Additional Math Pages & Resources

Friday, September 30, 2011

Battery Math, Part IV

We've been considering the math of batteries this week. Today we're looking at battery chargers. All of them work in a similar way - they push electrical current into your battery until its chemistry is returned to the original state. Beyond that, they vary in almost every way possible.

Here are a few chargers I found around my house and my office at AnsMar Publishers, home of Excel Math curriculum. You may find an assortment around your house too. Why not check? And while you are at it, count your batteries ... you will be amazed!

Assorted regular battery chargers
Apple iPhone
Blackberry phone
Warehouse Pallet Jack Charger
Forklift and Truck starting charger
Car charger in use (my car, sigh)
High-tech new charger
Here's a new charger I bought to refresh my wife's batteries. It can test batteries, deplete them until empty, recharge, cycle down and up multiple times, etc. There's a display for each cell being charged. I hope it's an improvement over the twenty-year-old chargers (first photo) we've been using.

Most newer chargers indicate when a battery is fully charged. There may also be options, as shown on the big Craftsman charger. We can adjust the current, and the voltage, and the charging time. You can also see actual charging current on the ammeter.

Some devices use external power supplies along with internal charging circuits (mobile phones), and others have removable batteries that you put into an external charger (cameras).

Most chargers "pump electricity into a battery" (restore its chemistry) by providing voltage slightly higher than the battery  generates. So if you have a 1.5 volt cell, you can charge with 2 volts, and the battery would "receive" electrical energy. This has to be done carefully, to avoid overheating, damage or fire. So chargers incorporate many safety devices.

Electropaedia, a UK-based website, provided this handy advice on 3 main functions of a charger:
  1. Getting the charge into the battery - Starting the charging process
  2. Optimizing the charging rate - Stabilizing the charge
  3. Knowing when to stop - Terminating the charge
I recommend you visit this page, scroll down, and watch the animated battery charging.

They suggest an analogy for battery charging - filling a glass with beer. If you pour too quickly, the beer foams up and creates a mess, and you can't get much liquid beer into the glass. If you pour slowly, it gives the beer a chance to settle in the glass with a manageable amount of foam. And so it is with electrical charging of a battery - slower is better - unless you are waiting anxiously for the Auto Club to start your car!

The choice of "how to charge" depends on how much time, money, sophisticated circuitry, charging voltage and current available. Since more and more devices are becoming portable, and they all take batteries, recharging is a common household chore today. Knowing how and when to charge each battery properly is asking too much of the average person (us) so the trend is to build the intelligence into the charger.

Here are some charging options your smart charger might employ:
  • constant voltage
  • constant current
  • taper current
  • pulsed charge
  • negative pulse (discharge then charge)
  • IUI charge
  • trickle charge
  • float charge
  • random charge
  • slow charge
  • fast charge
And I think that's just about enough information on batteries and chargers for this week - going farther will take more than elementary math.

Thursday, September 29, 2011

Battery Math, Part III

Yesterday we looked at common, household batteries. Today we will investigate rechargeable batteries and learn how they differ from regular cells. We'll keep our analysis at the elementary math level, which is all that most of us possess. I think the math we teach in Excel Math curriculum gives most people all the skills they need for life's math challenges.

A rechargeable battery means the chemical changes that occur during use (discharge) can be be readily reversed when electrical power (charge) is applied. In contrast, regular batteries involve irreversible chemical changes.
  • In rechargeable NiCad batteries, Cd(OH)2 and Ni(OH)2 formed when the cell is working are converted back to Cd and NiOOH when the cell is recharged.
  • Regular Carbon-Fluoride-Lithium batteries (used in cameras) generate energy by converting (CF)n and Li metal to Carbon and LiF. The (CF)n is not recreated when the battery is charged, but the cell decomposes, creating fluorine gas (unsafe and unusable).
  • Every battery variation uses different chemistry, and these are readily available if you search the web. As I'm not a chemist, we will stop with just the two examples above.

A rechargeable battery should be able to discharge and charge thousands of times without degrading, overheating or short-circuiting. Developing these batteries is much more difficult than creating a regular battery, and the quality of the materials is more important. That's why they cost so much to purchase! However, your overall cost of operation should be lower IF you use your battery-powered device regularly.

Another factor in the cost of batteries is the capacity (or current-generating ability). That is measured using a unit called milliamp hours. This is partially dependent on the quality of the chemistry of the battery, and also on the quantity of materials inside. If you weigh batteries with different ratings, you will find that the highest-rated models weigh the most. Some battery cases are half-empty.

The capacity and quality of rechargeable batteries affect the price in unpredictable ways. I just compared about 20 different brands of batteries, all from the same Internet vendor, all in 4-packs. After checking some reviews, I decided to stick with brand-name batteries in my comparison.

The range of capacity on rechargeable AA batteries runs from under 1200 up to 3100 milliamp hours. The weights ranged from 15 to 30 grams. The power voltage varies from 1.25 to 1.65 volts depending on the chemistry.

When you see the chart below, it is readily apparent to you which is the best deal? Does my display present the relevant information in a clear and understandable fashion? [click on image to enlarge]

Despite the ratings printed on the side, many batteries cannot be charged to their full rated capacity, nor do they hold the charge and provide power to your camera, flashlight, etc. in the same way. For this reason, you should use matching sets (age, brand) when powering a device.

There are many different sorts of battery chargers, and they are evolving with battery chemistry. If you have a 10-year-old charger, it might be time to look into a new one. Maybe we will do that in tomorrow's blog post.

Or in English, Let the buyer beware! I suggest that you read reviews on Amazon or your favorite rating site BEFORE you buy a load of expensive rechargeable batteries. There are many disappointed customers of bargain-price batteries.

Wednesday, September 28, 2011

Battery Math, Part II

Batteries are tiny chemical generating plants that produce electrical current at a given voltage while they proceed to consume themselves via chemical reactions. Yesterday I reviewed a battery tester that might help me save money on replacement batteries. I'm thinking that our elementary math skills are more than enough to tackle the intellectual challenges posed by "dry cells."

We consume and discard hundreds of millions of batteries a year without thinking too much about them, except "Didn't I just replace these recently?" If we get fed up and decide to switch to rechargeables (Tomorrow's blog!) we will be thinking "Boy, these rechargeable batteries are expensive!"

These are the most common battery sizes in the US.

(For a full discussion and range of battery data, refer to the battery manufacturer's web pages or this Wikipedia article.)

Batteries are usually made up of one or more generating units, known as cells. Each cell produces between 1.25-3.0 volts, depending upon their chemistry. These cells can be stacked inside a larger case, or inside the user's device if more power is required. The full battery designation identifies the size, shape and terminal layout of the battery and its chemistry.

For example, a CR2032 battery is always LiMnO2 chemistry, 20mm in diameter, 3.2mm in thickness and always produces 3 volts. I know from physical inspection that a CR2016 battery is thinner (1.6mm) but it's the same diameter and produces the same voltage.

The voltage produced by a battery cell depends on its chemical makeup, not its size.
  • NiCd (nickel cadmium) and NiMH (nickel metal hydride) produce about 1.25 volts per cell
  • Mercury batteries produced about 1.35 volts
  • Zinc-air batteries produce about 1.4 volts
  • Carbon-zinc "regular" batteries produce about 1.5 volts
  • Alkaline batteries produce about 1.5 volts
  • Zinc-manganese dioxide produce 1.5 volts
  • Zinc-silver dioxide produce about 1.55 volts
  • Lead-acid batteries produce about 2 volts per cell
  • LiMnO2 (lithium manganese dioxide) produce about 3 volts per cell
  • 9 volt LiMnO2 (lithium manganese dioxide) produce about 10-11 volts when new
NOTE: This might be more clearly displayed in a table. I'll let you do that on your own.

If you put multiple batteries + end to - end (in series), as in a flashlight) you get higher voltage. This will give you a brighter flashlight.

If you use larger batteries (or put several in parallel) they can produce more current. This will allow a flashlight to run for a longer period of time.

  • D cells produce about 16500mAh on a load of 4.7Ω resistance, at 70°F
  • C cells produce about 7800mAh on a load of 10Ω resistance, at 70°F
  • AA cells produce about 2450mAh on a load of 24Ω resistance, at 70°F
  • AAA cells produce about 1120mAh on a load of 160Ω resistance, at 70°F
  • CR2032 cells produce about 200mAh on a load of 15,000Ω resistance, at 70°F
  • LR44 cells produce about 150mAh on a load of 625Ω resistance, at 70°F
  • 9 Volt cells produce about 565mAh on a load of 510Ω resistance, at 70°F
Notice that the specifications imply that each battery is happiest with a different kind of load. The battery is chosen to suit the device it's powering!

I found an elaborate discussion about battery life with two similar (but unequal) devices using the same battery. Here's a simplified, elementary-math version of the discussion:

An SR44 cell has a 150 milliamp/hour rating. If we assume we can use up 90% of its working life, we get 150 x  90% = 135 milliamp/hours. That's equal to 135,000 microamps.

Well-designed Device A draws 4 microamps when working and 2 when resting. If used 1 hour/day and resting 23 hours/day, the calculation for battery life is:

(1 hour x 4 microamps) + (23 hours x 2 microamps) = 50 microamp hours/day. Battery life is 135,000 ÷ 50 = 2700 days = 7.39 years

Poorly-designed Device B draws 18 microamps when working and 17.5 when resting. If used 1 hour/day and resting 23 hours/day, the calculation is:

(1 hour x 18 microamps) + (23 hours x 17.5 microamps) = 420.5 microamps per day. Battery life is 135000 ÷ 420.5 = 321days = .87 years

Tuesday, September 27, 2011

Battery Math

In the United States we classify batteries in mysterious ways. Rather than using numbers like 12 volt, 500 amp/hrs as we do with automotive batteries, we just use letters.

We have these familiar labels that we rely on (although some batteries do have fine print listing ampere-hours):
  • AAA
  • AA
  • C
  • D
  • 9V
I'd like to investigate a bit of the elementary math involved in checking for a bad battery.  

Why? you might ask.

My wife uses lots of rechargeable and disposable batteries in her job as a PE teacher. Portable music players, amplified megaphones, walkie-talkie radios, etc are all part of her portfolio of equipment. So the batteries are always going dead, dying, leaking, being lost, getting into the wrong piles ... you name it.

There are some clues for the user when a battery has lost its charge, but they are not crystal clear:
  • device stops working completely [is device broken? contacts not being made?]
  • device works intermittently or weakly [is device failing? is battery contact weak?]
  • battery leaks or bursts inside device [clearly battery is bad]
  • battery indicator says it's bad [rare, undependable]
You might attach a voltmeter to a battery and see if you get voltage at the terminals. Nice thought, but due to the way batteries work, they show full voltage when there is no load applied - and that means during a voltage check. You need more elaborate equipment to fully load-test a battery. Most people don't have that capability.

I used to have a small tester like the one you can see above this paragraph. It was a free gift from a client 15 years ago, but it became unreliable and finally it broke completely.

In an effort to improve our situation, I looked for a better battery testing option than the cheesy $5 tester at my local electronics store. This is what I found - the $75 ZTS Multi-Battery Tester.

In order to understand how it works, I checked out the patents behind this invention. In essence, the inventors tested HUNDREDS of samples of 25 different TYPES of batteries. They distilled their finding down to tables that included the NO-LOAD VOLTAGE, the VOLTAGE shown at various PERCENTAGEs of charge, and so on. Here are a few:

Then they came up with a device that has a set of studs on top. You set the battery on one stud, connect a lead wire to the other end of the battery, and the machine "wakes up". It notices which stud you are using, checks the reference table and compares the no-load voltage, applies a small load, measures the load voltage, compares what it sees to the tables in its memory, and makes a preliminary judgement. Then it applies a series of other loads in rapid succession (pulses) and checks its memory again. A clever idea and well worth a patent, in my opinion.

Eventually, in a few seconds, it issues its findings by lighting the proper LED at the top of the device. The graphic above shows a battery at 60%, and the one below shows 80%.

I'm happy to say that we battery consumers can do all this with elementary math, as taught in our Excel Math curriculum.  No rocket science is required. Here are all the batteries we had put in the recycle box, ready to throw out! But our new tester says they are 80% or 100% charged!

Total value = $34.68. In the first afternoon I have saved almost half the price of the tester.

Monday, September 26, 2011

Majoring in Arithmetic

Arithmetic is the most fundamental realm of mathematics, used for counting and calculations. It involves number sense, quantity, combining numbers and separating numbers - the traditional operations of addition, subtraction, multiplication and division. These lead to integers, fractions, decimals and related concepts like factors, powers and roots.
This definition of arithmetic comes from a useful book called The Words of Mathematics.

arithmetic (noun, adjective): from the Greek arithmos "number", from the Indo-European root ar- "to fit together." A related borrowing from Greek is aristocrat, presumably a person in whom the best qualities are fitted together. Arithmetic must once have been conceived of as fitting things together, or arranging or counting them. An arith-métic (emphasis on 3rd syllable) series is one in which each term is a fixed number apart from adjacent terms, just as the counting numbers of arithmetic are equally spaced.

It's ironic that the word is related to aristocrat, when today it means basic, or common. It's not really a respectable discipline. We think of the word as child-like, primitive, trade-oriented. It's like Home Economics. Auto Shop.  (Please don't take offense, home-ec and shop teachers!)

Telling your friend that your daughter is at Harvard and "she's majoring in arithmetic" would come across badly, like saying "my son is in the 95th percentile of wood shop students".

In most of our educational system, we prize the high achievers. We want everyone to be in the Gifted class. All students to be College-Prep. All our children to be above average. Garrison Keillor has exploited this tendency in his comedy about a mythical Midwestern town:

where "all the women are strong, all the men are good looking, and all the children are above average," has been used to describe a tendency to overestimate one’s achievements and capabilities. The Lake Wobegone Effect, where everyone in a group claims to be above average, is observed among drivers, CEOs, stock market analysts, college students, parents and education administrators.

A more grandiose term for this human tendency is Illusory Superiority. If you follow this link to the Wikipedia page on Illusory Superiority, you see that we educated folk ought to speak like this:

Illusory superiority is a cognitive bias that causes people to overestimate their positive qualities and abilities and to underestimate their negative qualities relative to others. This is evident in intelligence tests, performance on tasks, and the possession of desirable characteristics or personality traits.

Thus we see ourselves as better than others - not that they are bad, mind you - but just not as good as we are. If we are being gracious, we might not claim to be smarter than anyone else, but:
  • "primus inter pares" (Latin: the first among equals)
  • "All animals are equal, but some animals are more equal than others" (George Orwell, Animal House)
Around thirty-five years ago, just after I graduated from college, an SAT survey was given to about a million students. A full 70% classified themselves above the median in leadership ability, 85% put themselves above the median in their ability to get along well with people, and 25% rated themselves in the top 1% of students.

Optimistic, the positive thinkers would say. Delusional, the pessimists would say. Who is right?

I graduated first in my class in college, the valedictorian, yet for the last 10 years I have been "majoring in arithmetic". Woe is me.

Friday, September 23, 2011

What number am I thinking of? Part IV

This week's blogs have considered NUMBERS. The fonts used for numerals. Computer-generated or handwritten shapes. Things we write on paper or type on a screen to express value.

Today I am changing gears slightly. I want to consider NUMBERS that are used to express identity. Not Social Security numbers, or bank accounts, or even driver's license numbers. But License Plate Numbers, or as they are commonly called in England, Number Plates.

The fonts used to create the numbers on these plates are distinctive in many ways because of their public function. They are not usually employed in math calculations!

Although in some places you are able to choose license plate numbers or characters, you don't get to choose the shapes of the numbers. That decision has been made for you. The numerals below are typical of those on US license plates, although each state can choose its own designs:

Here's a different font used in Ontario, Canada. Notice its height and the lower x-height of the 4? The roundness of the bowl on the 9 compared to the squarish bowls of the US font above?

Fonts used in North America tend to be narrower and taller than those in Europe, due to our standard license plate size, which is taller than the horizontally-oriented European plates.
  • US plates 12" x 6" (305mm by 152mm)
  • European  20.5" x 4.5” (520mm by ~ 112mm)
Because US plates have less width, condensed characters are used to fit into the available space.

Like US states, European countries can create their own fonts. Here's a common German font called DIN 1451, that was in use for most of the 20th century. Notice there are two variants in the font set (6 & 9).

Here are fonts used on UK, French and German plates. I have one or more of each country's plates in my garage collection. All three of these illustrations are to the same scale.

Can you spot the differences? Are you German, French or from the UK? Can you tell me which is which?



Here's a license plate font from New Zealand. What's the most obvious difference from the numerals we have looked at so far?

If you want to see license plate designs and fonts from all over the Western world, you can go to a resource site created by Leeward Productions.

If you want to see actual license plates, then go to the License Plates of the World site.

If you want to see how license plates are manufactured, here's a quick tour for you.

If you'd like to download a license plate font onto your computer, you can find one here.

I've been working on this post for two days, and it's starting to get to me. If I don't take a break from these license plate fonts soon, I'm going to need:

Thursday, September 22, 2011

What number am I thinking of? Part III

Over the past few days we've been investigating the shape of our numerals.

I'm marveling at our visual ability to identify an arbitrary shape and relate it to a quantity of items. This is no trivial challenge - it's a foundational skill that kids must develop in order to understand mathematics.

Excel Math assists students in drawing (writing) numerals in Kindergarten and First Grade. We believe that physically creating the shapes (playing with counting blocks, etc.) is critical in developing a deep understanding of mathematics. There is a hand-eye-brain coordination/connection that is reinforced by actual physical activity of creating the numerals in an answer (as opposed to clicking on the correct answer on the screen).

Here's a typical 3rd grader's Lesson Sheet ... [click the image to enlarge]

Handwriting and penmanship are little-valued today, so many teachers don't push too hard for neatness and precision in making numerals. Is it important in life? No doubt you have your own opinion on the importance of this skill.

I think I have average-quality handwriting. Here's my own personal font, written quickly on a notepad and captured with my iPhone. Not exactly a straight and level baseline, is it?

Here are the personal fonts of four other Excel Math employees. One woman, three men. One leftie, three right-handed. I put the pad down in front of each and asked them to write 0 through 9.

 What can we discern by comparing these 5 sets of numerals?
  • Nobody did a "handwriting" style 2 with loops.
  • Only one person made a closed top on the 4. Only two had the cross-bar actually cross the upright
  • Several did the top of the 5 as a separate stroke, the rest made an s-like shape.
  • Three of the 6s are reclining, with the bowl up and to the right, not under the top
  • Two people crossed the 7's, in European style. 
  • The 8 is made in one stroke by a couple people, and others created two separate loops.
They all drift up a bit to the right. Most start out widely spaced at the left, then get more crowded as the writers realized their 9 might fall off the right edge of the notepad...

I found an interesting website that provides a selection of handwritten fonts that you can install on your computer. Here is a selection of the numerals from these fonts:

You see the same sort of variations in these "professional" handwritten fonts that you do in our numerals written by our company employees, even though these are very carefully done.

Maybe I could make some money by setting up my own handwritten font! I'll reinforce my understanding of the numbers, and make a few bucks on the side. Maybe if I do a little practice to get the hang of it ... I'll write numbers ... maybe all the way up to 100!

My hand hurts. I quit.

PS - We use Helvetica on our student Lesson Sheets, and MS Comic Sans Bold for the answers in the Teacher Edition. The rest of the material in the Teacher Edition is in set in ITC Stone Sans. This note is in Courier.

Wednesday, September 21, 2011

What number am I thinking of? Part II

Yesterday I was wondering at our ability to recognize numeral shapes (numbers) in so many different fonts and handwriting styles. Here are 3 fours. I mean three 4s. You can see the variations. Our eyes and brains decide these are all the same shape anyway, ignore the differences, AND determine that they are equivalent to IIII.

I can only imagine the complexity of the tasks that our brains manage when we deal with numbers on a page. It's a wonder that our young students can do this at all - don't be frustrated if your child struggles to process things you take for granted.

I found an information-packed document from FontShop that labels some of the visual aspects of characters. Be sure to check out the education section of their website if you are interested in learning more about fonts.

Here are things your eye notices, but you might never have known about the vertical dimensions of characters (numerals):
  • Ascender is the top of a lowercase character (the topmost point of l)
  • Baseline is where your numerals "sit"
  • Caps Height is the height of the uppercase characters (the height of L)
  • Descender is the bottom of a lowercase character (the bottom point of q)
  • x-Height is the height of lowercase characters, like v
Here are things your eye notices, but you might never have known about the horizontal dimensions of characters (numerals):
  • Compressed, Condensed and Expanded are relative degrees of width of a character
  • Italic characters are shaped more like handwriting, and may be slanted
  • Light, Regular, Bold and Black are relative degrees of weight of a character
  • Oblique characters are slanted, usually with the top to the right of the base; similar to italic
  • Roman is the upright style of characters; NOT slanted
  • Sidebearings (inner and outer) are the white spaces (attached or belonging to the character) on each side of the strokes
  • Weight is the thickness of the lines that make up the characters
  • Width is how wide the characters are. The widest character is an "m"
Not all dimensions concentrate on the strokes that make up the characters themselves. We need white space between the numerals in order to read them quickly and accurately.
  • Kerning is the space between specific pairs of characters; usually adjusted by hand  
  • Tracking is the spacing between characters, words or numerals on a line or in a block of text
  • Leading is the distance between baselines in a block of text
How do you determine the size of a font? This is too complicated to cover in depth, so let's just say:
  • Font size describes the vertical baseline-to-baseline distance for which a font was designed. Fonts are chosen by size, because baseline distance is more relevant for layout design than dimensions of specific characters.
  • Font height is the height in mm of tall letters such as k or H. The font height is often about 72% of the font size, but it can vary.
Amazingly complex, isn't it?  This is why we teach kids mathematics, so they can understand the numbers about numerals!

When you draw characters by hand, you can expand or squeeze them any way you like. But printers, typographers, web designers, etc. have to measure the size of the characters. That should be simple, right? Wrong. These guys use units called picas and points, but alas, not all points and picas are the same:
  • 1 point (Didot) = 0.376 mm = 1/72 of a French royal inch (27.07 mm)
  • 1 point (ATA) = 0.3514598 mm = 0.013837 inch
  • 1 point (TeX) = 0.3514598035 mm = 1/72.27 inch
  • 1 point (Postscript) = 0.3527777778 mm = 1/72 inch
  • 1 point (l’Imprimerie nationale, IN) = 0.4 mm
  • 1 pica (ATA) = 4.2175176 mm = 12 points (ATA)
  • 1 pica (TeX) = 4.217517642 mm = 12 points (TeX)
  • 1 pica (Postscript) = 4.233333333 mm = 12 points (Postscript)
  • 1 cicero = 4.531 mm = 12 points (Didot)

Tuesday, September 20, 2011

What number am I thinking of? Part I

About 2 years ago I did a Blog Post entitled "Take a Number." I'm borrowing these two images from that post - images that include all the numbers available on my Mac, in every font. Take a look. [click on images to enlarge them]

What strikes me when I study these images carefully is the variation in shape of the characters.
Some numerals are solid; others are outlines. A few are made up of almost all straight lines (see the spacy font at the bottom of the right column). Many are all curved lines.

The loops may be round, oval, filled in with dots, crossed by slashes, filled in entirely, or there may be no loops at all (see the black font in the middle of the first column).

Font designers go to great lengths to differentiate their work from what has gone before, but must also retain the essential character of a font family AND fit the individual shapes into that family. If they can demonstrate the uniqueness of their font design, they can patent it. Here are a few recently-patented fonts:

None of these fonts look like fingers held up for counting, or like ASL signs for the numerals:

yet our students manage to learn them the fonts (or hand signals), draw or write them, and use them to represent quantities.

As they gain experience from Kindergarten to Third grade or so, we expect that kids can recognize any Arabic numeral shapes, even when they vary dramatically when hand-written or in decorative fonts. Plus Roman numerals, too.

We'll think about this a bit more tomorrow.

Monday, September 19, 2011

I just want you to call

I just want you to call.
Remember (once upon a time) when we picked up the phone handset to talk?
Listened for a dial tone, spun the dial or pressed the number keys.
Said hello, had a conversation and put the handset down to disconnect?

Things have changed.
A. The phone has to be plugged in and powered, or charged up.
B. You have to have a signal.
C. You need a phone account with credit or minutes on it.

NOTE: I'll concede that in earlier days, when away from home, we had to have BOTH a working pay telephone AND the change in pocket, or a calling card or credit card.

D. Many phones contain the numbers you want to call, so you can look up those phone numbers in the phone book IF you put them in earlier.

E. You have to press SEND for it to connect and dial.

F. You must make sure your hands-free device is (still) working and it's connected to your phone.

G. You need an assortment of cables and chargers in every car and corner of the house

And all this is needed if you get your phone from the phone company!

If you get phone services from a cable provider, satellite company, etc. there will be other requirements - like having the cable connected, the power on, the modem active, the modulator free of the framwitz resonator, and so on.

In an effort to simplify my life and improve my home communications, I have been using Skype for several years. We decided this weekend to investigate some alternatives - perhaps there might be a half-dozen more providers of low cost communication.

I was wrong!

Let's for the moment exclude the Social Media (Facebook, Google+, Twitter, etc.) and texting (MMS) on our mobile phones. And the mail.

We now have AT LEAST all these verbal and visual communications providers:

  1. Apple FaceTime
  2. Apple iChat
  3. Barablu
  4. Callcentric
  5. ComBots
  6. FaceFlow
  7. Fring
  8. FullyVoip
  9. Goober
  10. iCall
  11. Jajah
  12. Jaxster
  13. Lycos Phone
  14. Messenger
  15. Nimbuzz
  16. Nyoombi
  17. ooVoo
  18. OrganIP
  19. RebelVox
  20. Rebtel
  21. Skype
  22. Scydo
  23. SightSpeed
  24. SIPhone
  25. Tango
  26. TelCentris
  27. VBuzzer
  28. Veribu
  29. Viber
  30. VOIPBuster
  31. Vopium
  32. VSee
  33. Walkie Talkie
  34. Wengo Phone

Exponential growth is a term that loosely means things are growing much much faster than we expected. I think it can be used safely in regard to complexity and telephony services!

The chart below shows LINEAR growth, CUBIC growth and EXPONENTIAL growth.

Exponents are discussed in 5th and 6th grades in Excel Math.

If you can, call us.
Use any technology you want. We'll tell you about Excel Math and why we think elementary schools and kids benefit by using it.

Friday, September 16, 2011

How Do We Make Sense of these Numbers? Part II

Yesterday I showed how we can create a graphical view of data to better understand what's going on. I made a triangle graph, which requires some serious mathematical thinking. Today I will try to do the same data on a pie graph (without the help of a spreadsheet).

Here's the data describing the home loan situation in the US today:

A. 26.0 million owner-occupied homes are paid off
B. 50.0 million homes have a mortgage
    B1.  11.0 million are underwater (mortgage is greater than the house is worth) and
    B2.    2.5 million are near or on the line (less than 5% equity)
    B3.  36.5 million have equity (mortgage is at least 10% less than the house's value)

The trouble with displaying data on a triangle is the difficulty of determining the area of the triangle, and calculating the proportional area of that total which represents a value. My triangle yesterday had sides of 10, and was placed on a grid, both of which helped in the calculations. Our calculation was also aided by a constant that comes from the formula for the area of an equilateral triangle. Simplified, that formula is [length of side squared x .433].

I'll bet you are wondering how we will calculate the area of a pie-shaped slice in a circle. If not, you should be. But that's the wrong question. What we want to do is show a proportional section of a circle, and unlike the triangle, calculating the area is not the easiest way. We can use a proportional number of the 360 degrees in a circle instead of calculating area.

That was definitely easier than calculating the area of segments of the triangle. I was able to get the angles without much trouble, because we had sets that together made 90 or 180 degrees. I hope you enjoyed for graph-making without software or spreadsheets, using data from the authors of those news reports.

If you are a loan broker you now know the segment you want to reach first.

Thursday, September 15, 2011

How Do We Make Sense of These Numbers?

I read this in the newspaper today:

The average interest rate for 30-year fixed-rate mortgages dropped to 4.17%, eclipsing the previous low of 4.21% in October 2010 ... Refinancing activity remains lacklustre given the backdrop of tighter credit standards at banks and the fact that so many homeowners owe more on their mortgages than the value of their houses ... However, about 53% of borrowers with equity in their homes (20 million homeowners) are paying more than 5.1% on their mortgages; about 36% are paying more than 5.5%; about 17% are paying more than 6%.

These are so many percentages closely squeezed into one paragraph that it's almost impossible to understand what's going on. Shall we see what an elementary math education can do to make sense of it all?  We need more information.  I found a different article published 2 days ago in another paper - and more percentages:

About one-third of the country's 76 million owner-occupied homes have no mortgage because they were purchased for cash or their loans have been paid off; about 10.88 million homes, or 22.5% of those with a mortgage, were “underwater” and 2.2 million or 5% of those with mortgages had less than 5% equity, bringing the total of under or near-under to 27.5% of total mortgaged homes.

I think we have enough data to sort things out a bit. 76 million homes; one-third with no loans. Let's display this data clearly. I'll make sure the numbers line up and add up:

A. 26.0 million are paid off
B. 50.0 million with a mortgage
    B1. 11.0 million are underwater (mortgage is greater than the house is worth) and 
    B2.  2.5 million are near or on the line (less than 5% equity)
    B3. 36.5 million have equity (mortgage is at least 10% less than the house's value)
Some don't need mortgages (A), some can't get them (B1 & B2), and some could benefit from them (B3). Why aren't the B3 people benefiting from lower interest rates? Why aren't loan officers chasing them?

Let's plot this out to visually grasp it!

Maybe I have lost all of you.

This is darn hard work. Brain, a calculator, and some lines on the screen. No fancy spreadsheets doing this work for us today. But losing your home is even harder to bear. And so is having your life savings lost by a bank that loans money recklessly.

Make sure your kids are paying attention in their math classes!

Wednesday, September 14, 2011

Imagination = Intervention?

We've been talking about the use of real math in the imaginary worlds of famous detectives.

Staying on that general theme, I've recently reviewed a math curriculum called Number Worlds™. The authors conceived an imaginary math universe with five different Lands, where young students can explore and develop what math teachers call "number sense".

I am a big fan of alternative worlds (Star Trek, Narnia, Oz, Middle Earth, The Borrowers) and I publish math curriculum, so I read on with interest. Here's my take on it:

Object Land is a world of tangible objects.
Children manipulate and talk about one or more objects. They begin with 3-dimensional things they can touch, talk about and compare, then move to 2-dimensional items (pictures / diagrams), which prepare them for numbers. The activities and language that children encounter create a foundation for math.
Picture Land depicts quantities by groups of dots on dice, dominos or playing cards
Number words are connected to items with dot patterns (half-way between real objects and abstract symbols / numbers). Kids build a mental understanding of the relationships between sets. For example, they may notice that a domino with 5 spots is similar to one with 4, but with one extra dot in the center. Numerals appear in this Land as an alternative way to represent quantities.
Line Land is marked by numerals appearing along a horizontal path; farther to the right usually means more (a ruler)
Numbers represent both a sequence of positions along an ordered path (ordinal), and names given to set sizes of different magnitudes (cardinal). Kids discover that addition or subtraction of objects is equivalent to movement forward or backward along a line. They also move from small countable objects to abstract numbers and numerical operations.
Sky Land is a place where numerals appear along a vertical path; higher usually means more (a thermometer)
Students discover movement up and down in a vertical rather than horizontal plane. They use the language of height (increased size). Right/left, backwards/forwards, up/down are eventually connected to greater/lesser.
Circle Land is the home of numbers that appear in a circular arrangement; clockwise usually means more (a clock)
Many natural processes are cyclical: waking and sleeping, the rising and the setting of the sun, the waxing and waning moon, etc. This Land introduces a cycle, or path that returns to itself. Students develop spatial understanding and get hints of geometry.
Each Land has its own operations, vocabulary, and symbols. The curriculum gives kids the chance to explore each Land's terrain, understand how the Land works and how to speak its language; move from one Land to the next; and finally to share what they have discovered.

This all sounds very logical. And fun. I'm getting into it. I'm imagining those worlds. I'm wishing I had thought of this!  But then suddenly!

Woe is me!

I read the fine print, and I lose heart.  

Sigh. Groan. Gnashing of teeth.

The publisher declares: A prevention/intervention curriculum for Pre-K to 1st grade kids who have fallen 1-2 grade levels behind their peers.  

What are they saying? These worlds aren't real? They're imaginary AND remedial? 

Suitable only for students who by age 5 are already 2 grades behind their peers? How can that be?

Imagination = Intervention?  versus Pragmatic, practical, real = Advanced Placement?

That is so wrong! C'mon kids - let's go. Beam us up, Scotty, we gotta get out of this place.