Additional Math Pages & Resources

Friday, May 28, 2010


Finishing off this week's one-word blog topics, today's subject is NOT a new drug for attention-deficit disorders, but a term we use in the physical sciences (and sometimes in math). Here's the symbol:

It means that things are stable. A reaction is not going on; things are not changing; everything is at the same temperature, etc.

In economics, it's a word used to describe a constantly-maintained balance between supply and demand, so prices aren't fluctuating, there are no shortages, and no inflation. No surplus, no deficit.

In populations, equilibrium is reached as one person dies and another is born at the same moment.

Utopia might be a synonym for equilibrium. I hope equilibrium represents the state of affairs somewhere. It certainly doesn't reflect reality here on earth.

Reality contains some events called "Accident" and "Greed" and "Selfish" and "Disaster" which result in "Inflation" and "Stagflation" and "Deflation" while shaking up any equilibrium that ever accidentally happens.

Here's a final graphic for you, to represent the relationships of today's topic:

A shorter word that sums up much of this concept is SHARE. Or EVENLY DISTRIBUTE.

In case you were wondering why we would have to worry about equilibrium in the classroom, here is Kid Math expressed through the words of a Lazy Town cartoon character named Stingy:

"One for me and one for you
and wait - give that back! 
and one for me and another for me 
and one more for me. 
And this broken one for you."

Thursday, May 27, 2010


We have mentioned Deficit, Surplus, Thrift and have now come to Waste. This is one of the subjects that educators would like to see handled properly in math (or indeed ANY) curriculum. It's an antonym (opposite) of Thrift.


Not, want not, they say. Uh, OK, you say. But actually we do. A lot. But waste is a complex concept.
Check it out here if you have some time [to waste?]

Waste is a verb:

I wasted my time.

Waste is a noun:

We have too much nuclear waste.

Waste is sometimes an adjective:

Out in the waste land.

Educators are interested helping students to avoid several particular meanings of the word, such as

"Use inefficiently or inappropriately" and also "Spend thoughtlessly or extravagantly"

Some story problems in Excel Math are intended to reinforce a positive messages:

Three boys picked up trash on the beach. Byron picked up 457 pieces, Rico picked up 158 pieces and Miguel picked up 45 pieces MORE than the other two combined. How many items of trash did Miguel pick up from the beach?

A few help students learn efficiency and thoughtful purchasing. Here's an example:

Serena is planning a trip. She checked several airlines to compare prices and flight times. What information does she NOT need to buy her ticket:
1. the day of the trip
2. how long she will stay
3. the type of seat she wants
4. the length of the runway

Sometimes we tackle ecological issues. Our product is in the schools for years, so we tend not to deal with extremely contentious, political or time-sensitive issues. Let's use this as a sample:

In 1995 Arizona's butterfly population was estimated at 200,800,000. In 2001, the estimate was 189,950,000. Which choice best shows the decrease in the butterfly population?
1. 10 billion
2. 10 thousand
3. 10 million

We want students to consider ways to use their time - do video games waste time? Is volunteering a positive use of time? Here's an example:

Laurie volunteers two hours a day, five days a week assisting seniors at a nursing home to go to the park. How many hours does Laurie volunteer in eight weeks?

Wednesday, May 26, 2010


Continuing our elementary education in money-related math terms, today we come to the word Thrift.

As I explained yesterday, the educational framework in California and most other states requires that we teach kids to live as responsible citizens of society. That means in the classroom we do not recommend "Eat, Drink and Be Merry, for tomorrow you die" as a guiding philosophy.

We are expected to have a more responsible attitude. We are positively required to teach Thrift, which means "restrained or disciplined spending habits" and "wise management of money and resources."

The dictionary suggests that being thrifty means "thriving by industry (work) and frugality, which leads to prosperity". Notice that we are in California, not Scotland. At the moment, it's hard to say this approach is working for our state's finances!

Back to the classroom:

We practice calculating unit pricing - here's a typical problem from our 6th grade book: 

One store has apples for $.79 per pound. A second stores has apples for $1.50 per kilo. A third store sells apples for 25¢ apiece. Which is the best price per apple?

We point out that the best unit price doesn't always mean the best deal for the customer, especially with perishable goods. You might throw food away because it spoiled before you could eat it. But let's say you were eating out, and you need to calculate the bill with a tip. Excel Math is here to help:

Rami and her friend had a lunch bill that came to $22.50. Rami paid the bill plus a 20% tip for the waiter. How much money did she pay? 

We explore many facets of wise shopping - comparing prices, calculating the unit price, applying discounts, understanding sales taxes, tipping on a dinner check, etc. We must respect both being a laborer and being management, so we don't quibble about whether to tip the server 10%, 15% or 20% on lunch!

Ian deposited $170 in a bank account that pays 5% annual interest. How much interest did he earn after one year? 

We provide a basic overview of savings accounts, and interest paid on deposits. When I wrote this problem 4-5 years ago, I thought that 5% was a pretty low interest rate. Now I'd like to find a bank that would pay me so much!

We also tackle foreign exchange rates, a great math subject! Here's a fairly recent problem (but today the exchange is 1.43):

One British pound equals 1.87 US dollars. How much money in US dollars would you receive in exchange for 8 pounds?

If you never learned to do these calculations in school, you could pick up a copy of 6th grade Excel Math and teach yourself how to be thrifty. Let's end with a complicated problem:

4 oz for 16¢                       2 lbs for $1.44                6 ounces for 30¢ 

Here are the prices for a certain product. If you want one pound of this item, how much money will you have to spend if you chose the cheapest option?

Tuesday, May 25, 2010


Yesterday we talked about deficit. That implies overspending and subsequent need for credit, use of math to justify an unbalanced budget, etc.

Today we'll investigate the opposite term, surplus.

In my imagination, that implies khaki-colored stuff piled on tables in a warehouse. Some jackets. Some tents. Some pipes and brackets and un-definable hardware. Like this:

Of course, we don't get surplus stuff without some one, some where, at some time, having bought too much, which leads to a deficit (in their budget) and a surplus (in this store).

Surplus also implies discounted prices. Somebody (first the army, now the store owner) has more than they want, and they lower the prices to motivate us to buy (more than we need or otherwise want).

We often call this urge to get a good deal "being thrifty" but it can lead to the opposite - buying more than you need because the price is attractive. That leads to a deficit in the family budget.

Did you ever notice people buy more at a "thrift store" because things are so cheap?

In California Curriculum Standard #60042 we (publishers and teachers) are required to teach "thrift, fire prevention and humane treatment of animals and people."

Exactly why thrift, fire prevention and humane treatment are linked together I cannot say. I only have to worry about the math portion of this assignment.

PS The definition of thrift is extended to include avoiding waste and/or littering.

Monday, May 24, 2010


This week I am going to focus on single-word subjects for the blog.

Here's a word that's in the news a lot lately. DEFICIT.

What does it mean? An excess of expenditure compared to income.

What is its opposite? The opposite term is surplus, or more income than expenditure.

Was it just invented by our current government? No, not at all.

Is it a math term? Yes.

It's related to those symbols I covered a few months ago - less than and greater than.

There are lots of situations when you might run out of money but that DON'T reflect deficit spending. Those are called a cash-flow crisis. Among other solutions, you might require credit. Here's how it happens:
  • Teachers are often paid on a 9-month schedule. They can run out of money in the summer unless they have saved up during the year, or have credit so they can borrow.
  • Small but growing businesses might have to order lots of goods to restock their open shelves, but don't sell those products quickly enough to pay their suppliers on time. They need to borrow.
  • Churches and charities often receive the majority of their income at the end of the year when donors are doing their tax planning. They have to borrow to make it through the summer or fall.
Here's a real-life example of deficit spending and a cash-flow crisis.

A local restaurant opened and was immediately popular. But it nearly failed in the first few months. Why? Because the owner had over-spent before opening and used up all the credit he could get. He had to pay the lenders back on a fixed schedule.

Even though every table was full and there were people waiting for seats, he couldn't handle the unexpected costs of buying more plates, glasses, pots and pans to keep up with the orders. And there wasn't room for more dishwashers to clean things fast enough. The stress in the kitchen caused the chef to leave, and the owner couldn't do everything himself.

You might say they had a plate-flow crisis!

The solution was to get an experienced chef who (1) was used to this sort of thing, and (2) had the courage to turn down a few bookings, and (3) would defer taking a salary for awhile until the restaurant could stabilize its cash flow.

There are other solutions to a cash flow crisis besides credit. You can come up with ways to get people to pay BEFORE you have to spend any money on delivering products. Publishers know the beauty of subscription products. The customer pays up-front, and you deliver the subscriptions over time. But you have to be sure you don't spend all the money and find yourself unable to deliver the last issues!

I'm probably boring you. You may have an attention deficit.

Friday, May 21, 2010


Is math boring? Is it drab? Does it put you to sleep? Are the questions endless and repetitive?

Why not break out of the mold and add some spice and creativity to your life? Do some mathart!

Here's my reaction to the 250 billion post yesterday:

I call it the Exclamaquest sign.

Here are a couple pieces of art from Zim Olson, who has a great math website that you should visit,

Math Art Non-Identity

Apple Vector III

You could read about math art here and here if you don't know where to start. Some people think that fractal art is what math is about, and it's pretty amazing in some respects.

But it just gives me a headache. I prefer the old-fashioned subjects, like this space scene. Is it math?

Heck, why not try a piece of math art yourself? I admit, I did this.

create avatar

Thursday, May 20, 2010

Two Hundred Fifty Billion

A few months ago I did a blog on imaginary words that convey the sense of size or value.

Here's a real phrase that seems unimaginably large.

250 billion. 
Two Hundred Fifty Billion. 
Dos Cientos Cinquenta Mil Millones. 
Or as the Greeks would say, διακόσια πενήντα δισεκατομμύρια

After yesterday's post on saving 1.5 trillion calories in our food by 2015, I came across a story about four global companies in the food business. Between them, they sold 250 billion US dollars of products. In one year!

A shorthand description for the companies is ABCD.
  • Archer Midland Daniels - $ 70 billion of grain, palm oil, ethanol
  • Bunge - $ 42 billion of soybeans, plant oils
  • Cargill - $ 116 billion of sugar, cocoa, corn, wheat, fertilizer, animal feeds
  • Louis Dreyfus - $ 20 billion of grain, cotton, orange juice, coffee, rice, sugar
Wow! That's a lot of food. It's hard to even imagine how large that number 250 billion is.

Let's compare this 250 billion to other 250 billion numbers

1. Gross sales of food by 4 big companies
2. National debt of Mexico
3. Gross National Product of Denmark
4. 2500 metric tons of gold
5. Number of hours of television watched by all Americans last year
6. Forecast for online sales of products in the USA by 2014
7. The number of microflora bacteria in only 1 gram of probiotic supplements!
8. The following is the most math-ish example of 250 billion that I have found today:

Alan Chaumont and Tom Muller checked all the prime numbers up to 250 billion and found only 21 of them are Elite Prime numbers. They said,

"A prime number p is called elite if only finitely many Fermat numbers Fm = 2^(2m) + 1 are quadratic residues of p. (huh?) Previously only the interval from 1 to 1 billion was systematically searched for elite primes and 16 such primes were found. We extended this research up to 250 billion and found five further elites, among which 1 151 139 841 is the smallest and 171 727 482 881 the largest."

Dennis Martin took their results and went on up a few hundred billions farther, but found no more elite primes.

I'm about to stop counting now. My final example?

9. The number of whirly-gig novelties created out of beer and soda cans for sale at flea markets!

Wednesday, May 19, 2010

Vow to cut 1.5 trillion calories by 2015

I saw this calorie headline today. Is is real, or meaningless? I vote for meaningless!

The article went on to say the Healthy Weight Commitment Foundation is composed of 80 companies that produce or sell us about 20-25% of the total food and beverages we consume (these are "processed" foods, not vegetables). They will reduce calories compared to what was in our foods back in 2008. Presumably they are going to reduce the calories (sugar? fat?) in their meals or drinks.

The Robert Woods Johnson Foundation will be counting those calories to make sure the reduction happens. The companies have also sponsored a kid's website EAT PLAY BURN to encourage physical activity.

In the imaginary world of slim-trim 2015 Americans, what does a 1.5 trillion calorie cut mean?

Let's analyze the numbers:

We will probably have about 330,000,000 people in the United States by 2015.

Currently we are told to eat 2000-2500 calories per day. We produce much more. Some gets wasted but much gets eaten. Let's estimate 3000 to make the numbers easier - it doesn't matter exactly in this case as we are trying to get a rough idea of the scope of the problem.

Multiply these two numbers, and you get a lot of zeros! Calculators aren't much help because their displays don't show as many numerals as we need. You can use scientific notation if you want to, but I like to see the zeros to get the real impact of the number.

330,000,000 x 3,000 = 990,000,000,000 or American eat about a trillion calories a day.

Here's how we count those zeros:

000 thousands
000,000 millions
000,000,000 billions
000,000,000,000 trillions

So the headlines mean that these companies which supply one-quarter of our calories will sell:

1 trillion x 365 x .25 = 91 trillion calories 

in a year. The promise is to cut that to 89.5 trillion calories. Will they sell us less food? That's not their idea. They will just take some calories out of what they already sell, by changing the ingredients.

What's this reduction represent as a percentage of daily intake?  About 2 percent.

You can easily reduce your own intake by 60 calories a day and achieve this reduction today! Why wait til 2015?  

But how much is 60 calories? you may ask.

Here are some examples of food items that represent 50-75 calories:
  • 1/2 banana
  • 1/2 packet of mayonnaise
  • 1 apple or orange
  • 1 slice of bacon
  • 1 slice of processed cheese
  • 1 egg
  • 4 ounces soda 
  • 2-3 packets of catsup
  • 6 french fries
  • 9 raisins
  • 10 almonds
My conclusion? It's not going to take rocket science to save us 60 calories a day. But it will be easy to eat and drink those calories back on if we're not careful ...

Tuesday, May 18, 2010

How many boxes? They weigh how much?

The continuing saga of using elementary math when we grow up ...

Some of our local customers like to pick up their Excel Math curriculum to save our 10% shipping fee. We're happy to help you if you'd like to do that.

Before you come, be sure to call and let us know so we can have the order assembled and ready to go. And just for fun, take a quick look at this 2-minute video. I admit the graphic and filmwork are not up to professional standards!

We have had moms come in to purchase a school's worth of curriculum using the PTA budget. They are always disappointed when we tell them we can't put boxes for 20 classrooms into their car.

How much could that order possibly weigh? they ask. About 25 pounds a box, I said in the video.

That's a good question for elementary math. If you've had Excel Math as a student, you would know how to solve this problem. Here's how I would do it.

How many boxes make up this order?
  • 2 for teacher editions and spare books
  • 9 sets of 22s = 18 boxes
  • 9 sets of 30s = 27 boxes
  • Total 47 boxes
How much weight does this order represent?
[I got some real weights from Dave, our supreme warehouse commander]
  • 6 Grades x 3 classes per grade = 18 classrooms x 2.5 lbs per Teacher Edition = 45 lbs
  • 3 Grades x 3 classes per grade x 22 kids per class = 9 x 56 lbs (22 set) = 504 lbs
  • 3 Grades x 3 classes per grade x 30 kids per class = 9 x 73 lbs (30 set) = 657 lbs
  • 6 Grades x 3 individual sets/grade extra (for newcomers) = 18 x 2.5 lbs = 45 lbs
  • Total weight 1251 lbs plus (if you want it on a pallet) 40 lbs = approx 1300 lbs
  • Driver and passenger (helper) at 150 lbs each = 300 lbs
    It will fit in a Ford Transit Connect van, with a maximum capacity of 1600 lbs. IF you and your helper don't have too much for lunch!

    Monday, May 17, 2010

    Ten percent with a minimum of $5

    Ten percent with a minimum of $5

    Long-time Excel Math customers know that this is our pricing for shipping our curriculum to customers in the continental United States. I should clarify - This is what we charge. But it's not exactly what the shipping costs! Alas, if only it was that easy!

    We have lots of opportunities to use math in calculating the shipping prices of our products.

    Because we publish books, the products are heavy. That means shipping is expensive, or slow, or both.

    Let's say you purchase a Teacher Edition for $29 and nothing else. You live near Boston in code 01238. We are in 92064. The book weighs about 2.5 lbs depending on the grade level.

    • We charge customers a minimum of $5 for shipping and spend $1.00 for a box or envelope
    • Fedex Ground ships in 5 business days for $7.75
    • USPS will ship by Parcel Post in 2-9 business days for $9.92
    • USPS sends Priority Mail in 2 days in a flat rate box for $10.20
    • USPS Media Mail [book rate] goes in 8 days for $3.16
    Depending on the volume we ship each day, there might also be a pick-up charge for them to send a truck to us. Which shipping approach do you suppose we choose for this order? 

    Let's look at another example. If you decide to add a student individual set of lessons for $17.00 to your order,  the total is $46 and the weight is now 5 lbs.

    • We charge the same $5 for shipping these two books but use a bigger box  
    • Fedex Ground will ship it in 5 business days for $10.54
    • USPS will ship by Parcel Post in 2-9 business days for $12.34
    • USPS sends Priority Mail in 2 days in a flat rate box for $10.20
    • USPS Media Mail [book rate] arrives in 8 days for $3.94
    Again if we chose the lowest-price shipping (Media Mail) we pay for the box. If you chose Priority Mail, USPS provides the box. If you are in a hurry, you can ask us to ship a faster method as long as you are willing to pay for the increase in shipping.

    Most of our orders go on pallets and 10% is a good approximation for our cost of trucking these pallets to schools. Depending on the carrier and the distance, the trucking companies charge us by the pallet, or per mile, or by the box, and in some cases for special handling for delivery to school sites. Don't forget, we leave the pallet behind too, and we have to buy those (used) from a supplier.

    We try to work out the most reasonable approach for each school district. For example, with our Kodiak Alaska customers we use a combination of media mail, freight barge, dog sled, etc.

    You could say that shipping is our loss-leader attention-attracting feature. Depending on where you live and which product you buy, we lose a little on one-book orders and hope to offset that by smart handling of of volume shipments.

    And if you want to save the 10%, you can come pick up the products at our warehouse. More on that tomorrow...

    Buying and shipping math curriculum is not as simple as buying a DVD and getting it 2-day Air from Amazon. If that's what you prefer, we have the Projectable CD product for $49 (minimum shipping still $5.00)

    Friday, May 14, 2010

    Eight days a week we turn it up to 11

    Today, to finish our week of talking about exploring time in math lessons, I come to the concept of exaggerating to the point of hyperbolic excess. More than once. And so on. Ditto.

    Here are some examples of what I mean:
    Talking like this really isn't math but it's human nature. 

    How do kids know the difference? How do they learn when we are being serious and when we are kidding around in our use of numbers? I'm not an expert on youthful liars but I understand they can be pretty good by the age of 4-5. Some kids might be lying by the age of 3. How do we know? Because they tell outrageous fibs without knowing how adults judge the difference between truth and lies.

    Of course that's a whole subject in itself. Let's stick to the numbers. How does anyone know if numbers are accurate?

    We have a few tests and questions we can ask ourselves:
    • is the number consistent (does it appear the same way each time we see it)
    • does a number fit a pattern (use a check digit, or have the right number of digits, etc.)
    • is the number from a reliable source (person, program, machine)
    • is the number intended to be an estimate
    • is the number precise (to a reasonable number of decimal places)
    • is there a stated degree of precision (is it meant to be accurate)
    • has the number been rounded (from another value)
    • is the number arbitrary or actual (on a scale of 1-5 how do you feel? vs here are 3 rocks)
    • is the number a repeating fraction (.66666) with no end?
    • is the number irrational and not repeating or finite (π = 3.141592 ...)
    Mathematicians use a term "confidence" or "confidence interval" to express how well they believe a given number represents the real number.  I've extracted a nice formula from Wolfram Mathworld to clearly explain this subject of confidence interval.

    Here it is:

    I'm betting you'll look at this and think "They don't learn that in Excel Math!" You're 110% spot-on.

    Thursday, May 13, 2010

    Some things just can't go any faster

    We've been talking about time and math this week. Learning about minutes and seconds, months and years, etc.

    How can you use this? I'll give you a publishing example. In a former (business) lifetime, we wrote auto repair manuals. The major book each year included about 2000 pages to cover all the material, and we had to write those pages in about 5-6 months.

    How many people did we need?  Now that's a real math question.

    We averaged, if I remember properly, about 2 hours per page. That means 4000 hours for that book. Now a single editor worked about 40 hours a week, but didn't WRITE for 40 hours. At best, we got 32 hours out of a 40-hour week.

    So 4000 ÷ 32 = 125 weeks to write the book.

    For one person. That wouldn't work because we needed this book every year. In fact, we had to do it in about 6 months, because we needed to sell it while that year's cars were still being sold.

    There are 26 weeks in half a year - in fact to give ourselves a cushion, we assumed only 25 weeks.

    125 weeks ÷ 25 = 5

    We need 5 writers and some artists and some proofreaders to get our book finished in 6 months.

    Here are "some of the boys" outside the office. I didn't work there then, I was only 13 when this photo was taken. But when we left this building in around 1980, I got the sign off the doorway. It's in my garage somewhere ... Oops I digress.

    After some practice we came up with formulas (formulae) for writing our books. We could predict how to write any manual. At least we could until they came up with new-fangled things like front-wheel-drive and electronic fuel injection. All the calculations changed.

    One thing I learned was you can't speed up each and every project. No matter what the boss says. Not by putting on more people. Not by working overtime. Some things just take time!

    For example, 1. Ladies Shopping

    2. Hoping a mountain lion will get bored and get off my porch!

    3. Having a baby!

    Wednesday, May 12, 2010

    It's about time

    Yesterday's blog post was about teaching time in math class. Today I want to share a practical application for a grown-up person using elementary math to solve a time problem.

    We have to create a slideshow for a social event. Here are the facts:
    • We have collected roughly 100 photos. Well, 104 to be exact.
    • We have a piece of music we like that runs for 20 minutes and 48 seconds.
    • We have experimented with timing and feel that 15 seconds is adequate for each slide.
    Here are the questions:

    Q1. If I set the slide show to go exactly 15 seconds per slide, will my music be long enough?

    A1. 104 x 15 = 1560 seconds. 1560 ÷ 60 = 26 minutes. The music is NOT long enough.

    Q2. I can allow the software do the work of evenly dividing the slides across the timing of the music. What will the timing be per slide?

    A2. 20 minutes 48 seconds = 1248 seconds ÷ 104 = 12 seconds per slide.

    Q3. We need to show the program 5 times. Will the laptop computer batteries last long enough or will I need the power cord? I can put the computer to sleep in-between the shows. I estimate about 30 minutes total running time for each show, between start and sleep.
      A3.  In magazine reviews, the computer seems to have "about 4 hours" of battery life.
              4 x 60 = 240 minutes. 5 x 30 = 150 minutes. 240 - 150 = 90 minutes reserve power.
              I should be able to run the slide show on the battery.

      Extra Credit Question: How many different slides are shown in the first photo at the top of this post?

      Tuesday, May 11, 2010

      Late, Later, Latest.

      Late: Overdue, tardy, behind schedule. Formerly holding a job or position. Formerly alive.

      Don't worry, I'm neither formerly employed nor formerly alive. I'm just late with the blog today.

      Late is a state of mind. I hate to be late, so I am usually early. Last week I had a conversation with my sister about this. She said something to the effect that people who are always on time are boring. I disagreed (I had to!) and said we "on-timers" are just a bit more intense, and furthermore, we "watch the time" and try to treat other people and their time with respect. I got nowhere with this argument, but then that's the nature of brother/sister relationships (perhaps family relationships in general).

      Aren't family photos fun?

      Here are some of my relatives. Some of them are late, as in formerly alive, but we still remember them.

      What does all this have to do with math?

      Math is the subject area in schools that is responsible for teaching time.

      We teach time - seconds, minutes, hours, days, etc. We explain how 12- and 24-hour clocks work. Then we teach more complicated time scenarios with examples like this:

      Q: Kathy and Mike have agreed to meet their mother for lunch at 1 pm. Kathy knew it would take her 30 minutes to drive to the restaurant. What time does she need to leave her house in order to reach the restaurant on time?

      A: 12:30 pm.

      Feel free to use any information from the first problem in the second problem:

      Q: Kathy and Mike have lunch then walk into the theater next door to see a movie. The lunch takes 1 hour and the movie runs for 2 hours and 15 minutes.  If they met for lunch at 1 pm, what is the earliest time that Kathy could expect to return to her house?

      A: Lunch = 1 hour + Movie = 2.25 hours + Driving = .5 hours = total of 3.75 hours (3 hrs 45 min)
           3:45 added to 1:00 pm = 4:45 pm

      Once we've figured out this sort of thing, then we can talk about time zones. If you want to call your mom on Mother's Day and she's in Cleveland while you are in Phoenix, you better know when to call!

      We introduce longer time intervals. We explore the calendar. So we can solve problems like this:

      Q: Kathy was born on May 15, 1950. When is her next birthday and how old will she be?

      A: May 15, 2010 (this coming Saturday) she will be __ years old. Happy Birthday Kathy!

      And finally students have the ability to tackle questions like this:

      Q:  Mike was born 15 months, 16 days and 17 minutes after Kathy. When is his next birthday, how old will he be, and what day of the week will it be?

      A: I am not telling.

      Monday, May 10, 2010

      How Many Numbers Do You Have?

      Numbers are everywhere around us. Imagine what it would be like if you didn't understand numbers! You would be lost.

      As I look around my house, the most visible appearance of numbers is on the keypads of our phones and television remotes. I decided to do a census and found lots of numbers staring back at me!

      2 remotes for Samsung Monitors
      1 remote for Sony television
      1 remote for Sony DVD player
      1 remote for stereo integrated amplifier
      1 Sony short wave radio 0-9 keypad
      3 Apple computer keyboards
      1 Dell laptop keyboard
      2 cordless telephones
      2 mobile telephones
      2 calculators
      1 desktop telephone
      1 wall-mount telephone
      1 combination pad for garage door opener
      1 keypad on microwave oven
      1 keypad on regular oven
      4 dials on stove
      1 dial on toaster
      1 dial on washing machine
      1 dial on dryer
      1 dial in refrigerator for temperature
      1 scale on outdoor thermometer
      1 dial on digital kitchen scale
      1 dial on bathroom scale
      1 number dial on wall thermostat
      1 number dial on hot water heater
      4 various scales on measuring cups
      1 scale on syringe for measuring small amounts of liquid 
      Number dials on most clock and watch dials

      I'm exhausted just looking around - there's no end to the numbers.

      Thank goodness I was taught how to work with them.
      And I can help younger people learn 2 work with numbers 2.
      So they know where and when to use 2, to, two, or too.

      Friday, May 7, 2010

      One is the Loneliest Number

      Almost breaks your heart, doesn't it.


      Standing alone. By itself. On a precarious base. Likely to fall over at any moment and become an underscore, or perhaps a dash.

      What are the possibilities for a lonely one? Is it just a simple vertical line, signifying the lowest whole number? Ignored and passed over as we and our schoolchildren strive for the highest value possible? Is it best considered only as the first of three digits in a perfect score?

      Not likely. Not in my business, which is really publishing, not just math.

      Typographers certainly give the one plenty of attention. Look at all the variations! These are all the same size, I simply changed the font. These are the versions of a one that I have on my computer.

      Can we take another look, you ask? Certainly.

      Thursday, May 6, 2010

      Counting from one, or zero?

      What makes the most sense to you? Counting up from one, or zero?

      Look how it pans out visually ... Here's counting from one, in sets of tens.

      Here's counting from zero. See the difference? Each row starts a new set, or decade of numbers.

      Does this matter? I hear:

      No it doesn't. The values are the same. Kids don't care. We all start counting at one.

      Then I hear:

      Why yes, it does. Look how much more sensible it is to consider all the Twenties together, the Thirties, etc.  These groups are like decades (sets of 10 years). We need that zero!

      We need that zero. Well some of us do, but some don't. Let's agree to disagree. Which raises a new question: Do we need zeros before the 1, 2, 3?

      We do need these zeros if we want our data to be sorted correctly on a computer. Otherwise, 5 will be sorted up into the 50's decade group rather than between 4 and 6. Oops, I mean between 04 and 06. If you've ever lost a file or folder after changing its name, you might look around. It could still be there, but placed with an unexpected set of neighbors.

      In computer (and roulette) terms, 0 and 00 are not the same. In math terms, they are the same.

      Wednesday, May 5, 2010

      The value of nothing

      Nothing. Zero. Nada. Zip. Nil. Zilch. Nought. Zippo. Null.

      Notice the Zs and Ns? What's that all about? Why are N and Z associated with the symbol or value of nothing?

      I think about it and all I come up with is a goose egg.

      Cardinal numbers are 1, 2, 3 and ordinal numbers are 1st, 2nd, 3rd.

      Do you know the ordinal form of zero?

      Zeroth, Noughth.

      There's a mouthful.

      In Excel Math, we teach kids about numbers. Zero isn't taught first, because it gets its meaning from its relationship with numbers that DO indicate value. Here it is with its friends:

      No one has really worked out the "right" way to use zero. It stands a bit alone from the rest of the numbers, as you see here.

      The British number the floors in their buildings assuming the ground floor is zero, the first floor up is one flight up the stairs, the first floor down is one flight down. The rest of us in the English-American world consider the ground floor as one, the first floor up as two, etc. Which makes more sense?

      (Personally, I think the British system works well, even after having wandered up and down corridors with my luggage a few times!)

      The development of the word seems to have followed this path:

      The idea of a placeholder (indicator of place value) was conceived by Hindu Indian astronomers, passed through to China, shared and transferred via Arabic traders. They gave it the name sifr (little circle that keeps the place). Those traders brought the concept and the word to Venice, where the Italian form of the work became zefiro.

      Zephiro was an existing term that meant west wind (zephrus in Latin and Greek; zephyr in English). Once given this mathematical use of nought, it was later abbreviated to zero. In French the word was cifre (cipher).

      There are lots of philosophical ponderings on the value of zero in our modern world. Just do a search on the term and you can read a few. This is my favorite, from Alfred North Whitehead:

      The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish.

      Tuesday, May 4, 2010

      Permutations make me lose my marbles

      Combinations and permutations are confusing concepts.

      Let's see if we can clarify the difference:
      • Combinations involve making choices from multiple options (so do permutations)
      • Combinations don't care in what order the selections are made (permutations do care)
      • Combinations could include things like scoops of ice cream on a cone (any order is ok)
      • Permutations exist when cones and scoops are chosen (cone is chosen first or sticky hands...)

      Now I'm not sure that's a good example! If you are selecting ice cream scoops you don't normally have to worry about their order, or if they are co-mingled by melting. Let me try another one.

      If you are selecting numbers and letters for a password, you can select them in any order but you have to be able to re-enter them in the SAME order or the password won't work.

      This is an "ordered combination" of letters and numbers. A permutation.

      Sometimes a password must begin or end with a number, and no repeating (33333) or sequential (12345) combinations are allowed. For security reasons, the software might also exclude your name, or your birthdate, because they are too easily guessed by a person who wants to hack into your information. These are permutations limited by externally-imposed rules.

      There are lots of formulas and strategies for calculating combinations and permutations. Most of them exceed the limits of elementary math. We want to play with the concept, understand the distinctions, and happily getting the answer is key, not extrapolating general rules from a single example.

      In Excel Math, most of the work is done by simply listing the possibilities, like this:

      We start with the question and select the first letter of each color R, B, Y, G, W. If you have brown and black and blue you need another key letter, but any letter will do. Then we put the letters together in a string like this:

      Remember that order matters when you are counting permutations. So Red-Yellow is NOT the same as Yellow-Red. If we didn't care about order we would have asked how many combinations we could make.

      I like to line up my pairs in columns so I can make sure I get all the choices and no duplicates. Yes, there are formulas for calculating the answer, but getting the answer is key in today's blog, not extrapolating general rules from a single example ... by the way, if ice cream isn't your thing, and scarves are unnecessary in your climate, you can use marbles for your permutations.

      Monday, May 3, 2010

      It's impossible to say at this time

      I suppose the title of this posting must be explained. We tend to think that math can give us concrete, specific answers to well-asked questions. Sometimes it can, but not always.

      Here's an example - how long is the coastline nearest your house? Start at one border of your state or country, and go across (up/down) to the other border. What is the length?

      You can't say. It depends. It depends on the tide. It depends on the length of your ruler. It depends on the size of the grains of sand or rocks you chose to include or exclude. It depends on how you treat rivers and estuaries and other coastal features. It depends on your definition of littoral.

      Take a pair of dividers and step your way along a map of the coastline (as this man is doing in the etching from the Gutenberg library). You will find that the length of the coast is inversely related to the gap between the pointers of your dividers. To be more specific, as the distance between the pointers approaches zero, the length of your coastline approaches infinity. L.F. Richardson was the first to clearly state this problem - about 60 years ago.

      How can this be? Can reality be changed? Do the dimensions really shift as we measure? No. Reality isn't changing, only the way that we measure it!

      Here are a couple maps of England, Scotland and Wales that I found in Wikimedia commons. The first is measured with a 50km ruler and the second with a 100km ruler.  I've counted the segments to save you the trouble.

      Measuring the coastline with a 50km ruler gives an estimated coastline of 3425km, while measuring with a 100km ruler results in a total of 2750km.

      How can you use this in real life? I suppose it could be applicable to waistlines too.

      If someone rudely asks your waist size, you might say "It's impossible to say at this time ..." and go on to explain the coastline paradox.