Garbage Truck Math

This blog is about using math (learned in elementary school) when we grow up. Of course, since I have never fully grown up, I tend to focus on simple things that amuse me. Like garbage trucks! There is some strange fascination that kids men have with these complicated and  noisy machines - it's a bit like train and plane spotting.

You might know about the major types of garbage trucks, but if not, here they are:

• Front loaders - which pick up "dumpsters" and throw them over the cab into a storage area on the back of the truck. The pictures below show a front loader.
• Side loaders - these pick up containers (or people do the work) and toss the refuse into the side of the truck.
• Rear loaders - this is the old-fashioned kind of truck where people lift and toss the garbage into the space in back.
• Roll-off trucks - these trucks drop empty waste containers in your driveway, then come back and take the full container away
• Miscellaneous - there are other specialty trucks that can lift heavy or bulky stuff, suck up icky wet messes, and so on.

Most trucks today have compactors inside, which compress the garbage tightly so the truck can hold more before it has to unload. Then they drive to the landfill or recycling center, tilt up, and the compactor pushes the waste out the back.

This poor guy from EDCO (a local company) broke down on the street outside our office. I got the chance to take a couple nice pictures with the body in a partially-tilted-up position so they could work on the hydraulic system.

You need lots of math in the garbage-collection business. At the very least, you should know:

• your customer count
• the average number of trash containers your customers put out
• the variation in loads based on season (holiday wrappings, summer gardening, etc) 
• holidays on the calendar, so you show up on the right days when the cans are out
• what load (weight or volume) capacity for each of your trucks 
• the strength of the machinery so you know how much of a load the trucks can lift
• how many miles the route is, including to/from the landfill
• the turning radius and maneuverability of the trucks compared to the route
• the air pressure to put in the tires of the truck
• fuel consumption so they don't run out of fuel, and 
• how often to maintain trucks so they don't break down when full of garbage!

Here are some resources if you have time to look at a few more garbage recycling trucks:

Classic Refuse Trucks
Garbage Trucks

You can even download garbage truck driver simulation games, to practice your trash collection skills:

Garbage Truck Video

The Fastest Way to Board, Part III

In the past 2 days we've covered some theories on how to board an aircraft most efficiently. The goal is to get on board quickly, overcoming the delays due to people standing in the aisle while someone ahead of them stows carry-on bags in the overhead bins.

It appears the most efficient way is to start with window seats first. We can also stagger the rows that are boarding to reduce conflict.

Inventor Rob Wallace in Australia has come up with a system he calls "the flying carpet".  A seat map design of the aircraft is painted on the boarding lounge floor. People are asked to step into the space that represents their seat on the actual plane.

The catch is that the standing space (about 7 x 18 feet) is only large enough for one person every 2 or 3 rows and 1 or 2 persons on either side of the aisle. Only 30 will fit on the map at a time; the rest have to wait at the sides. 

The first person to step into a space blocks the adjoining spaces before and behind, so no one who will be sitting adjacent to her / him can board at the same moment. The other people in that row have to wait.

When the mat is full, the gate agent says go, and the first group of 30 board the aircraft. The next 30 step onto the map in spaces that represent their seating position, and then they board together.

This process separates people adequately for quicker loading of the overhead bins. It does so outside in the relatively spacious boarding area rather than inside in the crowded aircraft. Agents don't have to call numbers or try to line people up manually. One mat design will fit most aircraft, as unused spaces don't matter.

We will have to see if it catches on - so far he has run into the "not invented here" response at the airlines, even though he claims a plane can be loaded in half the time.

Personally, I prefer the Persian or Russian magic flying carpets which eliminate the need for those pesky aircraft altogether. Apparently they worked like this:

Flying carpets were spun on a loom like an ordinary carpet; the difference lay in the dyeing process. Artisans discovered that a certain clay, 'procured from mountain springs and untouched by human hand', acquired anti-magnetic properties if superheated at 'temperatures that exceeded those of the seventh ring of hell' in a cauldron of boiling Grecian oil. The Earth itself is a magnet and has trillions of magnetic lines crossing it from the North to the South Pole. The scientists prepared this clay and dyed the wool in it before weaving it on a loom. When the carpet was ready, it pulled itself away from the Earth and, depending on the concentration of clay, hovered a few feet or several hundred feet above the ground. Propulsion was possible along the magnetic lines, which acted like aerial rails. Although they were known to the Druids in England and the Incas in South America, only recently are physicists beginning to rediscover the special properties of these lines...

Perhaps if we spent all the money saved from using Wallace's flying carpet, we could figure out how to make these magic carpets again ... in the meantime, you could enjoy a flight across Seyed Alavi's flying carpet if you go to Sacramento airport.

The Fastest Way to Board, Part II

I'm continuing the topic from yesterday, on using math to find the best way to board an aircraft. I'll start with a quote:

Dr Eitan Bachmat, doing research at Ben- Gurion University said: "What we suggest is that airlines just don't bother too much with annoying their customers and either board randomly or, if they want to do something, let the window people on, then the middle and then the aisle seats. Playing with the rows seems to be futile."

I read a study from Boeing, who wants to sell profitable planes. Since Boeing has aircraft handy, unlike most researchers, they used some and filled them repeatedly with volunteer passengers. Boeing's research indicated that loading times can be improved by:

• using doors nearer the middle of the aircraft (possible on some jumbos)
• two doors (ditto but dual jetways are quite rare in airports)
• two aisles (only on wide-body planes)
• loading from windows in toward the aisles

The back-to-front method only works best on jumbos (747 types) where two aisles, 40+ rows and lots of room diminish the conflicts.

Boeing says loading rates have slowed from 20 passengers/minute in 1965 to 10 passengers/minute in 2000.  Deplaning has also slowed, from 27 passengers/minute to 17 passengers/minute.

What has caused this massive drop in efficiency over 25 years? Based on what I have read and observed:

• More carry-on baggage (resulting in more fussing around rather than sitting down)
• Higher load factors (more crowded planes)
• Longer planes (uh, Boeing didn't say this)
• Tighter seating, fatter people, and thinner aisles (politically incorrect observations ...)
• General lack of attention from passengers (cell phones, cups of coffee, etc.)

I boarded Concorde once, eight years ago. Just over 100 passengers loaded in under 5 minutes. How? No overhead bins to speak of, no carry-on luggage allowed, only 4 seats per aisle and everyone excited and focused on getting in our seats. We used the random method.

 Back to the original question - can we use math to summarize our findings?  Of course we can. Who would believe a blog post without a chart or graph? So here's one I have created, from a research project that compared different boarding methods:

The best result was alternating both seat letters and rows, so no conflicts arise in loading the overhead bins. For example, the gate agent would say:

Row 5 A and Row 7 F
Row 9 A and Row 11 F
Row 13 A and Row 15 F
please board now.

That approach is fastest for people once they are on the plane, but sadly it's very confusing to people in the gate area. So most researchers discard that option. This report recommended either "multiple small groups" or "seat letter groups".

Can you see a consensus appearing? Three of the expert reports suggest the gate agent say something like this:

Now boarding all rows for windows seats only please. Seats A and F only.
then

Now boarding all rows for center seats only B and E and window seats if you haven't already gotten on.

then

Now boarding all rows for aisle seats C and D. Last call.

Yes there are issues with people sitting together, but hey, it's only for a few minutes and you can cheat if you have to - a few defectors might even speed up the process.

The Fastest Way to Board, Part I

The point of this blog is to explore ways that adults can use the math they learned in elementary school. Today we take on an irritating process most of us have experienced - boarding a commercial aircraft. My main qualification to tackle this complex subject is personally accumulating 3 million frequent flier miles, including 2,286,450 miles on American Airlines alone.

What's the best way to do the boarding process? Can Math Help? That is the question.

Here are some options:

1. No assigned seats - everyone rushes for the seat they want
2. No assigned seats - order imposed outside the plane and freedom of selection inside
3. All seats assigned - board in random sequence
4. All seats assigned - board from back of the plane to the front
5. All seats assigned - board in small groups alternating at back, then front, then back, then front.
6. All seats assigned - board the window passengers first, then center seat, then aisle
7. All seats assigned - board one side of the plane first, then the other
8. All seats assigned - board using "flying carpet" method (more on this later)

Here are the goals:

1. First and business class offered priority
2. Frequent fliers offered next priority
3. Families with children, seniors, wheelchairs, etc. given extra time
4. All passengers moderately satisfied
5. Plane loaded faster ($30-75 per minute value to airlines)

With about 28,000 flights per day in the US, only 5 minutes saved boarding each flight would decrease airline costs $4.2-10.5 million per day and result in less aggravation to 3.5 million passengers. Multiply that by a year, and we're talking real money - billions! A real incentive to improve this tangle:

From Speeding up the airplane boarding process by using pre-boarding areas by Albert Steiner and Michel Philipp

Sadly, here are some problems:

1. Crowded planes increase the boarding time but full flights are more profitable for airlines
2. Less carry-on luggage decreases boarding times and may increase revenue (airlines like this) but carry-ons save passengers time and money (passengers like this)
3. Boarding times are influenced by other airport processes - security checks, shopping, cabin cleaning, etc. which impact passenger boarding
4. The aircraft itself influences boarding due to number of doors, number of aisles, aisle width, number of seats per row, bathroom locations, overhead bin sizes, etc.
5. Passengers ignore these methods and "do their own thing" at least 10% of the time
6. Weather plays an unpredictable role as the number of coats, hats, etc. impact the loading times
7. The human factor - flight attendants, gate agents and passengers all impact boarding times by helping or hindering
8. Although more than half of passengers travel alone, each plane has roughly 20 parties of 2 and 9 parties of 3 passengers sitting together (rarely are there larger groups).

A quandary arises as we notice there are a few passengers who willfully disregard the assigned process. These people (a) please themselves, (b) usually speed the boarding process for everyone, but (c) irritate other passengers. Common sense ("no one's going now") and selfishness ("so I'm going now") work together to overcome inefficient procedures and baffle the people doing simulations.

So how do we find the best method?

1. We can do observations: watch people, film them and time them
2. We can do simulations: create software or mechanical simulators to run through scenarios
3. Trial runs with real agents and passengers
4. We can think hard, then apply trial and error and develop our own method.

Visit this site to run simulations and see which methods look best to you.

More tomorrow!

It's not clear to me, Part V

Friday's here already and I'm still working on the subject of things you can see through, but not clearly.

Two days ago we were able to compare how color changes the way light comes through an ashtray, or vase, or drinking glass. Although we didn't get deeply into the physics of this, it seems light is composed of rays of various wavelengths, and when it passes through colored glass, some rays are blocked, and some colors don't come through. [Click here to see the colors again]. In general though, the glass remains transparent (see-through).

Today we look at texture and how it affects glass. Here's another 9-image grid - each picture was taken through a different piece of textured glass [click on the image to enlarge it].

These textures were created to give windows an interesting appearance, and to enhance privacy. A textured glass like #4 is normally used in a bathroom to prevent people from seeing in. Examples #7 and #8 wouldn't be as good for privacy, but they do distort the view a bit.

Taking a step back with the camera, you can see how the textured parts of this glass obscure the houses and things in the distance, while still letting in plenty of light and giving your eye something to look at in the foreground. These are the same pieces of glass I photographed in samples #1, #2 and #3. What can we call this effect?

1. Opacity means a substance is completely impenetrable to (absorbs) visible light
2. Transparency is light passing through a material without diffusion (you can see clearly through it)
3. Translucency is when light comes through a substance but is diffused (you cannot see clearly through it)
4. Absorption means a material allows some wave lengths through and not others; this results in colors
5. Refraction is light being redirected or bent as it passes through the glass
6. Reflection is light being bounced back rather than passing through glass

Glass with surface textures or internal elements tends to refract and reflect the light. The glass become translucent rather than transparent. For a more detailed discussion, check this Wikipedia article.

What does this have to do with math (you might ask)? Plenty. In geometry, you learn about rays, angles, parallel, perpendicular, reflection, etc. Without this knowledge, there can be no clarity about glass.

Colored art glass displayed underwater, in an aquarium where light is passing through both water and glass (that is a baby Moray eel peeking out at us).

It's not clear to me, Part IV

This is a sample of a new "living photograph" created by a light field camera. After it loads on your computer, you can click on various parts of the image and it will focus on that part of the image. (This is actually a Flash video that simulates what an actual graphics processing program will be able to do with images captured by this light field camera.)

For example, if you click on the red part in the center, the image will sharpen up and you will be seeing "through" the glass panel at the person on the street. The glass will vanish.

If you click on the grey rectangle at the top left corner of the picture, the focus will shift back into the room and you will no longer see the person on the street clearly. You will see the glass that's in-between you and the street.

I did read most of inventor Ren Ng's 203-page doctoral dissertation, to save you the time and effort. I found a formula that might explain the after-the-shot focus trick for you:

 Or maybe it won't explain things - this math is WAY above my head.

Here's my take on this concept - a conventional camera allows you to capture the light that hits on the sensor (or film). You as photographer have to be sure the light is correctly focused when you capture it.

In essence the Light Field camera captures all the light that enters into the body of the camera. The processing software allows you to move back and forth in space along those light rays that are inside, thus changing the focus and depth of field.

Ng quotes photographer Henri Carter-Bresson:

The creative act lasts but a brief moment, a lightning instant of give-and-take, just long enough for you to level the camera and to trap the fleeting prey in your little box.

With this new digital light field camera we can solve that problem by catching lots more "prey in your little box". Then we can reach in and grab the specific thing we want.

He finishes by saying:

... the main price to pay for this new kind of photography is recording and processing a lot more data. Fortunately, these kinds of challenges map very well to the exponential growth in our capabilities for electronic storage and computing power.

I will have to ask the boss about getting me a new computer and camera!

It's not clear to me, Part III

This blog is about using math in adult life. The last two posts have dealt with transparency, glass and some aspects about glass that we can understand with even elementary math training.

Today I want to consider colored and shaped and textured glass. We can still SEE THROUGH this glass, but our seeing is affected by the features of this kind of glass. Here are a few small pieces (crystals for old watches) so you can imagine the impact of color and shape, corners, curves, arches, etc.

After some consideration, I went around my house grabbing everything made of colored glass, and shot these pictures out in the back yard. What can we learn from about light, refraction, translucency, opacity and so on? Take a close look at these 9 samples [click on the image to enlarge]:

Here are some thoughts:

1. Too much red in the glass means I can't see green anymore (although the trees are visible, their color is changed)
2. Dark blue means no green either
3. Curved surfaces distort and reflect the image, and bring in light from other angles
4. A folded seam causes the image to be out of focus, along a specific line
5. Rippled glass makes everything seem warped and the sizes get confused
6. Prisms or cut and slanted faces are less transparent, more reflective, and vary in color
7. Green seen through green is still green; this glass seems to glow
8. A bulls-eye effect is created when looking through the bottom of a cylinder
9. Too much color or bubbles and dirt make glass translucent rather than transparent

I learned that my camera has much more trouble focusing on a reflective glass surface than my eyes. With selective cropping, I can see through small portions of small items (as if I was holding it to my eye). And the more that's going on with the surface, texture, color, and so on, the more difficult it becomes to see through it.

In other words, it's not clear to me.

It's not clear to me, Part II

Yesterday I did a post on glassware - specifically, how people in my family tend to grab the same glasses when they come to our house. In the process I took pictures of a variety of glass objects. I called the post It's Not Clear To Me as a pun, but then I started to think about the terms transparent, translucent and opaque.

I did some research and learned about refraction too. Here are the differences in these terms:

1. Opacity is the measure that a substance is impenetrable to (absorbs) visible light
2. Translucency is the property of allowing light to pass but diffusing it (you cannot see clearly through it)
3. Transparency is the property of allowing light to pass through a material without diffusion
4. Refraction is the degree to which light gets redirected or bent as it passes from one material (air) into another (glass)

In the process of taking the pictures, I noticed some glass is sharper, clearer, more sparkly than other glass. It is more transparent, more refractive, perhaps. Why?

These two glasses are both from Germany - high quality glass - washed in the same dishwasher, polished with the same dish towel - yet they are definitely different.

Some glasses are known as lead crystal - glass with up to 25% lead added to the molten silica (sand) instead of calcium. Addition of lead changes the properties of glass:

• makes glass heavier (as lead is 5x heavier than calcium)
• lowers the melting point of glass
• makes liquid glass less viscous (more runny) which makes it easier for air bubbles to escape during cooling
• results in a softer surface which is easier to cut
• gives it a higher level of refraction - it is more "sparkly"
• makes a ringing sound when struck or rubbed

Any kind of glass slows down the light passing through it. The refractive index is a scale that allows us to measure this drag on light (making it bend, or turn). Here are some materials and their refractive index numbers:

0.00 = vacuum
1.00 = Air at standard temperature and pressure
1.33 = water (liquid)
1.38 = human eye (cornea)
1.50 = ordinary glass
1.58 = polycarbonate (plastic)
1.70 = lead crystal glass
1.76 = sapphire crystal
2.15 = cubic zirconia
2.41 = diamond
3.02 = mercury ore
3.96 = silicon

No wonder a diamond sparkles! Here are some well-known companies making lead crystal glass objects: Arc, Baccarat, Mikasa, Steuben, Swarovski, Waterford.

It's not clear to me

 What is the meaning of color? 

What is the color of light? 

Why are some things clear, some things translucent and others opaque?

Some sharp-edged and others softly spontaneous?

 In our house there are approximately 200 drinking vessels. Cups, mugs, glasses and so on. We have our favorites, and I've noticed when my relatives come to visit, they grab the same glasses that we like. They do this year after year. No matter if it's mom, uncle, sister or even friends - only a few pieces get used. Here are the top five:

I could do a scientific inquiry - study the geometry of the cuts in the glass decoration; analyze the color of the glass (did you notice they are different?); compare the weight of the base, thickness of the walls, and the capacity of the vessel; make a study of the shape of the glass compared to the size of the hands; decide whether temperature of the fluid makes a difference; or discern if diameter alters the ability of a nose to sniff the beverage.

But even knowing all these things might not tell me why my family likes these best out of the 200 they have to choose from. Since these glasses are my favorites too, I have often hidden them in the back of the cupboard to reduce the risk of shaky-handed, elderly family members. But to no avail - they will push their way through rows of other glassware to sneakily extract these (while knowing I fear for their safety).  No matter what the beverage - water, milk, orange or tomato juice, wine - these glasses do most of the work.

How do we explain it? Quantity vs Quality? Aesthetic vs Efficient? Cold hard logic vs Touchy-Feelie Emotion?

My wife says "Stop trying to analyze and apply math to everything. They choose those glasses because (1) they look nice and (2) they feel good in your hand."

Any transparent / translucent / opaque portable container (cupped fingers of your hand to ladle to jug) can convey a liquid to your lips. Making the process enjoyable is something else altogether.

453 for me

I'm glad this is Friday. I have tired of my chosen theme this week, which was to write a blog talking about each number from 449 to 453. I have been doing math, researching history, combing the regulations that govern our society, and so on.

First the math:

453 has only two pairs of factors: 453 x 1 and 151 x 3

453 Grams is one gram less than a pound.

Now let's look at 453's historical and social meanings. I'll start with the most emotion-laden example first:

Sec. 453 of the Social Security Act states: the government shall create and operate a Federal Parent Locator Service, for the purpose of establishing parentage or establishing, setting the amount of, modifying, or enforcing child support obligations. The FPLS shall obtain information about or help find any individual obligated to pay child support, and/or against whom an obligation is sought. This may be shared with those to whom the money is owed or who have parental rights and/or custody. (FYI, I have no children, parental rights or obligations)

Model 453P is a pink umbrella stroller (very useful for parents).

Danner 453 represents a light, comfortable hiking boot (suitable for men who owe child support and occasionally push an umbrella stroller).

Opus 453 of Wolfgang Mozart is his 17th Piano Concerto. There are another 120 pages beyond this one (in case you'd like your child to learn to play it ... the lessons will cost a lot).

Number 453 of the Top 1000 Rock Songs of All Time is Joe Cocker's The Letter (written by Wayne Carson Thompson):

"Gimme a ticket for an aeroplane,
Ain't got time to take a fast train.
Lonely days are gone, I'm a-goin' home,
'Cause my baby just a-wrote me a letter"

 (no more child support, cause his baby just wrote him a letter and he's a-goin' home.)

Class 453 is a US Patent category for Coin Handling ideas. "includes machines and appliances employed for the purpose of handling coins (or checks or tokens similar in shape to coins and used as substitutes therefor), for facilitating the making of change, or for transferring change from a clerk / cashier to the customer."

Route 453 is a main road in at least 4 different countries. Here's a road sign from Japan's highway 453:

Part 453 is an intermediate setting wheel from an ETA 2892-A2 automatic-winding watch movement. I have several watches that use this part, and I can assure you it's much smaller than it appears in this illustration. It sits just under the date display window (here showing 20).

IH 453 was a special automatic, 4-way-leveling combine made by International Harvester to harvest crops that grow on hills or on sloping ground.

Bobcat 453 is a small skidsteer loader with lifting arms. By having each side's wheels driven independently from the other, skidsteer loaders are capable turning in their own length. They are compact and agile little multi-use tractors.

DR-453 is the Major Disaster Declaration issued by FEMA in Oklahoma on November 26, 1974 due to severe storms and flooding. Relief assistance was given to these counties: Canadian, Creek, Dewey, Grant, Lincoln, Logan, Major, Oklahoma, Osage, Pawnee, Payne, Tulsa, and Washington. Rumor has it that Bobcat 453s may have been used in the clean up.

452 for you

This week I am focusing on the sequential numbers of my posts (as counted by Google Blogger software) and looking for mathematical or historical and cultural significance in those numbers.

Here's what I have learned today about 452:

• It is an integer, even number, rational number and a composite number.

• Here are its factors: 452 x 1, 226 x 2 and 113 x 4 (113 is a prime number)

• Here are some of its multiples: 452, 904, 1356, 1808, 2260, 2712, 3164, 3616, 4068, 4520, 4972, 5424, etc.

• 452 is CDLII  (or CCCCLII) in Roman Numerals

• 452 is NOT an area code for US telephones, but is the international code for Vietnam

• It's the number of a very nice Lego Lunar Exploration Set. I don't have one but it looks fun.

• It's the model number of an enormous vacuum cleaner that you can attach to your riding lawn-mower (this image looks a bit like the Lego set, doesn't it?).

•  It's the model number of a steam locomotive which has a vacuum attached to its front end; currently being used to clean up tracks in India. [Click image to enlarge]

• A P0452 code is a diagnostic trouble code in your car. It's related to the fuel vapor recovery system in your fuel tank. A fuel tank pressure sensor responds to changes in the tank's pressure or vacuum so your engine management system can detect leaks in your car's fuel system. A loose fuel filler cap is a common accidental cause of this code.

(I appear to have gotten distracted from the purpose of this blog, which was to show the mathematical implications of 452 ... have I uncovered some mystical relationship between the word Vacuum and the number 452?)

• 452 is the Gulfstream serial number for this private jet, which is currently for sale.  It does not have a central vacuum system.

• 452 is a bus route in London, England.

And finally, here's one last vacuum-free 452 relationship:

• Regulation 452 is the Family Medical Leave Act (FMLA) which allows up to 12 weeks of unpaid leave every 12 months to eligible employees (employed for a total of 12 months and worked 1,250 hours in the year prior to the date of leave). The employer must have at least 50 workers within a 75 mile radius of the employee's workplace. Primary reasons for FMLA leave are (1) employee has a serious health condition; (2) to care for a serious health condition of spouse, child, or parent; or (3) maternity or paternity leave.

What is a 451?

This is the 451st post in the daily Excel Math blog. (If this is your first visit, I am using the numbers of the postings as themes this week.) Although # 451 seems like a lot, I noticed that today's issue of the Financial Times newspaper is # 37,644!

So what things can we do today with the number 451? What can it mean?

• It's the title of a book by Ray Bradbury: Fahrenheit 451

• It's a consulting company: 451's analysts focus on IT innovation, providing insight into the market and competitive dynamics in emerging technology segments. Vendors, investors, service-providers and end-users rely on 451 to assist their strategic and tactical decision-making.

• It's an agency: 451 Marketing is a 20-person, Boston-based, in-bound consumer public relations, social media and search marketing agency that focuses on blogger relations, search- and social-optimized press releases, social media accounts and developing keyword-rich client content.

• It's a former Internet Press: 451press.com was originally launched as a multiple-author, multiple-site, blog network back in 2006 but it shut down without a profit in July of 2009.

• It's a restaurant: Zed451 has redefined fine dining - where for $48, engaging chefs serve you tableside with steaks, ribs, seafood, poultry and game, while a hefty bounty of soups, salads, fresh vegetables, cheeses and breads awaits at massive Seasonal Harvest Tables, all within a warm landscape of natural elements with a stylized edge.

• It's a Federal Regulation: Part 451, Awards: authorizes US agencies to pay cash, grant time off, and incur expenses for honorary recognition of an employee. It requires the Office of Personnel Management (OPM) to regulate this recognizing and rewarding; covers funding cash awards for members of the armed forces for the adoption of their suggestions, inventions, or scientific achievements; and delegates authority to OPM (from the President), so cash awards may be given to those who would not otherwise be eligible.

• It's a tax topic: Topic 451 describes individual retirement arrangements - personal savings plans which allow you to set aside money for retirement, while offering some advantages in deferring taxes. It describes setting up IRAs with banks, financial institutions, mutual funds, or life insurance companies.

• It was a year: 451 is the date of the Fourth Ecumenical Council. Although they settled 30 other issues, Roman Catholic and Orthodox delegates split over a definition of the essential nature of Jesus, some saying (1) in Jesus Christ both Divinity and Humanity are united without separation, confusion or alteration, while opposing points of view held that (2) Jesus' Humanity is absorbed by his Deity or (3) the Divinity and Humanity of Christ were separate, as if two persons living in the same body.

• It is a medical code: ICD-9-CM 451.0 is a set of 5 medical codes that specify a diagnosis of Phlebitis or thrombophlebitis in the legs (diseases of the circulatory system) on an insurance reimbursement claim.

Sheesh! This is fascinating, but I'm going to quit sitting in this chair and go out for a walk before I get a 451!

Nine times five is forty-five; forty-five times ten is four hundred fifty

The "times tables" are emotionally associated with elementary math. Mostly the association is, Do I HAVE to learn this stuff? I hate it! WAH!

You know the times tables - here's one for you if you have forgotten:

• 1 times 4 is 4
• 2 times 4 is 8
• 3 times 4 is 12
• 4 times 4 is 16
• 5 times 4 is 20
• 6 times 4 is 24
• 7 times 4 is 28
• 8 times 4 is 32
• 9 times 4 is 36 
• 10 times 4 is 40

When I searched on times tables I got 19.8 million hits, and when I searched on how to learn times tables I got 26.2 million, and how to teach times tables gave me 4.77 million. Wow!

Is multiplication so very mysterious? Or so very hard? Or so very lucrative for teachers, tutors, trainers, book publishers, software developers and musicians?

These seem to be key phrases that are common to the sites I viewed:

• math is fun
• life will be easier; ease the pain
• the best way; the latest research shows ...
• flash cards, games, songs, pictures
• rote learning is wrong, boring and dull
• memorization (this is or isn't a dirty word, depending on the site)
• calculators make learning this unnecessary

One site told me what their researchers had learned:

1. NO ONE enjoys learning times tables - it's like cleaning teeth or using sun lotion, you JUST HAVE TO DO IT.
2. Bribery works. Students learn faster if there is something they value at the end of it; preferably money.

Here is a table that goes from 1 to 20 in each direction. Somehow I can't imagine my mom handing me this and saying, Learn these math facts, and I'll give you some money!

I suspect she must have said Learn them, or else! So I did.

 My mom at her graduation from teaching college back in 1954

Nowadays many moms say the same thing mine did. One mom in a million might say, Don't bother with that. Here's some entertainment instead, have a few laughs! [click to enlarge]:

Artwork by Jacob Borshard
If you are good at searching the web you can find two more of these tables ... 

I'll bet very few moms, either now or then, will offer money for learning the times tables. But you can always ask ... just protect your backside and don't blame me!

If 20x20=400 and 7x7=49, why doesn't 27x27=449?

This is my 449th post in the Excel Math blog, where we investigate ways that adults can use math they learned in elementary school.

As I have been approaching the 450th blog, I started to think about what I could write about. Maybe squares and square roots? (although squares are often tackled in middle school math).

In my head I started to calculate the square root of 450 - it's more than 20 squared (400) and less than 25 (I know from memory that's 625). I got my calculator out and learned that the square root of 450 is ~ 21.21.

Suddenly I dropped all thought of 450 and decided to focus on getting blog post #449 finished. This is today's deadline; this blog is a daily task and tomorrow can take care of itself. (That comes from Matthew 6:34 in the Bible and doesn't help me solve #449, so maybe I'll save it for blog 634). Now back to the math.

Maybe 449 has an even square root? This sounds right to me:

• If 20 x 20 = 400 and 7 x 7 = 49, then 27 x 27 = 449 Right?

No!  Why?  Because. No matter how badly you want it to be. Not even if you wish on a star, it isn't.

Sadly, the square root of 449 is ~ 21.19, just a tiny bit lower than the square root of 450.

Math answers do not exist to please us - they exist as a consequence of the rules of numerical behavior. Who set the rules? The people who throughout history have "proved" how math works.

It's true that

• Twenty times twenty equals 400, and
• Seven times seven equals 49

However,

• Twenty-seven times twenty-seven equals 729
• Twenty times twenty-seventy equals 540
• Seven times twenty-seven equals 189
• Those two numbers added together do equal 729 (it's a long way from 449)

Why do those two calculations add up to the square of 27 and the other numbers didn't?

That's too long an answer for this blog, but I can say this:

Just wishing for something doesn't make it so. Even if we wish fervently.

If it did, we would all score higher on math tests ... get admitted to any program we apply to ...  reach our sales targets ... and in card games we would always win.

Fervently: exhibiting particular enthusiasm, zeal, conviction, persistence, or belief; having or showing emotional warmth, fervor, or passion

Sensitive, Specific, Accurate and Precise

We don't use these four terms in Excel Math. They are used in more advanced statistics than we teach, but they interest me because they are related to the ratios which were the subject of other blog postings this week. To prepare for this post I read through a number of articles on statistical terms. You might check out this Wikipedia entry, or this one.

Imagine a medical test used to screen people for a disease. We think of the test results being either positive (she has the disease) or negative (he doesn't have the disease). Unfortunately, we can have more than two potential results - we can have four:

• True positive: Sick people are correctly diagnosed as sick
• False positive: Healthy people are incorrectly considered to be sick
• True negative: Healthy people are correctly diagnosed as healthy
• False negative: Sick people are incorrectly considered to be healthy

Sensitivity describes the ability of the test to identify positive results (people who DO HAVE the disease). We express this with a ratio:

A high-sensitivity test finds almost all sick people. If you are lucky enough to get a negative result in a high-sensitivity test, it's very likely you don't have the disease.

Example: 100 true positives ÷ (100 true positives + 1 false negative) = 100 ÷ 101 = .99

NOTE: A test which always indicates positive (despite the health of a subject) may have 100% sensitivity but some of its results may be untrue. Sensitivity does not necessarily mean a test is accurate.

Specificity describes the ability of the test to identify negative results (people who DON'T HAVE the disease). Here's a ratio for specificity:

A high-specificity test finds almost all well people. If you are unlucky enough to get a positive result in a high-specificity test, it would be very likely you have the disease.

Example: 100 true negatives ÷ (100 true negatives + 5 false positives) = 100 ÷ 105 = .95

NOTE: A test which always indicates negative, (despite the health of a subject) may have 100% specificity, but some of its results may be untrue. Specificity does not necessarily mean a test is accurate.

Sensitivity and Specificity are related to but they are different from Accuracy and Precision.

Accuracy means the degree to which the test results mirror reality. An accurate test gives us the best information about the state of the person's health. Here's one way to represent accuracy:

Precision means the test results are consistent and repeatable across various sample sizes and populations. A precise test means it gives consistent results across years, continents, gender, age levels and ethnic groups. Here's a ratio that represents precision:

Ideally, you might say that we want any test to be sensitive, specific, precise, and above all, accurate.

Even math tests?