two steps at once -- take the last two digits and subtract that two-digit number from the remaining digits.

There's another way to see why this works. The square of the base, 100, has 5 in its alpha. But the digits of the number in base gross are the same as those in dozenal, taken two at a time (just as the digits of binary taken two at a time produce quaternary and the digits of quaternary taken two at a time produce hexadecimal). So we take the dozenal digits in pairs and perform the alpha test (by subtracting them in pairs).

The usual method for divisibility by 5 at Dozens Online is called SPD, for "split, promote, discard," and it is based on this square alpha test. Here is a link to the SPD test, by the user treisaran:

(Note: At the above link, an upside-down 2 and 3 are used for ten and eleven, respectively.)

SPD simplifies the square alpha test even more. The "split" part is the same as for square alpha -- split off the last two digits. But we don't merely subtract these digits. First of all, it's pointed out that if the last two digits are already a multiple of five, we can just "discard" them -- since after all the sum or difference of two multiples of five is a multiple of five.

If the last two digits aren't a multiple of five, then we really only need to subtract the remainder those digits leave when divided by five, not the entire two-digit number. So if the last two digits are 1 more than a multiple of 5, then subtract 1, if 2 more than a multiple of 5, then subtract 2, and so on. At the link, we "promote" by subtracting/adding the same number to both the two-digit number and the remaining digits until the former is a multiple of 5, and then we discard them. At the link, 1 is added to both the last two digits and the remaining digits to produce a multiple of 5.

This is worth looking at some more examples:

Example: Is 1985 divisible by 5?

-- Split: 19, 85
-- Promote: 85 is one more than 84, a multiple of 5, so we subtract 1 from both:
             18, 84
-- Discard: 18

And 18 is a multiple of 5, hence so is 1985.

Example: Is 39266 divisible by 5?

-- Split: 392, 66
-- Promote: 66 is two less than 68, a multiple of 5, so we add 2 to both:
             394, 68
-- Discard: 394
-- Split: 3, 94
-- Promote: 94 is two more than 92, a multiple of 5, so we subtract 2 from both:
             1, 92
-- Discard: 1

And 1 is not a multiple of 5, hence neither is 39266.

One thing about SPD is that it requires knowing all the two-digit multiples of five -- beyond those that appear in the 5's times table (up to 100, not just 50).

SPD in Other Bases

If we count SPD, then 5 is no longer opaque in any base. We can show that in any base, we can resolve 5 using the divisor, omega, alpha, or SPD rules. Indeed, SPD for 5 generally works in base 2, 3, 7, 8, twelve, and also thirteen. The next bases with an SPD for 5 are seventeen and eighteen, but in those bases require memorizing too many multiples of 5. Indeed, the square of seventeen is about twice as much as the square of a dozen, so twice as many multiples of five are needed.

It's also possible to use SPD for factors other than 5, but this is tricky. It can be shown that SPD based on square alpha never works -- instead we need either cube alpha or cube omega. This means that we must divide the digits into groups of three rather than two. For bases like decimal and dozenal, 7 is cube alpha, which is similar to square alpha. But for bases 9 and eleven, 7 is cube omega, which means that we must do the opposite at the "promote" step (that is, if we add 1 to the last three digits, we must subtract 1 from the remaining digits, and vice versa).

Square omega rarely appears as a divisibility rule. This is because b^2 - 1 factors as (b + 1)(b - 1), so almost any factor of square omega is already a factor of alpha or omega. The only time we'd need square omega is for odd bases. In odd bases, either alpha or omega has the factor 4, while the other is also even and thus has the factor 2. Then 8 can be tested with SPD for square omega. If 8 already appears in the alpha or omega, then sixteen is testable with SPD, and so on. Of course, Dozens Online focuses on even bases, where square omega is useless.

Notice that the following link (from Art of Problem Solving, which I mentioned last December) gives us divisibility rules (in decimal) for 7, thirteen, and other primes:

Both 7 and thirteen are listed with cube alpha rules (but these aren't true SPD). Of course, using any cube alpha rule entails knowing the multiples of the factor up to the cube of the base.

The above link also mentions a "tail-end" divisibility rule that works for any factor that is coprime to the base (that is, it ends in 1, 3, 7, 9 decimal, or 1, 5, 7, b dozenal). I've used this rule myself when playing "the factor game" with myself -- take a stopwatch, stop it at any random time, and then mentally factor the number that appears. It's stated that this rule works great in binary, where every odd number ends in 1.

There are several bases where thirteen is testable via cube alpha, cube omega, or square alpha. But eleven is never testable with any of these rules. It can be shown that unless it appears as a divisor, alpha, or omega, eleven is testable only via fifth power alpha or omega. Fifth power alpha is basically useless unless we're in binary, where there are only three multiples of eleven to memorize. Notice that eleven could be testable in bases 4, 8, or sixteen by converting to binary first.

Other Uses of Dozenal

I remember reading a story about a college student who earned grades of B+ and B- in his two college classes, yet he was considered to have only a B- average for the semester. Here is the reason:

{a} (default decimal -- that is, all numbers are in decimal from this point on)

A = 4.000
A- = 3.666
B+ = 3.333
B = 3.000
B- = 2.666
C+ = 2.333
C = 2.000
C- = 1.666
D+ = 1.333
D = 1.000
D- = 0.666
F = 0.000

B+ = 3.333
B- = 2.666
Average = 2.999 = B-

This was posted to Math Forum as part of an argument that 0.999... is not 1. Notice that I actually taught my eighth graders that 0.999... = 1 -- see my September 9th post for more information. (Yes, I taught them about 0.999... on 9/9 -- that was intentional as it was back when I used the Pappas trick of using the date in my math problems.) The poster argued that if 0.999... = 1, then 2.999... = 3, and his average would be B, not B- as his college insists.

There are several issues here. First of all, notice that if the average falls between two grades, then the GPA is considered to be the lower grade, no matter how close to the higher grade it may be. Thus only straight-A students are considered to have an A average. A student who takes 100 classes and earns 99 A's and one A- has a GPA of 3.99666, which isn't quite 4.0, so it's an A- average. (As Eugenia Cheng would say, it's 4 - epsilon for some small epsilon.)

In a recent Simpsons episode, Bart jokes about Lisa having a 3.9999 average -- and I forget how many 9's Bart said. Notice that if there were infinitely many 9's there, then Lisa would have an A average since 3.999... is exactly 4. Of course Bart can't keep saying 9's forever. Thus there are only finitely many 9's and Lisa is stuck with the A- average -- assuming, of course, that she attends the college where grades are calculated this way.

Let's return to the original poster. Note that his B+ is worth 3.333, in other words, 3 + 333/1000. (Do you recall the special cousin from my last post, with her mixed number calculator?) There are only finitely many 3's there, not infinitely many. His B- is worth 2.666, in other words, 2 + 666/1000. And again, there are only finitely many 6's there, not infinitely many.

The average of 3 + 333/1000 and 2 + 666/1000 is 2 + 9995/10000 or 2.9995. There are only finitely many 9's there, not infinitely many. The last 5 didn't appear on the transcript, and so the original poster only saw 2.999.

Of course, we can argue all day over whether colleges should always round grades down. (Notice that a student with grades of A- and B really does have a B+ average, since the average of 3.666 and 3.000 is exactly 3.333.) The real question is, why did the original poster make the unwarranted assumption that there were infinitely many 9's after the decimal point? Why did he assume that a B+ is worth 3.333..., with infinitely many 3's after the decimal point, when only finitely many 3's actually appear on the transcript? And indeed, why would the college choose such a strange number of points as 3 + 333/1000 for a grade of B+?

Well, the answers to all of these questions is obvious. The intended number of points for a grade of B+ is actually 3 + 1/3, and all the grades are supposed to be 1/3 point apart. But the values on the transcript are calculated in decimal, so it uses 3.333 = 3 + 333/1000 for 3 + 1/3. The original poster instinctively knew that 0.333... = 1/3, and so he assumed that there were infinitely many digits after the decimal point. In other words, the source of the problem is that 1/3 doesn't terminate in decimal.

This whole problem goes away if we use another base that has three as a factor. By now, you should already know what base we're going to use:

{c} (default dozenal)

A = 4.0
A- = 3.8
B+ = 3.4
B = 3.0
B- = 2.8
C+ = 2.4
C = 2.0
C- = 1.8
D+ = 1.4
D = 1.0
D- = 0.8
F = 0.0

B+ = 3.4
B- = 2.8
Average = 3.0 = B

Many dozenalists stress the importance of three as a divisor of the base. On the other hand, five doesn't show up as much. Notice that if there were such grades as double-plus and double-minus, then decimal, with its divisor of five, works perfectly:

{a} (default decimal)

A = 4.0
A- = 3.8
A-- = 3.6
B++ = 3.4
B+ = 3.2
B = 3.0
B- = 2.8
B-- = 2.6
C++ = 2.4
C+ = 2.2
C = 2.0
C- = 1.8

and so on. But in reality, double-plus and double-minus aren't actual grades.

By the way, another term from our Dozens Online vocab list is "sevenite." This is related to repeating "decimals" in other number bases. A "sevenite" in is a prime p such that both 1/p and 1/p^2 repeat the same number of digits. In decimal, the smallest "sevenite" is 3:

1/3 = 0.333... (one digit repeats, 3)
1/9 = 0.111... (one digit repeats, 1)

Dozenal has no small sevenites -- its smallest sevenite is more than 12^3. The only other bases in the (near-)human range with single digit sevenites are 5 (in base 7) and 3 (in base 17), according to the official list of sevenites:

For example, in base 17, we have:

1/3 = 0.5b5b5b... (two digits repeat, 5b)
1/9 = 0.1f1f1f... (two digits repeat, 1f)

The name "sevenite" comes from the fact that 7 is a sevenite in both bases 18 and 19. Moreover, not only do 1/7 and 1/7^2 repeat the same number of digits in those bases, but so does 1/7^3.

Dozenalism often crosses over with music theory. Recall that on Pi Approximation Day, I posted some musicians who used the digits of pi in dozenal to create a song.

Here, dozenal is chosen because our current music scale is 12EDO. In previous posts, I wrote about other scales such as 17EDO, 19EDO, and 31EDO. Musicians of those scales would then end up using pi in base 17, 19, or 31. Neither 17 nor 31 make any version of the list of best bases. Meanwhile, base 19 does make the best bases list when alpha rules are given importance. (But base 19 could be tuned up instead to vigesimal, where all of those alpha rules become divisor rules.)

Dozenalism also crosses over with calendar reform. I've written before about Timothy Travis, who once created a calendar in dozenal. His calendar contained only six days per week.

By the way, I noticed that today, the famous MTBoS blogger Sarah Carter posted her weekly "Monday Must Reads" series, and I couldn't help notice the following:

There are 6 days in a Zen week.
The first day of the Zen week is called Aquaday.
The second day of the Zen week is called Dolphinday.

Recall that some members of Dozens Online use "zen" as short for "dozen." So if I didn't know any better, I'd thought that this was a link to a dozenal calendar! In reality, it's a link to a complicated math puzzle -- in decimal. (By the way, earlier I wrote about a calendar based on 11 and 33. The original idea for this calendar ultimately goes back to our twelftyist, Wendy Krieger.)

Some dozenalists are also tauists. It appears that dozenalism draws people who wish to come up with alternatives to the standard -- whether its the base, musical scale, calendar, or circle constant.

Before we leave dozenalism, here's a link to a relevant Numberphile video. It was first posted on December 12th, 2012 -- that is, 12/12/12:

Lower Bases

Now let's get to the smaller bases and the lower range of the human scale. As we've said before, the problem with binary is that the numbers grow too long, too fast. Recall that this post is labeled as "traditionalists" because we'll learn more about arithmetic in decimal by imagining how we would learn it in other number bases.

Back in the days of the old California State Standards, first graders were expected to learn about numbers up to 100, second graders to 1000, and third graders to 10,000. (This is why Day 100 was traditionally a first grade "holiday.")

We have an obvious pattern here:

1st grade: 10^2
2nd grade: 10^3
3rd grade: 10^4

To make things simple here, let's assume that 4th graders learn up to 10^5. (In reality, I think at some point students learned about numbers "in the millions.")

Notice that this doesn't quite match up with the new Common Core Standards. Even though the limit for second grade remained at 1000, suddenly 100 became the kindergarten limit. (This is why Day 100 is now a kindergarten "holiday.") For some reason, the limit for first graders isn't something like 200 -- instead it's 120. I'm not sure where 120 came from -- unless Wendy Krieger somehow sneaked twelfty into the Common Core Standards (all the way from her home in Australia)!

By the way, before we look at the smaller bases, let's look at Krieger's favorite base of twelfty. She doesn't use 120 different symbols. Instead, she uses something she calls "dozen-dicker" -- she alternates between bases 12 and 10. (The old Babylonian system would be a "half dozen-dicker" system to represent sexagesimal.)

Here's how we count by tens in Krieger's base twelfty:

10, 20, 30, 40, 50, 60, 70, 80, 90 (ninety), V0 (teenty), E0 (elefty), 100

According to Krieger, a native twelftyist wouldn't call 100 "twelfty." Instead, she'd call it "one hundred," since that's what it looks like. But for some reason, Krieger uses "one thousand" to refer to the square of the hundred -- that is, 10000. One hundred thousand is one "cention," and one thousand thousand is one "million." To keep track, Krieger usually writes her million as, since it's important to keep track of the pairs. The symbol for Krieger's base twelfty is:

{ca} (default twelfty)

If we wrote this as {ac}, it would become "dicker-dozen." Krieger writes that she often uses a "dicker-dozen" table in order to multiply in base twelfty.

{a} (default decimal)

But let's get back to small bases. To keep things simple, we cast away "twelfty" and just use rising powers of ten for the grades:

1st grade: 10^2
2nd grade: 10^3
3rd grade: 10^4
4th grade: 10^5
5th grade: 10^6

According to the pattern, kindergartners only have to count to 10^1, or ten itself. In practice, they were often taught to count to 20. The California Standards also mention 31 -- probably because of the calendar, but notice that it's approximately 10^1.5, to continue the pattern with powers of ten.

But to keep things simple, let's just write:

Kindergarten: 10^1
1st grade: 10^2
2nd grade: 10^3
3rd grade: 10^4
4th grade: 10^5
5th grade: 10^6

The whole point of this exercise is -- let's assume that in other number bases, students also learn to count to the corresponding powers of their base. So in senary (the lower end of the human scale), this gives us:

Kindergarten: 6^1 = 6
1st grade: 6^2 = 36
2nd grade: 6^3 = 216
3rd grade: 6^4 = 1296
4th grade: 6^5 = 7776
5th grade: 6^6 = 46656

We see here that senary fourth graders can only count to 7776, while decimal third graders can count all the way to 10,000. Now let's try octal:

Kindergarten: 8^1 = 8
1st grade: 8^2 = 64
2nd grade: 8^3 = 512
3rd grade: 8^4 = 4096
4th grade: 8^5 = 32768
5th grade: 8^6 = 262144

In no case is an octal student in one grade beaten by a younger decimal student. This is why senary is sometimes considered too low, with octal as the smallest human-scale base. (Base 7 fifth graders just barely beat decimal fourth graders, so base 7 is definitely a human scale, albeit an odd, base.)

Higher Bases

In higher bases, the main concern is whether we can memorize the tables well enough to use the Stevinian algorithms.

I can already hear traditionalists asking, what exactly are these "Stevinian algorithm" -- is it something that traditionalists want us to learn instead of the standard algorithms? Well actually, the Stevinian algorithms are the standard algorithms! They are named for Simon Stevin, a Flemish mathematician who first came up with these algorithms. (This was about a century after the lattice method appeared, according to Pappas.)

Here are a few links to threads discussing the multiplication table in different bases:

I wish to pay special attention to bases 14 and 16, as these are near the upper human scale limit. We see that some posters consider 14 human scale, but not 16:

Bases 8-15: «Natural-scale» bases, and I would say 10, 12 and 14 stand out as particularly practical bases for civilization.
Bases 16-30: «Higher natural-scale» bases. There is a transition here...

Double Sharp:
2. Human-scale bases {7, 8, 9, 10, 11, 12, 14, 15}, from septenary to pentadecimal, excluding the very resistive tridecimal {13}. These are the bases that can be wielded with current algorithms, and would take less than twice the time it does currently to teach multiplication. At my most pessimistic the upper limit shrinks down to duodecimal, but usually I think {14, 15}'s close kinship should help a little. The best bases here are probably {10, 12}.

In fact, here are more threads discussing the multiplication tables in bases 14 and 16:

Double Sharp:
Here is a line-by-line approach to tetradecimal multiplication:

The 1 and 10 times tables
1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, 10
10, 20, 30, 40, 50, 60, 70, 80, 90, a0, b0, c0, d0, 100

These are very, very trivial.

The 2 and 7 times tables
2, 4, 6, 8, a, c, 10, 12, 14, 16, 18, 1a, 1c, 20
7, 10, 17, 20, 27, 30, 37, 40, 47, 50, 57, 60, 67, 70

These are divisor rows and are thus very simple. The first is in a sense already covered in the doubles in addition. A low example is 6+6, because the natural way to express c on the fingers is to show 6 on each side. A high example is b+b. Normally in decimal you could guess based on the high evens from 12 to 20, but perhaps our heptadactyls from the initialisation would see that the obvious way to express b on just one side is 7 toes plus 4 fingers, leading to 8 on the hands plus 10 on the feet giving 18.

The opening of the sevens row then follows from the limbs easily till 20, and it is obvious how it must continue.

And let's skip to the bottom of this post:

The squares
1, 4, 9, 12, 1b, 28, 37, 48, 5b, 72, 89, a4, c1, 100

These usually come easily.

The 9 and b times tables
9, 14, 1d, 28, 33, 3c, 47, 52, 5b, 66, 71, 7a, 85, 90
b, 18, 25, 32, 3d, 4a, 57, 64, 71, 7c, 89, 96, a3, b0

These rows are very difficult, but almost everything in them has already been covered, so the only difficult fact becomes 9*b=71.

Here's a similar thread for hexadecimal:

We notice that Double Sharp first lists divisors of the base as easy, and then later on mentions squares as easy. In an earlier link, icarus gives "easily memorized product maps" -- these are rows which include multiples of the base.

We see that in bases like 14 and 16, neither Double Sharp nor icarus learned the entire table at once, but focuses on the easier products first. Common Core recommends the same in decimal -- which is why there are so many references to tens, doubles, and so on.


I wish to end this traditionalist post by linking to another heated Barry Garelick thread:

It's one of the few threads where one commenter actually stands up to the traditionalists:

Aaron Blackwelder:

You are critical of the Edutopia article that promotes progressive trends in education but do not offer any research-based counter arguments to substantiate your claim that traditional drill and kill, sit in rows, do worksheets, compliance-based education is superior to 21st Century models that are based on the work of Piaget, Montessori, and Gardner.
I’m wondering if you’re expecting your work to be taken seriously.
Here's a response by a traditionalist:

Tara Houle:

Furthermore, it is not Barry’s responsibility to defend what he’s written here. Rather, the onus is on those proposing these changes to offer up successful examples of why these changes ought to be made. If you could please provide examples of any jurisdiction that has had more success with pbl/inquiry based learning, especially with arithmetic, over conventional methods, I’d be happy to read it. But please…if we are to consider that education is a profession, I would ask the courtesy of providing empirical data that supports these illustrations. Nothing less will do.

Today's post is all about learning arithmetic. The original Edutropia article doesn't mention a grade level, but if it's suggesting project-based learning in the primary years, then I must actually side with the traditionalists.

Plus, I notice that the original article mentions controversial classroom management methods. Since I've been struggling with management during my first year, the last thing I need to read are such controversial methods.

I have yet to decide when my next post will be, but hopefully I'll wrap up one of the many loose ends that I've set up in my summer blogging!

Saturday, July 22, 2017

Pi Approximation Day 2017

Today is Pi Approximation Day, so-named because July 22nd -- written internationally as 22/7 -- is approximately equal to pi.

It also marks three full years since my very first post here on the blog. Yes, I've declared Pi Day of the Century -- March 14th, 2015, to be the ceremonial start date of this blog. But the chronological start date was July 22nd, 2014. I like celebrating both Pi Day and Pi Approximation Day -- including eating pie both days -- and so I celebrate both of them as launch dates for the blog.

Last year, I used the anniversary to create a FAQ about my blog and myself. I do the same this year -- and this year's FAQ will reflect the changes that have happened to my career since last year. As usual, let me include a table of contents for this FAQ:

1. Who am I? Am I a math teacher?
2. Who is Theoni Pappas?
3. How did I approximate pi in my classroom?
4. What is the U of Chicago text?
5. Who is Fawn Nguyen?
6. Who are the traditionalists?
7. What are number bases?
8. Why wasn't my first year of teaching as successful as I'd like?
9. What did I do about cell phones in the classroom?
10. How should have I stated my most important classroom rule?

1. Who am I? Am I a math teacher?

I am David Walker. I'm trying to start a career as a math teacher, but I must admit that my career isn't off to as good a start as I'd like.

Last year was my first as a teacher at a charter middle school, but I left that classroom in March. And this year, I was just about to start at a charter high school, but I was told that enrollment at the school is declining, and so they decided not to hire me as a teacher.

By now, I fear that I won't be hired to teach this year at all. If I'm not hired, then I'll try to return to substitute teaching. But this will make the launch of my teaching career that much more difficult.

So the answer to this question is no, I'm not currently a math teacher, but I'm hoping that I'll be one again soon.

2. Who is Theoni Pappas?

Theoni Pappas is the author of The Mathematics Calendar. In most years, Pappas produces a calendar that provides a math problem each day. The answer to each question is the date. I enjoyed, but only rarely wrote of, the Pappas calendar during the first three years of this blog, 2014, 2015, and 2016.

Last year, as I was about to begin my first year of teaching, I came up with the idea of starting each class with my own Pappas question -- a Warm-Up question whose answer is the date. But this quickly fell apart when I found out that our school curriculum provided a "daily assessment" question that I was required to give as the Warm-Up. Of course, the answers to these questions weren't just the date.

I was disappointed when Pappas didn't publish her calendar for 2017. So in January and February, I had neither my own Pappas calendar in the classroom Warm-Up, nor any questions directly written by Pappas for my home calendar.

In March, my local library held its biannual book sale. For fifty cents, I found a book actually written by Pappas, The Magic of Mathematics, published in 1994. And so I decided to blog about one page from her book everyday that I post for the rest of this year, until 2018, when it appears that she will publish her actual Mathematics Calendar once again.

I wanted to retain the tradition of incorporating the date in choosing a page to blog about. And so I decided that the page number will be the same as the (Julian) day count of the year. For example, today is July 22nd, the 203rd day of the year, so I'll blog about page 203 today.

This is what Theoni Pappas writes on page 203 of her Magic of Mathematics:

"Ever wonder, irritatedly, how it is you receive so much junk mail -- catalogues you never requested, offers for 'super' buys? How were your name and address accessed? Welcome to the electronic age and the loss of privacy."

This page is in Chapter 8, "The Computer Revolution." It is the only page of the section titled "Cryptography: Anarchy, Cyberpunks, & Remailers."

Keep in mind that Pappas wrote the above in 1994, when the Internet was in its infancy. Imagine how much easier it is to access personal information now that the Internet is in its prime.

It also reminds me of the old show Square One TV. In a Mathnet segment produced a few years before Pappas wrote her book, the villain was a woman named I.O. Privacy. She used personal information on a computer to send mailers to those who were most likely to spend a weekend on vacation at Pocono's Paradise. When the victims were out of town, she would send her henchmen out to their homes and rob them. Oh, and the culprit's initials, I.O., stood for "Invasion Of."

So Pappas warns us that living in the digital age has its drawbacks as well as its advantages. Here's how she ends this section:

"Many feel that if governments can use methods and devices to encode important information, an individual also has a right to use similar methods to insure personal privacy."

3. How did I approximate pi in my classroom?

Since today is Pi Approximation Day, I should write something about approximating pi. I won't let my disappointment -- of thinking I'd be hired at a new job only to be denied -- get in the way of celebrating the special day.

The dominant approximation of pi in my classroom was, of course, 3.14. I didn't quite reach the unit on pi in my seventh grade class -- Grade 7 being the year that pi first appears in the current Common Core Standards. But I did reach a unit on the volume of cylinders, cones, and spheres in my eighth grade class.

The main driver of my use of 3.14 as the only approximation of pi was IXL. I used this software to review the volume problems with my eighth graders. The software requires students to use 3.14 as pi, even though I provided them with scientific calculators with a pi key. For example, the volume of a cylinder of radius and height both 2 is 8pi cubic units. IXL expects students to enter 8(3.14) = 25.12, even though 8pi rounded to the nearest hundredth is 25.13. IXL will charge the students with an incorrect answer if they enter 25.13 instead of 25.12. And even on written tests, I didn't want to confuse the students by telling to do something different for written problems, and so I only used 3.14 for pi throughout the class.

But there was another problem with our use of calculators to find volumes. Some of the calculators were in a mode to convert all decimals into fractions. Thus 25.12 appeared as 628/25 -- and I couldn't figure out how to put them in decimal mode. This actually allowed me to catch cheaters -- only one of the calculators was in mixed number mode. Here was one of the problems from that actual test, where students had to find the volume of a cylinder of radius and height both 3:

V = pi r^2 h
V = (3.14)(3)(3)(3)
V = 84 39/50

so it displayed 84 39/50 instead of 84.78 or 4239/50. So any student who wrote 84 39/50 on the test -- other than the one I knew had the mixed number calculator -- must have been cheating.

Here's an interesting question -- suppose instead of catching cheaters, my goal was to make it as easy as possible on my students as well as my support staff member, who was grading the tests. That is, let's say I want to choose radii and heights carefully so that, by using 3.14 for pi, the volume would work out to be a whole number (which appears the same on all calculators regardless of mode). How could have I gone about this?

The main approximation of pi, 3.14, converts to 314/100 = 157/50. It's fortunate that 314 is even, so that the denominator reduces to 50 rather than 100. So our goal now is to choose a radius and height so that we can cancel the 50 remaining in the denominator.

We see that 50 factors as 2 * 5^2. The volume of a cylinder is V = pi r^2 h -- and since r is squared in this equation, making it a multiple of five cancels 5^2. It only remains to make h even to cancel out the last factor of two in the denominator. Let's check to see whether this works for r = 5, h = 2:

V = pi r^2 h
V = (3.14)(5)^2(2)
V = 157 cubic units

which is a whole number, so it works.

If we had a cone rather than a cylinder, then there is a factor of 1/3 to deal with. We can resolve the extra three in the denominator by making h a multiple of six (since it's already even). Let's check that this works for r = 5, h = 6:

V = (1/3) pi r^2 h
V = (1/3)(3.14)(5)^2(6)
V = 157 cubic units

Spheres, though, are the trickiest to make come out to be a whole number. This is because spheres, with the volume formula V = (4/3) pi r^3, have only r with no h. Thus r must have all of the factors necessary to cancel the denominator. Fortunately, the sphere formula contains the factor 4/3, and the four in the numerator already takes care of the factor of two in the denominator 50. And so r only needs to carry a factor of five (only one factor is needed since r is cubed) and three (in order to take care of the denominator of 4/3). So r must be a mutliple of 15. Let's check r = 15:

V = (4/3) pi r^3
V = (4/3)(3.14)(15)^3
V = 14130 cubic units

Notice that the volume of a sphere of radius 15 is actually 14137 to the nearest cubic unit -- that's how large the error gets by using 3.14 for pi. Nonetheless, 15 is the smallest radius for which we can get a whole number as the volume by using pi = 3.14.

Now today, Pi Approximation Day, is all about the approximation 22/7 for pi. Notice that if we were to use 22/7 as pi instead of 157/50, obtaining a whole number for the volume would be easier.

For cylinders, the only factor to worry about in the denominator is seven. Either the radius or the height can carry this factor. So let's try r = 1, h = 7:

V = pi r^2 h
V = (22/7)(1^2)(7)
V = 22 cubic units

For cones, we also have the three in 1/3 to resolve. Let's be different and let the radius carry the factor of three this time. For r = 3, h = 7, we have:

V = (1/3) pi r^2 h
V = (1/3)(22/7)(3^2)(7)
V = 66 cubic units

For spheres, unfortunately our radius must carry both three (for 4/3) and seven, and so the smallest possible whole number radius is 21:

V = (4/3) pi r^3
V = (4/3)(22/7)(21)^3
V = 38808 cubic units

This is larger than the radius of 15 we used for pi = 3.14. But notice that there are many extra factors of two around in the numerator -- 4/3 has two factors and 22/7 has one. This means that we can cut our radius of 21 in half. Even though 21/2 is not a whole number, the volume is nonetheless whole:

V = (4/3) pi r^3
V = (4/3)(22/7)(21/2)^3
V = 4851 cubic units

To make it easier on the students, I could present the radius as 10.5 instead of 21/2. (Recall that the students can easily enter decimals on the calculator -- they just can't display them.) There is still some error associated with the approximation pi = 22/7, as the volume of a sphere of radius 10.5 is actually 4849 to the nearest cubic unit, not 4851. Still, 22/7 is a slightly better approximation than 157/50 is, with a much smaller denominator to boot.

With such possibilities for integer volumes, I could have made a test that is easy for my students to take and easy for the grader to grade. I avoided multiple choice on my original test since I didn't want to give decimal choices for students with calculators in fraction mode (or vice versa). With whole number answers, multiple choice becomes more feasible. Of course, the wrong choices would play to common errors (confusing radius with diameter, forgetting 1/3 for cones).

4. What is the U of Chicago text?

In 1991, the University of Chicago School Mathematics Project published a series of secondary mathematics texts. I was able to purchase both the Algebra I and Geometry texts for two dollars each at a local public library. One thing I discovered was that the U of Chicago Geometry text is based on the same transformations that appear in the Common Core Standards -- even though the U of Chicago published the text nearly two decades before the advent of the Core.

The common denominator is NCTM, the National Council of Teachers of Mathematics. This group was behind both the U of Chicago text and the Common Core Standards. Still, this means that one can satisfy most of the Common Core Standards by teaching out of the U of Chicago text.

There are newer versions of the Geometry text than the one that I own. I've read that while my old version introduces reflections in Chapter 4 and the other isometries in Chapter 6, in the new version all isometries appear in Chapter 4.

The U of Chicago text is not the only book that I mention here on the blog, not by a long shot. Many posts here refer to the Illinois State text. This is mainly because I was a math teacher, and I used the Illinois State text when teaching at my old school. 

To supplement the Illinois State text (mainly when creating homework packets), I used copies of a Saxon Algebra 1/2 text and a Saxon 65 text for fifth and sixth graders.

To learn more about some of these texts, here are some links to a webpage by Cathy Duffy. Duffy's goal is to help homeschooling parents choose a textbook for their children. Here is her review of the U of Chicago text:

and the Saxon series:

5. Who is Fawn Nguyen?

For years, Fawn Nguyen was the only blogger I knew who was a middle school math teacher. Back when I was a middle school teacher, I enjoyed reading Nguyen's blog, but now that I've left, it's not as important for me to focus on her blog over all others.

Nonetheless, let's take a look at Nguyen's blog today. She hasn't posted in two months, and so the following link is to her most recent post:

This is a computer program that takes two whole numbers as input. Then the computer draws a rectangle whose dimensions are these two numbers.

Afterward, the computer performs the following algorithm -- inside the rectangle, it draws the largest possible square that fits inside the rectangle. The side length of this square will equal the smaller dimension of the rectangle. If there is a rectangle left over after cutting out the square, then the computer performs the same algorithm on the remaining rectangle. The algorithm terminates when the final rectangle is itself a square.

The name "Euclid's Algorithm" reveals its origins -- the Greek mathematician proved that if the sides of the original rectangle are whole (or rational), then the algorithm terminates. Euclid referred to the side length of the final square as the "common measure" of the dimensions of the original rectangle, and nowadays we can think of this length as the GCF of the original whole number dimensions.

But not all lengths have a common measure. Some are incommensurate (irrational). The most famous example of a rectangle with incommensurate dimensions is the golden rectangle. If a square is taken away from a golden rectangle, the result is a rectangle that is similar to the original figure. And if a square is taken away from the remainder, the result is again a golden rectangle -- thus Euclid's algorithm fails to halt for this rectangle. The ratio of the dimensions of a rectangle is (1+sqrt(5))/2, which I mentioned in another post about a month ago as the golden ratio, Phi.

But today is Pi Approximation Day, not Phi Approximation Day, so let's try Euclid's algorithm on a rectangle with length pi. For simplicity, the width of the rectangle will just be 1. (By now we've left Nguyen's original post long behind, since her class only dealt with whole number lengths.) We can approximate pi by pretending that there is no remaining rectangle, which would then make the dimensions rational.

-- Our starting rectangle has dimensions (pi, 1).
-- The first iteration allows us to cut three unit squares from the rectangle. Ignoring the remainder would then imply that pi = 3 -- and actually, Nguyen does do the (3, 1) case. And indeed, this crude approximation pi = 3 is implied in the Bible.
-- The second iteration takes the remaining rectangle of dimensions (pi - 3, 1). Seven small squares can be cut from this rectangle. Ignoring the remainder would then imply that pi = 3 + 1/7 (since the seven small squares would then have the same length as one of the large squares). This, of course, is equal to 22/7, which makes today Pi Approximation Day. The Greek mathematician Archimedes was the first to use this approximation.
-- The third iteration takes the remaining rectangle and cuts out one more smaller square. Ignoring the remainder would then imply that pi = 3 + 1/8 = 25/8 (since it's now as if eight small squares can fit on the side of the unit square). The ancient Babylonians used this approximation.
-- The fourth iteration takes the remaining rectangle and cuts out 15 smaller squares. Ignoring the remainder would then imply that pi = 333/106. This is less obvious, but if we were to take Nguyen's program and enter (333/106, 1) -- actually (333, 106) has the same effect -- we would see three large squares, then seven medium squares, then one square of its own size, and finally fifteen of the smallest squares. There is a theory that this approximation also appears in the Bible -- here's a link to a Reddit post (written exactly six years ago on Pi Approximation Day):

-- The fifth iteration takes the remaining rectangle and cuts out one more smaller square. Ignoring the remainder would then imply that pi = 355/113. Again, we can enter (355, 113) in Nguyen's program and see that there is one more smallest square than there was for (333, 106). This approximation was known to the ancient Chinese.
-- The sixth iteration takes the remaining rectangle -- which is very thin, because 355/113 is such a great approximation. Indeed, the remainder is 292 times as wide as it is long. And so we normally cut off the approximation at 355/113.

The rational approximations 3, 22/7, 25/8, 333/106, and 355/113 are called "continued fractions." We can use continued fractions whenever we need to find a rational approximation. (Calendar reform and microtonal music are two topics mentioned on the blog that can make use of continued fractions, though I didn't explain how in any of those posts.)

I hope Fawn Nguyen posts again on her blog soon!

6. Who are the traditionalists?

I use the phrase "traditionalists" to refer to those who oppose Common Core for mathematical -- not political -- reasons. The traditionalists tend to believe that many of the topics in the Common Core Math Standards are taught too late.

My own relationship with the traditionalists is quite complex. I tend to agree with the traditionalists regarding elementary school math and disagree with them regarding high school math.

For example, they believe that the standards in Grades 4-6 that require the use of the standard algorithm for arithmetic should move down at least one grade level. I tend to agree with the traditionalists on this point here -- most of the reasons for opposing direct instruction, rote memorization, and drill don't apply to the youngest students. I remember once learning about two nonstandard algorithms for addition -- the Left-Right Method and the Plus-Minus Method -- on an old 15-minute math show for sixth graders, "Solve It," that used to air on our local PBS station back when I was that age. For mental math, these nonstandard algorithms are great, but for a young elementary student first learning to add two-digit numbers I prefer the standard algorithm.

On the other hand, the traditionalists also prefer that high school seniors take AP Calculus in order to open the door to college STEM majors. In particular, some STEM colleges here in Southern California, such as Cal Tech and Harvey Mudd, require calculus for admission. Even colleges that don't formally require calculus are likely to send students rejection letters if they apply without having calculus on the schedule, especially if they apply to STEM majors. Therefore they want the classes leading up to senior year to prepare students to take AP Calculus -- including Algebra I in the eighth grade. They oppose Common Core because its eighth grade math standards do not comprise a full Algebra I class. I only partially agree because I don't want seniors who can't handle AP Calculus or eighth graders who can't handle Algebra I to be labeled failures. Just as AP Calculus can open the door to STEM, it can close the door to a non-STEM career if students are forced to spend too much effort trying to pass their high school math classes.

OK, so I support the traditional pedagogy with regards to elementary math and oppose it with regards to high school math. But what's my opinion regarding traditionalism in middle school math -- you know, the grades that I'll actually be teaching? In middle school, I actually prefer a blend between the traditional and progressive philosophies. And the part of traditionalism that I agree with in middle school is making sure that the students do have the basic skills in arithmetic -- since, after all, that is elementary school math, and with elementary math I prefer traditionalism.

I've referred to many specific traditionalists during my three years of posting on the blog. The traditionalist who is currently the most active is Barry Garelick. Here is a link to his most recent post:

Garelick teaches middle school math right here in California. In this post, he criticizes the famous businessman Elon Musk for encouraging Project-Based Learning instead of traditional math lessons.

He highlights two commenters in this post. One of them is SteveH, a traditionalist in his own right. I consider SteveH to be a co-author of Garelick's blog since he posts there so often.

But today I wish to focus on the other commenter, Richard Phelps. Here's what he writes:

My high school physics class — in the early seventies — comprised project after project. Teacher called it “Harvard Physics.” Each class period was filled with setting up equipment and building contraptions, in small groups. Essentially, it was a “lab only” course. I learned less in that course than in any other over 4 years of high school, and I’m including gym in the comparison. Typically, an entire class period was devoted to delivering just one fact or concept — “authentically” — that could have been simply told to us in less than a minute or, with some discussion, in just a few minutes — un-authentically. Moreover, so much of our attention was focused on building the contraptions and getting them to work that, in most cases, the one factoid we were supposed to learn was lost in the morass of mostly irrelevant information.

Some traditionalists are satisfied by returning to traditional math while conceding that projects are at least useful in science. But apparently, to others like Phelps, even science should be project-free. To Phelps and other science traditionalists, students should spend most of science class either taking notes based on a teacher's lecture (which could include a lab demonstration, but only the teacher is performing the lab, not the students), or answering questions out of the text. Anything else, to the science traditionalists, is not learning.

Here's my problem with what Phelps is saying -- I don't know what state he is in, but I know that Garelick is a fellow Californian. Here in the Golden State, the two major university systems (UC and CSU) have what's known as the a-g requirements for college admission. Let's look at the a-g requirement for science:

Laboratory Science ("d")
Two units (equivalent to two years or four semesters) of laboratory science are required (three years are strongly recommended), providing fundamental knowledge in two of the following disciplines:
[emphasis mine]
-- Biology
-- Chemistry
-- Physics

So we see that the UC and CSU systems expect high school students to take laboratory science. A student who takes the lecture science courses that Phelps prefers would not be admitted to a UC, since the UC explicitly states that it expects incoming students to have had laboratory science.

Last year, I was hired as a math teacher at my middle school. But I suddenly found out that there was no science teacher, and I unexpectedly had to teach science as well as math.

Meanwhile, the Illinois State text that I mentioned above is strongly project-based, and I was required to have students complete a project once every two weeks. It is undoubtedly a curriculum of which the traditionalists would disapprove.

Now here's the thing -- one day, I tried to pass out a worksheet for science, and my eighth graders complained about it. I wrote about this in my January 27th post:

-- On [January] 13th, I decide to print the students a Study Island worksheet on Human Interactions. The idea is that this lesson would bridge the gap to the Green Team next month. But then partway through the lesson, the students complain that it was too boring and refused to work on it. One girl who had transferred in from another school told me that her old school had a real science teacher who gave interesting projects, such as an Edible Cell Model. I agree to try something similar in my class.

The girl told me that my lessons weren't "fun" enough. When I tried to explain that lessons should not be "fun," but instead should allow them to learn a lot (as Garelick, SteveH, and Phelps would argue), she countered that she'd learned much from her previous class, yet she nonetheless had fun then.

I suspect if I were to tell this story to a traditionalist, he'd tell me that this was insufficient evidence that science classes should be fun. "So," our hypothetical traditionalist would ask, "when you finally gave the project, the girl was engaged, right?"

Well, let's find out:

-- On [January] 17th (after the three-day weekend) I introduce an Edible Molecule Project to be completed at home, just like the Edible Cell Model from last year. But the students reject this as well, telling me that they can't do it since they haven't learned anything about molecules yet. This includes the new girl, who probably never wanted to do the project in the first place. She only brought it up in order to tell me how her old teacher was a much better science teacher than I am (which is true, since she was a genuine science teacher and I'm not).

"Oh," our hypothetical traditionalist responds, "there were other issues in the class, and so this doesn't show that science teachers should give more projects."

And part of the problem was that as a natural math teacher, I rarely covered science, and so it's understandable that the students would reject something that was unexpected. I suspect that if I had covered science the entire year (as I should have), my science lessons would have been accepted -- regardless of whether they were project-based or traditional.

I plan on writing about my eighth grade class -- indeed about one particular student. Earlier I wrote that I'd post it soon, and in fact I'd had nearly the entire post already prepared for posting. But that was the day I found out that I wasn't going to be hired at the new school. The major disappointment weakened my will to post, especially about teaching -- and besides, I would need to spend extra time applying for teaching jobs. Instead, I wrote about topics from books written by both Theoni Pappas and Eugenia Cheng.

Notice that the Illinois State curriculum covers both math and science. As I think about my science failure more and more, I now realize that I should have dropped the Illinois State math projects and covered only the science projects. Indeed, this is essentially what my counterpart did at our sister charter school. I could have met the two-week project requirement with only science projects and prepared the students for lab science "d" in high school to boot, and this would have provided me with a loophole to keep the math class more or less traditional.

By the way, I wonder when Phelps believes that students should have their first lab science class. I'm certain that he'd admit that Ph.D candidates need to do original research in labs. So it remains to be seen whether Phelps would accept lab science for lower division, upper division, and MS candidates.

7. What are number bases?

I write about number bases today because they are related to both the Pappas topic (on computers) and the traditionalists (and you'll find out why soon).

Pappas explains:

"The binary system (base two), which uses only 0s and 1s to write its numbers, held the key to communicating with electronic computers -- since 0s and 1s could indicate the 'off' and 'on' position of electricity."

Earlier in the book, Pappas counts from zero to twelve in binary:

0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, ...

As she explains, the first mathematician to work in binary was the famous Scottish mathematician John Napier. This was before computers, so he used a chessboard to display his binary numbers.

Pappas writes:

"For example, to add 74 + 99 + 46, each number is written out in a row of the chess board by placing markers in the appropriate squares of the row so that the sum of the markers' the number they represent. 74 has markers at 64, 8, and 2 since 64 + 8 + 2 = 74."

In modern binary notation, we would write 74 as 1001010. Pappas proceeds:

"After each number is expressed on the chess board, the numbers are added by gathering the markers vertically down on the bottom row. Two markers sharing the same square equal one marker to their immediate left. So two '2' markers produce one '4' marker. Working from right to left, any two markers sharing the same square are removed and replaced by one marker in the adjacent square at the left. At the end of this process, no square will have more than one marker. The sum of the values of the remaining markers represents the sum of the numbers."

Here's the connection to traditionalism -- notice that this is actually the standard algorithm for adding in base two. The line about "any two markers sharing the same square are removed and replaced by one marker in the adjacent square" corresponds to "carry the one" in the standard algorithm.

And so we see that by learning about other number bases, we can shed more light on how young students learn arithmetic in base ten.

There are other number bases besides base two, of course. There is, in fact, an entire message board devoted to base 12, or dozenal:

This website is the de facto home of number bases in general. Indeed, the members of this site come up with their own words to describe the properties of number bases. This actually deserves a post of its own, and indeed I'll be making such a post soon. I admit that lately on the blog, "soon" essentially means "whenever." Well, let's just say I'll post about number bases before we reach the last page of the Pappas chapter on computers (page 221, which corresponds to August 9th).

8. Why wasn't my first year of teaching as successful as I'd like?

I've been reflecting on this for the past four months. And I've realized something -- whenever I have a bad day in the classroom, whether it was this year or back when I was a sub, the reason for that bad day can be summarized in a single sentence:

The students were led to believe that they could do whatever they wanted, then they resisted when they found out that they couldn't.

I remember my first day of substitute teaching, years ago in an eighth grade math class. Several of the students refused to do any work, and I kept nagging at them to do it. The students eventually rebelled, and during one period, they pulled a classic sub trick -- they all jumped up at the same time. I asked myself, what was I thinking making the students do work when I'm just a sub? And so I thought it was better not to expect too much from students -- at least the students wouldn't rebel against me the way they did that first day. I was wrong to believe this -- yet it permeated my entire line of thinking when I was in the classroom, all the way until this year.

The real reason the students jumped up that was that I didn't make it clear from the start of the period that I expected them to be quiet and do work. When I was taking roll, the students continued to talk, yet I didn't interrupt roll to tell them to stop. From this, they concluded that they could do whatever they wanted, and so they were shocked when I tried to punish them for not doing their work. If I had made it clear that there was to be no talking even during the roll, they probably would not have jumped up when I told them to work.

I don't write about classroom management often on the blog. This year, however, I wrote extensively about management in several posts labeled MTBoS (Math Twitter Blogosphere).

In last year's FAQ, I wrote about E's, S's, and U's -- conduct grades used by the LAUSD. My charter school didn't use these conduct marks, yet I still mentally divide my students in into three groups corresponding to these conduct grades.

Also in that FAQ, I wrote about a participation points system that I would use this year. I was hoping to use this as my primary management system, but it failed. This is what I wrote:

At the start of each unit (that is, right after the test), each student has two Participation Points. I will award a point each time a student gives a correct answer, to a Warm-Up or any question that I ask during the main part of the lesson.But if a student fails to participate or otherwise misbehaves, I'll deduct a point. If the student's Participation Point total drops below zero, I will begin assigning consequences, beginning with a minute of detention for each additional point below zero. I expect these detentions to be the most effective in my eighth grade classes, because based on my school's block schedule, Math 8 always meets right before nutrition or lunch, so students will want to avoid those detentions. (Math 7 also meets before nutrition on certain days of the week.) Failure to show up to detention results in a doubling of the detention time, and further consequences occur when the total detention time exceeds a certain amount. 

The problem was that in all three classes (but especially sixth grade), there were many students who were smart, yet very talkative. They would run up their Participation Point total by answering questions, and so when they lost points, it was never enough to earn them a detention.

Since the students weren't punished, they thought that it was okay to talk. Soon it was similar to the worst days as a sub -- since they believed talking was OK, they were shocked when it was time for them to work, and I ended up yelling at them. My biggest fear from last year was realized -- all of the S-students became U-students.

When I came up with the point system, I wanted to make it as easy as possible. So I gave them a single score to represent both academics and behavior. This is what Eugenia Cheng warned about in her Beyond Infinity, when she wrote about using one variable to represent two dimensional data.

And so I should have kept track of academics and behavior separately. I could still have awarded points for correct answers. But I should never have deducted points for bad behavior. Behavior needed to be addressed with a discipline hierarchy separate from Participation Points.

9. What did I do about cell phones in the classroom?

I addressed this as part of last year's FAQ. I wrote:

And so, at the end of any detention earned due to cell phone use, I will require the student to say "Without math, there wouldn't be any cell phones" before releasing them. And if by chance I must confiscate a phone, I will require the student to say the same before returning the phone.

And yes, I really did make them say this last year. But I had a completely separate problem with cell phone use this year.

I would see a student apparently typing something on a cell phone. I would tell her to put the phone away, and then she'd claim that she didn't have a phone out. Then I'd say that I see her with a phone out -- only for her to reveal that she'd been pressing on her cell phone case, not the phone itself.

The idea here is that I can't reliably tell whether a student has a phone case or actual phone out, and so the only fair thing to do is let students use phones whenever they wanted. The argument was that it's far better to let ten students use phones with no punishment whatsoever than to punish even one innocent student for merely having a phone case out.

Of course, students aren't entertained by playing on phone cases -- they are entertained by playing on actual phones. The sole purpose of playing with the case was to "neuter" the no cell phone rule -- by exploiting a loophole to prove that the rule is "unfair."

In reality, a student playing on a phone case isn't innocent. Such a student is trying to create a classroom where she can use her phone with no punishment. Her goal obviously isn't to make herself smarter, learn a lot of math, or become a valedictorian.

Thus a student who plays with a cell phone case deserves punishment -- yet I was powerless to do so, since nowhere in my rules was it stated, "no cell phone cases."

This leads to our last FAQ item:

10. How should have I stated my most important classroom rule?

This is what I wrote in last year's FAQ:

Rule #3: Respect yourself and others.

Students respect themselves and each other as well as me, the teacher, by following all rules and allowing others to learn the material.

This rule worked for Fawn Nguyen, the teacher from whom I got this rule. But it didn't work in my classroom at all. Of course, a student playing with a phone case isn't respecting the teacher, but I needed a rule that would require the student to put the case away immediately.

Here's a much better rule:

Rule #1: Follow all adult directions.

And so if I say, "Put the phone case away," the student couldn't counter that phone cases were against the rules, because my direction was to put the case away.

If I ever find myself in the classroom again, this will definitely be my first and most important rule.

OK, here are some Pi Approximation Day video links:

1. Draw Curiosity

Notice that this video, from last year, actually acknowledges Pi Approximation Day.

2. Reed Reels

This video mentions that even Legendre -- the author of the geometry text we've been discussing in some of my recent posts -- contributed to the history of pi. In particular, Legendre proved that not only is pi irrational, but so is pi^2 (unlike sqrt(2), which is irrational yet has a rational square, namely 2).

This video also mentions a quote from the mathematician John von Neumann: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” We've seen Eugenia Cheng, in both of her books, say almost exactly the same thing.

3. Fact Retriever

This video gives ten amazing facts about pi.

4. Numberphile

No Pi (Approximation) Day is complete without a Numberphile video. In this video, we learn that pi is not just irrational, but is also transcendental -- that is, there doesn't even exist a polynomial f, with integer or rational coefficients, such that f (pi) = 0 (unlike sqrt(2), which is algebraic). This is even stronger than Legendre's result that pi^2 is irrational.

5. Sharon Serano

Well, I already gave ten facts about pi, and so this video is twice as good.

6. Vi Hart

No, I can't avoid another one of my favorite math YouTube posters, Vi Hart. Of course, in this video she argues that there's nothing special about pi at all

7. Danica McKellar

I've mentioned McKellar's video in my past Pi (Approximation) Day posts, and it's a favorite, so I'm posting it again this year.

8. A Song Scout

This is another pi song based on its digits. Unlike Michael Blake's song (listed below), it is in the key of A minor rather than C major.

9. Michael Blake

I like the idea of using music to teach math, so here's a favorite, Michael Blake's "What Pi Sounds Like." I tried to find a way to incorporate songs such as this one into the classroom.

10. The Ancient Melodies

Going back to number bases, this song uses the digits of pi in dozenal, or base 12. Dozenal was chosen because we commonly use the 12EDO scale in Western music.

Bonus. Radomir Nowotarksi

This song is actually based on the Fibonacci sequence, not pi. I mention it here because in a previous post, I wrote about a Numberphile video which featured a song where the notes (1 = C, 2 = D, 3 = E, up to 7 = B) were based on reducing the Fibonacci numbers mod 7. I never found that video, but here Nowotarski is doing the same thing.

Nowotarksi uses the Lydian mode in this song rather than the major scale. This means that note 4 corresponds to F# rather than F. Like Michael Blake, Nowotarksi places a chord on each note, but these are different chords due to the use of F# over F. Where Blake uses a D minor triad (D-F-A), Nowotarksi uses a D major triad (D-F#-A).

I've mentioned that many songs based on digits are manipulated so that they end on Note 1, which is the tonic of the scale. But Blake's song ends on Note 5, since five is the last digit before the first appearance of zero (which he was tried to avoid). Song Scout uses G# for both zero and seven -- and his song actually ends on G#. The dozenal song changes the last digit from Note 5 to Note 4 so it could end on Eb (which serves as a tonic in that section). The Fibonacci song repeats, and each section ends on Note 7.

And so I wish everyone a Happy Pi Approximation Day.