Additional Math Pages & Resources

Friday, April 30, 2010

Littorally Landlocked

I'd like to build on a previous post this week.

Littoral means the shoreline (plus some distance above it and below it). Now imagine a country, next to a large body of water -  we say it possesses littoral territory.

But how much territory? How far out into the water can the country claim ownership? The so-called Territorial Waters. When you go outside these waters you are on The High Seas.

We talked about territorial waters months ago in a blog, where I told you that technically a ship in territorial waters should set its clocks to match the time in the country that claimed the water. When you are on the high seas you can more or less do what you like. But if you commit a crime like piracy, ANY nation can come after you.

Pirates and slave ships on the high seas can be targeted by any nation's navy, under the legal doctrine of Hostis humani generis which means "enemy of mankind."

This concept of territory became a topic of interest in San Diego 30-40 years ago when we had  a large tuna-fishing industry. As countries along the South American coast began to claim fishing rights to the water out 200 miles, it restricted where our fishermen could hunt. And it has caused a few international incidents.

Under United Nations treaties, territorial waters extend a maximum of 12 miles from the mean low tide line of the adjoining shore. This ownership applies to waters BELOW the surface and to the airspace ABOVE the surface. Here's a diagram from Wikipedia to show you what I mean.

You can see that like littoral zones, there are many fine distinctions about a country's waters - internal, territorial, contiguous, exclusive economic, international, etc.

But maybe this means nothing to you, because you live in one of the 44 land-locked countries. That means you have no direct access to the sea. You have no fishing rights. You have to bring in goods by plane or be subject to tariffs imposed by your neighbors.

Doubly-landlocked countries have at least two borders to cross before they can get to the beach! Only Lichtenstein and Uzbekistan have that problem. San Marino and the Vatican are both completely surrounded by Italy and only have that one route (cross Italy) to get to a body of water.

You might think that the USA has only Mexico and Canada as neighboring countries. That's true if you only count "land" neighbors. But 21 countries are considered to be our water neighbors.

Who are your nearest neighbors? Go here to find out.

Assuming your country isn't landlocked, how much coastline does your country have? That's another problem, and we'll save it for next week.

Here's are some startling facts - Canada has 10 times the coastline of the United States, even though the area of Canada is only 10% bigger than the USA.

Final fact  - Norway has more coastline that the USA, even though it's only 1/30th the land area.

Thursday, April 29, 2010

Combinations galore

Combinations. They are taught in math as a prelude to probability. How many of these times how many of those equals this many to choose from.

We normally give examples in clothing - 3 pairs of pants, 5 shirts, 2 kinds of shoes and 3 hats.
How many combinations can you make?

3 x 5 x 2 x 3 = 90

But I saw an article about a new hamburger chain in town that offers 321,120 combinations!

As I often tell my wife, more choices aren't always a good thing.

Here's the menu from the website of the burger place, in case you are already hungry. (looks suspiciously veggie to me)

Step 1 Meat choice = 4 x 3 x 3 = 36 variations choose 1
Step 2 Cheese = 12 variations choose 1
Step 3 Toppings = 30 variations choose 4
Step 4 Sauce = 21 variations choose 1
Step 5 Bun = 3 variations choose 1 IF you chose a bun in Step 1

36 x 12 x 21 x 3 = 27,217 choices WITHOUT getting into the toppings.

Since we have to choose 4 out of 30 (including premium priced ones) we have a LOT of choices.

If we divide 321,120 (their number) by 27,217 (our number) to get 11.8 which in not the number of topping choices, but it is about what the burger will cost you in dollars.

Math can't solve this without knowing more about the assumptions of the newspaper writer and the actual menu at the restaurant!

Wednesday, April 28, 2010

How can I get more blog traffic?

This is quite possibly the most frequently asked question in the blogosphere. How can I get more traffic?

I KNOW how I could get more traffic. I would have to start out by writing about things that are more interesting than math! But then I could add:
  • Photos of an attractive person of either gender, in a fast-looking car. Like this VW GX3 Trike, and this Lotus M250,  both promised (and shown) but never offered for sale.

  • Food is always popular. I would tell you how to make this healthy salad with heirloom tomatoes, extra-virgin olive oil, marble potatoes, spring mix greens and artisan balsalmic vinegar. Not forgetting the home-baked bread made with stone-ground hard winter wheat from Bob's Red Mill:

  • We could mention a few celebrity names - they're always good for a few clicks:
  • Miley, Ricky, Dolly, Billy-Bob, etc.
  • Put in some song lyrics (I think lyrics is the most-searched-on word of all time):   
  • There is a city of gold ~ Far from the rat-race that eats at your soul ~ Far from the madness and the bars that hold ~ There is a city, a city of gold       
  • Start a discussion on a notorious topic:  
  • Wanna buy some Goldman Sachs housing bond CDOs?
  • Throw in some tips on Windows7: 
  • Migrating from Win98, NT and 2000 in 20 minutes or less!
  • Do a review of the iPad:
  • Getting the iPad's virtual camera to work correctly
  • Cover some sporting events:
    Curling, anyone?
Then I would have to add a few punchy jokes. And some winning stock picks. And horse racing tips.  I'd link to lots of other sites too. Some of the links would take my readers to places that astonish and amaze them. (First I would have to find those places, though ...)

Of course, I would link to a few movies on YouTube. Here's one I took myself. It involves fast cars, designers dressed in black, bright lights, snappy drum playing, smoke and mirrors, wow!

This is a lot of work. And there's no room for math. You came here for math, right? To learn how to use that stuff you studied so hard to learn all those years ago ...

I look at the viewings of my old posts and wonder, Why are these the most popular?  The stars appear to be:
  1. Why does ink cost so much?
  2. Scoville Units (hot chili peppers)
  3. Some are more equal than others

Tuesday, April 27, 2010

An inch here, and inch there - what does it matter?

Today's blog is based on a conversation with my pal Gary, as we sat and looked at the new window in my kitchen. It was replaced last year, but due to some fabrication issues it will have to be replaced again. I think this time I will ask for it to be enlarged a bit. This one is a couple inches too small in each direction. Some of the opening is "wasted."

You might wonder, What difference does a couple inches make in an existing window opening? It's still a window isn't it?

Here's a perfect time to use elementary school math to see the difference.

This photo shows the original window, from 1954 (sorry about the dirty dishes).

We will need the dimensions, so here they are on a drawing.

The old window was about 76 x 36 inches which works out to be 19 square feet of window area. I'm going to adjust for the center posts which are an inch wide (so each of them obstructs about 36 sq in of area). The total glass area is 18.5 square feet.

Here's a picture of the new window going in. You can see the gap around the top of the new window frame. It's difficult to remove the old steel window frames, so we left them in the walls. The decrease in size is measured from the old frame's inner edges, not from the opening in the wall.

Here's the drawing with the new dimensions. These indicate the size of the glass.

The new window has 13.77 square feet of window opening. The pillars in the middle are each 3 inches wide compared to 1 inch for the originals, so we subtract a bit more there.

An inch here and an inch there reduced the window to 74% of its original size (13.77 ÷ 18.50). I think we can do better. We now (after breaking out the old windows) know the sills are level and straight, so we don't need so much extra space around the new window frame.

If we could get half the space back, what would it do to the opening?

73 x 33.5 = 2445.5 - 124 = 2321.5 ÷144 = 16.12 square feet or 87% of the originals.

Is 87 halfway between 100% and 74%? Yes.

Monday, April 26, 2010

Please take this littoraly

Last week we reviewed a number of math words. Over the weekend I ran across another word that could have math implications - littoral.

The word first appeared in our newspaper because a new ship arrived in San Diego Bay, the USS Freedom. It's called an LCS (Littoral Combat Ship) and it's a 400-foot surface vessel intended for operation in the littoral zone (close to shore).  Here it is:

The USS Freedom can sweep for mines, chase drug runners, fight off pirates, and rescue sailors in distress. It can cruise for 3500 nautical miles without refueling and reach speeds of 47 knots (52 mph)! That's why the front deck is so clean - otherwise everything would blow off.

The very next day I read about a community in Brittany (Northwestern France). The author said that oysters are a normal source of protein for "French living on the littoral."

How does math fit into oysters and Navy ships? Well, how wide is the littoral zone? How close to shore? Who determines what's close? Is it how close a 400-foot ship can get to shore? Well, like many things, it turns out to be a very complex subject.

Littoral could be as simple as being from the point of highest tide to the point of the lowest tide (although on a flat beach, that can be miles). Or it could be much farther out, especially if you don't want to be grounding your new ship in the mud. Luckily the USS Freedom's draft is only 13 feet.

Here's a Navy drawing of the littoral zone. They consider it to be from the high tide region out to where the water is about 200 feet deep. It includes the beach, the back shore, the foreshore, the shoreline, the nearshore, the offshore and so on. Marine biologists generally consider the littoral zone to be much narrower.
Here's an oyster. It lives in the inter-tidal and sub-tidal zones (both within the littoral zone), and is not happy about being disturbed by these big Navy ships.

Here's a Google map of our region's littoral zone. There are many underwater features; some of them are shown here.

Friday, April 23, 2010

Inverted and Recurring Words

These math words popped up during yesterday's blog: Inverse [or Inverted] and Recur [or Recurring].  Let's invert the order and define Recur first, okay? (Is this a clever way to define inverse, or what?)

Recurring means continuing, on-going, repeating repeating repeating

In math we could use it like this:

Give the value 2/3 in decimal form. Answer = .66666666  In this case, the 6 is a recurring number.

We could use it like this in a family:

Dave and Katy had a recurring argument over the size of Katy's mobile phone bill!

Or like this:

In her recurring dream, she would always have to give a speech to a large group of people ...

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Now let's get back to inverse. It means turn over; opposite; reverse, negative, "the other way".

In math, we can use it like this:

Give the inverse of the fraction 2/3: Answer = 3/2  Another term for this is reciprocal.

Notice that the product [multiplicative inverse] of these two numbers is one. That means zero can't have a reciprocal, because 0 times a real number is 0, not 1.

Or we can use inverse like this:

Give the inverse of the real number -5: Answer = 5

Notice that the sum [additive inverse] of these two numbers is always zero.

This discussion has reminded me of two more words that we sometimes use in the math curriculum - most often describing coins. These are obverse and reverse.

Obverse means turned to face you. Reverse means turned to face the other way.

We would use it like this:

Which side of a coin is the obverse? The face or "heads" is always the obverse side. In the case of coins without a portrait (the Euro) the obverse is the common side, shared by all variations of the coin.

Which side of a coin is the reverse? The back side or "tails" is always called the reverse.  

These photos show a Roman coin that my friend Ken located in a field in Dorset, England. He gives most of them to the farmers who own the fields, or to museums.

Thursday, April 22, 2010

Reversing and Transposing Numbers

Reversing and transposing are both common math mistakes, but they are not the same thing.
  • Reversing means writing the mirror image of a number.
  • Transposing means writing multiple numbers out of order; changing the order of arrangement
Walking through our next-door-neighbor's parking lot this week, I noticed a 4 written in reverse on the curb:

NOTE - do you know how hard it is to set a number backwards on a web page? Not easy, I assure you!

I was surprised that the street painter would have the stencil upside down while painting. Writing numbers backwards is one thing, but painting a number stencil the wrong way around is a bit careless. Or maybe it was done on purpose? Who IS 24 and why don't they like her/him?

Kids often write their numbers backwards, sometimes even upside-down. To help alleviate that, we give them plenty of chances to practice. You may see this in our K and 1st grade materials:

Practice helps a lot, even for dyslexic folks. Maybe the curb painter is dyslexic and meant to paint this instead:

or this

( That introduces another math term - inverted. It means upside down, on its head. )

You can do a "reverse look-up" of a telephone number - you put in a number and get out the caller's information, rather than putting in a name and getting the phone number. This function is especially useful when you are getting those irritating calls from a person or business that always hangs up.

(Reminds me of yet another math term - recurring - things that keep happening over, and over, and over, and over, and over ... like those phone calls.)

There are lots of complex ways of transposing numbers. One is changing the arrangement of data in a spreadsheet from rows to columns. I know there are functions that help you do this in spreadsheet software [copy/paste special/transpose], but I often find myself re-entering data because it's faster for me to retype small blocks of text than to remember the software process.
  • Move data from rows to columns and vice versa
  • Move a term from one side of an equation to the other (as long as you reverse its sign)
  • Move music to another key (higher or lower)
If you are very clever, you can reverse transpose in a spreadsheet. That means re-sorting the order of the columns, then turning them into rows, or vice versa. In this usage it does not mean writing the numbers backwards.

I'm not going to tell you how to do this. The Excel Math blog isn't supposed to be about Excel spreadsheets. We are an elementary math curriculum, since 1976. Here's a brief history.

Wednesday, April 21, 2010

Numbers for Names, Part 3

I think it's time to finish this run on numbers and names, but first, I found a company called

E8's ownership includes 1% owned by Moover Toys. Their 1% of E8 cost them $88,888.88, thus 100% of E8 is $8,888,888.88. Do you think the people in this company have a thing about the number 8?

The company website says's mission is to provide the most advanced, intuitive, user-interfacing platform in the world based upon the unifying supersymmetry of the exceptional Lie group E8 and its corresponding 248 dimensions.

I don't understand any of this. It's not elementary math. But I like this artwork. Click for a larger version:

Here's a description of what it might mean (condensed from the original here):

The Lie group E8 consists of 240 points that are tightly and symmetrically packed in 8-dimensional space. This object has 696,729,600 symmetries. In comparison, the 8 points at the corners of a 3-dimensional cube have only 48 symmetries.

We can't visualize objects in 8 dimensions, but we can draw 2-dimensional projections. For example, if you shine a flashlight on a cube, its shadow could look like a hexagon. If you orient the cube correctly, the shadow looks like a regular hexagon (6-sided figure; all sides equal and all angles equal).

This diagram does the same for the E8 root system. The light is "shining" on these 240 points so the 2-dimensional shadow they cast is as symmetric as possible. The E8 has 60 symmetries: 30 rotations and 30 reflections. The 240 points wind up in 8 concentric rings of 30 points each. Those are the black dots.

The lines are "shadows" of the lines that frame this shape, back in the imaginary 8-dimensional space. Each line connects a point to its nearest neighbors among the other 239. Each of these 240 points has 56 nearest neighbors - they are very tightly packed!

Here's how the picture looks on my giant monitor, zoomed to 1200% of its original size! Click for the larger version.
PS - I was unable to confirm the 60 symmetries ... I started to count, but ZZZzzzzzzzzzzz

Tuesday, April 20, 2010

Numbers for Names, Part 2

Yesterday we talked about fictional characters with numbers for names. Today the subject is websites whose names are numbers. Think of some proud parents naming a new anything - a town, for example:

Or a song:
Well, imagine the excitement on 15 March 1985 (25 years ago), when the first .com Internet domain name was registered by Symbolics, Incorporated, a computer firm in Cambridge, Massachusetts.

The logo above is the original company's design. They changed to the logo below when a new parent company ( bought this domain address last year.
 By now, (2010) there are now about 84 million domain names. Most of the good words are used up! And so are the easy numbers. Now you need to search, get creative assistance, etc. to find a domain name. There are some rules and some blocked off addresses, so not everything you think of will work.

Try searching the lowest numbers, and see what you find. is the lowest numerical value I found that went to a working website. Their motto? The first address for outsourced IT.

Nice pun. I like it!

Here are a few more sites with numbers (some of these translate or divert you to another site):
  • goes to a site for the latest James Bond movie.
  • goes to the Disney site for 101 Dalmations.
  • takes you to the Fidelity Investments websites, where you see 401(k) retirement plans.
  • goes to a site that explains how to do 1031 property exchanges (1031 is a tax regulation).
  • takes you to a website from Drake, the tax software publishers.
This is kind of a depressing theme, seeing as we have just finished paying our 2009 US taxes...

NOTE - Most website addresses we type in are words, or a combination of letters and numbers, but the "real" address for every site is one that only computers can recognize and remember: or 2005:0db8:85a3:08d3:1319:8a2e:0370:7334

Let's end with this - some Internet addresses are static (permanent) and some are dynamic (assigned when needed). Just like real people's addresses and names.

Have you or anyone you know changed their name or address to suit new conditions or relationships?

Monday, April 19, 2010

Numbers for Names, Part 1

Can you think of characters from literature (okay, movies are good too) who were given numbers instead of names? If you are a math whiz, and you have a child on the way, perhaps he or she could use a number-name. Here are a few for you to chose from:

Number One was the name given to Majel Barret in the pilot for Star Trek (the first series).

Later Star Trek character William Riker was called Number One by Captain Picard, but that was his title, not his real name. Number 1 was also a title used by Ernst Blofeld, a James Bond villain.

Number 2 is the second-in-command to Doctor Evil, in the three Austin Powers movies. And second-in-command in the Prisoner (see below).

Johnny 5 was a robot in the movie Short Circuit.

Fifth is the name of a character in Stargate SG-1.

Number 6 is the only name given to the Prisoner, in the Sixties television series.

007 of course. Need I say more?

Seven of Nine is the name of a lady captured by the Borg in Star Trek Voyager

Nine is a little rag doll, created in the movie named Nine

Thirteen is the name used by a woman who works with Doctor House in the House TV series.

Agent 99 is the assistant to Maxwell Smart, in Get Smart

One Eight One (181) is a prisoner in Peter Seller's comedy Heaven's Above 

Most of these numbers are used for prisoners or secret agents or aliens - all hiding (or having hidden) their true selves. Are names, not numbers, the best means to express a person's true humanity?

Numerologists would probably say no. They are people who find mystical or esoteric meanings in the relationship of numbers to physical objects or people.

Numerology is to mathematics what astrology is to astronomy, or alchemy is to chemistry. As science began to be applied to a subject, the pre-scientific approach was often branded as questionable and arbitrary - something only to be regarded with suspicion.

I'm skeptical of that suspicion. Let's see now, does doubly negative suspicion equal support?

Friday, April 16, 2010

How Many Go Into One?

How many multi-tasking functions and features can be crammed into one device? Can math help?

While researching this subject, I found the
  • Three-in-One Pen Tool (Adobe Illustrator)
  • Fidelity Four-in-One Index Fund
  • Five-in-One painter's tool
  • Six-in-One wire stripper tool
  • TrueTemper Seven-in-One planter's buddy garden multi-tool
  • Eight-in-One fingernail manicure set (sure to be taken by security in the airport)
  • Nine-in-One calculator, box cutter, memo pad, laser pointer, and so on
  • Ten-in-One Circus sideshow with freak shows, wild animals, etc. 
  • Eleven-in-One Spinal Tap Volume Control (I made that up, just to see if you were paying attention.)
  • Twelve-in-One Web Page, where the owners created 12 films in one semester of school

I tried to find a thirteen-in-one device, but only came up with this passage: In each of the planes of existence, the thirteen in one, the mystical number ... shall direct the unfoldment of being.  Uh, hmmm. Well. We'll cut this off and get back to the subject at hand.

Is there any limit to the number of items in one? I suppose not. But a multi-function device is limited in its utility if and when when the functions overwhelm the user. More is not always better.  Here's my candidate for king - the 80-in-one memory card reader.

In theory, this reader can replace a separate reader device for each of the following memory cards:
  • MMC/ MMC 4.0/ HS MMC/ HS RS MMC/ RS MMC/ RS MMC 4.0/ MMC Mobile
  • SIM (mobile phone) Card 

Do you think there are really 80 different kinds of memory cards? Who verifies these claims?

That's the champ for now, I guess, unless you grant me a bit of poetic license:
Could it be Disney's One-Hundred-in-One Dalmations?

PS - Yes, I know it's really One Hundred and One Dalmations!

Thursday, April 15, 2010

How old?

Yesterday the blog was about Constitutions, partly because I knew I was going to see The Rivalry - a play about the Abraham Lincoln-Steven Douglas debates. This play was written by Norman Corwin who has been called the Poet Laureate of the Radio. Check out his site or read his biography.

Norman is still around - he will be 100 years old in 2 weeks! He came to see the play yesterday at Lamb's Players Theatre. Here's a picture of him, telling us about how it feels to see something he wrote, performed on stage 50 years later.

Telling other people about this experience has been interesting. The first comment from everyone has been How Old? 100! 100? Really?

Those comments got me thinking. What would he have seen in the past 100 years?
Statistics (here's your math for the day!)
100 birthdays
5200 weeks (almost a mile's worth of weeks, so to speak)
36,500 days
110,000 meals

18 Presidents
Taft, Wilson, Harding, Coolidge, Hoover,
Roosevelt, Truman, Eisenhower, Kennedy, Johnson
Nixon, Ford, Carter, Reagan, Bush
Clinton, Bush, Obama

Countless Wars
Mexican Revolution, Haiti, Dominican Republic, Russian Civil War/Revolution
World War I, World War II, Korean War, The Cold War, Vietnam War
Gulf War, Somalia, Bosnia, Kosovo, and so on

1910-20 Motion pictures, Crossword puzzles, zippers, radio, toasters, fortune-cookies
1920-30 band-aids, robots, traffic lights, speakers, rockets, penicillin, electric shavers
1930-40 tape, jet engines, parking meters, Monopoly, tape recorders, radar, copy machine, helicopter
1940-50 television, computers, aqualung, atomic bomb, microwave oven, mobile phone, Velcro
1950-60 Credit card, bar code, solar cell, hovercraft, Barbie, pacemaker, computer chip
1960-70 Audio cassette, video disk, contact lens, hand-held calculator, computer mouse, ATM
1970-80 microprocessor, VCR, Pong, ethernet, laser printer, cell phone, Walkman
1980-90 PC, Apple, Windows, Prozac, World Wide Web

Wednesday, April 14, 2010

We the people

Kon-sti-tu-shun. Con-sti-tu-tion. A written set of rules for a government. These rules keep the branches of the government in balance and balance the government's rights with those of its citizens.

The US Constitution starts out like this:

We the People of the United States, in Order to form a more perfect Union, establish Justice, insure domestic Tranquility, provide for the common defense, promote the general Welfare, and secure the Blessings of Liberty to ourselves and our Posterity, do ordain and establish this Constitution for the United States of America ...

and it goes on for quite a while.
Constitutions establish valid means of changing the government over time, such as voting in new leaders, establishing new rules via a legislature, etc. Many constitutions say a few of their rules are unmodifiable: the dignity of its citizens (Life, Liberty, Pursuit of Happiness), the citizenship of its leader, and so on. Of course, these rules can always be changed by a revolution or takeover from outside, or declaration that an emergency demands temporary suspension of the rules.

Perhaps due to a lack of education, I have assumed that a constitution is the foundation of every good government, except dictatorships where the dictator simply says "Let's do it my way. Now!"

If this kind of thing appeals to you, here is a website with constitutions from every country.

But reading this week, I learned that not all democratic countries have written constitutions.

(Here comes some math, finally!)

Countries without any written constitution - Israel, New Zealand, United Kingdom
Oldest surviving constitution - the sovereign state of San Marino; written in 1600
Longest constitution (most words) - Yugoslavia (now gone)
Most emergencies causing suspension of constitution - Argentina did this 52 times in 150 years
Most recent constitutions -Thailand (18 different ones since 1932; including 11 coups). For more details, see the chart below, created by Patiwat Panurach.

Who would have thought you could get a graph out of a constitutional crisis, eh?

One of the UK's elected leaders recently proposed they “consolidate the existing unwritten piecemeal conventions that govern much of the way central government operates into a single document.” Given the successful history of UK parliamentary procedure, I think the odds are slim to none, and he'll be voted out.

PS - Today is the anniversary of the day Abraham Lincoln was shot in 1865. It was the first time a US President was assassinated.

Tuesday, April 13, 2010

We just call it math

This blog is really about psychology. We just call it math.

When you eat lunch or dinner out with other people, there is always a chance to show off your math skills by dividing up the check among the diners.

Of course, some people insist on having accurate breakdowns of their specific dish, including the drinks, and bickering about setting the tip, etc. If you take this to extremes, it damages relationships. Quickly. Or you demand separate checks, which is a hassle for the server. But your group trusts one another, and you eat out regularly, it's easiest to just divide evenly.

Here at Ansmar Publishers, home of Excel Math, we use the even division approach. We take the bill and divide it amongst the number of diners. We don't miss an opportunity to poke fun at each other if one of us orders a particularly extravagant dish. Or steal some fries.

Today we went to Filippi's Pizza Grotto. They offer some of the best deals in town. The check for 4 of us looked like this:

$5.80 lasagna
$5.50 submarine sandwich
$5.50 submarine sandwich
$7.80 pepperoni pizza
$5.85 ice tea (3 x 1.95)
30.45 subtotal 1
  2.66 tax
33.11 subtotal 2
  4.89 tip (15% of $33)
38.00 total

Divided by 4 = $10 per person.

The sandwich guys might argue that they had a $5.50 sandwich that turned into a $10 lunch, but that's the way it goes.

Jim (our business manager) takes the extra 2 dollars back to the office and puts it into the "cookie fund" jar, where it sits until we have a birthday to celebrate. Then the money is spent on cake and ice cream (never cookies).

Monday, April 12, 2010

How does a math book come about?

Sometimes people ask me how we create Excel Math. Here's a quick overview:

We start by collecting state math standards and customer expectations.

Then we begin cutting, pasting, sorting and categorizing the new requirements. We compare to our existing material, figure out what we must delete (to make room for the new), and what we must add. We create Scope and Sequence documents - listing the concepts we'll cover in each grade. Then we survey some existing customers, asking "Here's what we propose, what do you think?"
Next we start copying page files from our archives to the active working area of our Mac computer network. Or we might start from scratch if there are many changes, or we change page orientation like we did with Kindergarten in 2009.

If we start from scratch, we create 370 page files for our 155-lesson format. That's front and back of 155 lessons plus the tests. All those pages are reproduced (at a smaller scale) in our Teacher Edition, so each file gets linked to master Teacher Edition files. We change all the copyright dates, page frames, etc. We back up all this material, then take a deep breath and dive in!

We create what we call bins for each new concept to be added. We create 10-100 problems to each bin. These are at the correct level of difficulty, including various ethnic names, a balance of males and females, interesting, timely topics that will appeal to kids of the right age group, etc. We organize, compare, check, proofread, save and print these bins. Here's an example of a problem bin:

Then we go into the page files, take out old problems, insert new ones from our bins, and rearrange the pages as necessary. We check the pages against the Scope and Sequence.

We find any problems out of order (where we might ask a question before the subject was taught) We move or adjust lessons to improve the spiraling sequence of problems. Where possible, we add graphics to improve the appearance and interest level of the pages. We check again, proofread inside the office and outside using contract workers and kids.

We add extra activities at the back of the TE, and Create A Problem items on the back of test pages. These cover concepts that are not calculation-based, or need more discussion, or are best taught by running around the school campus.

We create tests that reflect the concepts taught in previous 15-20 lessons, and make up test correlation charts showing which test question is assessing which lesson concept.

We then have both adults and kids do all the work, solve all the problems and review all the wording. We look at the files for missing material, adjust any political correctness issues, and improve the artistic appearance (there are no machine tools to perform this kind of work, it is all done by people).

Now we create high-resolution PDF files for our two printers. We send them the files or carry them over, depending on the size of the project.

They print the pages and collate them into sets of 10, 15, 22, 30, or 35 Lesson Sheets. Some pages are collated into individual student sets and we apply gummy-binding on the spines and a cardboard back.

We print and coil-bind the Teacher Editions.

The original page files are re-processed to extract the lesson-only part of the page and format it for screen display. This becomes our Projectable Lessons product. Those files are assembled and we create a DVD master for our disc duplication company.

That's about it!

Friday, April 9, 2010

Seven square miles surrounded by reality

I saw a curious bumper sticker on my way to work today. It said

Ocean Beach. Seven square miles surrounded by reality.

Subtle humor, eh? You might need to know Ocean Beach is a hippie community of people somewhat detached from reality. That gave me a theme -- math-based puns that we couldn't understand without some basic math knowledge. For example,

The United Kingdom and the United States - Two nations divided by a common language.

Ok, this one is a bit different:

A man sent in ten jokes to the newspaper, hoping to win a prize. But no pun in ten did.

Advice to all math teachers:

You must tell the truth, nothing but the truth, but not the whole truth.

Another one on math teachers?

A math teacher is someone who talks in someone else's sleep.

Moving on... the title of a book on elementary math:

Arithmetic Simplified, by Lois Carmen DeNominata
OK, we can do better than that one, can't we? How about this from Alice in Wonderland:
How many hours a day did you do Lessons? asked Alice.
Ten Hours the first day, said the Mock Turtle, nine the next, eight the next, and so on.
What a curious plan! exclaimed Alice.
That's the reason they're called Lessons, the Gryphon remarked, because they lessen from day to day.

Let's try a family math joke.

My life is all arithmetic, the young mother exclaimed to a friend. I try to lose weight, add income, divide my time, and try to keep my children multiplying!
Sticking with the theme of home and math, we can ask,

Why are turkeys so good at arithmetic? Because they are always counting the number of chopping days til Thanksgiving.

I know times are tough in the banking industry, and there's lots of turnover and turmoil. But we can still try to joke about them, can't we?

I went into the bank the other day and asked the teller to check my balance. He reached out and shoved me!

Well, how about more arithmetic?

How many times can you subtract 7 from 83, and what is left after you do? I can do it as many times as I want, and there is always 76 left.

Moving on to more complex things:

A statistician is a man who puts his head in the oven, his feet in an ice chest, and says that on average, he feels comfortable.

You know what? It's not that easy to make math-related puns that are funny. I've saved you from hundreds of bad ones. We'll end here:

Why did the chicken cross the Moebius strip? To get to the other ... uh ... hmmm

Thursday, April 8, 2010

Size of a Frequent-Flyer Mile

Yesterday I was just getting started on miles. We defined statute miles and nautical miles, metric miles and Roman miles. Even a country mile. But what about frequent-flyer miles?

A flyer-mile is not really a unit of distance, but a unit of currency. Airlines created frequent-flyer programs in an attempt to retain valuable customers. Miles are awarded (or earned) for paid travel on a given airline. Typically, you get one mile in your account for each mile flown on the airline.

Note - the miles granted are based on mileage calculated by the airlines. In most cases these distances are nautical miles between runways. For example, from SAN (San Diego) to ROC (Rochester, NY) the distance is 1952 nautical miles. But the question then arises - is this at sea level, or up in the air 30,000 feet? etc. etc.

When you have a bunch of miles saved up, you can redeem (exchange) them for free or discounted travel on the same airline. The pile of miles is an incentive for you to stick with the airline until you redeem them. If you have a certain number of miles in your account, you can become a special member of the airlines' savings program. You will get bonuses for flying in a premium class seat, buying tickets when a new route if offered, etc.

So if the flyer mile is a unit of value, not distance, what is the exchange rate to dollars, euro, etc.?

It's very hard to say. My research suggests consumer value is about 1.2 - 2.0 US cents per mile. That means if you cash in 25,000 miles to get a ticket which would have cost you $300, you got 1.2 cents of value per mile redeemed.

The value (or cost) to an airline is very much less than this, partly because only 11-12% of miles ever get redeemed. But again, it's hard to say. I found conflicting reports when I tried to learn how many miles are being hoarded by all of us who are flyers.

Is it 14 trillion miles worth $480 billion OR 10 trillion miles worth $165 billion?

As of today, I have earned 2,234,650 miles on American Airlines. Yes, that's 2.234 million miles. Ouch, my aching bottom! And I've earned lifetime Platinum status in the AAdvantage program.

Of those miles, 119,650 are still waiting patiently to be redeemed.

Here's a simple math challenge: at 1.2 cents per mile, what is my balance worth?

119650 x .012 = $1435.80

Wednesday, April 7, 2010

Many Miles, To Go

Most people in the USA know a mile is 5280 feet. We memorized it long ago, when we were in elementary school. It's equivalent to 1.6 kilometers. Of course, I'm talking only about a statute mile, defined legally by Queen Elizabeth I and the English Parliament about 400 years ago.

Another unit of distance is called a nautical mile, which is 15% larger (6076 feet, 1.15 statute miles, or 1852 m). This mile is equal to the distance of 1 minute of arc along any meridian (a line around the earth going through the poles).

(I realize this introduced the units of degrees, minutes and seconds of rotation, which we'll have to deal with another time!)

Does this diagram make it more clear? Sometimes a picture is worth 1000 words. We use nautical miles for navigating ships and airplanes because it's related to the size of the earth rather than the length of a foot. It's much easier for calculating distances on navigational charts.

These units are the most frequently used miles, but when we look back in history we find that the Romans started using a mile several thousand years ago. It meant 1000 paces (they defined a pace as one step with each foot; about 5 feet) thus a Roman mile was 5000 Roman feet.

An olde English mile was originally 8 furlongs, each with 625 feet. That's also 5000 feet. But since the feet were a bit larger, the old mile was about 1.3 of our current statute mile. A Scottish mile was not precisely defined but historians think it was about 10-12% longer than our current mile - or about 5920 feet. An Irish mile was about 1.25 of the statute mile, or 6720 feet. These old alternative mile usages were abolished in 1824 by an English Weights and Measures Statute.

The term metric mile is sometimes used in athletic competition, and it refers to a 1500 meter race (about 4920 feet).

We only teach the statue mile and nautical miles in Excel Math.

Our final definition today is the country mile. Let's put the definition in the form of a question:  

Q. How far is a country mile?  
A. Same as a city mile except there are no convenience stores along the way.

Ha ha ha. It really means a long, but undefined distance.

Tuesday, April 6, 2010

Run with the Renminbi


A hard word to say! It's a Chinese word that means the People's Currency. This currency was first issued around 1949 by the People's Bank of China. Its currency symbol is CNY but it is also abbreviated as RMB.

I've visited China three times totalling about 10 weeks, and we didn't use that R word while working, talking business or sight-seeing. We used the words yuán (yen) and the jiǎo (1/10th of a yuán). Yen means round and you probably know it's also the word used for Japanese currency.

Just as we have slang terms (calling a dollar a buck) the Chinese have multiple nicknames for their money too -  piece and feather are two of them.

The picture shows one yuán at the top, a half-yuán in the middle, and a jiǎo  (dime) at the bottom.

You may have seen lots of hollering in the news about the relationship of the dollar and the renminbi. Some people want the renminbi to be valued higher relative to the dollar, so that American goods will be cheaper for the Chinese to buy, and their goods will be more expensive. For a long time the exchange was about 8 to the dollar. Now it's about 6.8 to the dollar.

[NOTE - These arguments go on constantly about many currency pairs - the Euro vs Pound Sterling vs US Dollar vs Swiss Franc, etc.]

There are lots of viewpoints on this and we can't resolve them here. But I can say that the exchange rate in China used to be very different for a tourist than for a business. I mean the BLACK MARKET where you got a completely different exchange rate if you did money-changing on the street rather than in a bank. This situation exists in many countries.

The Chinese renminbi is accepted in some places outside mainland China, such as Mongolia, Thailand, Cambodia, Laos, Pakistan, Taiwan, Hong Kong, etc. - similar to how the US Dollar is accepted when Americans buy things along the Canadian or Mexican borders.

You need a little bit of math to work out exchange rates and it helps to have practice. Because we live near the US-Mexican border, lived in the UK for a year, and worked in Canada for 4 years, I can do most exchange rates in my head.

Years ago our whole family took a big vacation together and went to Canada. In Vancouver, my sister-in-law offered to pay for lunch. She went to the register, and came back to the table thrilled that the Canadian cashier took her US dollars "straight across" or 1:1.
At the time, a US dollar bought about $1.30 Canadian. Why was the Canadian merchant so obliging? Let's look:

Dinner for 14 people =     $185.00 Canadian dollars (symbol is CAD)
Paid US  $185.00 x 1.3 = $240.50 CAD

He made a quick $55.50 CAD bonus on the lunch.

If you want a hard mental challenge, go into a country, buy some of their currency, then try to re-use that money in a third place while keeping some sense of its value. This is not calculus or rocket science, but it will definitely have you counting on your fingers in the Hong Kong duty-free shop!

[Yes, the money-changers profit most from cross-border, cross-currency wheeling and dealing!]

Monday, April 5, 2010

Did You Feel It?

I don't really focus this blog on current events, but it's hard to focus when the ground is shaking!

There was a moderately-strong earthquake yesterday, centered about 100 miles east of us. Here's a view of the region, showing the locations of the 73,786 people who have contacted the US Geological Service about the quake:

The quake was about 7.2 in intensity, and originated near the star on the map. It's a sparsely populated area in northern Mexico. Geologists are still working to determine the exact location.

To learn more about earthquakes in the USA and other countries, you can visit Did You Feel It? - a site run by the Geological Survey to collect data about earthquakes.

To learn more about the math of earthquakes, you can read this Excel Math story in Sixth Grade:
(Click the image for a larger version)

We didn't have any damage here at Excel Math (that we have noticed so far).

Thursday, April 1, 2010

Loosen up. Go Home. Smile.

In 1690, JOHN LOCKE said:  

The third branch [of Science] may be called Semeiotike, or the doctrine of signs ... the business whereof is to consider the nature of signs the mind makes use of for the understanding of things, or conveying its knowledge to others.

[paraphrasing to make this shorter, he finishes with]

... since the things the mind contemplates are not visible it is necessary that something else, as a sign or representation of the thing it considers, should be present. And because the scene of ideas that make up one man's thoughts cannot be opened to the immediate view of another, to communicate our thoughts to one another (as well as record them), signs of our ideas are necessary.


Yesterday we talked about tightening bolts. When pondering what math subject I could address today, I thought about how we use signs and symbols as short-hand, to save on writing out full descriptions.

Here are a few less-commonly known symbols, used to save manufacturers the high cost of translating instructions into many different languages. They indicate which screws or bolts you can safely remove to take the back off a piece of equipment.

A similar set of symbols is used in public places. This meeting place symbol is used for emergencies - when you have to run out of the building due to a fire or earthquake. The spot where you gather to make sure everyone is safe should be marked like this:

I just heard we are taking off work early today, so I invented these symbols!

 I'll finish with this, and leave it to you to work out the meaning.

If you get stuck, try the Dictionary of Symbols by Carl Liungman

Pounds per foot, or foots per pound?

This topic follows the one on bolts, because it's all about Torque. Torque is a force applied to rotate an object around its axis. Knowing a bit of math helps you use a torque wrench.

Torque is commonly used to describe how tight a bolt should be, or how much twisting/pulling power is generated by a car engine.

(A full discussion of this would take us way out of the range of elementary math and into zones that require an engineering degree.)

But our bolt blog reminded me that the degree of tightness you apply to a fastener was once described in ft/lbs or lbs/ft. Now we are more likely to use a Metric unit called Newton meters or N•m. In either case, it's a measurement of force applied to a lever of a given length.

Remember the blog on rate? We discussed heart beats per minute and respirations per minute? This is the same kind of thing - one Newton of force per meter, or one pound of force per foot.

Here's a typical torque wrench. Yes, it's one foot long.
  • The head is on the left, with a switch that enables it to work in either direction
  • The head is capable of accepting sockets that drive fasteners or nuts. 
  • The handle is on the right, with a feature that lets you adjust the torque to be applied. 
  • A CLICK / SNAP feature tells you when the fastener is tight.
The sound and feel mean you don't have to be looking at the wrench when it hits the proper tightness. You can be groaning with your eyes closed and still notice that it's tight enough. SNAP. CLICK. Stop pulling now is what those indications mean.
An older design uses a needle and scale, so you can see the force you are applying, and stop yourself at the proper tightness. But you have to watch what you are doing.

What happens if the bolt or nut is too tight for your torque wrench? If you pull or push too much, you will break the wrench. If you need extra force, use a wrench with a higher rating and longer handle.

You can also make an extension and lengthen the arm of the wrench so you will have more leverage.

CAUTION - the extension MUST be at the head of the wrench, not the handle. If you put a pipe on the handle to extend it, you can break the wrench or bolt. Yes, I do know this from personal experience!
Not like the the picture above! You have to do it like the photo below!

In this example, the total length of the extended wrench is now 5 feet, so 10 pounds of force on the handle will give you 50 pounds of tightening force on the wheel nut.

If you weight 120 pounds and climb onto the handle, what will the force be at the wheel nut?

That's right. 600 ft/lbs.