Additional Math Pages & Resources

Friday, April 29, 2011

What To Do When Your Math Is Just Not Good Enough

Sometimes problems come your way in life, and the math skills you possess just don't seem to be enough to get you through the problem. Since I create and solve dozens of math problems a day, I'm pretty good at it, but today I am baffled.

I recently inherited some stocks. See if you can follow their trail ... starting with an iron ore mine in Labrador.

Bison Petroleum & Minerals was a Canadian exploration company founded in 1960 to consolidate 4 other companies. In 1987, the name became United Bison

United Bison then became Nalcap Holdings on the basis of 1 Nalcap share for every 7.7 United Bison shares.

In 1968 Pacific West Realty Trust was created. In 1988 PWRT investors swapped their holdings for stock in Asiamerica Equities Ltd., owned by Stone Mark Capital,  a shell company on the Vancouver Stock Exchange.

Asiamerica then bought controlling shares in Nalcap Holdings (formerly Javelin International Ltd. and Bison) and also gained control of Constitution Insurance Co. of Canada.
At the end of 1991, Asiamerica changed its name to Mercer International to reflect the company's move away from investments. Mercer acquired pulp and paper mills in East Germany and British Columbia.

By 1994 management decided to split the pulp and paper business from financial services and mining. The Nalcap assets were spun-off as Arbatax International Inc. (iron ore royalties, insurance, and real estate), on the basis of 1.5 Arbatax for every 1 Nalcap share.

Arbatax eventually became MFC Bancorp, a Swiss and Hong Kong finance and banking business.

In 2002, MFC distributed the assets of Mymetics, a Swiss bio-medical research company, to its shareholders in lieu of cash dividend. MFC shareholders got .95 shares of Mymetics for each share of MFC. Mymetics does vaccines research but only trades for 15 cents a share.
MFC next changed its name to KHD Humboldt Wedag which took over MFC's NASDAQ listing.

KHD then split into two separate companies - Terra Nova Royalty Company which moved to the NYSE ($8 a share), and KHD which trades in Frankfurt ($7 a share).

KHD spun off some other business and investments of MFC to shareholders, creating a thrid company Mass Financial, which did no US business but traded in Vienna ($6 a share).

In 2010 Terra Nova did a one-for-one stock swap to acquire Mass Financial back.

Question 1: What are my handful of shares in Pacific West Realty Trust from 1973 worth?
Answer 1: It is difficult to say at this time.

Question 2: What are my small stack of shares in Arbatax from 1996 worth?
Answer 2: It is difficult to say at this time.

Question: What are my huge pile of shares in Mymetics worth?
Answer: About seventy-five bucks.

How do I find out the answers? Here is some advice I was given by our estate attorney:  

You don’t need to worry about it, they do. All we have to say is “Sell All Shares” on the Stock Transfer forms and they will cash out and send you a check.  

Thank goodness.

Thursday, April 28, 2011

Mathematician's Home Tool Kit

In our Excel Math curriculum we provide instructions for teachers on how to use our material. At the back of each Teacher Edition, in the Manipulatives Section, we include a list of items that are useful to have handy during class. These lists are specific by grade level and the concepts learned in that grade.

Last night it occurred to me that it might be helpful to have a math tool kit for home use. What would I include in my tool kit?

While researching, I came across many lists of things could go into a kid's math tool kit: pattern blocks, six inch ruler, dice, coins, cubes, number cards, animal cards, dry-erase marker and eraser, counting tokens, etc. Here's a list from a school in India.

Searching Google bought up these entries:
  • Math Toolkit resources help educators teach the content ...
  • Mathematics Improvement Toolkit is a collection of professional development resources that allow teachers to support students ...
  • Mathematics Adoption Toolkit is a data-driven format for reviewing instructional materials ...
  • Math Tool Kit is a wealth of activities that connect concepts in a contextual content of construction ... (What did that mean?)
  • Maths Toolkit offers a dynamic environment in which teachers and pupils can explore numbers and relationships ...
  • Math ToolKit is a cross-platform C++ class library for creating computing intensive applications ... (whoops)
I didn't mean a "virtual math tool kit" as described above. I meant real, usable tools for real life. For kids or adults. Here's my $150 math tool kit:

12 foot / 3.6 meter tape measure with both standard and metric units
18 inch ruler / 50 cm straight ruler with both standard and metric units
12 inch combination square and level
digital caliper for measuring small things
set of plastic compass, protractor, triangle, square, etc.

0-5 lbs / 2.3 kg digital scale for food and/or postage
0-330 lbs / 150 kg scale for people
set of standard and metric measuring spoons and cups

digital probe thermometer

0-60 psi / 04 bar tire pressure gauge

Desktop calculator with 8 digits display  (and/or calculator on mobile phone)

CASE $10
Plastic tackle box or crafts supplies box with lid and latch

Wednesday, April 27, 2011

More numbers with special meanings

Yesterday I gave some examples of numbers that mean MORE or OTHER than their numerical value. The alternative meanings are discovered in real life rather than in the classroom. I called this number slang.  Here are a few more examples:

  • 7 or Seven - a Lotus car driven by "the Prisoner" in a TV series
  • V70XC - a Volvo 4x4 station wagon
  • 99 - a Saab sedan
  • 100 - an Audi mid-size sedan
  • F150 - a Ford light-duty pickup truck
  • 450SL - a 4.5 liter V8-engined Mercedes-Benz
  • 500 - a Ford sedan based on the Taurus
  • 650 - a 5-liter BMW coupe
  • 900 and 9000 - large Saab sedans
  • 5000 - an early Audi
Cars aren't the only vehicles designated with numbers. Can you name the industry that labels its products like this?
  • B1
  • C5
  • SR71
  • E195
  • A320
  • S340
  • 747
  • L1011

 Or a company that labels its products like this?
  • 3 ~ 1.1
  • 3G ~ 1.2
  • 3GS ~ 2.1
  • 4 ~ 3.1
  • 4 ~ 3.3

Or the products associated with these numbers?
  • 0
  • 31
  • 40
  • 57
  • 409
  • 501
  • 7-11
I'll give you a summarized report on alphanumeric name branding mumbo-jumbo:

Alphanumeric brand names include a mix of letters and numbers (e.g., 7UP, A8, 3M). There are literally millions of registered and unregistered alphanumeric brands. Despite widespread use, the way consumers make choices among them remains unclear. Alphanumeric brands usually follow a sequence (e.g., Audi A3, A4, A6, A8). Consumers have different perceptions about these brand names, but many assume that as the numeric portions of the brand names increase, the product is superior or more recent. Consumers tend to prefer product options with higher versus lower numeric portions (e.g., X-200 versus X-100), even if they are objectively inferior. This numeric value effect varies by type of product, the degree to which consumers think about their decisions, and the availability of product information. It is common for consumers to decide most technical products with higher alphanumeric labels are improved (Pentium IV more advanced than Pentium III). However, many products do not get better as the brand number increases, and it may be difficult or impossible for the average consumer to figure out many brand names actually refer to (Bosch 500 SHV65P03UC dishwasher).

OK, I say let's 86 this topic.  If you want more, go to Wikipedia. 10-4, over and out.

Tuesday, April 26, 2011

Numbers with special meaning

Normally numbers represent a value, but in some cases they have additional, extra meaning. We don't normally teach these extra meanings in our elementary math curriculum. You learn them along the way, in real life - just as you do not learn slang in class, but on the playground.

The term slang means informal words and expressions with non-standard meanings. I hereby propose that today we look at a few examples of number slang.

For example ten hundreds plus four tens equals 1040. Kids learn that in math class. But 1040 also has a special meaning in the United States - it causes a leap in the pulse rate, clenching of teeth, forming of fists, etc. - It's the number given to the tax form we fill out each year.

This first grouping contains numbers with financial implications:
  • 1040 - annual tax form used by Americans to declare their tax liability to the IRS.
  • 1099 - a form used to declare that a person or company has paid taxable money to someone else .
  • 401(k) - section of the tax code describing a self-directed retirement plan.
  • 501(c)(3) - set of rules governing non-profit organizations which are tax-exempt due to their focus on religious, educational, charitable, scientific, literary, public safety, amateur sports, or prevention of cruelty to animals and children.
  • 10K and 10Q - forms and reports describing the activities of publicly owned companies.
Here are a few more well-known financial numbers:
  • Chapter 7 - not parts of a large book, but a group of rules governing the liquidation of a company that can't pay its debts.
  • Chapter 9, 11, 12 & 13 - the set of rules governing bankruptcy procedures for municipalities (9), companies (11), farmers and fishermen (12) and individuals (13) which enable them to reorganize and repay their debts.
 Some NON-financial numbers
  • 18 or 21 - the legal public drinking age (varies from place to place).
  • 42 - the answer to all questions in life, calculated by a computer called Deep Thought.
  • 666 - the sign of the beast described in the Bible book of Revelation 13:17–18.
  • 411 - in the USA, the number you dial for information or directory services.
  • 911 - in the USA, the telephone number for emergency services.
  • 999 - the emergency phone number in many countries.
  • 0118 999 881 999 119 7253 - fictitious emergency number revealed on an IT Crowd TV episode.
  • 362436 - a numerical representation of a woman's "ideal hourglass figure".
  • 867-5309 - the telephone number of a girl named Jenny on a 1982 Tommy Tutone album.
  • 555-12-3456 - typical 9-digit social security number used to identify American taxpayers.
  • 120/80 - the top number for nominal blood pressure for an adult.
  • 20/20 - normal quality vision; a number representing "visual acuity".

Monday, April 25, 2011

Way too many choices

Permutations and Combinations
These terms are related to the business of organizing members of sets in different ways.

In elementary school math, we might say Vanessa can wear a blue shirt or white shirt with blue jeans or white jeans. How many combinations can she wear? Vanessa can wear 4 combinations:


We have 2 choices of shirts times 2 choices of jeans, or 2 x 2 = 4

Permutations are similar but in those cases the order of the members of the set is important. For example, giving cabinet member jobs to 3 politicians - we have 3 jobs (1, 2 and 3) and could have 6 permutations:

123 and 132 and 231 and 213 and 321 and 312

We have 3 x 2 x 1 = 6 permutations.

When would this ever be important in real life?

Unlimited Options
I started today's blog with a title stating there are way too many choices! What did I mean? I'm working on our kitchen.  "Say no more" some of you will respond. The t-shirt choices of last week's blogs are nothing compared to the options you face in doing a kitchen...

Here are numerical values associated with some of my choices:

Kitchen sinks (1181 choices)
      Undermounted (547 choices)
             Stainless -> (497)
                  Twin Bowl (259 choices)
                       Width between 24-36" (215 choices)

Wall Ovens (334 choices)
     27 inch wide (95 choices)
          Convection (62 choices)
                 Single Oven (31) or Double Oven (31)
                        Oven alone (26) or with microwave (5)

Cooktops (486 choices)
     Gas powered (280 choices)
         Width between 24-36" (194 choices)
                 5 burners (76 choices)
                        Front Controls (61) or Side (15) 

Bottom Freezer Refrigerator (209 choices)
      Free-standing (175 choices)
             Counter-depth (109 choices)
                  Twin Doors (57) or Single Door (52)
                       Width between 24-36" (92 choices)

Built-In Dishwashers (224 choices)
      Width of 24"  (214 choices)
             Depth of 22-25" (141 choices)
                  Stainless inside (110 choices)
                       Food grinder (42) or Not (68)

Add to these choices the options in faucets, disposals, drawer and door pulls, hinges, etc. In addition to these "simple"  and "well-defined" choices in "appliances" and "machinery", we will also have to decide from a virtually infinite range of paint colors, wood stains, flooring options, counter-tops and lighting.

Most of the choices are dependent on the cabinets we opt for. Hence I conclude: Too many choices!

Thursday, April 21, 2011

Sets and Grouping, Part IV

For the past few days I have been demonstrating sets and groupings using t-shirts as my sample material. Today I'll do some math for you. Here's the basic question:

If Duluth Trading stocked one of each combination of color and size of each variety, how many men's t-shirts would Duluth have?

(Use the 4 tables from Part I of this blog for your source of data. If you go to their website you might find that some shirts are out of stock, and some sizes discontinued, etc. but we will work from data in the catalog I used for the tables.)

I'd expect our typical student to do it this way:

Look at each line (lines are what I'll call a shirt model) in the 4 tables. Multiply the number of colors times the number of sizes to get the possible variations. Let's call these variation sums V. That's

 C x S = V

Add the products of our multiplication for all the lines (models) in that table to get a sub-total:

V1 + V2 + V3 ...  = ST

Take the sub-totals from each table and add them together for the grand total. This is simple math.

ST1 + ST2 + ST3 + ST4 = T

I typed the data into a spreadsheet (from my Duluth catalog - I don't work for them),  so I can create formulas to do the multiplication and addition work for me. Otherwise I would use paper to make notes as it's hard to hold the sum of a string of 13 numbers (table 1) in my head while I do tables 2, 3 and 4. So here are my sums:

ST1 = 572, ST2 = 110, ST3 = 22 and ST4 = 90.

We can add these in our heads. I'd do it this way 90 + 110 = 200 then 200 + 22 = 222 and 572 + 222 = 794. I did this out of order because I could see the subtotals were going to be easier.

Duluth would need to have 794 shirts in stock just to have one of each variation.

Now imagine we will buy these 794 shirts and give them to the local Rescue Mission so they can put a shirt on the back of every needy guy in town. Assuming we couldn't get any discounts, what would it cost to buy all these shirts

Could we come up with an average price per table by adding the prices in each table and dividing by the number of shirt models? Let's try that. There are 13 shirts in table 1, 5 in table 2, 3 in table 3 and 3 in table 4. I get table average prices of: 

ST1 = $17.12, ST2 = $21.30, ST3 = $21.83 and ST4 = $21.50. Add the sub-totals and divide by 4 = $20.44. This is our average price per shirt. Multiply by 794 to get $16,229.36.

We'll have to ante up more than 16 thousand dollars for the shirts using this approach.

Will it be different if I added all the numbers and prices first, and divided by the total shirts instead of using the 4 separate tables? Let's see ... the sum of shirt prices is $459 divided by 24 models = $19.13. This is our new average price per shirt. Multiply by 794 to get $15,189.22.

Wow! I can get the same shirts for $1000 less.

Will it be different if I do the math on each model (multiplying colors x sizes x prices) and then add those products? Indeed it will.

This approach results in a total of just $14,347.00 for all the shirts.

Notice the total is an even number of dollars (no cents). Because all the shirt prices are even or multiples of 50¢ and 794 is an even number, I think this total is more accurate than the ones with 22¢ or 36¢ at the end of the number. Why?

Later we can divide the total price by 794 to find an average price per shirt of $18.07.

All of these calculations were done properly by the spreadsheet. I checked with a calculator. But the totals on the first two approaches were wrong! Why?

These different yet inaccurate approaches are taken by hopeful students (and business people) all the time. If you were the generous t-shirt donor, and it was your $15,000, would you be interested in who was doing the math with your donations? I would be. Let's hope it's a former Excel Math student!

Wednesday, April 20, 2011

Sets and Grouping, Part III

This blog reviews how we use elementary math in our everyday, grown-up lives.

Yesterday we grouped shirts by fabric type and pockets/no pockets. We were able to use Venn diagrams (overlapping circles) to demonstrate how items can be members of more than one set at a time.

Today, as you can see, we are looking at t-shirt colors. More specifically, how t-shirts can be grouped into sets, by color.

Notice that the primary colors (black, blue, etc.) are sorted alphabetically, not in "order of color."I have decided to group these colors using my own eyes, sensory perception, and color names that I like. If you are colorblind and/or have art training, you might divide the color groupings differently. You could use PMS colors, or some other color labeling scheme.

The middle column is men's shirt colors. I decided to contrast this by adding a right column with women's colors. Men can chose from t-shirts in these colors:
  • 1 black
  • 1 burgundy
  • 3 browns (including camel)
  • 7 blues
  • 1 chamois
  • 4 gray (including graphite)
  • 4 green (including olive)
  • 1 orange
  • 2 red
  • 1 white
Women chose from these colors:
  • 1 black
  • 2 brown (including mocha)
  • 5 blues
  • 1 gray
  • 3 green
  • 3 orange (including sienna)
  • 1 purple
  • 1 red
  • 2 yellow (including chamois)
  • 2 white (including natural) 
Color designers make the choices in concert with the fabric manufacturers. Nowadays color choice is expected, and universal. Who could conceive of consumer products only offered in black and white?

But that's not always been the case - certain colors are more expensive, difficult to produce, harder to protect from fading, etc.I have neon-yellow, fluorescent-green and bright-orange emergency shirts that I keep in my car.

(If you're a long-time reader, you might remember the blog on (in)visibility. )

No doubt these are more expensive colors to produce, with limited appeal. I've noticed that Duluth no longer offers those yellow and green colors in t-shirts.

In math class, we often use color as an example of items that can be sort, grouped, or used to create combinations and permutations.

(If you can't recall the difference, you can go here to get our math glossary.)

I found an interesting resource for color lovers (and web page designers) put together by Mark at Techyuva. You can see how math very quickly gets mixed up with color... and you can download some color charts.

Many years ago we introduced this girl with the extensive wardrobe to our 3rd graders. Students calculate how many kinds of outfits she can create with her patterns and colors. See if you can spot some of her mis-matched socks... [click the image for a larger view)

Tuesday, April 19, 2011

Sets and Grouping, Part II

Today we will be using elementary math (as taught in Excel Math curriculum) to group items by common characteristics. Yesterday I showed you my collection of data about t-shirts from Duluth Trading Company.

I looked at my spreadsheet to see if I could see how to organize the shirts. I decided to draw a Venn diagram to illustrate a few sets of items that share common features:

I organized the t-shirts into 4 sets. I counted the number of shirts in each set and put those numbers into yellow circles  (2 different ones), a green circle and a blue circle based on the type of fabric used to make the shirt.

Within those 4 fabric sets, I separated the shirts with pockets (beige small circles) from those with no pockets (grey small circles). Since shirts either have pockets or they don't, all shirts in a fabric set fit into either the pocket set or no-pocket set.

Now I have added ovals. The pale yellow one indicates that there's a larger set which includes shirts made of pure cotton (yellow) and shirts of blended cotton-polyester. Similarly, there's a pale green oval to include shirts made of pure polyester (green) set and the cotton-polyester blend shirts.

Notice that there are two sets of circles in the cotton oval, and two in the polyester oval. But there are only 3 fabrics included in these two ovals, not 4, because one fabric (the cotton-polyester blend) belongs to both.

The nylon fabric (blue) is by itself. No shirts are made in a cotton/nylon-spandex blend or a polyester/nylon-spandex blend.

How can we include the blue circle in a larger set? Let's try grouping by pockets or no pockets, instead of fabrics.

The dark blue lines show the set of shirts with pockets. The red lines show the set of shirts with no pockets. These two sets transcend the fabric choice sets. They include the nylon-spandex shirts.

Set relationships take a lot of thinking. It's fairly straightforward to draw these groupings and to shade the diagrams appropriately, once you see the relationships.We give kids opportunities to do this in class.

It's harder when you are using Adobe Illustrator, various degrees of transparency, and trying to get all the layers in the proper order from top to bottom so you can turn them off and on!

Come back to see more sets tomorrow.

Monday, April 18, 2011

Sets and Grouping, Part I

One of the interesting things about marriage is learning how differently people think about things. I tend to say I'll have some juice, while my wife says Would you like some orange juice, tomato, grapefruit, cherry juice? With or without sparkling water? Ice or no ice? In a glass or the bottle? Plastic glass or glass glass? Straw or no straw?

More choice is not always better - especially if you hungry and dining with a crowd of people, the server is handing out menus with 50 options, and reciting the 10 daily specials ...

That brings us to the elementary math skill of decision-making by categorizing. We make life more manageable if we can slice, dice, sort and arrange our options into meaningful categories or groupings. In the math world we call these groups "sets."

About 18 months ago I did a blog on an outdoor work clothing website called Duluth Trading Company. If you are indecisive, stay away. If you like choices, this is your place!

NOTE: I have no business connection with DTC although I have purchased about 6 shirts from them.

In the next few blogs I will show how we help kids understand sets, using DTC catalog options are our raw material. The illustrations below show their offerings of MEN'S SHORT SLEEVE T-SHIRTS. Nothing more. No dress shirts, no long sleeves, no women's shirts. I typed these details into a spreadsheet and separated this huge inventory into categories based on type of fabric.

Using our elementary math skills (taught in Excel Math) we can:
  • decide how many combinations of shirts we could wear
  • calculate the total cost of buying one shirt of each style
  • investigate the types of fabrics and speculate on the heaviest or lightest shirts
  • speculate on the inventory requirements or the number of SKUs that DTC uses
  • sort by fabric, collar, sleeve, pocket, color, length, girth and overall size 
Stay tuned for more t-shirt math - and if you need a shirt, go shopping.

Friday, April 15, 2011

Phooey on Flats, Part III

This week I'm sharing how we teach reasoning in math class. We are using "find the best way to reduce problems caused by flat tires" as an example of a societal problem. ( Read the two earlier blogs in this series )

Here are the problem-solving techniques we teach in 4 through 6th grades:
  1. Reasoning using logic
  2. Reasoning using patterns
  3. Reasoning by trial and error
  4. Reasoning by asking questions
  5. Reasoning using a possibility chart
  6. Reasoning by process of elimination
  7. Solving problems using deductive reasoning
  8. Estimating which answer is most reasonable 
  9. Reasoning by examining evidence and making notes
  10. Reasoning by working backwards from a given solution
  11. Determining if there is enough information to solve a problem
Which process might be most useful in deciding on a flat-tire solution?

Re-read the list, then keep them in mind as we review some real flat tire incidents. I will code these 4 different solutions red (useless), black (not sure) and green (helpful):

A. My vehicle has a mini-spare tire. Driving across the desert on paved roads, I hit a small rock and cracked my alloy wheel. The impact damaged the tire irreparably. I changed the wheel, put on the mini-spare and drove home. It took 10 days to get a new wheel and tire.

RUN-FLAT - wouldn't have helped if the rim was broken.
TIRE INFLATOR - wouldn't have helped if the rim was broken.
ROADSIDE SERVICE - the rim was ruined so my spare was essential. There was no mobile phone signal for 50 miles.
TPMS - it would have warned me that the tire lost air (but I learned that as soon as I hit the rock).

B. In the morning my tire was flat. I put on the spare. I could see a huge screw sticking out of the middle of the tread of the flat tire. I marked the spot, pulled out the screw and went to the store to have the flat fixed.

RUN-FLAT - the screw probably wouldn't have mattered to a run-flat tire; it would have held air.
TIRE INFLATOR - would have re-inflated the tire after removing the screw (or I could leave it in).
ROADSIDE SERVICE - someone could have changed my tire for me.
TPMS - it would have warned me that the tire lost air (but I could easily see it).

C. My wife was driving down the street. She realized a rear tire was flat when someone honked and pointed. She limped the car 2 blocks to a parking lot and called me to come and change the tire. The sidewall had been damaged by a pothole.

RUN-FLAT - probably wouldn't have prevented the sidewall damage.
TIRE INFLATOR -  it wouldn't have worked on the sidewall damage.
ROADSIDE SERVICE - someone DID change her tire. Me. This worked!
TPMS - it would have warned her when the tire was damaged.

DI was about to depart on a cross-country trip when I saw my front tire was low. I pulled out a screw sticking out from the corner of the tread, and the tire went flat. No tire stores in town had a new tire, and they wouldn't repair mine. Rather than continue without a spare, I bought an inflator and filled the tire. It's been holding air for 9 years (now as a spare).
RUN-FLAT - none available on my elderly car and wheel.
TIRE INFLATOR -  it DID work on the sidewall / tread puncture.
ROADSIDE SERVICE - someone could have changed my tire, but I would still have had to inflate it.
TPMS - it would have warned me that the tire was leaking.

Don't worry right now about choosing the best solution of the 4 solutions. The question is Which of the 11 math reasoning methods are we using?

[Click here for my answer]

Thursday, April 14, 2011

Phooey on Flats, Part II

This week we are looking at the problems of flat tires. You should read the first blog in this series if you haven't done so already.

There are a variety of solutions that reduce flat tire problems. We will be evaluating these using reasoning techniques taught in Excel Math. We may not come up with a single definitive solution - but of course the answer often depends on how you phrase the question.


Advantages of run-flat / self-sealing / self-supporting tires
  • Driver and passengers aren't inconvenienced by (most) tire damage.
  • A driver can continue for 100-150 miles at 50mph with a damaged run-flat tire.
  • No need for a jack or spare tire.
Disadvantages of run-flat tires
  • Weight of the tire is 10-20% higher than a regular tire
  • Price is 20-30% higher than regular tires
  • Ride and handling often inferior to normal tires
  • Tires usually can not be repaired after having a flat
  • Few dealers have equipment to change run-flats
  • Limited availability of tires in stores (only 3% of tires sold) so it may take days to get
  • Specialized wheels may mean owner is locked into one type of tire

Advantages of an inflator with sealant
  • Smaller and cheaper than a regular or mini-spare
  • Wheel doesn't need to be removed for re-inflation
Disadvantages of an inflator
  • Might not work if the sealant has been stored for a long time, or if it freezes
  • Won't help if tire or rim are badly damaged
  • Residue from inflator may ruin equipment or the tire valve or pressure sensors
  • Inflator gas may be flammable and dangerous for technicians 

Advantages of a roadside service
  • Someone else does the hard work of changing or replacing your tire
  • Convenience for the driver and passengers
Disadvantages of a roadside service
  • Cost and complexity of the system
  • Unpredictable contact (phone may not be available)
  • Wait time for the technician to arrive
  • "Big-brother is watching me" issues 

Advantages of TPMS
  • Gives advance warning of a tire "going flat"
  • Reduces the likelihood of 75% or more of flat tires
  • Reduced tire wear and increased energy efficiency from keeping tires inflated correctly
Disadvantages of TPMS
  • Additional cost to vehicle manufacturers and owners
  • Another warning light for owners to interpret
  • Inconvenience of false alarms
  • Need to replace sensor batteries
  • High potential for damage when tires are replaced
  • Unlikely to last the lifetime of the vehicle
Now that you have seen the four primary solutions to flat tires, what do you think? How would you go about selecting a solution? Tomorrow we can do a case study and try one of our elementary math problem-solving techniques.

In the meantime, enjoy the famous Michelin poster introducing Mr Bibendum, who says:

"Now We Must Drink, that is to say, 
To Your Health, Gentlemen!"
The Michelin tire swallows up the obstacles 
(road hazards, nails, glass, etc.)

Alas, a few years ago, in a frenzy of political correctness,
Michelin put him on a diet, took away his drink and cigar, 
then gave him a smile and a green plant.

    Wednesday, April 13, 2011

    Phooey on Flats, Part I

    For the next few days I will focus on flat tires (on your car, not your bicycle).  

    You might ask, How can elementary math be helpful with a flat tire? Fair enough; here are some processes that Excel Math covers as we help kids learn "math". They can be used when considering flat tires.
    • Reasoning using logic
    • Reasoning using patterns
    • Reasoning by trial and error
    • Reasoning by asking questions
    • Reasoning using a possibility chart
    • Reasoning by process of elimination
    • Solving problems using deductive reasoning
    • Estimating which answer is most reasonable 
    • Reasoning by examining evidence and making notes
    • Reasoning by working backwards from a given solution
    • Determining if there is enough information to solve a problem
    [Speaking of observation, did you notice I arranged these concepts in order of length rather than any other logical sequence?]

    Am I just assuming that reducing flat tires is a valid subject for logical problem-solving? Do you need data to convince you a flat tire may be a clear and present danger?

    How many flat tires occur every year?
    • US - 220 million flats per year; about one flat per passenger car per year.
    • US - 23,000 vehicles are damaged due to blow-outs, to the point they must be towed.
    • US - 1.2% of traffic fatalities are due to tire-related accidents.
    • US - tire blow-outs are most frequent on light trucks in southern states during hot months.
    • UK - One flat tire (tyre) per 20-25,000 miles traveled.
    • UK - 30-40% of drivers had a flat in the last year. 
    • UK - flats comprise 10% of total vehicle breakdowns.
    • EU - flat tires account for 28% of breakdowns of 112,000 commercial vehicles in 2010.
    • On average, a driver will have 5 flat tires in their driving lifetime.
    What types of flats might occur?
    • 25% are blow-outs; sudden loss of pressure with dramatic damage to the tire.
    • 25% slow leaks that happen when driving; often result in a ruined tire from driving on it.
    • 50% discovered when the car is parked; inconvenient but not normally ruinous.
    Problems related to flat tires:
    • It is dangerous to stop to change a tire on a freeway and/or at night.
    • It can be difficult and dangerous to raise the car with a jack.
    • Wheel nuts can be too tight to remove.
    • Alloy wheels can stick to steel hubs making it difficult to remove the wheel.
    • It can be hard to line up the wheel and the hub when replacing the wheel.
    • If you have a spare it's usually under the stuff in your trunk.
    • The dirty flat tire has to be lifted and stored in the trunk.
    • If you get another flat before your tire is repaired you will be stranded.
    • They always happen at an inconvenient time.
    That should be enough to get us started. Tomorrow we will look at possible solutions.

        Tuesday, April 12, 2011

        Home Owner Math

        Many people see home ownership as a chance to make a fortune or (more likely nowadays) a chance to lose a fortune. I'd like to spend a couple minutes looking at the economics of owning a house - with the assumption we only use elementary school mathematics.

        This is not a lesson on how to buy a house. I won't tell you about the cheapest mortgage, the lowest down payment, how to get rich quickly, how to save on your taxes, or how to profit from foreclosures.

        Instead of following those typical themes, I'll show why even with a hot market area, with low mortgage rates and minimal expenses, you may not make money on a house. Which doesn't mean you shouldn't own one. Just don't give up your day job when you buy one.

        Look here:

        Now let's add some real numbers to illustrate my point [click the image for a larger version]

        Grab a calculator and let's do some math.

        This house has doubled in value. Over 20 years the gain is about $375,000 appreciation in value. Add that to the original $375,000 cost of the house and it's now worth $750,000. If they could possibly sell, the owners might net $725,000.

        Over those 20 years there could also be a savings of up to $60,000 in income taxes due to various deductions.

        Even if you are a tight-wad, have a low-interest-rate loan and no association fees or mortgage insurance, the expenses over 20 years amount to $833,800.

        That's an average of $3475 a month, which could rent a very nice house. Keep in mind the owner's mortgage payment was $1800 a month.

        This does not include discretionary work, furniture, appliances, remodeling, gardening, etc.

        I am ignoring the "lost income" from the $100k down payment. If it had been invested at 5% over the past 20 years, the $100k would grow to $265k - a gain of $165k. At 7% it would have produced a gain of $287k.

        Did the home owner make a profit?

        Monday, April 11, 2011

        Seafood Math

        This blog is focused on explaining how we (grownups) can use the math learned in elementary school. Since our company's Excel Math curriculum is used by thousands of schools to help kids learn math, we think about this all the time. Today, following a seafood-filled family reunion, I thought I'd look at the math related to this important dietary choice.

        Do you like this fish painting? It was done by a very artistic person at my wife's elementary school.

        Seafood means foods that come from the both salt and fresh water. The term includes plants (seaweed) and animals (fish, shellfish) which can be either caught in the wild or farmed.

        You might wonder if you really need math skills to eat seafood ... let's see:
        • Wikipedia tells me there are 32,000 species of fish. (number with separator)
        • We only eat a few varieties of the total number of fish species. (small but indefinite quantity)
        • Seafood is the primary source of protein for a billion people. (large number word)
        • Seafood is about 15% of the total world supply of animal protein. (percentage)
        • The Fish Society in England has a huge shellfish and caviar assortment for £1500/$2400. (foreign currency conversion)
        • At least a dozen types of seafood can be eaten raw. (collective number)
        • Sashimi is raw seafood cut into pieces about 2.5 cm (1") wide by 4 cm (1.5") long by 0.5 cm (0.2") thick. (cubic dimensions, in metric and standard units)
        • Folklore says not to eat oysters unless there is an R in the month. (calendar)
        • Sardines have been eaten by people for thousands of years. (calendar number words)
        • Many seafood recipes say, "Start with a whole fish". (complete; not fractional or decimal)

        Because seafood is highly-perishable, it is often shipped live, refrigerated or frozen. Alternatively, fish can be salted, smoked, dried, or canned. We had plenty of fresh seafood during our family get-together:

        If you want to learn more about sourcing seafood, you can look here for videos from Taylor Shellfish, one of the largest providers in the United States.

        Wednesday, April 6, 2011

        How Much Are You Really Worth?

        This blog is about using elementary math to make sense of the world around us.

        Yesterday I did a blog on the value of vehicle inspection programs - Pennsylvania compared the total cost of its programs, as borne by 11 million people, versus the life of 127 people who might have died IF there had been no vehicle inspections. A big IF.

        There was a number put on a human life - $5.8 million for the Value of a Statistical Life (VSL). Where did that come from? You can read the full report here.

        Bur first a different definition: VSL is a statistical term describing the cost of death prevention in certain circumstances -  the cost of reducing the average number of deaths by one. 

        I was looking at my Social Security statement recently and I am quite sure that the money I have earned in my working lifetime (since 1967) does not equal $5.8 million! Are you interested in your worth as estimated by your government?
        • Office of Management and Budget says $1-10 million
        • Food & Drug Administration says $5-6.5 million (value of a drug or treatment)
        • Environmental Protection Agency says $7 million (value of a life damaged by pollution)
        • Department of Transportation has ranged from $2-5 million
        • Department of Agriculture uses $5-6.5 million
        I found the following paragraph uses a fascinating line of reasoning!

        ... there has been no adjustment (to the VSL) for growth in real incomes, but research indicates that as people grow richer they are willing to pay more for safety.
          Potential damage associated with vehicle accidents includes the personal dis-utility of death or injury and a variety of economic losses (to victims and others), including property damage, traffic delay, lost productivity, and the costs of police, investigation, medical, legal, and insurance services. The benefit of preventing economic loss  to society as a whole, apart from victims and their families, should be counted.

          VSL includes lost after-tax earnings, suffering and lost quality of life and a separate "productivity" component. Avoiding losses should be treated as the entire benefit to potential victims of accidents and their families. In contrast, reductions in property damage, medical expenses, traffic delay, and other social costs associated with fatal accidents should be treated as added benefits to society as a whole, and not included in a victim's benefits.

          Sheesh! This is complicated.We are not only worth something on our own, but our avoidance of dying saves the society money that's not allowed to be counted. Hmmm. What about money generated for our employers by our inventions, or copyrighted products we create? Who counts that?

          It seems that the VSL number does NOT really calculate the output (or the consumption) of one person. For that we have to create a new number, the Consumer Unit (CU). And it appears that capitalism isn't the only economic theory needed for this - we also might need to consider Karl Marx's viewpoint on individual output and consumption.

          Go here if you want to read more about this interesting subject. Take your calculator!

          It looks like these numbers that represent personal worth have significant influence on policies like the value of an education, cost-avoidance of keeping young men out of prison, etc.

          Tuesday, April 5, 2011

          How Much Are You Worth?

          I had the chance to read over a study of Vehicle Inspection Programs this week. In summary, the state of Pennsylvania did an evaluation of their existing annual vehicle inspection process to ask:

          • Do inspections reduce the number of fatality or injury accidents?
          • Does the inspection period influence the level of accident reduction?
          • Do inspections influence the mechanical condition of cars?
          • Are vehicles in states with mandatory inspections in better mechanical condition?

          Pennsylvania concluded that their inspection program is cost-effective (without necessarily answering the 4 questions above). How did they determine this?

          Much of their statistical work goes far beyond the level of elementary math I assume we all share, but we can do some pondering on the topic. Here are the some numbers presented:
          1. 11 million inspections performed annually
          2. 16,000 inspection stations
          3. Average cost of inspection between $16-23 with average of $20
          4. Vehicle owner time spent on inspection is valued at $9-34 with an average of $17
          5. Inspections result in 114 fewer crashes a year, and 127 deaths saved
          6. A person in the USA is worth 5.8 million dollars
          7. Cost of the actual repairs was not considered worth tracking
          Let's do a few math problems.

          Cost of the inspections:

          11,000,000 inspections x ($20 fee paid + $17 lost time) = $407,000,000

          Revenue to inspectors:

          11,000,000 x $20 fee = $220,000,000 ÷ 16,000 = $13,750 per inspection station

          Value of a life equals:

          $407,000,000 ÷ 127 lives = $3,204,724.41 or the value of a life lost in a car accident is $3.2 million.

          How do they determine the value of a human life? Pennsylvania says that if we are each worth $5.8 million (see the study link above), when we die in an accident, we have lost 39 years that we would have otherwise lived, out of a life expectancy of 75-80 years.

          If we can find a solution for the problem (3.2/5.8 ) = (39/x) we can find out how old they assume the average killed driver is. I come up with 41-44 years of age.


          Pennsylvania's study concluded the cost of accidents range from $737 million to $1084 million and the cost of the inspection program equals $267 million to $621 million, therefore the inspection program is worth continuing.

          Other thoughts:

          There are many things that could be argued about in this analysis. But I'd just make one point - 20 states have some sort of safety inspections, and 30 states do not. Vehicle inspections have been done with varying levels of effectiveness since 1926. The issue has been studied for a LONG time.

          The AAA in 1967 found no proof that vehicle inspection is effective in reducing accident or death rates. NHTSA in 1989 decided there was no evidence that inspections reduce crashes. Pennsylvania in 1981 concluded that accident rates of vehicle with annual, semiannual, and no safety inspection programs were equal. Norway in 1992 found no statistically significant differences in accident rates between the three groups.

          Folks, walk around your car, check the tires, lights, and wipers - and fix them if they need fixing! And drive safely. You are worth $5.8 million!

          Monday, April 4, 2011

          How Many Examples Do You Need?

          We learn by example.

          Someone shows us how to dive into a pool, we try it; they show us again and eventually we are doing spectacular dives ourselves. I saw this "learn by example" process last week at the Furnace Creek Resort pool, in Death Valley. But instead of beautiful dives, it was a group of young men developing their skills at doing cannonballs into the pool, sending huge geysers of water up and over the decking. The fuss didn't bother me - I was on a mini-vacation...

          You might have thought lifeguards would be stopping the festivities - but no. Instead there were dozens of adults encouraging the frivolity. Moms, dads and strangers were giving helpful advice, pointing out flaws in the kids' form, and judging height and volume of splash. Younger kids were squealing every time water flew. It was a community effort and unique in my poolside experience.

          One young man was having trouble. Was he resistant to advice? A slow learner? A klutz?

          A couple other men and I discussed how we could help, but before we interfered, one of the moms came over. We learned he was new to the USA, having recently come from Iraq. He came to the desert with his sponsors for a taste of home. He had never been in a pool before. He couldn't swim. His English was fine. His muscles were toned. He was friendly. But he was obviously frustrated. Yet not willing to give up.

          Wow! Suddenly our attitudes changed. He was no longer a slow learner, but someone whose courage was being applauded. Would you dive into the deep end of a pool if you couldn't swim? Would you jump, grab your knees, lean back and close your eyes, hoping for a mighty splash before you drowned? Would you do it 15-20 times, despite younger, faster learners around you? Despite laughter at your expense?

          It occurred to me today that many kids and adults have the same issues with math. We fear diving in. We know we will flounder around, catch heat from our peers, get the answers wrong, get water up our noses - and have to do it again and again before we get it right.

          In addition, we might not know English. We might be new to the USA, new to our school, in front of an uncaring peer group; stumbling over non-metric units of measure and local slang.

          Excel Math is written for kids like our cannon-balling fellow from Iraq. Lots of chances to learn, lots of chances to succeed. Simple language. Low-stress approach. Friendly products.

          Missed the big splash this time? No worries, it will spiral around again and you'll have another chance. Here's a visual summary of our approach. [Click image for a larger version]

          You can read more at the Excel Math website.

          Friday, April 1, 2011

          The Math in My Friday

          Today I'd like to list a few of the important math-related questions that face me going into the weekend:

          1. How much will it cost to ship an ill-advised purchase back to the supplier?

          Fedex tells me that 9 pounds shipped in 4 days insured for $1500 will cost me $22.50.

          2. How many calories were in the piece of pecan dessert I ate on my morning tea break?

          It appears that the bars might range from 250-360 calories per slice. I had a small piece so I'm going to hope it's 250 calories.

          3. Friends are driving from Hesperia to Julian to San Diego. How far is that? How long will it take and will they arrive for dinner by 6?

          Thanks to Google Maps I know it's about 150 miles from Hesperia to Julian (4 hours) and about 65 miles from Julian to San Diego (1.5 hours). Add some time for looking around and I think we'd better plan dinner later than 6 pm.

          4. I have to get a new track and rollers for the hall closet at my house. I want four 2-wheel roller assemblies and a 4-foot double track that is suitable for 1 3/8" doors. How much will I have to spend and can I get that stuff at the Home Depot near the office?

          Thanks to my telephone and a cooperative Home Depot phone operator, I know that they have the parts I need for $11.98 - with tax $13.03. I went over to the store on my lunch hour and got the parts. Total time about 15 minutes including driving over and back, plus 5 more minutes to pick up the 97 cents that fell onto the floor next to my seat.

          5. I am looking at the code for the web blog and I'm troubled by the amount of HTML being embedded in and around my text. Is there an easier way to do these multi-colored question/answer blocks? Can I simplify the coding?

          Indeed I can. But it's too complicated to explain here. I'm guessing I've cut about 30 characters per paragraph from the code.

          6. I have been working as the executor of my mother's estate, and I need to close her credit union accounts and transfer all the remaining money to another account. How many branches do they have where I could get this done? How long will it take me to accomplish my tasks?

          Only two branches in a whole state? Wow! Guess I have very little choice. I am on hold ten minutes ... now waiting again, but I am doing the blog while waiting, so it's not that irritating. Ten minutes so far with no progress ... aha, now they say they will get the answers and then call me back. This one isn't answered yet.

          7. Time for lunch. How long will it take to make a toasted cheese sandwich while I finish up the blog?

          Not quite as long as I thought. Good thing all the doors were open to take away a bit of smoke ...