Additional Math Pages & Resources

Friday, January 29, 2010

How high?

Do you like to be up in the air, living on the edge, hanging over the rail with your camera, capturing it all? Riding the Stratosphere Roller Coaster, at the top of the Empire State Building, climbing the Eiffel Tower, dangling over the edge of the Grand Canyon's North Rim, enjoying the overlook at Bridal Falls?

Or are you like a couple of us at Excel Math - just a little happier with someone holding onto your arm while you close your eyes and back away from the edge, thinking thoughts of sea level and flat prairies? If so, read on.

I don't care all that much for mountains. I like floating on a river. Canal boats are a slow and relaxing way to spend a holiday. You simply putt along at a walking pace, through gentle countryside. Right? Thinking math thoughts, like how many hours of cruising can we do on one tank of fuel, where's the next pub ... la dee dah.

This is the life, we're thinking, then suddenly the boat enters a skinny aquaduct (water pipe) that goes up in the air! Or to be more precise, the ground disappears and we find ourselves in a metal channel stuck atop some stone pillars. The ground is 130 feet below us!

Here's another view, for the height-challenged reader.

While researching this blog, I read about a gal who was riding her bike along the path. She wobbled and fell over into the canal. It was so deep she couldn't climb out, and had to push her bike through the water, all the way to solid ground at the end. Oh boy. Imagine her pulse rate!

I had all sorts of math thoughts in my mind while preparing this blog, like how much water is flowing through the canals, miles of canals in England, Scotland and Wales, number of boats in the canals, etc.

Then I remembered going to the The Falkirk Wheel, a unique water lift that scoops boats out of a pond and hurls them into the air! Or to put it another way, you cruise into a pipe and it suddenly drops you 80 feet down into a pond! Here's an animated timelapse photo that's from the Wikipedia media library. My shots were too shaky!

But after remembering this traumatic, high-altitude canal adventure, I thought I'd just keep the blog simple. Here's another shot in some canal lock, somewhere in the UK, showing me at the elevation I prefer. Not too high. Can't fall too far. Lots of other people around. No math required.

Have a nice, flat, sea-level day!

Thursday, January 28, 2010

Well-diversified, risk-controlled portfolios

The title of this blog came straight out of a letter I received today from my retirement fund. They have hired a new firm of investment advisors. The terms of that agreement are defined in the letter.

It gives us a chance to see what we can do with math, as grownups concerned about retirement funds, over-paid bankers, volatile stock markets, etc.

This particular advisory firm is joining 5 other advisor firms, replacing one that is dropping off. The five who are staying on will receive the same fees as they did before. These advisors receive fees equal to 0.14% of the fund's average annual net assets, paid in installments at the end of each quarter.

QUESTION: If the value of the fund at the end of December 2009 was $34.9 billion, what were the approximate fees for the year?

We don't know the amount of the fund balance for each quarter, so we'll just use the annual balance for our calculations. Even though this is not entirely accurate, it's close enough for us.

                   x .0014

ANSWER: Just a bit less than $50 million, plus adjustments.

That charge is adjusted up or down by a factor based on the performance of the fund over a 60-month (5-year) period compared to the performance of the Russell 3000 Index over the same time period. The fees paid to the group of 6 advisors were reduced by $2.7 million (.01%) at the end of 2009 because they failed to meet their targets.

  - 2,736,000

That means they got $46 million, split roughly 6 ways.

That drops the average fee (per advisor company, not each person) down to $7,687,333.

The new advisor is expected to manage 8.5% of the total portfolio; about 35-45 stocks,  There's an expected turnover of about 40% in those stocks, meaning four out of ten will be sold and replaced with something else.

QUESTION: How many stocks are expected to be sold this year by the new advisor?

35 x .4 = 14
45 x .4 = 18

ANSWER: 14-18 stocks will be sold and replaced with new stocks.

QUESTION: How much will the new advisor firm be paid for the 8.5% percentage of the fund being managed, assuming they meet the performance target?

                        x .085
     $2966500000    or $3 billion being managed.
                        x .0014
ANSWER: The new advisor firm can earn about $4 million dollars.

Isn't math fun, and lucrative? Well that's another question.

How much did this fund earn last year? It increased 27.17%
But the fund advisors were also rated based on their performance over 5 years, which was 0.9%

Ah well. Not always so good for them ... or us!

Wednesday, January 27, 2010

How long can you eat?

Yesterday's blog was on calculating the length of your shoelaces, and I went to lunch immediately after finishing it.

As I looked at the spaghetti on my plate, I wondered if I could calculate the length of spaghetti noodles. In fact, how long is the total length of the spaghetti noodles in an average serving?

This requires some facts. So I grabbed a noodle and (lacking a ruler) laid it out on an Excel Math Lesson Sheet.

The noodle is just a bit shorter than an Excel Math Lesson Sheet. That means it's about 14 inches.

I could have dumped the whole plate out on my desk, lined up all the noodles and measured them.  But I was hungry and I kept eating away while I thought about how to figure out the length of a bowlful of spaghetti.

Measuring might have been easier than calculating their length from published sources. It's a bit like the exercise we went through several months ago wondering how many worms are in our worm farm . By the way, the worms are doing well. We moved them indoors for the winter.

Back to the spaghetti. By surfing the Internet I learned:
  • a side serving is 2 ounces by weight; a main serving is about 4 ounces.
  • a side serving is a bundle of (uncooked) noodles about the diameter of a US quarter dollar
  • a serving of cooked pasta is a pile about the size of a tennis ball
  • noodles expand when cooked because they are only 12% moisture before cooking
  • the average American eats about 20 pounds of pasta a year
  • an ounce of pasta is about 100 calories
  • a fine spaghetti noodle is about 1.5-2.5 mm in diameter and 10-12 inches long
Some of these facts came from the pasta manufacturers, and others from the World Food Programme, a United Nations website that is dedicated to ensuring that food supplies are safe, of good quality and can contribute to an acceptable nutritional and health status for all population groups.

Did I learn anything that could help me calculate how long the noodles in a serving would be? Well, I remembered a lesson we did on coins ...

I pulled the graphic off my hard drive. A quarter is about 25 mm in diameter. If we know the diameter of a noodle, maybe I can come up with something... and here it is.

Your own digital pasta noodle comparator. Noodles of 4 diameters, fitted inside a ring the diameter of a quarter. Just choose your noodle size and count the number that fit inside the ring. Multiply by 4 to get the whole set of noodles that will fit.

You could do it with math too, without this fitting process. But it would take more than elementary level math, so we'll leave it for another time.

Tuesday, January 26, 2010

How long should they be?

We're talking shoelaces today. How long should shoelaces be? We want them easy to tie but not so long that they drag on the ground, or you trip on them.

Several years ago I discovered the Ultimate Shoelace Website. Ian Fieggen, who created the site, has hosted more than 10 million visitors and can tell us almost everything we need to know about shoelaces. Except two things:

(1) Which shoes we are wearing today, and (2) the pattern of the laces through the shoes.

Here is his advice on selecting the length of shoelaces.

The length of a shoelace depends on five key factors:
  1. Lacing Method. The diagram shows uses Criss Cross Lacing.
  2. [P] - Pairs of Eyelets. The diagram has 6 pairs of eyelets (12 eyelets total).
  3. [H] - Horizontal spacing between centres of adjacent eyelets, measured with the shoe tight on the foot.
  4. [V] - Vertical spacing between centres of eyelets, or from the top of one eyelet to the top of the next eyelet.
  5. [L] - Length of each shoelace end (with which you tie your knots), measured from the middle of the knot to the end of the lace. About 250 mm (10") is about right.
The length of the lace should be (H + square root of (H squared + V squared) x (P-1)+L) x 2.

We don't do squares and square roots in Excel Math. That's algebra. So you could just put your numbers into his online shoelace length calculation engine: HERE

Or alternatively, measure your old laces.

If you don't have any, or your old ones broke, you can use a thin piece of string to lace up your shoes, then pull out the string, measure it, and go buy that size shoelaces.

If your shoelaces come undone all the time, you might be tying a granny knot. Here's the page for you.

On his page, Ian says teachers spend lots of time tying shoes. My wife is a PE teacher and she confirms his statement. Maybe a couple dozen pairs a day (in addition to her own).


Ian has written a book called Laces, and he included a sample pair of laces on the cover so you can practice!

Monday, January 25, 2010

Will you eat anything?

In my last blog, I expressed my desire to stay away from processed foods.

Today we will take a quick look at the ingredients in my favorite cereal (about the only processed food in our house). I won't tell you the name of this cereal, but it's very popular, so you've probably had some yourself. I was surprised to see how long this list was:

Whole Grain Oat flour
primary ingredient
Corn flour
adds a bit of flavor, nutrition and color
Whole Wheat flour
adds nutrients, color and structure to a product
Rice flour
thickening agent, also adds taste
flavoring and preservative
Calcium carbonate
calcium mineral supplement; it's what makes your water "hard"
Disodium phosphate
keeps powders from caking; used in laxatives!
Reduced iron
nutritional supplement for iron
B vitamin
Zinc oxide
nutritional supplement for zinc
Butylated hydroxytoluene - antioxidant to keep fats from going rancid
Yellow color 5
also called Tartrazine; a very commonly used food coloring
Yellow color 6
also called Sunset Yellow; an artificial coloring with an orange hue
Thisamin mononitrate
B vitamin
Pyridoxine hydrochloride
B vitamin
B vitamin
Folic Acid
B vitamin
Cinnamon flavored adds
a few more ingredients:
spice made from ground-up tree bark
Caramel Color
coloring made by heat-treating sugars
carbohydrate derived from starch; contributes flavour, colour and crispness
Red 40
petroleum derived food dye also called Allura Red AC
Blue 1
also known as Brilliant Blue FCF
Natural Flavors
Our government says this about labeling food and natural flavors: the essential oil, oleoresin, essence or extractive, protein hydrolysate, distillate, or any product of roasting, heating or enzymolysis, which contains the flavoring constituents derived from a spice, fruit or fruit juice, vegetable or vegetable juice, edible yeast, herb, bark, bud, root, leaf or any other edible portions of a plant, meat, seafood, poultry, eggs, dairy products, or fermentation products thereof, whose primary function in food is flavoring rather than nutritional.

There have been many other flavoring variations of this cereal, in addition to the basic formula. They include Cinnamon, Raisin, Honey Graham, Vanilla Yogurt Crunch, Chocolate Oat Crunch, Maple and Brown Sugar, Multigrain and Baked Apple. Of course, each contains a few (or a lot) more ingredients.


Here it is. A new website called THE GOOD GUIDE is researching the contents of products and rating them on a variety of scales. This gives us a numerical value (MATH!) for comparing foods.

The cereal described above receives a 7.2 rating which places it near the very top in cereals. It's good at nutritional performance. Sadly it also contains a few food dyes that we could argue are unnecessary.  Of course we haven't seen the cereal in its native color! These dyes are permitted here in the US, but are banned or being phased out in some other countries.

The Good Guide's very top rated cereal gets a score of 8.0. It contains:

Organic puffed millet
an edible grass seed commonly used in bean bags, hacky sacks, pillows and bird feed

Frankly, that doesn't sound very tasty to me! But people have been eating it for more than 10,000 years. We now harvest more than 30 million tons of millet a year - the largest producers being India, Nigeria, Niger and China.

The nutritional value of millet is similar to wheat, with about 11% protein. Dough made from millet won't rise, so it's used for flat breads or mixed with other grains.

People used to say "Mikey will eat anything" but I'm not sure he'll eat puffed millet ...

Friday, January 22, 2010

How Sweet It Is?

There's a worldwide sugar shortage. Almost. Egypt has enough. America is ok. China has plenty and apparently the European Community is swimming in the stuff. But the rest of the world is short.

You might say sugar is a luxury, but it's also considered a primary source of calories. It's an ingredient in bread and other staple foods. So we have a problem if most of the world is short on sugar this year.

Alternative sugar sources include sugar palms, agave nectar, maple syrup, honey and other natural sugars. But these options combined are not enough to make up for the expected 2010 shortfall.

NOTE: I am excluding high fructose corn syrup and chemical sweeteners from today's discussion, even though some of them are called "natural."

How does a sugar shortage like this happen?

Is it a math question? Or one of politics, intrigue, chemistry and crops? Yes is the answer to all the questions.

  • El Niño rains in Brazil have restricted harvests and watered down the cane
  • El Niño dry monsoon has cut production 40% from a normal year in India
  • Mexico, China, Russia and Central America have had smaller than normal crops
  • Some traders expect demand to exceed supply by 13.5 million tonnes
Europe has enough surplus that they could export 600,000-800,000 tonnes more sugar this year. However, due to a tariff cap established in 2004, they are limited to exporting 1,370,000 tonnes.

This was set up to protect sugar producers in other areas of the world. Negotiators didn't foresee a season when all other production combined would not be able to produce enough sugar to meet demand.

There are other consumers of sugar besides people, too, such as this ant!

As a result of fears about prices, they have risen  150% in one year to 29.82 cents per pound in New York's trading centers.

Will this hit us in the pocket? Will sugar cause our cost of living to rise?

Australians don't think so. They have lots of sugar, producing about 4.5 million tonnes a year and exporting most of it.

The average Aussie eats about 100 lbs (45 kg) per year. That's in raw form, packaged foods, soft drinks, etc. The wholesale cost of that sugar is now $25 per year ($2 per month) up from perhaps $15 a year ago. Even if it doubled in price, sugar isn't that big a cost for people in developed countries.

China sold 852,000 tonnes of sugar in the last month to supply countries in SE Asia, from a total crop estimated at 12,000,000 tonnes. Its prices in the three recent sales were around US $700 a tonne.

Notice my use of tonne. That's a metric ton. One thousand kilos. That's 2205 lbs, versus an "American ton" of 2000 lbs. So how much per pound is that Chinese sugar? Is the Chinese price above or below the NY price?

$700/2205=.317 or about 32 cents.

Notice how many numbers there are in this story? We'd be lost without a basic knowledge of math!

And we would be lost without sugar. Of course, I know lots of people eat too much sugar, especially in the United States. Or perhaps more accurately, you could say we eat too many sweetened foods.

My preference is to completely avoid all processed and packaged foods, and soft drinks. I have to add sugar myself if I want any. That way I can keep track of my sugar consumption, and enjoy a cotton candy at the county fair.

This photo shows a man making a fun treat originally called Fairy Floss. The process of spinning sugar into a fluff was developed in 1897. It was renamed Cotton Candy in the Twenties.

Thursday, January 21, 2010

Time for a new telephone?

Time and Telephones (and math) go together.

Back in the day ("Once upon a time"), we used to pay for every moment of every phone call. Not like today, with these so-called unlimited calling plans, though I caution you to read the fine print if you have one...

Some helpful watches were even marked at 3 minute intervals. See this counter at 3 o'clock? It has bigger lines at 3, 6 and 9 minutes, so you would know when to get off the phone.

Of course, not everyone had a fancy timer watch. Some folks bought tabletop timers. Here's a Smiths timer made of Bakelite, sold to budget-minded callers in the UK. You put it on the desk or table and switched it on before your call. The timer has scales for the 12, 20, 30-second and one-minute rates.

It didn't tell you when each rate was in effect, you had to remember that yourself. I recall that in California, we had cheap rates until 8 am, then day rates until 6, evening rates until 9, then cheap rates again all night long. Since we lived on the West Coast, we could only call relatives back east in the morning. Otherwise they were sleeping.

In researching today's story I found a watch and clock site that listed an extraordinarily cool device called a Tel-O-Time!

It fit on top of the rotary dial telephone, and was wound up whenever you dialed.  It conveniently provided - in the center of your phone dial - the time, the date and a red 12-minute timer to keep track of your phone call.

I think this is a pretty neat device. It clips on the phone, winds up with no conscious effort, and helps you save money on calls. One small problem - it became obsolete with Touch-Tone dialing. Oops. A bit like payphones are today with all the mobile phones we have. 

Next I ran across the TimeTimer. This count-down timer is built with a bright red disc that gradually shrinks in size as the time elapses. This way you can see the interval decreasing without having to do any calculations.

It's marketed to teachers of autistic kids, not to phone callers. But I liked the look. It would be great for long conference calls.

Do you wonder how it works? 

Here it's going from 40 to 35 minutes.

Of course today the phones usually display the time on a digital display. Mine shows a clock and calendar until I start to talk, then it displays the elapsed time of the call.

What led me to do a blog on this subject? Well, I wasn't really thinking in 3 minute intervals, I was actually thinking about some news I heard yesterday. It's about time. About time for what? you ask.

Time for my father-in-law to get with the Twenty-first century. He has finally purchased a new telephone, to replace his DIAL telephone. Good thing I didn't get him a Tel-O-Time for Christmas!

Now maybe he will start calling whenever he wants to, and to stop trying to squeeze all the conversation in before his 3 minutes are up!

Wednesday, January 20, 2010

Bring two Number 2 pencils

Does that phrase strike fear in your heart?

Yes, we're talking about tests today. But not taking one, so relax.

In Excel Math we provide all sorts of questions, but on quarterly tests in the upper grades we ask students to do bubble-in answer sheets. This is to give them practice with that most horrible of all challenges - the standardized testing answer sheet. Getting the right answer is only one of the tasks. Putting it in the right place is another. And fully shading that little box is a third.

Here you go. I said we're not taking this test, and to prove it, I have turned on the answers.

On this type of test you must make a selection from multiple choices, then find the letter of the answer, move to the right, locate that letter inside a circle, and darken the circle with your pencil.  You're probably familiar with this, but it's not a trivial conceptual skill, and it is terrifying to some people.

Why does the education community add this challenging element to tests? To make tests easier to grade.

We the People often use the same technique on ballots. A few years ago there was a huge outcry during our local mayoral elections. No hanging chads for us here in California. But we learned that for a write-in ballot to count, we had to write in the new name on a blank line and ALSO darken a box on the ballot.

If either of the pieces of data on that line were missing, the vote didn't count. Since the box was right there next to the name, several thousand voters didn't bother to color it in. The write-in candidate would have won, but all those votes were invalidated. Regardless of a voter's opinion of this particular candidate, it seemed a bit unfair. But this interpretation turned out to be a state law (if in doubt, throw it out), which further inflamed folks who rant about BIG GOVERNMENT.

I suppose this comes down to the age-old question:

Do machines serve the people, or do people serve the machines?

In both of these cases, the voters or students serve the machines owned by the people administering the ballot or the test.

So when you come to class, or go to vote, be prepared. Bring two Number 2 pencils in case you break the point on the first one. Read the instructions. And fill the boxes as requested, not just any old way you like.

Tuesday, January 19, 2010

Miles and miles and miles and miles

Following last week's post on red lights, I thought I would mention AARoads, a website that takes pictures of every road.

AARoads is not as interested in navigation as it is in documenting how a road looks, what the signage is like, and so on. Their folks drive around taking pictures of the roads. Really. I am not kidding. That sounds like a job that will never be finished, doesn't it?

Here's the southbound tip of California State Highway 209 (now decommissioned) from AARoads. You can just see the old Cabrillo lighthouse in the distance.

Can you guess how many miles of roads there are in this country? How about your county? Or city? Or even your own driveway?

Let's see what I can find out.
  • A quick search suggests there are 4 million paved road miles in the USA.
  • Approximately 1.5 million miles of public road are unpaved.
  • Around 1 million miles of roads are in urban areas, and 3 million are in rural areas.
  • There are 8.4 million "lane-miles" in the USA, adding multi-lane sections to the total.
  • Around 2.3 million lane-miles are in urban areas, and 6.1 million lane-miles are in rural areas.

How about the county? That's a bit more complicated to understand. Our county of San Diego says,

The 2000 miles of roads in San Diego County's system are recorded in an official document known as the Road Register. Many roads in San Diego County are maintained by other agencies. Freeways and State highways are maintained by Caltrans. Private roads are maintained by property owners. Public roads within cities are not in the County system.

There is only one privately-owned toll road in the county, the South Bay Expressway. It has 9.5 miles of roadway.

Click here for a real-time traffic map of San Diego County from Caltrans.

The City of San Diego cheerfully claims,

San Diego has a well-developed and relatively uncongested highway system. Four major interstate freeways and six state highways serve the City. The average daily round-trip commute ranks fifth-best of the 20 largest metropolitan areas in the US. Since 1980, more than 1,000 miles of streets and highways have been added to the San Diego region. About 50 more miles of new and upgraded freeways are planned for development by 2010.

Ansmar Publishers, home of Excel Math, is in the city of Poway, which has a total of 130 roads covering 180 miles. They are swept every other week and maintained on a 7-year rolling schedule.

Ansmar has some pavement of its own - 34,002 square feet of it, to be precise, mostly taken up by the 69 parking places we were required to provide. Come by some time, visit us and enjoy the parking!

My own driveway is about 1000 square feet. It's a complicated shape, so it's hard to calculate its square footage, but when we repaved it with bricks we worked out the area from the number of bricks used.

Monday, January 18, 2010

Dry as a bone isn't quite good enough

Do you ever wonder where all our trillions of dollars in taxes go?

The National Oceanic and Atmosphere Administration is a division of the US Department of Commerce. They watch and forecast the weather for the rest of us. They monitor the atmosphere, the oceans and fish, and do lots of research in the water and air. They have some satellites. And they teach us about what they do. In the process the 10 thousand employees of NOAA spend about 4 billion dollars a year.

Keeping track of the weather costs about a billion dollars a year. We get lots of useful data from NOAA. Today as San Diego is facing a severe set of storms, so I'd like to talk about the Palmer Drought Severity Index (PDSI), a tool developed by Wayne Palmer in 1965.


The PDSI describes relative dryness or wetness in the USA - that is, long-term moisture shortfalls or excess. It's not just variations in rain. The index is calculated for hundreds of climatic regions in the United States (see above map). They consider the weekly precipitation total, average temperatures, water capacity of the soil, etc. plus previously-recorded history for the past 78 years.

The PDSI is a primary tool for predicting the scope, severity, and frequency of abnormally dry or wet weather. It can help forecast disasters. It is considered by people concerned with irrigation, reservoir levels and potential forest fires.

The equation for the index was originally derived from studying the monthly temperature and precipitation in 13 instances of extreme drought in western Kansas and central Iowa. Palmer gave a value of -4 for these droughts. He assigned a +4 to extremely wet conditions. The result is 7 categories of wet and dry conditions (as well as decimals in between the whole numbers).

The index is a combination of the current moisture conditions, combined with a fractional value from the most-recently-calculated index. This means the scale includes the effect of duration of the drought or wet spell as well as the intensity. The moisture sum is the product of climate weighting and moisture departure. The weighting factor means that the index can be comparable across various regions - for example, Kansas and Florida.

The moisture departure factor means the difference between water supply and demand. Supply is (1) precipitation plus (2) stored soil moisture. Demand is (1) evaporation, (2) water needed to recharge the soil, and (3) runoff which keeps rivers, lakes, and reservoirs at a normal level.

The duration adjustment to the drought (or wet spell) is determined by calculations based on different historical wet and dry spells. A week of normal or better rainfall is welcome in an area having a long drought but may be only a brief respite and not the end of the drought. We can predict based on decades of experience with weather in that region.

One thing the PDSI doesn't do too well is account for water held in what we Westerners call the snowpack, so other factors need to be used along with the Palmer scale if you are talking about weather in the Sierra Nevada or Rocky Mountains.

Here's the January 2010 rainfall forecast for the continental US. It looks like San Diego will have more than normal rainfall in the next month.

Friday, January 15, 2010

What is it worth?

This is a fundamental question in a capitalist society. A mathematician's delight, too.

What is it worth? or perhaps the past tense: Was it worth it? or future tense, Will it be worth it?

It's the premise that underlies many television shows, from The Price is Right to Antiques Roadshow.

You look at an object, guess the price, then learn the real price, and scream Wow! or mumble Darn.

You ask your credit union - How much can I borrow to buy this old house? And they send out an appraiser.

Before sending your children to college, but after looking at the cost of tuition, you wonder Is it worth it?

WORTH is an great word. I like this definition: 
The value of an object in relationship to a purpose.

The way or the context in which an item is used determines much of its value. For example, on Antiques Roadshow, where the objects are normally collectible rather than utilitarian, values are quoted "at auction" or "in a retail shop" or "for insurance purposes."

In the car valuing business, databases and books are created by companies like in the US, Glass's Guide in the UK, and many others. I mention these two specifically because I have worked for each of them. We used surveys, research, statistics (and a bit of plain old good judgement) to determine approximate values for used cars. We qualified those valuations by asking questions like mileage, age, condition, options, color, location, etc.

Some sources go to great lengths to define categories of pricing:
  • Auction Price - what a dealer will pay to get a car for inventory (also called wholesale)
  • Trade-In Price - when handing it off to a new or used car dealer
  • Dealer Price - a retail price including profit asked by a dealer
  • Private Sale - average price in a private transaction between two  drivers (non-dealers)
  • List Price - the published asking price from the manufacturer, through an authorized dealer
The research is compiled into products known as PRICE GUIDES (not price lists).

It's not just the free market that worries about worth. The IRS has set up rules to determine what the value of a car is if you donate it to a charity. You can't just go to Edmunds' site and look up the price, then take that value as a credit against your income. The deduction depends not only on the beauty of your pristine or junky automobile, but also on what the charity does with it.

WORTH = The value of an object in relationship to a purpose.

If the charity gives your car to a poor person, that's one value. If it keeps your vehicle and uses it in primary mission of the charity, such as a food bank carting food around, that's another value. And if the car just goes right to the scrapyard, that's a third possible value. Woe to you if you misrepresent the value to the IRS. The rules have been exploited in the past, and they now check very carefully.

The cost to produce an object is not directly related to its value. It plays a part in the way initial prices are set, but sometimes it's very costly to produce something that has relatively little value to prospective buyers. What value does a gold-plated hair dryer have to a bald man?

WORTH = The value of an object in relationship to a purpose.

Most of us are always looking for a better deal, and we can be slightly tiresome in those efforts. A favorite story of mine goes like this:

Sir David Brown was the long-time owner of Aston Martin cars. A friend sidled up to him one day at a social event, saying, Sir David, I hate to impose on our friendship, but I wonder if, considering the long years we've known each other, you might be able to sell me an Aston at cost, rather than retail. 

Brown reportedly replied, I'd love to, it will be £2000 over the list price, and the first car I haven't lost money on in 10 years.

Thursday, January 14, 2010

How Do We Know Whooo is making that noise?

Sound consists of traveling waves of alternating compression and rarefaction in an elastic medium (such as air), generated by a vibrating object (sound source).

What is the most remarkable sound you hear on a regular basis?

(Don't say the sound of silly questions being asked by the Excel Math blog, because these are silent!)

I suppose it depends on where you spend most of your time. In my neighborhood, it seems like a tie between non-stop dog barking and airplanes departing from the airport. But there are birds out there too.

Last night, at dusk, I heard a very unusual noise. Looking up and squinting into the dark, I could just make out a pair of hawks circling and crying to each other. Are they mating? Just saying Good Night?

We've lived in the same neighborhood for 20 years, and have been privileged to see families of hawks almost every year, despite losing some of the trees in which they nest. I'm not equipped with bird-watching camera lenses, but here are a few images:

Click here to hear them.   Earlier this week I found a really neat site called NatureSounds run by Doug Von Gausig. He's been recording birds and other sounds around the Southwest for a long time. That's where the hawk cries came from.

This has raised questions in my mind:
  • How do we (people) know what's going on around just by hearing faint noises?
  • What is it in our ears and heads that makes hearing so precise?
  • How can we recognize a familiar bird's voice?
  • How can we possibly distinguish one bird's voice from another?
Are these even math questions? Do they involve frequencies, vibrations, speed of sound, etc.?  Yes, indeed they do. It's math alright. So what do math people do (versus simple bird-ologists)?

They record the sounds using a parabolic (math) microphone. Then plot (math) the sounds onto a graph (math) and then analyze (math) them to see if you can learn more. This is called a Sound Spectrogram (math).  Here's what a Barn Owl's song looks like.

Since most of us like to hear birds, we can go to Cornell's Sound Archive and learn as much or little as we want about sounds from their millions of recordings. Or just click my local owl to hear him.

Whoo's really excited about bird songs? Here is a page with hundreds of links to bird song recordings.

Wednesday, January 13, 2010

Every single light was red!

Have you ever felt this way? I have. Today.

Coming to work this morning it seemed like every light was red, starting with the one nearest my house. It was letting only 3 cars through on each cycle instead of the 15-18 it normally allows. Of course that meant we all had a long time to sit and think through our plans for the morning commute.

I decided to count the total signals on my normal route. There are 14. Of those 14, 11 were RED when I arrived.

Lucky me! I got time to think about the subject of my blog, I had time for a nap, time to plan my kitchen remodel, time to compose a note to my mother-in-law, etc.

Regarding this blog, though, I'm now interested in the words we use to describe vague quantities. Like those red lights. What's the most accurate phrase (mathematically speaking) for 11/14ths red lights?

I think these all might qualify:

All the lights were red!! (said in outrage, no one expects you to mean every single one ...)
Nearly every light was red!
Red lights to the max !
Many of the lights were red!
The bulk of the lights were red!

Majority of the lights were red!
The consensus of the lights was "Red"
There was a preponderance of red lights.
There were scads, heaps, bucketfuls, loads and oodles of red lights.
The streets were teeming with red lights.
The lion's share of the lights were red
There were (was) a whole bunch of red lights.

Most of the lights were red
A lot of the lights were red

Zillions of red lights!
Countless red lights! (not really true, because I counted them ...)

Umpteen lights were red (I like this slang word. It's more fun than saying eleven out of fourteen ...)

The following don't convey the appropriate fraction of redness

Several of the lights were red
Some of the lights were red
Occasional red light
A couple of red lights
Sundry red lights

This kind of numerically-representative talk is part of what we teach kids in math class. Except Umpteen isn't in the curriculum. Yet.

Umpteen falls into a category that Wikipedia calls Infinite and Fictitious Numbers. It means "innumerable but many", "a relatively large but unspecified number, employed for comic effect."

Yep, there were umpteen red lights this morning on my way to work!

NOTE: This is a sculpture in London, not a real traffic light!

Tuesday, January 12, 2010

Orange Juice Jitters

This morning I poured myself a glass of orange juice. I drank it in between bites of toast. That would normally be the end of my thinking about orange juice. But I saw this story headline in the newspaper:

Surprise Fall in the Price of Orange Juice

Without a good math background, you'd be hard pressed (indeed you would be squeezed) to figure out what they were talking about. Here are some details presented in the story - we'll call this group A:
  1. The price for frozen concentrated orange juice fell 13% from the previous Friday.
  2. The price for frozen concentrate fell 19.3 cents.
  3. The price for frozen concentrate is now $1.3185 per pound
  4. Several previous price drops this season were slowed by a 20¢ per day fluctuation limit.
  5. The USDA expects Florida's crop to decline by 17% this season compared to last.
  6. The USDA expects Florida's crop to be 135 million boxes.
  7. Florida is the world's second-largest source of orange juice (after Brazil)
  8. A box of oranges weighs 90 pounds, or 41 kilograms.

I decided to do a bit more research before applying math to the frozen orange juice business. (OJ novices can go back to my post in November for more insights.)  I hunted around on the Internet and learned more. We'll call this group B:
  1. The price of frozen concentrated orange futures is based on fixed lots of 15,000 pounds, to be delivered in March.
  2. Accuweather's forecaster predicted 6% of the total 2009 crop would be lost due to this weekend's cold weather in Florida.
  3. Temperatures have to stay below 28 degrees F for more than 6 hours to cause damage.
  4. The price on Wednesday was a 2-year high of $1.4965.
  5. The price on Thursday was $1.4115
  6. The price on Friday had risen 7 cents.
  7. Last year's crop was 162.4 million boxes
  8. Prices may rise to a peak of $1.5400.
    Too much information! may be what you are thinking. Too many numbers!

    You're right. Bullet points are useful in reports and overhead presentations, but presenting information as I have here makes it hard to distinguish the meaning and the categories of data. A chart would be nice, but so would a more meaningful display of the data. So let's rearrange our 16 items:

    Pricing History
    B4 Wednesday's price was 1.4965
    B5 Thursday's price was 1.4115
    B6 Friday the price went up 6 cents (did it rise to 1.4715?)
    A1 Monday's price fell 13% from Friday's level (did it fall to 1.2802?)
    A2 Monday's price fell 19.3 cents (or did it fall to 1.2785?)
    A3 price is now $1.3185 (apparently neither of the above were correct)

    Production Facts
    B7 2008 crop total output was 162.4 million boxes

    Future Speculation
    A5 USDA expects crop to decline 17% (would it fall to 162.4x.83=134.792?)
    A6 USDA expects crop to be 135 million boxes (apparently so)
    B2 Accuweather expects 6% to be lost (is that 6% of the 135 million boxes?)
    B8 prices might climb to a peak of $1.5400

    Reference information
    A4 several recent price drops were slowed by a 20¢ per day limit on trading
    A7 Florida is 2nd largest source of juice
    A8 a box is 90 lbs or 41 kilos
    B1 prices are based on March delivery of 15,000 lbs of juice
    B3 temperatures have to be at least 4 degrees below freezing for at least 6 hours to damage the crop

    Too much information! may be what you are STILL thinking. Too many numbers!

    What have we learned? The price of juice is volatile, especially in winter. What I want to know is how much the price of my morning juice will change if the futures prices change. This data doesn't tell us. But you might be able to get there from here if you want to check out today's USDA forecasts.

    Monday, January 11, 2010

    Flip, Slide or Turn?

    Our study of horizontal, vertical and diagonal would not be complete without recognizing that there are ways to move a shape to another orientation or position. This process is called TRANSFORMATION. There's a genre of popular children's toys called Transformers. They do flips, slides and turns. Let's see what those words mean.

    We call these moves by special names, and they follow agreed-upon rules, so all of us who modify a shape can do so in a repeatable fashion. Why bother? Because mathematicians (people) like consistency.

    If I ask you to create a right triangle:
    • of a certain size
    • with the right angle to the right side and the hypotenuse at the top
    • using 1 point black lines
    • fill the figure with green
    • slide it 2 inches to the right
    • rotate it 90 degrees
    I want your work to look like the figure to the extreme right. The red dots are there to indicate a point of rotation. The line shows the "hinge" about which it is reflected

    You might be doing this with pencils. No problems for you, especially if you have an eraser!

    Or you might have software, like the Adobe Illustrator program I use to create Excel Math problems. If you use software, you'll be hunting around the menus frantically wondering what to do. The process to follow goes like this on my machine:

    1. Create a triangle by using the polygon shape tool set to 3 sided-figure. Click and there it is!
    2. Move the vertices around to get the right triangle shape you want.
    3. Fill with Mint Julep Green (I can't match that green with browser-friendly Internet text colors).
    4. Click and Drag (slide/move) it to the right.
    5. Go to the palette (menu) and select Object/Transform/Rotate and enter 90, press ENTER

    To read more about flips, slides and turns, visit our April 30, 2012 blog post and download a free math worksheet. These are the transformations we teach to elementary school kids in math class. There are others.

    For example, shear slants things. Scale changes the size. Transform each lets you select whether to change angles, size in vertical or horizontal directions, etc.
    Finally, although these are plane figures (on a flat surface) we can still arrange them in "vertical space" or layers. That involves some instructions regarding whether to put this shape in front, middle, back, etc.

    Friday, January 8, 2010

    Center of Gravity

    My last few posts have dealt with concepts known as vertical, horizontal and diagonal. Because vertical is defined as the direction of the pull of gravity, it raises the question, On what does the gravity pull? Or where? We had to invent a concept called the center of gravity.

    For the purposes of calculation, we define this as a point where the mass of the item is concentrated This is NOT necessarily the geometric center of the item.

    For example, this photo clearly shows that the center of gravity on this car that was NOT in the center of the hoist!

    Can we use math to find out what went wrong? I think we can. First we need some facts.

    This is a Lotus Elise.

    The car weighs 1700 lbs.
    The length is 150 lbs.
    The wheelbase is 90 inches.
    About 32 inches of overhang on the front, and 28 on the back.
    The front wheels carry 590 lbs of weight (empty gas tank).
    The back wheels carry 1060 lbs of weight.

    OK, that should be enough.

    Here's an outline of the car, with a big black blob showing the location of the heavy engine. Two-thirds of the weight of the car is on the rear wheels when it is unladen (no people, little gas).

    The red spots show where the car is lifted by a jack when you have a flat tire.
    The orange in the back shows the alternative lift point when you have to raise the entire car. Notice how much farther back it is on the car, underneath the engine.

    I've simplified the drawing by taking out the car and using a wedge instead. This is an estimate to show how the weight is distributed. It allows you to see the vertical red line (center of the car, and center point of the flat tire lifting points). The vertical orange line shows the center of gravity, the point about which the weight is equally balanced.

    To raise this car safely, you need to position the rear lift on the orange position.

    Or as the shop found out, when you take off a front wheel the car falls off the hoist.

    Or a wall may fall down.

    Thursday, January 7, 2010

    Are you Leaning, or Diagonal?

    We touched on the horizontal and vertical dimensions of life yesterday. Today I'd like to discuss the diagonal. (You can see I don't have enough fancy font attributes for all these math terms.)

    A diagonal is a straight line connecting two non-adjacent vertices of a polyhedron. Or as a layman might say, a slanting line across the middle to opposite corners.

    When you are used to vertical and horizontal, diagonal is a bit different. It seems weaker, or less stable, or something. Shall I give you some examples?

    Fabrics can be cut and sewed in a diagonal way across the warp and weft threads. This technique is called "on the bias."

    That's not the same as having diagonal stripes on your shirt which is pretty rare, I think. No wonder this man's looking confused!

    Then there are cutting pliers called diagonal cutters, or dikes. And finally, I found a place that recommends diagonal bookcases. You don't need bookends, because the books are already tipped over.

    One thing a diagonal does is ADD STRENGTH to a square or rectangular structure. We teach this in some activities where we ask kids to build things with straws and string.

    Here's a tandem bicycle that I built 30 years ago which successfully employed many small diagonal tubes in an effort to improve rigidity. Notice that the diagonals connect sides of the main frame rather than the exact corners. This is due to the complexity of joining the tubes in the corners.

    This building by Frank Gehry is leaning, NOT a diagonal!