Excel Math

Take a Hike, Part I

2012-02-02T06:50:00.000-08:00

The highest point in the city of San Diego is atop Cowles Mountain (pronounced coles) at 1591 feet above sea level. In recent years, Cowles has become a popular hiking trail for backpackers, runners, hikers, and dog walkers. After it was featured in numerous magazines and reviewed on Internet sites, Cowles went from a few neighborhood fans and avid hikers to regular visitors of all ages and fitness levels.

Today let’s see how elementary math can help us compare the length (in miles) of various trails on this mountain. If you need a refresher course, check out some Excel Math Curriculum samples.

In the meantime, grab your hiking boots and walking stick as we stroll through a few miles of beautiful scenery and enjoy the view from the top. Here's a view from the base of the mountain:

This picture is a bit misleading because the trail actually veers to the left at the top of this peak. From there you head up the final mile to the top. This 1.5 mile trek from the parking lot on Golfcrest Drive to the summit is one of the most well-traveled routes up the mountain. The base of the trail is complete with restrooms and a water fountain plus a bulletin board with snake alerts and maps. Cowles Mountain is one of the most popular hiking destinations in San Diego County.

When you reach the top, be sure to touch the stone plaque, or (some people will say) your hike isn’t considered “official”. Some people like to jog down and then hike back up again for more of a workout (and a total hike of 6 miles).

Another way up Cowles Mountain is to take the Barker Way service road from La Mesa. This is a wide path (in most areas) but very steep and gravelly. Bikes are permitted on this road, but not on the trail. Many people bring their walking sticks for this route. Some also bring along their apps to calculate the distance they’ve walked, the time it took, their speed, and even calories burned.

Barker Way also has a trail that joins the first trail from Golfcrest about a mile up. This trail is much less traveled and becomes quite narrow in spots. But it’s very well marked and is about 1.6 miles to the top. You start off on the service road then take a trail off to the left, crossing a small bridge, and then following 25 switchbacks, joining with the Golfcrest trail about 1.05 miles up. Here's the trail about halfway up Barker Way:

Another route up Cowles Mountain starts in Santee at Big Rock Park. This trailhead tends to attract rattlesnakes at certain times of the year. Perhaps that's why it seems to have fewer hikers than the other trails. From this point to the summit is about 2 miles.

An even less traveled trail up to the top of the mountain begins on Mesa Road. In many of its segments, the chaparral is so high and thick that the trail seems to be passing through a tunnel. This trail is about 2.2 miles to the summit.

Once you arrive at the top, you have a 360 degree view of San Diego county that includes Point Loma, La Mesa, Santee, El Cajon, Mt. Cuyamaca, and surrounding areas. You can even see the Coronado Islands in Mexico on a clear day. There are a couple of maps at the top of Cowles Mountain to help you get your bearings and to show some of the points of interest visible from the summit.

If you feel like a longer hike, from the top of Cowles Mountain, continue down the service road and watch for markers indicating the path to Pyle's Peak. This is a more challenging hike with another 1.5 miles of switchbacks and gorgeous views. Keep in mind that after reaching Pyle's Peak, you will have to return back to Cowles Mountain and take one of the trails down to the bottom for a total round-trip hike of 6 miles.

Here's a recap of the trails and their distances in miles up the mountain:

Cowles Mountain is part of Mission Trails Regional Park, about 6,150 acres of both natural and developed recreational acres. So next time you make it to San Diego, be sure to explore some of the hiking trails and bike paths. Happy trails to you!

Mapping Out a Fond Farewell

2012-02-01T09:01:00.000-08:00

Today we say goodbye and congratulations to ExcelMathMike. After over ten years with Excel Math, first as a consultant and then as managing editor, Mike is now retiring. He leaves behind an amazing amount of content and creativity in the Excel Math Curriculum as well as on this blog.

The Excel Math Curriculum for Kindergarten through grade six is brought to you by Ansmar Publishers. This curriculum was written and created by Janice Raymond and is a proven method to help kids of all abilities learn elementary math skills. Over the past ten years, it's been edited, revised and updated by ExcelMathMike, the managing editor. He has made the lessons easy for teachers to use while providing substantial math content for students at each age level. With over 50,000 math problems to his credit, let's take a look at his impressive numbers.

Mike has written . . .

600 blog posts
6650 pages of Teacher Edition lessons with answer keys
2590 pages of Student Lesson sheets with CheckAnswers
2220 pages of Spanish Student Lesson sheets
1860 projectable lessons
400+ pages of Summer School Teacher Edition lessons
525 pages of Summer School Student Lesson sheets
252 pages of Manipulatives
204 pages of Activities and Exercises
50 Scope and Sequence pages
46 Glossary pages
9 Placement Tests in English and Spanish
various scripts for editing computer files
numerous resources for teachers and parents
more website downloads than we can track

and has created . . .

11,000 files for Excel Math Projectable Lessons
9,000 files for print products
2649 image files for this blog
700 files for ExcelMath.com
160 files for the Excel Math webstore
124 Stretch Questions
120 Create A Problem exercises
25 downloadable sample lesson pages
30 pages of a Sample Packet
20 movies
Score Distribution and Grade Charts
more files than he'd care to admit

Not only has Mike been creating math curriculum, he's been making it accurate, teacher friendly, politically correct, organized, error free, relevant and age appropriate. Whew! This sounds a bit overwhelming. Good thing he's getting some well-deserved time off.

Here's an aerial view of the Excel Math office where Mike's been working:

Mike has already created a new blog so you can follow his exploits and view his numerous collections at Curating Cazalea. We wish him all the best in this exciting new phase of his life.

This blog at ExcelMathMike.blogspot.com will continue with new writers and topics. Visit us again tomorrow when we take a hike to explore distance and altitude.

Counting Up and Out

2012-01-30T15:23:00.000-08:00

The Excel Math blog provides content about everyday events that can best be handled with a knowledge of elementary mathematics. It's brought to you by Ansmar Publishers, home of Excel Math. It's written by ExcelMathMike, the managing editor.

Our blog is monitored by a variety of web user tracking services. They let me know if anyone is reading what I write. The services include FlagCounter, SiteMeter, StatCounter and Google Analytics.

Here are some sample summaries. You can click on the images to enlarge the display. The data tells you a little bit about a lot of people who come by for a short visit. You can see that in the long run, the numbers of visitors are going up.

Why don't the numbers all agree? Because that's the way numbers are. They just don't.

We have visits per day (whose day? starting when? ending when?), we have page views by"load", by "unique visitors" and by "returning visits". The counters started when I turned them on (or inserted some cryptic HTML code into the page). Since we didn't start them all at the same time, their counts can't be synchronized.

Reconciling inconsistent and incompatible results is the bane of the scientists and researchers. And it's one of the things that we teach kids in Excel Math. Double-check your work (using our CheckAnswer system) and do the problem in several different ways if it's important.

As they used to say on car ads,

Your mileage may vary.

Starting this week, the Excel Math blog will have a new author. I am retiring after 8 years and millions of math problems. In the past 30 months we've had 100,000+ page views, on about 600 blogs.

ExcelMathCarol is taking my place. Please be kind to her.

PS - I'll still be writing, at Curating Cazalea. Stop by and see me if you have time. Cheers.

Taxes and Fees Revealed, Part II

2012-01-27T11:55:00.000-08:00

Yesterday I introduced the subject of taxes and fees, specifically referring to hidden charges on airline tickets. Today we will expand on this subject, using only elementary math taught in our curriculum.

About 15 years ago I started a consulting business. I went down to the county government building, created a fictitious name for my business, and got a business license. Because I only had one employee (myself) I only paid $39 for my business license.

Each year I had to pay the $39 but in 2004 the city added a $25 collection "fee" in order to compensate themselves for taking my $39. This applied to landlord licenses, business licenses, etc. They charged us for the work of taking our money, nothing else.

$39 x 1.64 = $64 or a 64% increase over the existing charge!

Outraged small business persons protested and eventually sued the city. The first judgement was in favor of the city, but an appeals court over-ruled and overturned the fee. Click here to read the entire judgement, or skip that and read my summary in plain English:

We must decide whether a 2004 levy imposed by the City of San Diego, without a vote of the electorate, for the purpose of recovering the cost of collecting and administering a general "Business Tax," is a fee or is a special tax that should have been approved by a vote of the electorate. We specifically hold the levy is not a fee, but is a general tax that is subject to voter approval, and is void because it was levied by the city council without approval by a majority vote of the qualified voters as required by the California Constitution. Accordingly, we reverse the summary judgment granted in favor of the City.

We learned that the difference between a tax and a fee is based upon the purpose of the charge:

A charge that covers the cost of providing a service to the payer or regulates the payer's conduct is a fee.
A charge that raises revenue for general spending without conferring any exclusive benefit to an individual payer is a tax.

Notice the careful distinctions made on this summary of airline fees, taxes, charges and surcharges:

In addition to what you see here, you may also pay:

Federal 7.5 percent excise tax on the base ticket price
Segment tax of $3.70 per person per segment (a single takeoff and single landing)
International facility tax of $8.20 per person for flights that begin or end in Alaska or Hawaii
A 6.25 percent tax on the amount paid for transporting property (freight, pets) by air

Compared to the airlines, we have an easy time in the book publishing business - but we still have to keep track of the 6000+ municipal and state tax rates in the USA!

Taxes and Fees Revealed, Part I

2012-01-26T15:05:00.000-08:00

PREMISE

"The rich always get taxed at a preferential rate. They have rigged the system, know how to evade taxes, and don't carry their full weight. It's not fair."

We are going to examine taxation without representation today. Specifically, the way that governments and airlines levy taxes and fees on passengers. Thanks to new regulations, airlines must disclose all fees, taxes and other charges. This is excellent for detail-orient, math-saavy shoppers. And it might provide some evidence that the rich pay lower tax rates than the poor. Or not. Let's take a look.

EVIDENCE: CASE STUDY 1
Take a look at this fare breakdown provided by Excel Math's favorite low-price airline, who shall remain unnamed:

In this case, the fare from San Diego to Providence, RI is $268. Taxes and fees add $67. Divide the fees by the fare to get a 20% "tax" rate for a discounted ticket (bought by a "poor" or frugal person?)

In this case, the fare from San Diego to Providence, RI is $919. Taxes and fees are another $109. Notice the Facility Charge and Security fees are lower for the second flight (did it route through a different intermediate airport?) That's about a 10% "tax" rate for the full-fare ticket bought by a "rich" (or last-minute-emergency) passenger.

EVIDENCE: CASE STUDY 2

The next case shows a non-stop international flight to London. There are lots of extra fees and these do not include baggage, seat upgrades or other discretionary choices by the passenger. It's not shown on this graphic, but the total price in this case was $841 for a restricted coach ticket. The fare was $224 plus fees of $199 and an outrageous fuel surcharge of $418!

If you divide the fees by the fare it's 89% "tax"; if you divide fees by the total (including fuel surcharges), it's a 24% "tax".

Finally, here's a similar international flight to London, in Business Class. We see the "Air Passenger Duty" charge has doubled! This business class, completely flexible and refundable ticket is priced at $11,691 with $10,705 as the fare, plus total taxes and fees of $292 and "fuel surcharge" of $694.

If you divide the fees by the fare it's 2.7% "tax"; if you divide the fees by the total (including fuel surcharges), it's a 2.5% "tax".

CONCLUSIONS
More expensive tickets pay a lower percentage of taxes and fees, but I don't think we can use airline ticket prices to prove the rich are getting preferential "tax" treatment. We can conclude this:

Basic fees are a huge proportion of an inexpensive ticket, and a much smaller portion of a high-priced ticket. As a result, they obscure the "nearly free" fare on the first example.
We can see that "fuel surcharges" are not equal for every passenger, and not proportional to the fares - are they determined by the square footage occupied by each passenger on the aircraft?
Why does the UK charge a Business Class passenger twice as much to enter the country as a Coach passenger?

Full, Fuller or Fullest?

2012-01-25T18:04:00.000-08:00

Today I filled my car with gasoline and put 19.660 gallons into an 18.5 gallon tank. How? This is a test for your elementary math.

Please select any answers that apply:

The owner's manual is incorrect (the tank holds more than 18.5 gallons)
The gas pump metering is incorrect (I got charged for more fuel than was delivered)
Some of the fuel leaked onto the ground or elsewhere out of the tank

I am happy to report that any of these, and many other reasons may apply. Let's take the 3 choices one by one.

1. Technically the owner's manual is correct, BUT the tank does hold more than 18.5 gallons. All fuel tanks on US motor vehicles have an expansion space that accomodates fuel expansion and vapors. This ranges from 10-12% of the total space of the tank, so my car might have an extra 2 gallons or so of space. Normally this does not fill with fuel, due to automatic shut-off valves that sense when the tank is nearly full. I was holding the handle to fill it up, and the pump did not stop.

I know a bit about fuel venting, because in a former career, I created products like Mitchell Automechanics and an Emission Control Training Manual, and other fascinating books that deal with automotive evaporative control systems (they keep fuel vapors from polluting our air).

2. The station I filled up at was using a pump that does not adjust volume to compensate for temperature variations. A label on the pump says "This device dispenses gasoline solely by volume measured in standard gallons (231 cubic inches). It does not adjust for temperature or other factors which may affect the energy content of each gallon dispensed."

This means I should have received an accurate number of gallons, but I might not have received the full amount of energy that is contained in fuel at the standard 60 degrees F. My fuel may have been hotter or colder (holding more or less energy) than I expected. This is a complex subject to grasp - visit this site for a clear (if long) explanation.

Alternatively, the pump might have been defective and dispensing the incorrect amount of fuel. But it's unlikely, as they are frequently checked.

3. Some of the fuel did leak out of the tank and land on the ground. Not much, but a little. Let's say 3-4 ounces (half a cup), or a small fraction of a gallon. It also spilled down the side of the car and stained the plastic trim. The attendant at the station hosed down the fuel spill and gave me some wipes to clean the car. I didn't take a picture of the spilled fuel, but the car looks OK.

Let's end with a simple question:

Is it possible to fill a container with more than it was supposed to contain?

Yes. Absolutely. We prove this every Thanksgiving meal when we eat more than we should!

PS You might wonder if the fuel tank can expand and contract like your stomach does. To a limited extent, yes, it can change its shape and volume when subjected to enough pressure or vacuum.

Over-under-mis-use of health care

2012-01-24T16:25:00.000-08:00

I saw an interesting study today that reported on overuse of health care in the United States. I wonder if any of us "over-use medical resources". What do you think? What does this mean? How can it be measured? Does it mean we are too healthy?

And why is this a topic for the math blog? We define under, over, middle and similar terms on the first day in Kindergarten; these are fundamental concepts in math education.

It turns out the researchers defined their terms like this:

underuse is the lack of providing necessary care
misuse is the provision of wrong care or medical error
overuse is using medical services with no benefit or where harm outweighs benefit

Underuse seems to happen most often with lower income or English as a second language patients. It means people don't get medication that could prevent a heart attack, or an assistance device that could reduce injury.

Misuse includes unnecessary procedures, incorrect medication or gross error.

The researchers identified 172 articles from 1978 to 2009 that measured the rates of overuse of procedures (53 studies), diagnostic tests (38 studies) and medications (81 studies), then examined the use of 18 therapeutic procedures and 24 diagnostic services.

For preventive services, overuse rates ranged from 7% to 61%. Cancer screening overuse varied from 8% to 61% for screening colonoscopies. Prostate cancer screening using PSA was 16% to 36%. Cervical cancer screening reported that 58% of those screened should not have been. The overuse rate of mammography diagnosis was reported to be 30%.

For medications, antibiotics for upper respiratory tract infections (URI) were the most commonly studied. The overuse rates for antibiotics for URI ranged from 2% to 89%.

What's the harm in testing a bit too often? Won't we find problems sooner?

Not necessarily - there are significant risks:

Risks directly related to the test, such as the radiation exposure caused by imaging tests
Risk of a false positive, leading to mental anguish and unnecessary follow-up tests and procedures, each with its own set of health hazards
Risk that a condition will be identified that is not harmful but now will be treated

Although there are enormous costs associated with overuse (estimated as high as $200 billion a year, or 10% of total healthcare expenditures), there are also personal costs. Remember, testing is a very specific and personal matter. Some patients want to be absolutely sure; others can live with less information (what I don't know won't hurt me). There are no hard-and-fast rules for which tests are needed in each situation.

The reasons and motives for under- and over-utilization are murky. Liability and malpractice suits drive doctors to do more tests, for example, as does peer pressure among patients (Have you have a body scan yet, darling?). We really don't understand our medical system well at all, as the following quote indicates:

What is most striking about this report is how hard the authors searched for data on overuse of health care and how little they found. They reviewed 21 years of the medical literature and evaluated 114,831 publications, yet found only 172 articles that addressed overuse of health care. If we are to improve health care in the United States, we must know as much about overuse of health care as we do about underuse of health care. This article gets us started.

Sadly, there is no international, national or regional correlation between outlays of money, procedures and testing, and longevity or quality of life. [click on the map to enlarge]

Golden Graphics

2012-01-23T13:43:00.000-08:00

Today in the Excel Math blog we are going to use one of my favorite subjects - GOLD - to illustrate a couple of math skills that we feel kids should learn in elementary school. Here's a stack of gold bars, just to whet your interest.

I used this same illustration 18 months ago, in another blog on gold. You can check to see how your country's numbers have changed since then - are they saving or spending their gold?

Remember number lines and timelines? We start teaching them as early as Kindergarten.

Here's a quick, 5600-year review of mankind's encounters with gold. The data comes from a website operated by The World Gold Council. [click on the image to enlarge it]

Timelines help us see events in relationship with other things happening at various points in time.

Graphs visually depict changes over time. Here's a chart showing how the price of gold was stable for centuries. The value is displayed by the vertical axis and the years (timeline) are on the horizontal axis.

After many countries linked the value of their currencies to the value of an ounce of gold (1870-1900), gold prices doubled to about $35 an ounce. That price remained stable for another 75 years, until many nations de-coupled themselves from gold. Suddenly around 1980, lots of everyday folks began to buy and hold gold. Some won; others lost in this speculation. The chart below shows the price of gold took 25 years to re-attain its 1980 peak.

Sometimes the data you accumulate can't be displayed graphically like we did above, because the units are unrelated, or the scale is too great. So you construct a table. Here's an example:

I took the number of tonnes (metric tons, or 2204 lbs) of gold held by the top 10 gold-holding countries in the world's reserve banks. I added the country populations, and calculated the ounces of gold per citizen, and the value at today's price. Clearly the Swiss are ahead in wealth, measured by gold reserves.

I hope these graphics make you feel wealthy as well as math-literate!

An early morning math diary

2012-01-21T08:13:00.000-08:00

IN THE EARLY MORNING RAIN ... COMES THE EARLY MORNING MATH

2:41 AM Playful cat wakes me up. I get up, carry him to the kitchen, and lock him in. Then go back to sleep.

6:31 AM Aircraft begin to fly over our house, starting with CO 1293 at 6:31 AM. I hear the first one, then go back to sleep, while in the meantime, these flights leave (according to SD Airport's website):

DL 4706 6:32 AM

DL 978 6:33 AM

UE 5325 6:34 AM

AA 1592 6:35 AM

AA 3081 6:35 AM

US 150 6:37 AM

US 81 6:39 AM

I dream about a yellow Corvette Z06 I saw yesterday, driving home from work.

US Flight 3 at 6:44 AM takes off quietly.

6:49 AM Sunrise, although the heavy cloud cover makes it impossible to see, except by passengers on the departing aircraft. I continue to slumber.

AS 493 6:53 AM

DL 1864 6:55 AM

7:01 AM I awake and stumble out to the kitchen, free the cat, go out the front door. I walk 80 feet to the sidewalk and pick up my newspaper for Saturday, January 21, 2012. It's number 37,830 since the paper began. If you want to subscribe, you can call 1-800-628-8088 (in the USA).

7:04 AM The thermostat tells me it's 62º F in the hallway. Brr. I switch on the heater and it starts to warm the house towards the regulated temperature of 68º F.

7:05 AM I grab the silver MacBook 5,1 (#W891870Z8QR) laptop and begin this diary.

AS 573 7:10 AM

DL 1592 7:11 AM

DL 210 7:15 AM

WN 3942 7:17 AM

WN 841 7:18 AM

WN 1050 7:20 AM etc.

7:21 AM I hear water running. Either someone else is up, or the toilet valve in the back bathroom toilet leaks and it's running by itself.

7:25 AM My wife comes in and says the toilet is leaking. I add a new chore to today's list of things to accomplish.

7:31 AM I look out the window and it's raining outside, or very heavily misting. I check the local weather report from our weather madman a few blocks away - at 7:30 AM, 219 feet above sea level:

Temperature 52.1º F
Humidity 94.5%
Barometric Pressure 30.05 millibars
Wind speed 0
Light misting rain Rainfall rate 1/10th of an inch
Visibility 1 mile
Clouds overhead 700 feet

7:45 AM I decide I need some more art in this blog. I grab my camera, ask the cat to pose, and take two photos.

7:46 AM I download the 46 photos that were on my camera, select the last two of the cat, and edit them.

7:50 AM I remember that the rainfall rate was expressed as "less than 1/10th of an inch" and I wonder if he means per hour, per day, per week, or what? I also see High surf advisory in effect from 5 am Sunday (that's tomorrow).

7:52 AM Have a short discussion with my wife about some news items she's reading in the paper.

7:54 AM Can't decide which to photo of the cat to use so put both of them into this blog. I don't have Adobe Photoshop on this computer, so I fire up Illustrator 10.0.3, and I combine the two photos.

8:00 AM The clock is chiming 8 times as I think about the teachers who taught me elementary math.

8:11 AM I notice the battery level on the laptop is 22% remaining. I'd better hurry.

8:13 AM I finish editing and previewing this blog, and post it.

How Much Dirt, How Fast, How Far, Part III

2012-01-20T15:00:00.000-08:00

This week I have been overseeing the fleet of tractors that are moving dirt around on the vacant lots below the Excel Math offices. I've also been investigating the types of elementary math skills that tractor drivers might need.

Here are the pull-push pair, shown coming up on a "road grader" who is not actually in front of them, but on a similar path downhill about 100 feet away.

HOW MUCH WORK?
In the past two blogs I've provided some of the data related to the tractors, and how the contractor can determine how long it will take to move the earth. Where do the charts and raw data come from? From the manufacturer, Caterpillar. They decided how large the tractors could be, how much power was necessary, how much soil could be moved, how fast the tractors can go and how much fuel they burn, etc.

Let's assume

they can move 40,000 lbs (each) for each cycle or 40 yards of soil
a cycle takes 3 minutes
a pair can do 17 cycles per hour
they work 9 hours a day including overtime

That means they can move 80,000 x 17 x 9 = 12,240,000 pounds of earth or 80 x 17 x 9 = 12,240 yards per day!

If we wanted to, we could take the area of the lot (shown yesterday) and calculate the days required to move the soil. But we don't want to do all that work, do we? Besides, I can just watch and calculate from observation. Here's a view from the bottom of the slope as they head back uphill, empty.

HOW MUCH MONEY?
While engineers were designing tractors and measuring their earth-moving capacity, accountants were determining how much money the tractors consume - including original cost, the depreciation, maintenance, spare parts, tires, fuel, insurance and other items. They decide how many average hours a tractor will last - in this case 17,000 hours. Dividing the operating lifetime and the total cost adds up to at least $80-100 per hour per tractor (plus tax and operator salaries). That means the lifetime total spent on each of these monsters is at least $1,360,000.

Using Caterpillar's numbers, the push me-pull you team will cost our contractor about $1500 per day plus cost of the operators.

Would you like to see them in action? I thought so. I walked down and spent a few minutes filming the tag-team operation.

PS: The original pushmi-pullyu from Doctor Doolittle did not move dirt (yes, I read all his books and still have many of them).

How Much Dirt, How Fast, How Far: Part II

2012-01-19T14:38:00.000-08:00

Yesterday I introduced the Push Me, Pull You team of wheeled soil moving machines working down the street from Excel Math. Today we will see what kind of elementary math skills a tractor driver needs to master. Or in this case, a pair of 627F drivers.

I looked through a Caterpillar Handbook and found data that will help the driver calculate how long it will take two 627F machines to move some dirt. We also have to learn beforehand how hard the soil is, how many rocks are there, how much the tractors sink into the dirt, the grade or slope they have to climb, the distance they have to move the dirt before dumping it, and so on.

Then, using the data provided in the handbook, you can determine how long it takes to move dirt. It appears from these charts that our pair of earth movers should take a couple (2-3) minutes per cycle of load-travel-unload-travel. There are lots of numbers here, but it's only elementary arithmetic, not rocket science. [click on the images to enlarge them]

If you can't calculate how much dirt can be moved per cycle, you'll never be able to give a quote to the customer who wants to have his area scraped. You might not have enough days in the schedule, your tractors may run out of fuel, or be unable to safely climb or descend a slope loaded with dirt, or the dirt may be too hard for maximum productivity, or too dusty for the local community to tolerate, etc.

I suspect this gets done well in advance by engineers in the contractor's offices, with only a little input from the drivers. Using the data in the chart and simple math enables us to make an estimate.

In our case, the dirt is fairly soft, yet the contractor has provided a whole fleet! The push-me, pull-you pair, a bulldozer with blade and forks to loosen the soil, a water truck to cut down the dust, a scraper to do precision earth-moving, and a roller to compact the soil once it's been moved. At least 6 pieces of equipment, operators, and fuel, water and maintenance.

Here's their assignment - take the extra soil (about 20 vertical feet, so we need to move 10 feet of depth) from the green area and move it to the red area. Level and compact the soil on the entire lot. Ready? Go!

How Much Dirt, How Fast, How Far: Part I

2012-01-18T13:46:00.000-08:00

Today's math blog is for men and women who have always dreamed of driving a tractor. A big Caterpillar - a loud, grumbling, earth-moving, wheeled, push-pull tractor scraper. Like one of this pair of Cat 627F monsters:

That's right, they are a pair that work together as a team. Notice in the shot above that the rear tractor is connected so they can push and pull each other. Here's a better angle:

In the photo below, the back tractor has disconnected and the tow bar is raised.

Now you may be wondering why a tractor driver needs math. Let me show you. Today we will just collect some data. We will list some things to consider when choosing an earth-moving strategy (guided by the Cat manuals). Our 627F models are:

SPECIFICATIONS

Length (each) 50 feet
Weight 84,000 pounds (empty)
Horsepower 330 front and 225 rear
Fuel capacity 262 gallons (consumption ~ 40 gph)
Top Speed 32 mph

Now, what can they do?

CAPABILITIES

Maximum load 48,000 pounds
Maximum volume 44 cubic yards
Width of cut ~ 10 feet
Depth of cut into soil ~ 12 inches per pass
Depth of spread soil ~ 21 inches per pass

Here's what our team of scrapers can handle when it comes to the type of soil:

SOIL TYPES

Dry (0-10% moisture)
Moist (10-20% moisture)
Wet (20-30% moisture)
Sand
Sand and Gravel (rocks up to 3")
Rock (up 50% may be up to 12" diameter)

Larger rocks, harder surfaces, wetter soil, steep banks, very soft sand, etc. all fall outside the preferred conditions for our scrapers. Once we've defined the material we are moving and the grade, we have to worry about getting stuck in the sand or mud. So we consider the condition of the soil:

SOIL CONDITIONS

3% (no tire sinkage into the surface)
5% (up to 2" sinkage into soil)
10% (up to 5" sinkage into soil)
15% (up to 8" sinkage into soil)
20% (up to 12" sinkage into soil)

Finally we need to plan based on the optimal soil delivery distance - how far must it be moved? Very close, and we can just push it with a scraper. Too far, and we need a dump truck. In the middle, we can use our soil movers. Their maximum optimal moving distance is under 2 miles.

Tomorrow we will do some calculations!

Why Do I Need To Understand Math?

2012-01-17T11:39:00.000-08:00

Kids often share these comments and questions with their teachers and parents:

Why do I have to understand math?
I'll never use this!
I have a calculator or spreadsheet so I don't need anything else ... why are you torturing me?

We have several alternative responses

Ok fine, you're right, I'll get you out of math class - I never liked math either
Hush - do your homework
What's the real problem? Have you been unable to grasp this concept?

I prefer to show that math is an integral part of our lives. It's a language that kids need to learn, and neither torture nor an esoteric field of study. Today in the math blog we will look at an example I discovered while reading my newspaper. Even if you get your news from television or online, you will have seen the same story:

Capsizing of the Costa Concordia

I'm going to work my way through the details of this unfortunate ship running aground, as an example of math adults may need to know. First, the (time/number-filled) map of the incident, then some data from my newspaper:

Carnival Cruises and the Cruise Industry

2012 estimated industry revenue $30.2 billion
Carnival and Royal Caribbean control 74% of the global cruise market
Carnival cruises has 101 vessels (now 100 afloat and one on the rocks)
Carnival in 2011 had $15.8 billion revenue
Carnival in 2011 made $1.9 billion profit (before tax)
Carnival in 2011 carried 9.6 million passengers

Distance

Costa Concordia was 290m long (951.4 feet)
The ship was only 300 meters offshore (one boat length)

Capacity

3800 berths (in one place) and 3800 cabins (in another)
114,500 tonnes (metric tons = 1000 kilos ~ 2240 lbs)
500,000 gallons of fuel onboard, distributed between 17 tanks
17 total decks on the ship, including those not open to passengers

Costs

Costa Concordia cost €450 million ($575 million or £375 million or ¥44 billion) to build
The "collision deductible" is $40 million
At least 5 insurance companies cover the vessel, which is expected to be scrapped
The loss is expected to be in the range of $600-800 million
The liability coverage carried by Carnival is around $3 billion
Passenger "association" class action is already asking for €10,000 compensation per passenger

People

3216 Passengers (maximum capacity is 3780)
1013 crew (maximum capacity is 1068)

Finally, I conclude with another graphic, and a highly-entertaining exchange that I found at an online discussion site. Reader A comments on the graphic, and Reader B comes back with an unnecessarily-sarcastic but highly-amusing jab at Reader A's math illiteracy. This is why you need to study math while in school!

Friday Fun Photography

2012-01-13T10:39:00.000-08:00

Photography is a field that's highly dependent on mathematics. Photography is a combination of two Greek words meaning light + drawing (showing with lines) or "drawing with light".

Photography is the art and science of creating durable images by recording light or other radiation using a light-sensitive sensor or film. A lens directs and focuses light onto the sensor during a timed exposure - resulting in an electrical charge for each pixel of the sensor, or a latent image on the film. Later, other technology is used to develop, adjust and display the photograph.

With manually-operated film cameras, it was essential to learn about distances, exposure times, film speeds, depth of field, etc. You had to juggle knobs, adjusting rings and cables. With digital cameras, much of the technical content is different, but you still need to know a lot about light, time, distance, color, etc.

I searched for combinations of the words photography and math, and found dozens of interesting sites. I located (but did not buy) a book called Mathematics in Elementary Photography, but I couldn't find any books entitled Elementary Mathematics in Photography. I found plenty of lame lessons plans that didn't seem to do much for either photo skills or math knowledge. Here's one site I liked:

Functions Plotted onto Photos

I enjoyed the images, an example of which is shown below. The photographer takes the photos first, then she puts lines on them and discovers what functions represent those lines. Or vice versa. It's a chicken and egg kind of exercise.

You have to get into middle school and high school to calculate these functions and plot these curves. We DON'T TEACH THIS in Excel Math elementary school curriculum.

After finding very little of interest on the link between math and photography, I decided that today's blog will have a minimum of math and a maximum of photos. (I'll try to sneak in a tiny bit of math at the end).

Here is a short series of people looking at pictures on walls. My uncle David Freund took, developed and printed the photos and produced this exhibit. I lurked in the background, capturing reactions (and postures) of the viewers, and the contrasts of polished wood against flat white walls and the black and white prints. [click on the images to enlarge them]

Conveniently, most of the viewers were wearing black. Was that a subliminal choice due to the B/W photos, or a reflection of the dismal weather outside?

I did a bit of photography early this morning, warming up for this blog, so I will close with these...

Notice that I spent about a half-hour between the time I took the first shot (above) and the last of 12 shots (below). We DO TEACH calculating elapsed time in Excel Math.

Geographical Math, Part V

2012-01-12T11:14:00.000-08:00

This is the 5th installment on numbers describing US geography. I chose geography because it promised to contain lots of numbers (see yesterday's post), giving me a chance to showcase how we use elementary math.

Talking about territories and states and counties leads to the ultimate political subdivision - the citizen. As I was counting and sorting people who live in the US, I wondered, "Are there classes of citizenship for each kind of political subdivision?" Let's find out - and we'll throw in some numbers if possible.

CITIZENSHIP
In the United States, citizenship most often acquired in one of two ways - you are either a "natural" citizen or a "naturalized" citizen.

Citizens from birth are "natural" citizens.
Those acquiring citizenship after birth are "naturalized" citizens. They can take a class, pass a test and be sworn in as citizens (the most familiar way) or can be adopted en masse ( Congress granted citizenship to all Puerto Ricans in 1917, and all Native Americans citizenship in 1924).

If you were not born on US soil but were adopted, born of a surrogate mother, born "out of wedlock", conceived via artificial insemination, or have dual nationality, then you may face extra hurdles becoming a naturalized citizen.

Natural citizens can be divided into two groups; those having been born themselves in the US, or under its jurisdiction, and those that acquired their citizenship by been born to parents who are citizens of the US.

Here are some legal (Latin) terms that describe categories of citizenship:

People born on U.S. soil or its jurisdictions are citizens under the doctrine of "jus soli" or by "right of the land".
People born of parents who were citizens (but not on US soil), are citizens under the "jus sanguinis" doctrine or "right through blood".
A third phrase describes a way by which people who have married US citizens may be naturalized: the "jure matrimony" doctrine, or "right through marriage".

NOTE: These technicalities make a difference when you are traveling abroad. My former boss and his wife were asked to leave Canada after living there on a work permit for 5 years - due to Canadian re-interpretations of her citizenship, because she was born in a US embassy in Africa rather than on US soil!

You can expedite US citizenship by serving in the US military. In the past 10 years, 76,000 people have became naturalized citizens by serving in these 26 countries: Afghanistan, Bahrain, China, Hong Kong, Cuba (Guantanamo), Djibouti, El Salvador, Germany, Greece, Haiti, Honduras, Iceland, Iraq, Italy, Japan, Kenya, Kosovo, Kuwait, Kyrgyzstan, Mexico, Philippines, Qatar, South Korea, Spain, Thailand and the UK.

MULTIPLE CITIZENSHIP
In some cases people may end up with dual (or multiple) citizenships, because it is the country who defines (claims) who its citizens are. More than one country can define or claim you as its citizen.

NO CITIZENSHIP

If a country ceases to exist, or is in the midst of war or ethnic conflict, or rejects its citizens, many people may suddenly become stateless - with no citizenship in any country. The nomadic Romani people (gypsies), are denied citizenship in many countries. A child born outside either parents' home country, or a child with no birth certificate, may be stateless and end up in a refugee camp, like the "Lost Boys of Sudan". About 12 million people worldwide were stateless in 2011.

LOSS OF CITIZENSHIP
A few actions can cause you to lose citizenship, or you can choose to renounce US citizenship. Although we often hear about people coming to the US, we seldom hear about the citizens who renounce their citizenship. This US State Department page outlines the various categories for losing or renouncing citizenship.

US CITIZENSHIP STATISTICS

The Census Bureau uses the term native to refer to anyone born on US soil, or born abroad with a US citizen parent. In 2010, including the military and others living abroad, the Census Bureau said we have approximately:

267 million native (natural citizens)

16 million foreign born (naturalized citizens)

21 million foreign born (not citizens)

NOTE: The Census Bureau uses the term foreign born for naturalized US citizens, people who are undocumented, lawful permanent residents (immigrants), temporary visitors (tourists, foreign students, farm workers), and humanitarian immigrants (refugees seeking asylum).

Geographical Math, Part IV

2012-01-11T14:53:00.000-08:00

Greetings! I am jumping back into a topic that we started last week. This blog is about using elementary (but not easy!) math, as taught in our Excel Math curriculum.

People often think of the United States as being composed of 50 states. Period. Full Stop. No more. But amazingly, there are far more governmental subdivisions than we expect. And it turns out that counting them is like herding cats. To establish order on these cats, I decided to build a table as I help kids learn to do in math class.

Division	Inside US	Number	Continent	Island	On USA soil?
State	Yes	50	Yes	Yes	Yes
District	Yes	1	Yes	No	Yes
Commonwealth	No	2	No	Yes	Yes
Territories	No	20; + ships	No	Yes	Sort of; international
Jurisdiction Diplomatic	No	200 +	Yes	Yes	Yes, inside foreign country
Jurisdiction Military	No	~ 1100	Yes	Yes	No, inside foreign country
County/Parish/Borough	Yes	~ 3140	Yes	Yes	Yes
City/Town	Yes	~ 30,000	Yes	Yes	Yes
Reservation, Home Lands	Yes	~ 300	Yes	Yes	Sovereign nation inside US
Foreign Embassies (see list)	Yes	~ 150	Yes	Yes	No; diplomatic immunity

If you recall, in prior blogs I addressed most of these entities, except Reservations, Rancherias and related entities in Alaska and Hawaii. These subdivisions are unique in that they are considered "domestic dependent nations". They are within the US boundaries (domestic), have a relationship with us for defense, etc (dependent) but have self-determination or rule (nation). [click on map to enlarge it]

The US Consitution says “Congress shall have the power to regulate Commerce with foreign nations and among the several states, and with the Indian tribes” so even though reservations fall within state boundaries, tribal sovereignty is dependent on, and subordinate to, only the federal government, not states. Other judgements have clarified things: “the establishment of a reservation by treaty, statute or agreement includes an implied right to hunt and fish on that reservation free of regulation by the state.”

Chief Justice Thurgood Marshall stated “England had treated the tribes as sovereign, and negotiated treaties of alliance with them. The United States followed suit, thus continuing the practice of recognizing tribal sovereignty. When the United States assumed the role of protector of the tribes, it neither denied nor destroyed their sovereignty.”

I'm not an expert on Indian affairs but we do have more reservations in San Diego County than any other county in the United States. Here's a new chapel designed by a friend of ours, Kevin deFreitas, located on the Reservation of the Rincon Band of the Luiseño Indians, a few miles north from the Excel Math offices.

Evidence-Based Education, Part II

2012-01-10T15:43:00.000-08:00

Welcome!

Yesterday in the Excel Math blog I complained about the complexity of evidence-based decision-making. I said the process is too complicated. Its very complexity and complication drives out simple but valid options. I normally talk about using elementary math in everyday life, but any curriculum selection decision-making process transcends simple math. [click image to enlarge]

Paraphrasing Benjamin Graham very slightly:

Normally, mathematics is expected to produce precise and dependable results; but in the stock market, the more elaborate and abstruse the math, the more uncertain and speculative are the conclusions.

In 44 years of Wall Street experience, I have never seen calculations made about stock values or investment strategies that needed to go beyond simple arithmetic or elementary algebra.

Whenever calculus is brought in, or higher algebra, take it as a warning that the speaker is about to substitute theory for experience, and make speculation seem to be investment!

Keep this quote in mind, and substitute procurement for investment as appropriate!

Acquisition and Procurement
With a few clicks of the mouse you can investigate dozens of purchasing projects that end up being scrapped due to complexity. [ Notorious examples include US Air Force tanker aircraft contracts and the UK's NHS computer system, etc. ] I have participated in several smaller but (in)famous development projects myself.

In the Education business, we as a society (superintendent, school board, principal) appear to be going down the same path. We decide that those closest to the action (teachers, parents) can't make decisions on their own.

The process, by its very complexity, demands that decisions be made by professional evidence-sifters - pushed upwards to a higher authority, who makes a judgement affecting more than one school or district - perhaps even a whole state.

One advantage from a School District point of view is the implied credibility of this decision-making process:

Who would criticize decisions made on the basis of facts? (Can you go wrong buying IBM?)
Are you setting yourself up against our authority?
After all this, why would you debate our decision?

However, it is simplistic to think that decision-making is funded, pursued and concluded in a completely objective way. Even if we do set our biases aside, refrain from emotional influences and spurn illicit offers of personal enrichment, the "evidence-based" process implies

We eliminate smaller companies, resource-poor researchers, non-expert witnesses, or advocates from either extreme end of the spectrum
Few decision-makers will re-study something that has been panned before
Only the most popular products are worthy of study
Only the most readable papers get published
Only the most attractive ideas get funding

And we layer this evidence-based evaluation onto an already laborious adoption process (6 years in California)! If it takes longer to choose the curriculum than a student stays in a public elementary school, are we over-thinking it?
I'll close with an illustration from an earlier blog on objective measures for the subjective topic of pain. Please select the level of pain you associate with involvement in a procurement project:

Evidence-based Education, Part I

2012-01-09T17:00:00.000-08:00

This blog usually addresses ways that elementary mathematics can be used by adults later in life.

Today I am leaving the comfortable ground of solid calculation (2 + 4 = 6) and venturing onto the dangerous, thin ice of opinion (Honey, do these slacks make my bottom look big?)

I want to tackle the current trend of promoting evidence-based decision-making, particularly when it comes to curriculum like ours.

A reasonable desire for repeatable, reliable, rational results seems sensible. Evidence-based materials give us teaching methods and curriculum that really deliver what we want.

But wait! Do they?

The decision to use evidence-based decision-making is NOT necessarily based on evidence.
Evidence is always picked and parsed by our personal prejudices, pre-conceptions and pre-suppositions. Not all evidence is equal. Just watch any detective show on television, you'll see.

For example, here are some steps in evidence-based decision-making.

Formulate a well-stated question
Apply rigorous data analysis to test the statement of the hypothesis and justify its conclusion
Identify an adequate set of observation- and experiment-based resources that answer the question
Critically appraise the various solutions, assessing their validity, comprehensiveness, etc.
Verify that your findings are supported by a "critical mass" of scientific research; find valid and consistent data across evaluators, observers, multiple trials and cultures
Confirm that your choice has been accepted in a peer-reviewed journal or a panel of independent experts through rigorous, objective and scientific review
Apply your chosen evidence-based solution in your own trials in your own context
Apply your own professional wisdom and personal experience, while ensuring they are subject to the consensus view of your co-workers
Re-evaluate the application of the solution (if necessary, identify areas for customization and improvement)
Decide, purchase and implement

So to sum it up, a product or program may be considered evidence-based if

repeatable research shows that the program produces positive results as expected
results are proven to be due to the program, not accidental or due to extraneous factors
the evaluation itself has been peer-reviewed by experts in the field
the program is endorsed and listed by a federal agency and/or respected research group

I think we rarely (if ever) depend on this degree of research and evidence to make up our minds. Why? It's too complicated!

I will close with an example - each variation in the "well formulated question" mentioned in Step 1 may demand an entirely different curriculum:

How do I teach fractions to a group of 4th graders?

How do I most effectively teach fractions to a 4th grader?

How do I most effectively teach fractions to a small group of 4th graders who are native English speakers?

How do I most effectively teach fractions to a small group of 4th graders who are native English speakers, while they are living in Germany?

How do I teach fractions to a group of six 4th graders, none of whom are native English speakers?

How do I facilitate the interactions of a group of sixteen 4th graders so they can discover the concept of fractions by themselves and master this concept for their lifetimes?

Geographical Math, Part III

2012-01-05T13:59:00.000-08:00

This is the third of a series illustrating the use of math with geography.

Today we will tackle the subject of sub-divisions of the country that are within the US. As I am not a political scientist, geographer or history teacher - the only reason for discussing these items is to demonstrate the importance of knowing basic math (as taught in our Excel Math curriculum).

Here are some of the "administrative divisions" of the United States:

FIRST RANK
State
Commonwealth (covered in a previous blog)
Reservation
Territory (covered in a previous blog)

SECOND
County, Parish, Borough
Consolidated City-County
Independent City

THIRD
City, Town
Township

I invite you to visit this Wikipedia page, choose any state and select a list of settlements or towns. Or, if you prefer your data closer to the source (and harder to parse), go to this Census page. Here is a map of my state, California, showing the population distribution. [click on map to enlarge]

Our population was roughly 37,350,000 in 2010, and the subdivisions are shown below:

COUNTIES

California is divided into 58 COUNTIES, which are responsible for all elections, property-tax collection, maintenance of public records, and local courts plus the job of enforcing the law within all areas NOT inside an incorporated city. There is one CONSOLIDATED CITY-COUNTY in the state, San Francisco. Two counties have been consolidated into others, five areas have tried to split out from existing counties and failed, and at least a half-dozen proposals are pending to change county status. [click map to enlarge]

CITIES

California has 482 incorporated cities and towns; 460 call themselves CITIES and 22 call themselves TOWNS. The term TOWNSHIP is obsolete and neither VILLAGE or BOROUGH have ever been used in California. Only currently incorporated places are counted. Ghost towns and previously incorporated cities that have merged into others are not counted. Seventeen cities in California have dis-incorporated themselves and are not counted.

There is no end of applications for math when studying people that live within a certain area of the world. Imagine the complexity of analyzing all the communities in all 182 regions that have visited this blog!

Geographical Math, Part II

2012-01-04T15:18:00.001-08:00

This blog is focused on using math learned in elementary school to understand the adult world around us.

Yesterday I explained a bit about districts, territories and jurisdictions. These are places and people associated with the United States, but not within the US. This subject was only chosen to provide us with some numbers we can investigate - not because I have any special knowledge about it.

Wake Island

In case my previous blog did not provide enough math for you, here are some numbers that describe a few of the places mentioned yesterday:

Territory	Area Sq Miles	Land Sq Miles	Water Sq Miles	Water %
Puerto Rico	5,325	3,425	1,900	35.68
Northern Mariana Islands	1,975	179	1,796	90.93
United States Virgin Islands	737	134	604	81.87
American Samoa	584	77	506	86.75
Guam	571	210	361	63.22
Minor Outlying Islands	16	16	N/A	0.00

The data came from Wikipedia.

MATH LIES AHEAD

Here's a simple math question related to the chart:

Which columns are essential, and which are discretionary or can be derived from the other(s)? How little do we need to know to fully populate this table?

This is what we teach kids to answer for themselves.

In our data there is total area, land and water area, and percentage of water for the first 5 rows. We'll ignore the last row because the minor islands don't claim water area.

If we are given total area, we need either land or water area and then we will know everything:

Total - (what ever number you have) = (the number you don't have)

then

(Water square miles) ÷ Total = (Water %)

Or we can derive the unknown data another way. We can start with total and water percentage to learn everything:

Total ÷ (Water %) = (Water square miles)

then

Total - (Water square miles) = (Land square miles)

Or we can do it yet another way. We start with total and water percentage but use a different operation to derive the land area percentage and square miles.

Total ÷ (1-Water %) = (Land square miles)

then

TOTAL - (Land square miles) = (Water square miles)

Now is when I say,

Isn't math fun?

and you say,

Get a life, Excelmathmike!

and I say,

Ok, then. Hiho, Hiho, it's off to an atoll I go...

Geographical Math

2012-01-04T10:19:00.000-08:00

The United States of America currently has 50 states. As a math person, I think that simply equating USA = 50 is a mistake. Why? There are many other sub-divisions of this country, including counties, cities, reservations, etc. But I'd like to investigate the math of external sites known as districts, territories and jurisdictions.

DISTRICT

The District of Columbia lies within the US and adjoins several states. It is not a state, but a Federal District, run separately to "host" the US capital city of Washington, along with housing about 200 foreign embassies, UN heaquarters, etc. It has a mayor but Congress ultimately controls the district.

TERRITORY

Territories and rules for their governance were originally created to manage newly-acquired lands while the borders of the United States were evolving - such as the "Louisiana Territory." There were many territories during the period from 1789 to 1959; 31 of these became states. Not all who applied were accepted, nor did all the land in each of these territories become part of the US.

We had no offshore territories until 1856 but we now have a handful of island groups. Other territories are "virtual" and exist because of a contract, lease, flag, or temporary need (war). The courts have defined these types of territories:

Incorporated organized territories

The last incorporated organized territories became states in 1959 (Hawaii and Alaska).

Incorporated unorganized territories

Palmyra Atoll is owned by the Nature Conservancy. It's a national marine monument, protected by the Conservancy and the US Fish & Wildlife Service. Palmyra includes about 50 islands roughly 1000 miles south of Hawaii. Palmyra was linked with Hawaii when Hawaii was annexed in 1898 but was excluded during the statehood process.
US coastal waters out to 12 nautical miles offshore are considered territory.
US registered/flag-flying ships, whether Naval, Coast Guard, or civilian, are considered territory.

Unincorporated organized territories

Pacific Ocean

Guam is the southernmost Mariana island. It has been a US territory since 1898.
Northern Mariana Islands became a US Commonweath member in 1978.

Caribbean

Puerto Rico has been a territory since 1898 and became a US Commonwealth member in 1952.
United States Virgin Islands became a US territory in 1917.

Unincorporated unorganized territories

Pacific Ocean

American Samoa has been a territory since since 1898 and became self-governing under a constitution written in 1967.
Swains Island is owned by the Jennings family and was annexed by the United States in 1925; also claimed by others.
Wake Island is actually three tiny islands surrounding a lagoon; it is managed by the US Air Force.
Midway Island once had a Navy presence; now a wildlife refuge supporting 3 million birds.
Johnston Atoll, Baker Island, Howland Island, Jarvis Island and Kingman Reef are all uninhabited

Caribbean

Bajo Nuevo Bank uninhabited; also claimed by others.
Serranilla Bank hosts a Colombian naval station; also claimed by others.
Navassa Island uninhabited; also claimed by others.

JURISDICTIONS

The US and many other large countries occupy (with permission) small chunks of land within other nations where they place military bases, embassies, consulates, etc. International agreements mean these jurisdictions are considered part of the country that has been given permission to establish them. Some countries have exploratory outposts in places like the Arctic or Antarctica (and the moon?).

Extraterritorial jurisdictions

Guantanamo Bay Naval Base a strip of land on the edge of Cuba, has been in US control since 1903; but the legality of the lease is disputed by Cuba.
Military bases and embassies exist in about 200 countries all around the world, established by lease or agreement with the host country. Here's a list of US embassy offices.

These associated lands and populations explain why the number 50 doesn't fully describe our nation.

This blog has been brought to you by Excel Math, built for elementary school students in the US, its districts, territories and jurisdictions (and anyone else who wants to buy it).

Would you care for a piece of pie?

2012-01-03T15:32:00.000-08:00

Welcome to the Excel Math blog, where we investigate the world using math concepts that we learned in elementary school. Today I would like to address PERCENTAGES since 1% and 99% are in the news almost every day. I'm going to stay out of the emotional part of this debate and simply say that wherever you have people and pie, you have inequality.

Naturally, in our elementary math curriculum we do not directly address income and wealth with students, but we do discuss things like dividing up pieces of a pie. They have strong opinions about eating pie, just as we adults have strong opinions about sharing pieces of the economic pie.

Since we can divide any pie into infinite numbers of pieces, I'd like to ask, Why one percent versus ninety-nine? Where is the line? How do we define (point fingers at, or enter) the top 1%?

The following example data is very rough and rounded to the nearest even number. It was outlined by a professional pie slicer. If you have ever cut an apple pie into sections, you know that the crust may break, the filling ooze out, or an apple segment get snatched away by your brother. So it is in economics - there is no perfect line.

If you expect to qualify for the 1% using your earned income, you need about $300,00-$400,000 per year. (The average American family has about $55,000 of income.) You are probably a doctor, lawyer, upper level manager or executive, car dealer, or successful business owner. You've worked hard for 20-40 years, been promoted and invested wisely so your investments produce income.

If you prefer to enter the 1% using your accumulated (or inherited) wealth, you need more than one million dollars. Take all that you own, subtract what you owe, and place yourself in the ranks according to the following scale:

top 1.0% with $1.2 million
top 0.5% with $2.0 million
top 0.25% with $3.0 million

Folks who enter the top with wealth are similar to those who qualify with income - except they probably earned less, but saved more over a long time. Both sets earned their membership "the hard way" - by selling their labor and brains. But there are even higher ranks which are nearly impossible to enter by sweat and toil and time:

top 0.10% with $5.0 million
top 0.01% with $25 million

The top tenth of one percent did not earn their slice of the pie by salaries alone (which are taxed pretty rigorously by the IRS, no matter which party is in power). They built and sold real estate developments, dotcoms that went public, stock portfolios that soared, holdings in the military-industrial complex, or worked in the banking and investment industry, or received inheritances. Their advisors and accountants helped them invest legally in business sectors and strategies with relatively lenient taxation.

Eventually the percentages at the top get so small that they're unmeasurable. The LA Times said recently that the six family members who have inherited Sam Walton's estate now have about $90 billion between them - about as much of the whole pie as the bottom 30% of Americans. That doesn't make them bad people, by the way, nor does it mean you will never see them walking down a street in your neighborhood.

House near San Diego, former home of the richest woman in the world

And finally, Mexican citizen Carlos Slim, the richest man in the world, has nearly as much wealth as these 6 US families put together! There are many billionaires like him spread around the world - lucky for us, we are not all sharing the same pie.

For very detailed discussions accumulating wealth, see the Who Rules America website. I am indebted to them for some of the thoughts here, as well as Jack Bogle (formerly with Vanguard).

Multiplication Mysteries

2011-12-21T13:32:00.000-08:00

Yesterday I shared the discovery of a new material called Magic Gold. It's an alloy of gold and ceramic. In some respects, the fusion of these materials gives us more than the sum of the individual parts. Magic Gold is still gold, but lighter than gold, stronger than gold, and tougher than gold. It's a form of magical multiplication (I had to get math in here somehow, this is a math blog!)

Alchemy is a philosophical and magical tradition that seeks a philosopher's stone, an object that transforms base metals into gold or silver, and creates a drink that confers immortality - the elixir of life or fountain of youth.

Here's a poem from centuries ago:

Augmentation is of the Elixer indeed,
In goodness and quantity both for red and white
Multiplication is therefore as they do write,
That thing that doth augment medicines in each degree,
In colour, in odor, in virtue and also in quantity.

And why may you multiply this medicine infinitely,
Forsooth the cause is this,
For it is as fire, which kindled will never die,
Dwelling with you, as fire does in houses,
Of which one spark may make more fire this way,
As musk in pigments and other spices more,
In virtue multiplied, and our medicine right so ...

Keep in your fire therefore both morning and evening,
So that you do not need to run from house to house,
Among thy neighbors to seek or borrow your fire,
The more you keep, the more good shall you win,
Multiplying it always more and more within your glass,
By feeding with Mercury unto your lives end,
So shall you have more than you need to spend.

Did you notice the distinctly modern tone in this ancient poetry? Multiply what you have, magically, so you will have more than you need to spend for the rest of your life? Sadly, despite the promises of our 401(k) brochures, and the ready availability of alchemic symbols on the Web, magical multiplication eludes our grasp today.

As human beings, we are pretty good at division - diluting and sweetening (this blog's topics of the past few days). We can water anything down and make it go farther for less cost. It's a lot more difficult to multiply and get something out of nothing, because in non-magical math, zero times any quantity is still equal to zero.

In his Philosopher's Stone, Van Morrison articulates this eternal search, lamenting his unending task of writing new songs:

Out on the highways and the by-ways all alone
I'm still searching for, searching for my home
Up in the morning, out on the road
My head is aching and my hands are cold

I'm looking for the silver lining in the clouds
I'm searching for the philosophers stone

It's a hard road, Its a hard road daddy-o
When my job is turning lead into gold
I'm on the road again and I'm searching for
The philosophers stone

Even my best friends they don't know
That my job is turning lead into gold.

Up in the morning, up in the morning
When the streets are white with snow
Up in the morning, out on the job
Searching for the philosophers stone.

Dilution and Density

2011-12-19T12:19:00.000-08:00

In the most recent series of math blogs I have been talking about drinks - water, juices, tea, etc.

What are drinks?

Drinks are liquids that we consume to sustain or entertain ourselves. They are carefully-contrived blends of various substances (carbonation, fruit or vegetable juice, water, sugar, etc.) that are wet, drinkable and nutritious (or intoxicating), etc.

As I discussed earlier, whenever companies create blends or combinations, government gets interested. Blending may be used to disguise fraudulent or unhealthful products. Thus we regulate what vendors can claim or sell. We often need some elementary school math skills to understand these regulations and the products being controlled.

Today I would like to turn your attention to a different type of "blend". I invite you to compare cranberry juice and gold.

Pure cranberry juice is extremely tart, sour, bitter and acidic. It's nearly impossible to drink without dilution and sweetening. We need to mix the juice with sweeter fruit juices or add water and sugar. Most cranberry juice products end up being about 25-30% cranberry juice.

Now let's make a big mental jump from red to gold.

Pure gold (24 carat) is very valuable, very heavy and very soft. It's nearly impossible to use without dilution with silver, copper, nickel, etc. We need to mix the gold with other stronger substances, which make gold much stronger, a bit cheaper, and change its color. Many gold products are 18k gold, which is a blend (alloy) of 75% gold and 25% other materials.

ALLOY: A metallic substance made by mixing and fusing two or more metals, or a metal and a nonmetal, to obtain desirable qualities such as hardness, lightness, and strength.

OK, with all that background established, this week a brand-new gold alloy has appeared. It's called MAGIC GOLD. It was developed by Hublot, a Swiss watch company. I have visited the company, and met with its dynamic president, Mr. J-C Biver.

To clarify the magic, I will use a trick question "Which is heavier, a pound of feathers or a pound of lead?" Of course the answer is "A pound of one is the same weight as a pound of the other."

This reveals the MAGIC at the heart of MAGIC GOLD. It weighs less than any other alloy of gold, yet it is still 75% gold. Why? How?

Here is the pound of lead part of the question:

The Swiss Federal Customs Administration (FCA) manages any claims made about gold, and Hublot quotes them when saying:

MAGIC GOLD contains 750 grams of pure gold per 1000 grams of alloy (proportions by mass), thus FCA gave us permission to use a hallmark of 750/1000 (18 carat)
MAGIC GOLD is a composite (alloy) made of pure gold [density 19.32 g/cm3] and boron carbide ceramic [density 2.5 g/cm3]
The density of this new material is found with this equation:

((% gold x mass of gold) + (% ceramic x mass of ceramic) = (mass of the alloy)

((0.3 x 19.32) + (0.7 x 2.5)) = (5.796 + 1.75) = 7.546 g so density = 7.546 g/cm3

Now here is the pound of feathers.

MAGIC GOLD is by volume 30% pure gold and 70% boron carbide (proportions by volume).
A watch case in 18k gold that weighs 100g contains 75g of pure gold (75%)
The same watch case in MAGIC GOLD weighs 48.68g (about half of 100g) and contains 37.38g of pure gold (about half of 75g)
So, for the same volume of material, the watch case contains half as much gold by mass
Total weight of the case is also half, but the proportion of pure gold remains at 75%

Where is the MAGIC here? The MAGIC is that Hublot can make watch cases and cuff links, rings, etc. using half of the gold but adding lots of research, ceramics, temperature and pressure. The resulting substance is stronger, more scratch-resistant than the old stuff and it's still 18k gold.

Sweet, eh? Just like cranberry juice punch!

Pour boiling water ...

2011-12-16T17:47:00.001-08:00

In the past few blogs, I have outlined the things that we can do to make water more palatable by chilling, carbonating, mineralizing and flavoring it. I described drinks that involve various blends of fruit juices, but skipped sodas and sport drinks.

Today we will consider hot beverages - without which most people would stay in bed, or fall asleep at their desks.

Do we need any math skills to understand these drinks?

With a cup of hot coffee or tea, the groggy sleeper ingests some warmth, some caffeine, some flavor and some aroma - and perhaps a few calories. There are plenty of other hot drinks in the world - infusions and heated spirits and hot chocolate, but coffee and tea are the two favorites.

Neither of these beverages has much nutritional value. We get no calories, no fat, no protein, but coffee is apparently a good source of Pantothenic Acid, and a very good source of Riboflavin, while tea is a very good source of Manganese. Both contain caffeine, with coffee supplying 2-3 times caffeine as much as tea.

Neither of these beverages can be easily made at home, from scratch. You can't grow the plants or prepare the raw materials yourself unless you live in certain tropical regions. [click images to enlarge]

But looking at the production process or chemical content doesn't tell us why people drink either one. There's plenty of intangible value to be obtained from some human activities, even when that value can't be quantified by the scientists.

NOTE - For detailed nutritional statistics on brewed, un-blended beverages made with tap water, check here for coffee, and here for tea.

COFFEE
I don't drink coffee myself, so I can't speak to its intangible virtues. But I can certainly see people all around me who enjoy it. Historians tell us this has been the case for about 5 centuries. How much do these coffee lovers drink?

I found estimates suggesting half the US population of 310 million drinks coffee, and the average drinker consumes 3.2 cups per day, so ( 310 ÷ 2 ) x 3.2 = 496 million cups
Other sources told me 400 million cups a day

And all this because someone thought of picking, drying, roasting and grinding these pretty berries, before forcing hot water through them!

TEA
The tea drinkers of the world, and I am one, find pleasure in drinking tea. We have been doing so for about 30 centuries! We have been taking a short break with it, blending it, sweetening it, diluting it with milk and cream, washing down cookies and cakes with it, and so on. How much do we drink?

I found that tea drinkers comprise 66% of the 65 million people in the UK and each consumes about 3 cups per person daily. That means ( 65 x .66 ) x 3 = 128 million cups
The Tea Council says the number is 165 million cups a day

Just for fun, a comparison: the UK has about 1/5 the population of the US but drinks about 1/3 of the tea compared to the coffee we drink.

It's amazing that all this activity centers around pouring heated water over some leaves, isn't it? Ok, it's a bit more involved than that, but still, it's hot water and plant leaves. And some math.

Non-water, Non-Alcoholic, Non-sense Drinks

2011-12-16T11:43:00.000-08:00

For the last few days I've written about water and how we prefer to consume our water and other drinks (meaning a beverage which is NOT water and not primarily a food). We like them a bit cooler or a bit warmer than our body temperature, depending on the drink.

It turns out that a little elementary mathematics comes in handy in dealing with drinks, as we must consider heat, cold, volume, percentages of dilution, etc. Here are a few reasons why some math is necessary when shopping (or indeed when trying to sell juice). Imagine if these were applied to the proverbial Lemonade Stand!

[Code of Federal Regulations]
[Title 21, Volume 2][Revised as of April 1, 2002]
From the U.S. Government Printing Office via GPO Access [CITE: 21CFR102.33]
TITLE 21--FOOD AND DRUGS
CHAPTER I--FOOD AND DRUG ADMINISTRATION, DEPARTMENT OF HEALTH AND HUMAN SERVICES
PART 102--COMMON OR USUAL NAME FOR NONSTANDARDIZED FOODS
Subpart B--Requirements for Specific Nonstandardized Foods
Sec. 102.33 Beverages that contain fruit or vegetable juice.
(edited slightly for ease of reading)

(a)
For a carbonated or non-carbonated beverage that contains more than 0% and less than 100% fruit or vegetable juice, and, if the usual name uses the term juice, it shall include a qualifying term such as beverage, cocktail, or drink to advise the consumer that the product is less than 100% juice (diluted grape juice beverage or grape juice drink).
(b)
If the product is a diluted multiple-juice beverage or blend of single-strength juices, then the names of those juices must be shown in descending order of volume unless the name specifically shows that the juice with the represented flavor is used only as a flavoring (raspberry-flavored apple and pear juice drink). The presence of added natural flavors is not required to be declared in the name of the beverage.
(c)
If a diluted multiple-juice beverage or blend of juices contains a juice that is named or implied on the label (represented juice), and also contains a juice other than the named or implied juice (non-represented juice), then the product name shall indicate that the represented juice is not the only juice present (Apple blend; apple juice in a blend of two other fruit juices).
(d)
In a diluted multiple-juice beverage or blend of single-strength juices where one or more, but not all, of the juices are named on the label, and where the named juice is not the predominant juice, the product name shall indicate that the named juice is present as a flavor or flavoring (Raspcranberry; raspberry and cranberry flavored juice drink); or it shall include the amount of the named juice, declared in a 5% range (Raspcranberry; raspberry and cranberry juice beverage, 10-15% cranberry juice and 3-8% raspberry juice).
(e)
The common or usual name of a juice that has been chemically modified shall include a description of the exact nature of the modification (acid-reduced cranberry juice, or deflavored, decolored grape juice).
(f)
If the product is a beverage that contains a juice whose color, taste, or other properties have been modified to the extent that the original juice is no longer recognizable, or if its nutrient profile has been diminished to a level below the normal nutrient range for the juice, then the fruits or vegetables from which the modified juice was derived may not be depicted in pictures on the label.
(g)
(1) If one or more juices in a juice beverage is made from concentrate, the name of the juice must include a term such as from concentrate or reconstituted, in a type font no less than one-half the height of the letters in the name of the juice.

(2) If the juice is 100% single-species juice consisting of juice directly pressed from a fruit or vegetable whose Brix level is adjusted (raised) by the addition of juice concentrate from the same fruit or vegetable, it need not include a statement that it is from concentrate. However, if water is added to this 100% juice to adjust (lower) the Brix level, the product shall be labeled from concentrate or reconstituted.

Now that you know all this, have fun inspecting the labels of these juice-like drink beverages we found in the local market. The first is what I would call a brightly-colored, flavored soft drink:

After tasting this, I don't know what I would call it, but it doesn't taste like either Pomegranate or Blueberry juice. It's sweetened with sugar and with concentrated pear and apple juices.

This is some tart red stuff that looks like juice, tastes like juice, but can't be called juice.

This is the most complicated label and very tasty stuff it is too, but what would you call it? Do we benefit from these complex creations? It's not a hot summer day but a cool winter day - yet it still tastes very nice, in my opinion.

Ah, orange juice. Not from concentrate. Turns out it was the cheapest of all these choices, too!

Cold, Clear, Sparkling Water

2011-12-15T16:10:00.000-08:00

What makes water appealing to drink? Can elementary math help? Let's see.

Which type of water is most appealing to you (select 1 of 3 answers below):

A glass of luke-warm, 85° F rusty tap water, with a bit of chlorine odor?
A bucket of greenish water with a few particles drifting (or swimming) around the bottom?
A small trickle burping up out of a rusty fountain with greasy kid-prints on the push button?

NONE! I'd rather go without!, you say? I completely agree with you.

I wondered why we tend to like water that is a bit colder or hotter than we are ... lukewarm or tepid aren't very nice, either for bathing, swimming, or drinking. Too cold is difficult - chattering or cracking our teeth - and too hot burns our tongues or scalds us.

I read that drinking cold water burns calories, as the body has to work to heat the water up. That sounds a bit fishy to me, but you can see here how people try to explain it with math. Other researchers suggest that beverages 10 degrees or less below room temperature are more platable, so we drink more. Of course they are not talking about plain water, but:

"Fluid replacement beverages that are sweetened (artificially or with sugars), flavored and cooled to between 59 and 70° F, should stimulate fluid intake."

Some studies show cooling a beverage will increase consumption between 40 and 80%.

Of course there are calories in sugar-sweetened water (and you get cavities from it, say dentists) or chemicals in artificially-sweetened waters.

Others say that drinking warm water relaxes the mouth and throat, making it a healthier choice. (Yawn).

As long as we are talking additives, here are some numbers that define salinity for you (salt in the water). The unit we are using is parts per million:

drinking water - 100 ppm
limit drinking water - 1000 ppm
limit for agriculture irrigation - 2000 ppm
sea water - 30,000 - 50,000 ppm
brine / slush > 50,000 ppm

HOW DO WE MAKE WATER MORE ATTRACTIVE?

FILTER IT
Take out dirt, chemicals and colors. Make it clean and clear. There are many filter systems available - they fit in the garage, or under the sink, or on the counter, or in the refrigerator.

BOIL IT
Sometimes water tastes "flat." Air dissolved in water occupies up to four times the space that carbon dioxide consumes, so get the air out of it first by boiling it.

CHILL IT
Warm or room-temperature doesn’t taste very good. It doesn't carbonate well either - CO2 dissolves better as water temperature decreases - the closer to freezing the better.

PUT MINERALS IN IT
Mineral salts enhance flavor, which is why mineral waters work well with food, and they are more appealing internally as the correct salinity of water allows it to be integrated more easily into the body.

CARBONATE IT
Carbonation means bubbles of CO2 - adding a tingle and sparkle in your mouth to fire up your sensory and olfactory systems. The bubbles help lift the aromas to your nose and up the back of your throat.

ADD FLAVORS TO IT
There are lots of things you can add to water, but more than a very small amount means it becomes a "flavored drink" or simply dilutes the other liquid (not always a bad thing).

I'm thirsty - it's off to the Excel Math water cooler for me!

How to get into hot water

2011-12-14T14:14:00.000-08:00

A common idiom in English is "you're in hot water" which means you're in trouble. Why? Is the phrase derived from the old image of an explorer being cooked in a pot by a bunch of cannibals?

I think hot water is universally considered to be a very good thing, as any one who's bathed in luke warm or cold water can attest. Today in the math blog we'll investigate this subject - a moderate amount of elementary math is enough for the main issues related to hot water.

US Average Soil Temperatures

TEMPERATURE
Most of our water comes from pipes buried well below ground level, where it stays a relatively constant 40-70° F or so, depending on location and weather. Since our body temperature is around 100° F, we prefer to wash (if we can) in water that is a bit warmer than the water coming out of the pipes - say about 104–120° F.

FLOW
A water heater in our homes should be able to raise water temperature about 40-60° F and supply us with at least 5-6 US gallons per minute (gpm) or more. Why? Because sink faucets and appliances can draw up to 3 gpm (each).

You may know that plumbing codes restrict shower water flow to 2.5 gpm. But remember, this is for ONE shower head. If TWO or more showers might be going at the same time, you need a much greater supply of hot water, or there will be angry shouting in your house!

HEATING
A BTU is the energy required to raise one pound of water by one degree F and a gallon of water weighs 8.3 pounds. Assuming 100% efficiency, raising 40 gallons of 50° F water up sixty degrees to 110° F requires a fair amount of energy, as you can see here:

(40 gal × 8.3 lbs) × (110 − 50) degrees

332 lbs x 60 degrees

19920 BTU

19920 / 100,000

(natural gas is sold by the "therm" which is 100,000 BTU)

.20 therms

NOTE: For more on Natural Gas, check my first blog on the subject. Here's the second one, too.

OPTIONS

We have a number of choices for heating our water. None are without compromises:

A 40,000 BTU/hr gas heater takes about a half hour to heat 40 gallons, at 100% efficiency. At $1 per therm, the cost of the gas would be about 20 cents. The tank will give us at least a 16-minute shower (40 gallons ÷ 2.5 gpm = 16 min) or longer since we usually mix some cold into the hot water flowing from the tap.

A typical electric water heater with a 4500 watt heating element takes a bit more than an hour to heat the same tank of water for roughly $ .50.

A tankless water heater with 40,000 BTU/hr can only supply about 1.6 gpm. To supply 2 showers (5 gpm and 60° F temperature rise), tankless heaters need capacity of about 150,000 BTU per hour, which means larger gas lines or electrical wiring.

Combination boilers in the UK and Europe heat water for both the shower and heating the house. They are compact and efficient and can generate 100,000 BTU or more. Since many are located inside the house, some fancy plumbing or venting may be required.

Solar heating is "free" but not fast. It requires a solar array and piping. The details vary so greatly with location, climate, and sun exposure that I can't do any reasonable math on solar water heating. While the heat is free, solar requires large holding tanks, because it works best when we are least likely to be taking hot showers (in the middle of a sunny day).

Lukewarm water is no treat. Neither is a cold shower.

Hot water is a luxury we all appreciate, enjoy and (most of us) happily pay for!

Best Price or Lowest Payment, Part II

2011-12-13T14:12:00.000-08:00

This blog deals with issues in adult life that we are capable of handling with elementary math skills, as taught in our Excel Math curriculum.

Yesterday I touched on the difference between getting the lowest initial price (good if you pay cash or are buying for the short term) or the lowest monthly payment (fine if you are financing for the long term). My comments here apply most directly to houses and sometimes to automobiles.

INITIAL PRICE VS LOWEST PAYMENT
How can paying a higher price save you money in the long run? It can ONLY IF you are able to get better financing and thus a lower total cost, OR if money is declining in value (inflation). It might be worth paying a bit more IF you can refinance later for less, or IF you don't plan to stay very long in your house.

Let's look at an example. We borrow $100,000 at 5% or 6% interest. Here's our cost:

$100,000 at 5% interest over 30 years; 360 monthly payments of $537 = $193,300
$100,000 at 6% interest over 30 years; 360 monthly payments of $600 = $216,000

Now imagine that we have to pay a slightly higher price but we can choose from 3 different financing alternatives. Here's the first option:

$110,000 at 6% interest over 30 years; 360 monthly payments of $660 = $237,600

A total cost of $237,600 over the 30 years. Hmmm. What happens if I get a better interest rate?

$110,000 at 5.5% interest over 30 years; 360 monthly payments of $625 = $225,000

This loan at one-half percent less interest ends up costing $12,600 less.

$110,000 at 5% interest over 30 years; 360 monthly payments of $590 = $212,400

This deal at one percent less interest saves me $25,200 compared to the first deal.

What happens if I fail to get a good loan rate, and I have to pay higher interest, say 7%?

$110,000 at 7% interest over 30 years; 360 monthly payments of $732 = $263,520

A change of 1 or 2% on the interest makes a huge difference!

monthly payments jump from $590 to $732 for $142 per month increase
overall cost jumps from $212k to $264k for $52k increase

Shop hard for good loan rates, but don't overpay for the house or the loan "points".

SOME REFLECTION
In the good old days we used to buy at the very limit of what we could thought we could afford, counting on inflation and time to raise our salaries and raise the value of the house. We gambled.

We bet on the interest rates coming down to reduce the cost of re-financing that hopefully would coincide with our greater income.

We gambled against interest rates going up, taxes increasing, losing a job, cars breaking down and/or divorce. All things that occur frequently, if not with certainty. Any one of these factors could make your deal collapse.

Even in "the good old days" people gambled and lost. But not too many people, and not too often.

Why? Among other reasons, Banks and Savings & Loan Associations kept us from getting too close to the edge. They made us put 20% or more down in cash (so we had a lot to lose), they talked to us about taxes and maintenance and other unanticipated expenses. They checked our pay stubs. They looked us sternly up and down, made us shuffle and stammer and invent good reasons for why we should get the loan. They didn't let us buy cars and houses at the same time.

PROCESSING or HOLDING?
In the past 10-15 years, it got easier. Loan applications were "streamlined" and less attention was paid to the property, or the buyer, or the ultimate holder of the loan. Everyone in the line was handling the paperwork with "tongs" or "gloves" rather than putting their fingerprints on it.

The loan was like a hot potato. Everyone expected to be in the house, or the loan, or the paperwork for just a short time, get paid well for their services, and move on without getting burned.

I call this processing. We were all gambling - everyone from top to bottom - both the naive and the sophisticated players.

We forgot that houses are also homes, and that we live in them as well as invest in them. Bankers forgot they were not just loan originators, but also in the business of holding loans over time, collecting the payments so they could pay interest to the folks dependent upon that income - making their house payments with the income. In my mind, this is holding.

Best Price or Lowest Payment?

2011-12-12T13:47:00.000-08:00

What's more important to you when making a major purchase like a house or car - the best initial price or the lowest monthly payment?

You might object, saying, "Aren't they pretty much the same thing?", and the answer is "No, not really."

The initial price is driven by the price of other competitive properties or vehicles. It is somewhat related to the supply of money and its cost (interest rates). The monthly payment is related to the price, of course, but it is also driven by the supply of money and its cost (interest rates).

I have younger friends who when they hear what we paid for our house, say "You bought your house at just the right time!" As far as the initial price goes, yes we did.

But when I say we got a 30-year, fixed-rate loan at around 10% interest, they say, "Whoa! You bought your house at the wrong time!" As far as the monthly payment (based on loan rates), we bought at the wrong time (or so it seems today).

The red lines on this chart indicate when I got (or re-financed) my 3 real estate loans. At least I got better at this as time went along!

35 years of average home loan interest rates

It's interesting (for me, in hindsight) to see this but doesn't help you in deciding if now is the time to buy.

A Buyer most likely to be interested in getting the lowest initial price is a

short-term owner
putting down lots of cash
buying an asset in a stable or declining market

A Buyer most likely to be interested in the lowest monthly payment is a

long-term owner
putting down little cash
buying an asset in a stable or improving market

Today we are at a point in our local market where the prices of homes are reasonable, but interest rates are low. That makes payments low. But the housing market is still chaotic; prices are likely to continue to decline; therefore short-term or cash buyers are not so inclined to jump in.

For more information on this subject, and how it particularly applies to housing in the Southern California area, read this nicely-written article. If this is totally confusing to you, ask your kids to help. We teach interest rates and percentages in our Excel Math curriculum.

Self-imposed Deflation

2011-12-09T10:36:00.001-08:00

It seems most of us prefer to buy food that's on sale.

I noticed a story on BBC News today (my wife and I used to live in the UK, so we are attuned to the issues there). UK supermarkets are as brutally competitive as the stores here in Southern California, and the BBC story compared 4 big chain stores. Asda, Morrisons, Sainsbury's and Tesco each have their own strategy to catch your attention, build loyalty, and get you to open your wallet:

Tesco gives you vouchers if their food is more expensive than Asda, when you prove it using their Price Check online.

Sainsbury's automatically gives you a voucher at checkout if any brand-name item would have been cheaper at Tesco or Asda. They do the research.

Asda supplies a voucher if their prices are not 10% cheaper than Tesco, Asda or Morrisons when you use their Price Guarantee software online or via mobile phone app.

Morrisons is taking the high ground and trying to focus on offering good food from local suppliers.

Can they all (at the same time) be the cheapest place to do your shopping?

"Yes," says Peter Lunn, an economist in Dublin, Ireland, "because we are drawn to discounts."

When you come out of a supermarket, you will have a cart full of discounted items. Your combination of items at that store will probably cost less than the same combination in a another supermarket.

However, if you'd gone to the competitor's store for their discounts, the resulting unique combination of stuff which you would assemble there will probably be cheaper than at the first store.

When researchers study a "standard cartful" of groceries (and unlike you and me, ignore all the special offers) the total prices vary by only about 3%. Asda (owned by Walmart) was cheapest.

But each of us buys different things, and many grocery items are bought due to a sale, regardless of we intend to buy when we go to a store. We only enjoy buying things that seem reasonable or cheap to us. We avoid the expensive items, or get them elsewhere.

Imagine the frustration of managers at a supermarket that drops its prices, pulls in more customers, and yet still falls behind in total sales to the competition. That happened to Tesco this year. One analyst said, “With more products available for less, the amount of cash taken has understandably dropped, despite Tesco having successfully attracted more shoppers. They call this strategy self-imposed deflation."

Why do they do it? To attract and (hopefully) retain customers for the long term.

That's exactly why we at Excel Math keep our curriculum prices low - after trying some tricky marketing ourselves,

Discount pricing formula courtesy of Dave Coverly at Speed Bump Cartoon!

we've found that special sales prices don't always work!

Everybody loves a discount, but this is ridiculous!

2011-12-08T09:51:00.001-08:00

I got my auto insurance renewal this week. Like mobile phone contracts, or buying a cable entertainment package, understanding the details demands that we use mathematics. Will the simple operations we learned in elementary school be adequate ... or not?

At least I know how to construct a 3-column, 10-row table. That's useful:

Insurance Category	Car 1	Car 2
Liability	112	157
Medical	5	8
Uninsured Motorist	24	26
Comprehensive Damage	14	31
Collision Damage $500 ded.	63	94
Towing & Rental Car	4	4
Collision Ded. Waiver	11	11
Total	233	331
Miles Driven Annually	7500	12000

Despite my boring blue cars (Honda, Volvo) I still have a total premium of 564 dollars and no cents.

After looking at the total price, I dove into the 3 pages of fine print regarding discounts on my policy premium. Here are the possible discounts off the "list price" for my policy:

20% for a Good Driver discount (no more than one moving violation "point", no more than one accident) Applied
25% for being At-Fault Accident Free for five years (5% for each year, up to 25%) Applied
22.5% for Cross Sold Credit (this does not mean I am angry, but I have Homeowner's insurance with this same company) Applied
12% for Continual Renewal for six years without lapsing (2% for each year, up to 12%) Applied
30% for Physical Damage Deductible of $500 rather than $100 Applied
15% for Multi-car Discount when two or more cars owned by the same person are insured Applied
5% off Medical premium for Passive Restraints (airbags) Applied
15% off Comprehensive for Passive Anti-theft Devices (factory car alarm/security system) Applied
10% off for Educator Discount (my wife is employed by a school district) Applied
5% off for Mature Driver Improvement Certificate (I'm insulted by the very thought of this!) Not applicable
1.5% discount for Good Student Grade Point Average (B or above) Not applicable
Discount for Distant Student living more than 100 miles from home (without a car) Not applicable
Discount for low annual mileage driven Applied

Now I suspect you are beginning to think the same way that I was thinking. Are all these discounts additive?

DON'T THEY OWE ME MONEY?

If you add them all up, you get: 20 + 25 + 22.5 + 12 + 30 + 15 + 5 + 15 + 10 = 154.5%

Take 154.5% times $564 and that means they owe me $871. Right?

Alas, no. That's not how it works. Each of the discounts can apply to only one type of coverage, and the discounts have already been applied to the prices shown above. Elementary math is really no use at all in deciphering exactly how the discounts decreased my price.

HOLY MACKEREL - WHAT WAS THE ORIGINAL PRICE?!

As Bob in our office is fond of saying, "It is difficult to say at this time." Comparison shopping with other companies is your best bet, or you might ask your insurance agent (if you have one) to go through the details with you.

WHAT IF I CHANGED CARS?

I was thinking about getting a more interesting car for driving in my spare time. What do you think of the engine in this red Pantera? Do you suppose my insurance company will give me a discount, or no discount?

The Cold Hard Facts

2011-12-07T10:15:00.001-08:00

Yesterday I talked about the saturation of the US market, by television and then by personal computer. The statistics were fairly simple, appropriate for we who have only studied math in elementary or secondary school. This isn't a blog with "rocket science" analysis, just a place to investigate interesting subjects and a bit of the math behind them.

Today we take a look at the refrigerator - an appliance so common in America today that it's nearly impossible to imagine living without one. Can math help us understand the refrigerator?

My wife's great-grandfather Morris was a mining engineer in the northern Mexico desert 100 years ago. He loved to surprise visitors from the United States by offering them ice cream in the middle of summer. How did he keep it frozen? In an icebox - a large cave filled (during the winter) with blocks of ice and straw. Go here to see his book.

My wife's great-uncle John sold refrigerators in the 1930's. He was a top executive with Frigidaire, making his personal fortune selling "boxes full of cold air." These appliances were greatly appreciated in Los Angeles, a fast-growing, hot and dusty corner of the US. Refrigerators were much better than ice boxes. Here's Frigidaire's one-millionth unit.

When I was growing up, my uncle Dick was president of LA Cold Storage, a company with the largest refrigerated warehouse on the West Coast. Today they chill 5.7 million cubic feet! I guess refrigeration runs in our family, although personally I am not that fond of the cold - I really suffered when visiting this Swedish Ice Hotel!

Visiting the Jukkasjärvi ice hotel

Sorry for the detour - now back to our topic of the day.

In the United States, England and France, ownership of refrigerators after WWII rose from under 10% (late Forties, early Fifties) to over 85% (mid-Seventies). My guess it it's now over 100%, with some households having more than one.

Since US households all have refrigerators already, manufacturers try to encourage us to replace our old ones BEFORE they wear out, thus reducing electricity consumption. This chart shows how energy is being saved (red line) as we replace (blue line) our smaller (17 cubic foot) refrigerators with cheaper (green line), larger (21 cubic foot) refrigerators. The data came from an energy blog/newsletter. [click the chart to enlarge it]

To see how refrigerators are adopted by households, we need to look at other countries coming up the curve of household automation. Here's a chart from a study of appliances in developing countries. I found the study very interesting even though it had a fair amount of complex statistical analysis.

The data implies this sequence of major purchases after electricity reaches a household: television, followed by refrigerator, washing machine and air conditioner. This information supports what we Americans know from experience:

There's a close link between watching television and needing to drink a cold beverage, while the clothes are in the washer. Hence the development of a recliner chair with built-in drinks cooler.

La-Z-Boy "Chill" recliner with cooler

How Much Time Do You Spend Staring at a Screen?

2011-12-06T14:33:00.001-08:00

This blog describes and demonstrates how adults can use the math they learned as kids. I write it because my "day job" is creating Excel Math curriculum materials.

Today I want to talk about television and computers.

During the decade I was born (the 1950's), the population of the United States grew from 150-180 million. The number of households increased from 44-53 million. There were about 3.4 people in a household back then. Only about 1 out of 11 households owned one or more televisions. Our family didn't have one for much of that decade. We were late adopters. Consequently, I don't know much about the famous TV shows of the Fifties.

In 2010 the US population was 309 million people, divided into 116 million households. That's an average of 2.66 people per household, with an average of 2.8 televisions per household!

USA	Households Total (millions)	Households w/Television (millions)	Households w/Computers (millions)
1950	44	4	0
1960	53	46	0
1970	63	60	0
1980	81	80	5
1990	93	88	14
2000	105	102	55
2010	116	116	94

For people born in the 1980's, there was a similar revolution. The decade started with only a very small percentage of people with a personal computer. Not wanting to be "late" again, I got my first, a Radio Shack TRS-80 Model 100, in 1979.

Now more than 80% of all US households have at least one personal computer, and virtually all of them have internet access.

(Most households that have a current computer also have a few obsolete computers too. How many do you have sitting in the garage or basement?)

In my 2-person household we have one television. We have 7 computers - an aluminum MacBook, a G5 tower, a silver iMac, 3 iPhones and one Windows Laptop.

In my 1-person office, I have no televisions. Today I have 4 computers - a titanium MacBook, a G4 tower, a white iMac and a brand-new silver iMac. Plus my phone and an iPod.

But hey, I'm in the publishing business. This is what I make math books with. What's your ~~excuse~~ justification for your computer(s) and television(s)?

Price of a Brownie vs Value of an Avocado

2011-12-05T13:28:00.001-08:00

Oscar Wilde said a cynic knows "the price of everything and the value of nothing."

Today I am going to examine a similar proposition - the price of a brownie versus the value of an avocado. This is the sort of useful analysis that an adult can engage in, using the math skills learned long ago in elementary school.

A couple hours ago, Dave and I went up to the food court to get 5 of their 3-tacos specials for folks in our office. When we received the food, the lady at the counter said:

"We ran out of avocado on the last 2 tacos, so I gave you a brownie instead."

It was very nice of her, but raised the question of equivalence in my mind. What I mean is:

How is the value of a brownie related to the value of avocado in a taco?

The brownie was $1.75 as a dessert item, and the avocado was rolled into the price of the taco special. But let's estimate it as a $1.00 upgrade to the normal taco lunch. So we got $1.75 worth of brownie to compensate us for a shortage of $1.00 worth of avocado.

Now let's take a look at the numbers - can we determine who's getting the best end of this deal?

150 grams / 6 ounces	Avocado	Brownie	Ratio
Calories	150	675	1:4.5
Carbohydrates	13g	72g	1:6
Cholesterol	0g	108g	infinity
Protein	3g	6g	1:2
Total Fat	22g	42g	1:2

If you are hungry for a dessert, the brownie definitely wins in this comparison test. It's got more of almost everything, including subjective things like intensity of flavor. Why not? It contains lots of chocolate, sugar and fats. Everything except "health."

On the other hand, an avocado is filled with exotic things, like:

phytosterols, including beta-sitosterol, stigmasterol, and campesterol
carotenoid antioxidants, including lutein, neoxanthin, neochrome, chrysanthemaxanthin, beta-cryptoxanthin, zeaxanthin, violaxanthin , beta-carotene and alpha-carotene
non-carotenoid antioxidants, including epicatechin and epigallocatechin 3-0-gallate, vitamins C and E, and manganese, selenium, and zinc
omega-3 fatty acids (alpha-linolenic acid) and oleic acid
polyhydroxylated fatty alcohols (PSA)
vitamin K, vitamin B5, vitamin B6 and potassium
dietary fiber

Impressed?

Which would you chose: Brownie or Avocado?

Which would you say has the most value?

Did you base your decision on reason (and math) or emotion?

Counting Up to 180, Part III

2011-12-02T08:00:00.000-08:00

Today I am showing the final list of countries that have visited our Excel Math Blog.
On these pages we talk about the math(s) we learned in elementary school and show how those skills can be utilized by grown up people.

This list is in descending order, with the most visitors at the top. If you have come to the Excel Math blog from one of these countries, were you looking for assistance in helping your kids learn math, or were you looking for Microsoft Excel software?

Country

Unique Visitors

Last New Visitor

101.

Guatemala

November 29, 2011

102.

Palestinian Territory

November 28, 2011

103.

Belarus

November 12, 2011