Additional Math Pages & Resources

Monday, November 30, 2009

Divide evenly

Mathematics sometimes has a way of missing the point!  Take the phrase "divide evenly."

In math it means when you divide two numbers, there is no remainder or there is a remainder of zero.

That leads us to the term Factors. Factors are all the integers that divide evenly into another integer (whole number). They are "what you get" when you divide.

The Greatest Common Factor is the largest integer that will divide evenly into any two or more integers. For example, let's take 24 and 30. What's the largest number that can be divided evenly into both of these numbers?

1   yes, 1 always works as a factor
2   (24÷2=12, 30÷2=15)
3   (24÷3=8, 30÷3=10)
4   (24÷4=6, 30÷4 doesn't work)
5   (24÷5 doesn't work, 30÷5=6)
6   (24÷6=4, 30÷6=5)
7   (24÷7 doesn't work, 30÷7 doesn't work)
8   (24÷8=3, 30÷8 doesn't work)
9   (24÷9 doesn't work, 30÷9 doesn't work)
10 (24÷10 doesn't work, 30÷10=3)
11 (24÷11 doesn't work, 30÷11 doesn't work)
12 (24÷12=2, 30÷12 doesn't work) this is the greatest factor of 24
13 (24÷13 doesn't work, 30÷13 doesn't work)
14 (24÷14 doesn't work, 30÷14 doesn't work)
15 (24÷15 doesn't work, 30÷15=2) this is the greatest factor of 30

The greatest common factor is of 24 and 30 is 6. It goes into 24 four times, and into 30 five times.

A related concept is the the Least Common Multiple, or the smallest number into which two numbers will divide evenly. The least common multiple of 24 and 30 is a big number. Let's see if we can find it:

48   (2x24, not a multiple of 30)
60   (not a multiple of 24, 2x30)
72   (3x24, not a multiple of 30)
90   (not a multiple of 24, 3x30)
96   (4x24, not a multiple of 30)
120 (5x24, 4x30)

The least common multiple of 24 and 30 is 120. It's 5 times 24 and 4 times 30.

You see, in the pursuit of mathematics we have completely missed the entire point of this blog. After Thanksgiving, we had some apple pie left over. As I was cutting the remaining (large) portion of the pie into two pieces, my wife said Be sure you divide evenly. As if I might do anything else!

If trust and belief in the innate goodness of men does not exist at your house either, you might investigate some of these gadgets that claim they will improve the even division of baked goodies.

This one works if you are starting with a complete pie, and marks/slices it into even pieces. How convenient! You can get others for 6, 8, 10, 12, etc. Or you could make your own if there are 7 in your family...

In contrast, the following device starts with the assumption that any pie-slicing person would be cheating on the slicing. To foil him you bake a rectangular pie (and get more in the oven!) using a dish marked with clever reference lines.

The cut pieces (a) look almost like they came out of a round pie dish and (b) are evenly-sized. This took some serious thinking on the part of its inventor. Go here to buy one if you are impressed.

Apparently I am not the only person who's been accused of not dividing equally!

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Wednesday, November 25, 2009

Thanksgiving Math

Happy Thanksgiving!

Let's have a quick TG math lesson, okay?

1. We did not invent Thanksgiving, nor are we the only ones to celebrate. The first North American thanksgivings known to historians are:

  • 1565 by Spaniard Pedro Meñendez in St. Augustine, Florida in thanks for making a voyage safely
  • 1578 by Englishman Martin Frobisher in Newfoundland giving thanks for making it back in one piece from his Northwest Passage expedition.
  • 1598 by Spaniard Don Juan de Orñate in El Elizario, near El Paso, Texas
  • 1604 by Frenchman Samuel Champlain and French settlers along with First Nations (Indians) participants
  • 1621 by English settlers at Plymouth plantation with Indians
  • 1763 by Canadian citizens of Halifax after New France was defeated and its land ceded to the British Canadians
  • 1777 by American citizens led by George Washington after driving out the British troops
  • 1863 by Americans when led by Abraham Lincoln at the end of the civil war
  • 1942 by Americans when Congress finally set a specific date of 4th Thursday in November
  • 1957 by Canadians when Parliament set the date on the second Monday in October 

2. But we shouldn't take too much credit for the idea of giving thanks for safety in the wilderness. More than two thousand years ago, in ~ 420 BC, the Hebrew people celebrated their return from exile and the start of rebuilding the temple. Here is the record from Ezra 3:

When the builders laid the foundation of the temple of the LORD,
priests in their vestments and with trumpets, and Levites with cymbals,
all took their places to praise the LORD, 
as prescribed by David king of Israel.

With praise and thanksgiving they sang to the LORD :
 "He is good; his love to Israel endures forever."
And all the people gave a great shout of praise to the LORD,
because the foundation of the house of the LORD was laid.

But many of the older priests and Levites and family heads,
who had seen the former temple [destroyed],
wept aloud when they saw the foundation of this temple being laid,
while many others shouted for joy.

3. Being thankful is more than > and not equal to ( ≠ ) family + turkey + parade + football + snow. Thankfully!

4. We can be thankful for all we are, for all we have, and for what we are not and do not have.

5. We at AnsMar Publishers Thank You for being part of the Excel Math family.

Brad, Janice, Bob
Jim, Becky, Dave
Carmen, Mike, Darcie

What adults should know about Add

I'm not talking about A D D, Attention Deficit Disorder, but the verb add, as in to combine numbers.

What about this most primitive math operation? Is it too simple for us, as grown-ups? Do we know everything there is to know about addition?

I hope not. I think not. How about a quiz today, while we all wait for Thanksgiving to arrive!

First, a bit of review:

Word/Concept Meaning
Add; Addition Process of joining or combining two or more numbers
Count; Counting Repeatedly adding one to a set of numbers; ascending
Plus Added to; this symbol "+"; slightly above normal (B+)
Sum; Summation Calculating the total of two or more numbers
Multiplication; Multiply Repeated addition
Subtraction; Subtract The inverse of addition; taking one number away from another number

Math words in English are derived from Latin roots.

When adding, the number you start with was first known as the augend. (We use this same root word when we talk about somethings being augmented...)

The number being added to it was first called the addend; nowadays we use addend for all numbers being added.

The word augend means "thing to be increased"
The word addend means "thing to be added"
The word add means "give to"
The word sum means "the highest" which leads to an interesting fact:

Did you know?

People used to add columns of numbers upwards? Like this:

Did you know addition can be very tricky?

Measurements in different units are tough, as in (3 feet + 4 yards) = 180 inches

Did you know there are things you cannot add?

Different kinds of objects (3 squares plus 4 rectangles) ≠ 7 octagons

Did you know you can be asked to consider two opposite operations at once?

The speed of the car is controlled at 45 mph ±5 mph.

Did you know the secrets of adding Roman numerals?

First, change negatives to positives ( IX means -1 and 10, so you make it V I I I I instead )
Rearrange all the letters in order of highest value to lowest, then simplify the total.
(XIII + III + L + VII  ) = L + X + V + I I I I I I I I = LXXIII

Did you know you can "show" addition on a number line?

Here's an example.

For more on number lines, read our May 2 bog post, "Using Numer Tines to Add and Subtract" and download a free math worksheet.

Did you know about infix notation? 

We indicate addition with a plus sign between two numbers (2+2). This is called infix notation.

Did you know addition can be performed in various ways on a calculator or in programs?

Infix notation or direct algebraic notation is when you press 2 + 2 + 2 = and you see 6
Prefix or Polish notation is when you press +  then 2 ↵ 2 ↵ 2 ↵ and the display shows 6
Postfix or Reverse Polish notation is when you press 2 ↵ 2 ↵2  + and the display shows  6

Did you know about operands and operators? 

Addition is an operation (something that is performed). The two (or more) addends are also known as  operands (the things upon which a math operation is performed).

Did you know the four special properties of addition? 
  • commutative - the order of the addends doesn't make any difference in the sum. 2+3=3+2
  • associative - when three or more numbers are added, grouping them in different ways doesn't make any difference in the sum (2+3)+4 = 2+(3+4)
  • additive identity - the sum of zero and a number is that number. 5+0=5
  • distributive property - the sum of two numbers when multiplied by a third is the same as each of the two numbers multiplied independently and then added (3+4)*5=35 and (3*5)+(4*5)=35

Name 2 things you have learned about addition today:


Write a sentence using some of the words you have learned today:

In summary, I personally can count at least two positive things: I have added "making a tabular table" to my web blogging skills, thus augmenting the value of my posts. Plus, I learned about infix notation.

Tuesday, November 24, 2009

Soft, hard or in-between?

Choose your favorite pencil. Unless you are taking an official test, it doesn't have to be a #2 pencil.

There are at least 20 different pencil lead options to choose from. Most but not all pencil manufacturers adhere to this scale.

As with any industry, there are folks with specific needs. Artists tend to favor softer pencils, and architects or engineers lean towards the harder ones.

Sadly, pencil hardness numbers from one manufacturer do not directly correspond to those from another supplier.

Pencil leads are typically a mix of graphite and clay, but some pencils contain carbon, charcoal, wax, grease or other materials.

In addition to the choice of hardness, we have lots of other pencil variables to consider - color, length, shape, size, etc.


There are wood pencils, plastic pencils, mechanical pencils and so on. And, I discovered, a number of people who are very excited about pencils. Here's a site I liked a lot.

If you use a mechanical pencil, then you have to know what diameter your pencil uses, or the leads won't fit. Here are a few common sizes (all measured in millimeters; I wonder why? German?):

0.5 (writing size)


1. Chew on them

2. Give your ear something useful to do

3. Test the hardness of a coat of paint or varnish. You simply make a mark on the finish with successively harder pencils, until you find one that scratches the finish rather than marks on it.  TEST

4. Hold your clothes on the line. Here's a peg pencil, from designer Yuta Watanabe.

5. Prop up your iPhone on a pencil stand and watch the President give a speech.

6. Carry your annual calendar around on it. This is actually patented!


Erasers are made of rubber, vinyl, plastic, gum, etc. They work because the eraser is stickier than the paper, and the graphite would rather jump onto the eraser than stay on the paper. In the process, some of the eraser wears off, and leaves little flakes or pieces.

Some erasers are so abrasive that they wear out the paper - you have to be careful or you will have a hole rather than a clean spot!

The most popular erasers in the US have often been pink in color, but I started using this kind in a technical publishing job, and since then have favored it over the pink ones.

Here's a font called Eraser Dust. If you like the pencil look, you can download it for free.

No, we are not going to use this font on our Lesson Sheets ... you wouldn't be able to tell the kids' writing from our printing.

I like these stealth pencils shown on the PencilTalk blog!

Monday, November 23, 2009

The MOST essential math tool

No, not a calculator or spreadsheet! It's a pencil.

This portable, light-weight, battery-free tool can draw a line about 35 miles long or write a full year's worth of Excel Math answers and can delete its own errors.
I noticed lots of pencils when I posted about putting Excel Math into electronic form. In virtually every video clip we have of students doing math, there are pencils. No one does math with a pen. It's assumed that you will make a few mistakes and have to erase what you did the first time. Even for grown-ups.

Maybe I should have said the most important tool is an eraser! But the two go together in most cases.

Why do we assume that we'll make mistakes and want to erase and fix them? I don't know.  But it's not just math where this attitude prevails.
My wife (a teacher) is quite adamant about cleaning up and correcting her work, especially when doing crossword puzzles.

Pencils seem very simple. Wood wrapped around some grey stuff (lead?) that makes a mark on paper. But they're pretty complex.

The pencil was developed and refined in Germany, over 400 years ago. Friedrich Staedtler was making pencils in Nuremburg in 1662. Hymen Lipman patented the world's first pencil with an attached eraser 150 years ago. (Even our great-great-great-grandfathers and mothers made mistakes.)

Most pencils are made from wood with a graphite core. If you have a few minutes, you can watch pencils being made.

Just this month Staedtler started selling a new wood plastic composite pencil. It's made from wood, plastic and graphite pellets, mixed into a dough and extruded out through a die, like a piece of pasta. They call it the Wopex.

Pencils are made all over the world, but there are now only two companies left in the United States that produce pencils with US wood in US factories. Does the fact that those companies are headed by people named Weissenborn and Berolzheimer indicate a German connection? Yah.

You can read more about them here, on the Timberlines Blog.

By the way, pencils don't have lead in them - they are filled with graphite and clay. And good ideas, just waiting to be written down.

Pencilman and Writer's Block

Here's a fun video about those ideas.

Most of us just know about #2 pencils - what happened to #1 and #3? In a future blog we'll tackle how pencils are graded and rated.

Friday, November 20, 2009

The Answer is Blowin' in the Wind

Can you measure something that is always moving, never constant in speed or direction, and invisible?
That's the challenge of measuring the wind.

Scientists use an instrument called an anemometer. In 1450 Leon Battista Alberti, an Italian architect, mounted a hinged Plate up on a bracket and watched to see how far the wind would lift the plate. Over the next five centuries various types of moving blades were used.

The Robinson Cup Anemometer was developed in 1846. It had 4 cups that caught the wind and spun. Modern designs normally have 3 cups which makes the device more accurate. The cups move at 2-3 times the speed of the wind, depending upon the detailed design of the device. The Windmill Anemometer looks like its name, relying on a propeller spinning in the wind.

The Lind Tube Anemometer was completely different - a bent tube with liquid inside. The open end is pointed towards a wind source, and the other end indicates wind speed on a scale. This was refined by James Lind in 1775. Similar tube designs today are used because there are no moving parts out in the weather. The indicating end can move a needle that is safely inside a comfortable office.

Here are a whole bunch more classic anemometer designs!

There are other designs on the market now, using electronic sensors that derive the wind data from sonic or laser measuring. You can even buy a hand-held anemometer for less than $50. A Christmas present, perhaps?

Plate and tube anemometers measure the pressure of the wind - its ability to PUSH.
Cup and windmill versions measure the speed of the wind - how FAST it is moving.

What's the difference?

Pressure occurs when a force is applied on a given surface area. The symbol of pressure is P.
Speed describes the rate at which an object travels a given distance. You can use the symbol S.

The trouble with using speed (or a related term, velocity) is that there is no visible object moving.

The wind is not like a car driving down the road. The wind is moving but it's still here, if you know what I mean. Plus, its direction and speed are always changing, so we tend to talk about the average wind speed, or force.

A sudden increase in the speed of the wind can be called a gust. Our government weather service decided a gust must be [going] at least 16 knots per hour, and more than 9 knots faster than the lull, or weaker winds. And to be a gust, the burst of wind must last less than 20 seconds.

Speaking of direction, it's normal practice to describe where the wind is coming FROM rather than where the wind is headed. There are 32 points on this Compass Rose, and the directions are shown in colored letters. You would say Southeast by South, or North by Northeast, and so on.

Which raises the question: Do we refer to true North, or magnetic North when we describe the wind? In San Diego, the direction is off by about 10°. This is called magnetic variation or declination. I think on land we tend to ignore the magnetic variation, but if navigating at sea, it's important.

A British mariner and scientist named Beaufort refined a scale that described the wind's effects (what it is doing to the things we can see) and proposed that as the best means for talking about the wind. Dozens of books have been written on this subject, and hundreds of scientific papers. We'll tackle it another time.

Here's this morning's weather status map from the US weather service. There are lots of wind indications and warnings along the Pacific Coast. Notice the time is given in UTC, which is 8 hours later than our time in California.

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Thursday, November 19, 2009

Virtually everything is virtual, why not Excel Math?

People often ask me why we don't do our math curriculum electronically, so kids could just log on, solve their homework problems, and submit them.

Ah, where to start.

1. Technical Difficulties
Here is a typical page from Excel Math. I have copied and pasted into the Blogger Editor, and turned it blue for your viewing pleasure. What do you think?

I65H177F23E12G8,285D5,743244840168654321115861, 3 8 2  _________
2, 5 4 1  _________
3, 4 8 8  _________3.  3 x 5 =
4.  15 ÷ 3 =
5.  15 ÷ 5 =
6.  5 + 3 =5 4      5 9      3 94  x  (3 x 2) =(5 3, 5 1, 4 9, 4 7)(3 9, 4 5, 5 1, 5 7)    72 4 0x    45 0 6Fill in the chart.Spider  Legs82264 6
+   2 1 21,0 0 0
3,0 0 0 3,0 0 0
2 0 5 + 1,0 8 0
8,2 8 52 0 x   1 4 03 5
6 + 2 4
6 54 x 6 = 2 41,0 8 02 0 53,0 0 03,0 0 01,0 0 01 4 0 miles5 books3 0 ÷ 6 = 53 224 5
68351 5671 2 - 5 = 78 eggs4 0 ÷ 5 = 81,6 8 05 95 4
- 4 5  94 5- 3 92,0 0 02,0 2 43 9
2,0 2 4 1,6 8 0 + 2,0 0 0 5,7 4 38 7 + 8 2 33 2 + 1 4 0 1 7 7

Not so good, is it? The flow leaves something to be desired.

We can't just paste into a page as if it were a novel. The layout or page composition is critical to conveying the meaning. The spatial relationships of the numbers cannot easily be represented in typical web browsers.

2. Size and orientation.
In effect, we (like many book publishers) sell real estate, not words. Our products are built the way they are due to the availability of paper, the size of kids' desks, the budgets of schools, etc. If we go to a different kind of paper, we have to re-compose everything. We did this to Kindergarten last year.

The same thing happens when you shift to the screen. Here is the same page, converted to a PNG file and positioned in your browser. I went ahead and turned on the answers for you.

It's too hard to read, you say? True. [Click the image for a larger view] If you have a giant screen like I do, it's not hard to read.

Of course if you want to see it on your iPhone it's a bit smaller. But still readable!

So we may need to put less on each screen than we do on each page, and then you can easily see and solve the problems.

That will give us a few more screenfuls. How many more? The overall Excel Math 7-grade product line includes about 1100 lessons plus tests and some other stuff, for a grand total (not counting Teacher Editions) of 2600 pages. From experience in doing a few of these page conversions, I think that this would result in approximately 55,000 screenfuls.

3. Price and Convenience
We are a low-price alternative to the big textbooks. We know the Lesson Sheets are used in a variety of math contexts - school classrooms, homes, back seat in the SUV, on the bleacher bench at soccer practice, etc.

It's not just the price of the product that our users think about, but the price of convenience, equipment and training and support.

4. Tactile Considerations
Not to be confused with tactical. We think kids need to feel numbers, write them, count things, manipulate stuff as they develop their sense of mathematics.

We have lots of other reasons too, but this is enough for now. Lots of other publishers do have math online, but we prefer ours on paper. Personally, I'd prefer virtual dirty dishes to virtual math.

Wednesday, November 18, 2009

EZ come, EZ go

What is unusual about this paragraph? It looks ordinary. Study it - nothing is wrong with it. But it is unusual. Why? What suspicions form in your mind? What's odd about this string of words? Anything? If you work hard at it, a solution will dawn on you.

Give up? Or did you see the answer? If you didn't, perhaps this chart will give you a clue:

Letters in most languages are not evenly distributed throughout all the words, nor are they evenly used in a sentence or paragraph. If you study enough samples, you'll find some letters appear with much greater frequency than others. The English letter distribution appears in the table above.

Why does this matter?

It matters if you are a cryptographer - a person who does hidden or secret writing - a person who creates secret codes (or breaks or deciphers those codes). The study of the particular characteristics of your own language's frequency data, letter combinations, common patterns, etc. is known as cryptolinguistics.

This used to be the exclusive interest of governments (one against another) or subversive groups (against a government) but now is much more of a daily interest to citizens. We see it in two senses:
  • we want to protect our online transactions and privacy
  • we don't interference with things like watching movies or listening to music
That means we want our transactions to be protected, but we don't want to be hindered by Digital Rights Management (DRM) in our computers or televisions.

Here's one of 8 flow diagrams illustrating a logic path used to encrypt emails and emailed files.

The various symbols indicate a decision and action that you take depending upon how the data responds to your query.

When I was growing up, there were games that would help you learn to be better at creating and breaking codes. Scrabble™ of course is one of the games that helps you to understand how the English language works and it gives you an incentive to put together complicated words using letters that appear infrequently.

If you are good in math, you have excellent language skills, love Scrabble™, and tend to think in a logical manner, you might be a good cryptographer.

If you are like most people, you won't have a clue about decoding. Unless you have a secret decoder ring.

Secret codes are used all around you. The antique dealer marks items with both his cost, and a retail price. The cost is in code.

Most Excel Math™ products have a secret code on the back. Here's the code on the back of the 4th grade Teacher Edition on my desk:  9071809. Even though I am the editor, I can never remember how to decipher this code. It's put there by our printer. Can you do it?

Here is the answer to the problem posed in the first paragraph. Can you decipher it?
Bogstavet e vises ikke i første afsnit.

Tuesday, November 17, 2009

That one is too thick, the other is too sweet, but this is just right

Have you ever wondered how we decide what is the proper thickness, richness, viscosity or mouth-feel for certain beverages? Remember the old days when we made our own drinks from concentrate?

I recall being baffled by the variation in the instructions:

Orange juice - mix with 3 cans of cold water
Lemonade - add 4 1/3 cans of cold water (one-third!)
Hawaiian Punch - mix in a 5-to-1 ratio with cold water

(I once drank a whole bottle of Hawaiian Punch concentrate! I can't stand the sight of it now.)

Why is there a difference in how much water you add to reconstitute juice?

It turns out to be a complicated question related to the viscosity of frozen concentrate, the freezing point of fruit juice from which most of the water has been removed, price of the juice, the amount of sugar in the mix, etc.

Freshly squeezed juices are evaporated in a vacuum chamber before being frozen. The process reduces essences and oils (readily noticed by our taste buds). When the juice is reconstituted, it may be refreshed with those essences and oils. Other things are added, such as citrus pulp, calcium and vitamins. Water is mixed back in until frozen orange concentrate is about three times strength of fresh juice. Lemonade and Limeade are slightly different because they need extra sugar to be palatable.

If the subject of fruit juice intrigues you, and you have enough math under your belt, you could do a research paper like this one, which investigated pomegranate juice concentrate:

Fourier transform infrared spectroscopy and chemometric techniques were used to detect adulteration of pomegranate juice concentrate (PJC) with grape juice (GJC). The differences between PJC and GJC infrared spectra occurred in the 1780–1685 mm region. Analysis of the spectra was used to: (1) differentiate pure PJC from GJC and (2) classify adulterated (containing 2–14% vol/vol GJC) and pure PJC samples. Two components explained 99% of the variability in each application. Partial least square analysis of spectra could also predict %  acidity and solids in PJC with correlation coefficients of 0.9114 and 0.9916, respectively. Conclusion? FTIR and chemometrics provide a useful approach for authenticating pomegranate juice concentrate.

In other words, it's possible to identify and catch a supplier of concentrate who is "stretching it" with cheaper grape juice. These photos show unadulterated pomegranate juice, straight out of the juicer!

ISO Analytical is a small company with 6 employees. They can tell you if your juice comes from concentrate or is fresh from the fruit; or if it has been artificially sweetened. Often your taste buds make you suspicious, but because taste is so subjective, it's hard for taste buds to convince a jury!

CONCENTRATING: Freshly-squeezed juice commands higher status and price than juice from concentrate. Most fruit juices are made from concentrate, which involves removing water from the juice prior to shipment. On arrival in the consumer region, concentrate is diluted with local water to its original strength. Juice made from concentrate (but claimed to be freshly squeezed) can easily be detected by isotope analysis.

EXTENDING: There are economic incentives to adulterate apple or orange juice with less expensive sugar solutions, especially when a poor harvests result in a fruit shortfall. The simplest fraudulent method of extending fruit juice is to add inexpensive sugars and dilute with water to rebalance the sweetness. Luckily, the addition of inexpensive corn syrup or cane sugar to apple or orange juice can be detected by carbon-13 analysis.

ADULTERATION: Maple Syrup is the concentrated sap of certain species of maple tree. Pure maple syrup is a traditional sweetener, renowned for its unique taste and flavour. Maple syrup can be adulterated by adding cane sugar - the taste of a little cane sugar or corn syrup is virtually undetectable. The temptation to fraud can be strong, but the carbon-13 signatures of corn syrup (-11.29 ‰) and cane sugar (-11.85 ‰) are very different from maple syrup (-24.27 ‰). Thus adulterated maple syrup is easily detected with carbon-13 analysis.

This is a useful application of mathematics!

Monday, November 16, 2009

Just the facts, ma'am, just the facts

Some of us have an obsession with the basic facts.
  • Tell me what's in this food - give me the ingredient list PLEASE.
  • I don't want any sodium laurel sulfate in my shampoo, thank you very much.
  • I'd prefer the Insider Edition of the nightly news, if you don't mind.
  • Give me the raw data, I'll be the one to judge whether I do X or Y with it.
And so on. But what "facts" really make sense? Do numbers always help us? Not always...

This morning my wife left the house for school. She reappeared 2 minutes later.

The Volvo says Low Coolant. Stop Engine. So I stopped the engine, what do I do next?

This is when you give her the keys to your car along with a kiss, and say thanks!

She did what the car asked! Notice the car didn't provide many facts. Just advice. Since it was cold outside, the car thermometer said 42∘F, and the engine had just been started, I was pretty sure it hadn't overheated.

If the car was just giving me facts, it might have said this:

Engine Coolant Level is 6.2 Liters

Does that help anyone? It wouldn't help me or my wife, even though now (after looking it up) I know the capacity is 7.2 liters. I don't know if being down a liter would matter. What we really want to know is what to do with the facts. Here are some symbols with words. These also interpret the facts:

Notice these thermometers show engine temperature. They don't tell me if the engine coolant is low or not. They don't even really tell me coolant temperature, or make a prediction. They just report a range. I created some more informative symbols which are shown below. Notice they have temperatures too.

The pictures above are theoretical. Here are some real indicators:

The instrument cluster picture shows me that the car is at normal operating temperature. The sub-dial below the tachometer displays engine temperature.

The pictures on the right show two possible displays for a Volvo. Notice the left one actually shows temperature in degrees. However, there are some issues. It is missing the red zone, which to most people would indicate overheating. And 300° F seems a bit high, since boiling is 212° F.  This dial could only work in the US, because the rest of the world shows temperature in Celsius. The right dial is more universally applicable.

Just as an aside,  20 years ago I was on a committee of ISO, the International Standards Organization. We were charged with deciding on the information and warning symbols that would appear on all vehicle dashboards. Here are a few of those warning symbols.

These symbols do not give the driver facts, they simply indicate something is happening in the region of the car that involves oil, or temperature, or electricity. The red implies something WRONG is happening. Back then I argued that an oil can with a drop of oil showing means virtually nothing! Given the computer processing power our cars have, we could easily display:

The oil level is too low. If you don't stop now it will cost you $2500 for repairs!

I believe that would catch most drivers' attention. And they would stop. But Volvo chose a simpler approach that worked perfectly in our case. It gave a clear message before any damage was done.

Low Coolant. Stop Engine.  

What is the morale of the story? When driving maybe I don't want math. Just keep me and my car safe. 


I added a liter of water, for the first time in 50,000 miles. The message went away and I drove to work.

Friday, November 13, 2009


Today we look at the Nation's Report Card. Sheesh. I didn't even know we got a report card.

Who are they going to mail it to, George Washington?

Ha Ha. Sorry, I momentarily forgot that the math joke blog was yesterday.

The Report Card is a result of our our taxpayer dollars at work within the National Center for Education Statistics, a division of the US Department of Education Institute of Education Sciences.

Dr. Stuart Kerachsky is the Acting Commissioner of the agency. He has a PhD in economics and over 30 years of business and education experience. He oversees people who track the condition and trends of education in the US and other countries.

They tell us this fall we are spending $543 billion on 50 million students at 99,000 public schools, with another 6 million (11%) in private schools and 1.5 million (3%) being homeschooled.

  • One in seven students wears a school uniform.
  • One in twelve students leaves school before graduation.
  • One in twelve students in public schools are classified as having a disability
  • Kids are 50 times safer from violence when they are at school than when they are away.
  • 55% of students are transported to school, at a cost of about $750 per year.
  • Given 3.2 million teachers, we have 16 students per full time teacher.
I'll spare you the detailed numbers, but overall, the report card shows that our kids are doing about the same as they were 35 years ago (when many of us old codgers were in school). Average reading and math scores in 2008 were not measurably different from the scores in 1971.

Here's a typical math question for a 4th grader; 56% answered incorrectly!

You need one piece labeled X, one piece labeled T, and one piece labeled R to answer this question. Which of the pieces has an angle greater than a right angle?

1. Only X
2. Only R
3. Only T
4. Both R and T

OK, let's go on to another one.

Kylena made a design from 3 pieces and called it a Shy Dog. Each Shy Dog used 1 T piece and 2 X pieces.

How many of each piece would she need to make 26 Shy Dog designs?
If she only had 11 T pieces and 15 X pieces, how many Shy Dogs could she make? 
Explain and show your process of finding out the possible number.

Only 10% satisfactorily answered this question... did you?

Thursday, November 12, 2009

Figures don't lie but they can tell jokes

Numbers (math) are a language (a means of communication) we use to measure, describe and categorize the world around us. When I was in school many years ago, Marshall McLuhan coined this phrase:

The map is not the territory.

He meant that a map gives you an idea of how the terrain, streets, countryside, etc. might look, but a map is NOT the real place.

Likewise, I would suggest that numbers are not reality, they are a way to represent the things we see, but in and of themselves numbers are NOT real. They don't lie or tell the truth. They are not animate objects with a sense of morality. We can use them anyway we wish. And we do.

Numbers can be used in a humorous way. Here are some math jokes that might get a laugh from you elementary school mathematicians:

Teacher: How much is half of 8?
Boy in math class: Half down or half across?

Teacher: Down or across? What do you mean?

Boy: Half down gives you 3 and half across is zero.

Why is the number 10 afraid of the number 7?  Because 7 ate nine and 10 is next! 

What does the zero say to the eight? Nice belt!

What did the 1 say to the 7? Nice visor!

Ok, enough with the math clothing jokes. How about geometry? 

What do you get when two geometric figures collide at high speed? A rectangle.

Now onto teaching:

What did the Excel Math Lesson Sheet say to the textbook?  Leave me alone, I've got enough problems of my own.
There are three kinds of math teachers in the world. Those who can count, and those who can't.

Ha ha.

What's the best time to go to the dentist? Tooth-hurty

What did the Five say to the Two about the Vee?   I'd like you to meet my cousin from Rome.

I'd stop at this, but I just found a website dedicated to math stories and jokes used in the Simpson's TV show. You might enjoy it.


Wednesday, November 11, 2009

Happy Birthday, I guess

Happy Birthday!

Did you ever wonder about birthdays? I think about things like this:
  • Are people born on all days of the year? 
  • Or is there a day when it never happens?  
  • Are equal numbers of people born every day of the year?
How could we find out?

One way is to ask. That should be reliable, but there is some chance that a few people do not know their exact birthday, and that others might lie. But who could ask enough people, and expect to get an honest answer?
  1. Certain agencies of the government could ask (military, census, DMV, IRS, passport, etc.)
  2. Some companies, such as life insurance, could ask.
This guy who works for an insurance company analyzed over 400,000 birthday records.

Days of the Year

He found that people ARE born every day, there is NO day when women stop having babies, and NOT all days have an even number of births.

Days in the Week
There are two days of the week with fewer births - that's right, Saturday and Sunday. This is probably due to doctors taking those days off and not scheduling Cesarean-section births on those days. C-Sections are about 15% of the total births nowadays.

Months of the Year

In the United States, some months (July, August, September) have more births, and some have fewer (February, April) than expected. This is usually explained by the fact that people have more time on their hands during the winter months and are more likely to be conceiving children than during the summer months.


There are some famous birthday questions, such as how many people have to come into a room before two of them have the same birthday. If that sort of thing is interesting, here are some equations that attempt to simulate the problem. Let me know if you come to a conclusion!

I think this is beyond even my math skills. But the answer is around 23.

Here are some more of my birthday questions:
  • Why should we even care about our birthdays and have parties for ourselves? 
  • Why have parties for young children who don't know they are one year old?
  • Do you have to invite the whole class if you invite anyone from your class?

Happy Birthday from Scrooge 
 I'm perfectly happy with a bar or bat mitzvah, or a Quinceañera, or other rites of passage. It's this annual business that bugs me.

Here's a Happy Birthday present for you:

It says 22 people might be enough to have 2 with the same birthday, but 23 definitely works.

Tuesday, November 10, 2009

Guess What

Guess what? The new Blogger Editor Interface (August 2011) tells me that I had one post long ago (Nov 2009) that never went from draft to final. That means my count is off by one!

Not an Excel Math student. Yet.

This is intended to bump the count up to where it should have been all along!

What do you really want to know?

Asking a simple question and getting a complicated answer

Calculating fuel consumption is simple math. However, we often say gas mileage, or fuel economy, when we really mean fuel consumption. As when we ask, How much fuel does this thing burn?

In the US we calculate consumption in units called miles per gallon.

What was my consumption on my last fill-up? My tankful was 15.8 gallons. We traveled 330 miles. I estimate 16 gallons, 320 miles - that's 20 mpg.


Answering a simple question in a different way

In many countries using the metric system, the calculation is liters per 100 km.

What was my consumption on my last fill-up on vacation in Europe? A tankful of 49 liters took us 990 km. Or say 50 liters for 1000 km = 5 liters per 100 km.

Comparing results

How do these compare? 990 km x .62 = 614 miles. 49 liters = 13 gallons. That's around 47 mpg.

Asking a completely different question but getting the same old answer

Now suppose you ask, How hard is the wind blowing? And I answer, Weatherman says gusts of 8 knots (nautical miles per hour).  You reply, Does that mean my sailboat will go 8 knots?

NO. The speed of the wind varies all the time. You have to consider hull drag, your sails, your skill, the direction you want to travel, how large is your crew, tides and currents, etc. If you are Dennis Connor in an America's Cup sailboat, sure! Your sailboat can do 8 knots.

You ask,  Then can I sail 80 miles from San Diego to Santa Catalina Island in 10 hours?

NO. Because the wind is in your face, you're sailing into the waves, the course is never straight, it's often quite foggy, and it could take days to sail there as you tack back and forth. Or sit becalmed. So virtually everyone motors up to Catalina, even in a sailboat.

Well, how much fuel would I need? How many miles per gallon does a sailboat get?

Whoops, no odometer on the sailboat. Fuel consumption is measured in gallons (or liters) per hour.

What you meant to ask was, What do our engines burn per hour at cruising speed, and how many hours will it take to get there?

My local experts say you'll burn 1-2 gallons per hour, and you face 13-14 hours of cruising at 5-6 knots per hour. That means you better have at least 35 gallons of fuel.

Comparing results 

What is the fuel consumption of this sailboat in miles per gallon?  Assume 2 gallons per hour and 5 knots per hour. That's 2 gallons every 12 miles, or 6 miles per gallon.

Asking the same old thing again, and learning even more

Now suppose you ask me, What if we fly up there - how much fuel will we need? 

I answer, Do you think we measure in miles per gallon, liters per 100 km, gallons per hour, or what?

Having learned by now, you answer, What!

Right!  Some aircraft do measure fuel consumption in gallons per hour, but many use weight of fuel rather than volume (because volume varies as altitude increases).  So gauges are marked in pounds, and we think of burning pounds per hour.

Be prepared to learn more than you thought when you ask a simple question!

Math teaches you to state your questions clearly. What do you know already? What do you still need to learn?

Be prepared to learn more than you thought. And when lives are at risk, don't just do theoretical calculations, get some real-world advice.  When someone asks, Can we make it without refueling?

I would ask, Are you walking (swimming, sky-diving) if we don't?