How are your pipes?

I've talked this week about blood, breathing, snoring, and so on. Today we look at another disgusting topic - arteriosclerosis - material collecting on the inside of pipes, so liquids cannot flow through. Only in this case, it's not MY arteries we are concerned with, but the pipes in my house. I've already had my arteries cleaned.

The house is almost exactly the same age I am. We've had it for 20 years and since only two of us now live it, the house is aging gracefully. But! Earlier this week the sink in our main bathroom stopped draining. We had already tried all the normal (and safe) tricks - boiling water, vinegar, sticking a clothes hanger in the pipe, etc. NOTHING was going down.

Here's a diagram for the non-plumbers, and people in other countries who do plumbing differently.

I took off the trap. I couldn't get anything to go into the drain extension! And I couldn't undo it, although I did discover it was leaking through pinholes. So I squirted the joint (under the trim ring) with liquid un-wrenching juice.

Then I went up on the roof with a plumber's snake, and ran it 30 feet down the vent stack. I got a handful of stuff out. I went back down, and with a torch, giant wrench, pliers and a hammer, pulled pipes out of the wall. The wall pipe was completely clogged! You can see a drain extension pipe here. Notice the hair and the solid gunk that we had to chisel out of the pipe!

Here's where it goes into the wall, after I have spent a 10 minutes scraping it out with long screwdrivers, the plumber's snake, and my fingers. Yuk.

Finally, here's another look at the pipe. It was 90% obscured when I started.

 

So what do you suppose is the problem? Can we define it with math? Let's try. A water conservation site estimates tooth brushing, face washing and shaving all take about a gallon (each):

87,600 = Number of times people brushed teeth in this sink (assume 2 people x 2 times/day x 60 years)

43,800 = Number of times people washed their hands in this sink  (assume 2 times/day x 60 years)

15,600 = Number of times people shaved in the sink (5 times/week x 60 years)

  1,440 = Hand Laundry (once a month x 2 gallons x 60 years)

     200  = Amount of cleaning solutions, soaps, cleansers, etc down the drain (8 oz/week x 60 years)

148,640 = let's round this up = 150,000 gallons of mucky water down the drain.

Is it normal for sinks to clog? I checked a plumbing website and found this diagnosis:

In bathroom sink drains what sticks is a combination of face cream, shaving cream, toothpaste, and stray hairs. That stuff often gets caught on the pop-up and the lift lever just below it, but after years it begins adhering to the walls of the pipes. I've seen it accumulate to the extent that even the trap pipes are almost closed up. 

We had completely closed-up pipes! It's normal. So I took them out and put in some new ones. Aren't they pretty?

I can now expect a few trouble-free years and tens of thousands of gallons to flow down these pipes.

Not So Loud, or so seldom

Yesterday I mentioned a common personal health-related condition - a very unpopular practice that disturbs other people in your house.

SNORING
But sometimes I snore, and so it's an appropriate topic for this blog (if I can discover some math about snoring). How about some numbers about the frequency of this irritating habit?

Perhaps 30-50% of middle-aged or older people snore, compared to about 5-6% of children.

The most reliable methods to control snoring are lose weight, stop smoking, breathe through the nose, and sleep on your side. I have taken the last two of these as my preferred approach. Even though you are asleep most of the night I can tell you from personal experience that it is possible to train yourself to stop snoring by sleeping only on your side, and to keep your mouth shut to breathe through your nose.

Other approaches you might consider include:

• Chin straps to keep your mouth closed
• Nasal strips and nose clips
• Mouth guards
• Fancy pillows
• Surgery
• Drugs
• Ear plugs (not for you but for your family!

SLEEP APNEA
A related condition is called sleep apnea. You stop breathing for a long time (at least 10 seconds),  oxygen levels drop 3-4%, carbon dioxide levels rise, then the "stay alive" reflex kicks in and the body gasps for air to catch up.

People with sleep apnea rarely get enough quality sleep at night, so they are liable to fall asleep in the daytime, even while driving or talking. I have at least 6 friends who have been diagnosed with serious sleep apnea -- it was terrifying to be with them in the car.

The "Medical World" has only known of this condition for about 40 years. About 9-10% of men are thought to have have sleep apnea, compared to about 4% of women. Only 10% of sufferers have been diagnosed. Sleep clinics identify the condition by having you come in and sleep in their observation rooms, or by giving you a recording device to wear while sleeping.

This condition is taken very seriously by doctors, because sleep apnea sufferers have as much as 30% higher risk of heat attacks and a 240% increased risk of congestive heart failure. One study suggested that medical costs for undiagnosed sufferers of sleep apnea might cost twice as much on average (per year) than the cost of a non-sufferer.

The Continuous Positive Air Pressure (CPAP) machine is nearly 100% effective in reducing sleep apnea. The CPAP machine was invented about 30 years ago. ResMed, one of the world's largest suppliers of CPAP machines, is located just a few miles from our offices. I saw an article that stated they probably supply 30% of the machines in the US, and perhaps 50% of those around the rest of the world. It's more than a one billion dollar business for ResMed alone!

Although it works to reduce the effects and dangers of sleep apnea, the CPAP machine isn't a cure, and it's a nuisance. It's certainly not attractive to wear the face mask. At least 25-50% of the people who get the machines stop using them.

Note: If you fall asleep while reading this abstract, you don't necessarily have sleep apnea, you are just normal:

"Recent studies have uncovered high prevalence of undiagnosed sleep-disordered breathing, and its linkage to metabolic or cardiovascular disorders which represent increasing health hazard. However, the mechanistic links behind these disorders as well as their contribution to the experimental observations and treatment responses remain poorly understood. Therefore, the screening of clinical measurements still relies upon relatively simple diagnostic features, such as signal averages or event frequencies, which may represent suboptimal or surrogate markers of the underlying abnormality. Consequently, most patients are being treated with general therapies regardless of the cause of their key dysfunction. Combining experimental measurements with mathematical modelling has the potential to provide mechanistic insights into the individual factors underlying the disease progression, which may finally enable tailored treatment alternatives for each patient. This review depicts a number of modelling approaches to elucidate sleep-related dysfunctions of the human respiratory system, and how these models are being used to translate the measurements first into new ideas and then into testable hypotheses. Such model-based investigations can provide systematic strategies towards better understanding, predicting or even preventing these dysfunctions. Along with the brief description of the modelling approaches, we discuss their relative merits and potential implications especially for clinical research."

Not so fast, or so far, or so often!

"I spoke in error. Apparently there aren't as many miles of blood vessels in the body as I said yesterday. "

Have you ever had to make a statement like this? Or are you in the kind of business that keeps a stiff upper lip and ignores any misstatement, mistakes or misses?

There's no way to know which is really the best policy. In my case, I hate to make mistakes but I'm in a job where we produce a product used by millions of kids, and unlike most books, ours are read from cover to cover and each problem is solved and re-solved many times over. The mistakes pop up like helium balloons escaping from a birthday party.

As far as the blood vessels go, the number is an estimate. More research which I initiated by myself, has shown me that the textbook authors disagree - and why not, they're just people like me. Some say 100,000 miles; others 100,000 km (62,000 miles).  They're all guesses. That's my guess, and I'm sticking with it until I find someone with a blood-vessel-odometer.

And since I am not a doctor or an anatomist, I have nothing to lose by admitting I did not do enough research to be confident of my claims.

However, I have exhaustively researched the following fact: my wife takes about 14 breaths a minute. She's a PE teacher; has been teaching for 30 years, and she knows her body statistics.

My Math expertise tells me that 14 bpm x 60 x 24 hours is roughly 20,160 breaths a day. That's more than 7.4 million a year!
 
My wife is now 56 1/2. If she has taken 14 breaths per minute all her life (on average), how many times has she breathed?

Here's the calculation  56.5 x 365 + (56 ÷ 4 = 14 leap year extra days) x  20,160

Restated, that's ((56.5 x 365) + 14) x 20,160 = 20636 x 20,160 = 416,021,760 or roughly 400 million breaths!

Whew. I know we can control how fast we breathe, and how slowly we breathe (within limits). We just can't control how infrequently we breathe (hold our breath) because eventually our bodies take over if they need more air. Gasp!

Since I'm 3 years older, I've breathed at least 20 million more times than my wife has. I think I'll go lie down now, I'm exhausted. And to keep from snoring, I'll breathe only through my nose!

Who knows how the blood flows?

My fingers are cold. It's poor circulation, I think. Then I wonder - how much circulation should I have?

It's easy enough to learn that the average person should have 5-6 quarts (4-7-5.5 liters) of blood.

I know that my pulse rate is about 78 heart beats per minute.

According to some gory medical websites, every minute the blood in the body travels out to the ends of our fingertips and toes, and back to the heart and lungs. It's estimated that the combined length of all the blood-containing pipes (arteries, veins, etc.) in the body is about 100,000 miles. [who decided that - it's 4 times around the world!]

So the rate of flow is around 5 liters a minute, or (60 x 24 x 5 = 7,200) about 7000 liters a day.

A liter of blood weighs about a kilogram. So your heart is moving 7000 kilograms of blood in a day, up and down in and out your thousands of miles of arteries and veins.

That means 7000 x 2.2 lbs = 15,400 pounds of blood moving around in 24 hours!

No wonder my fingertips are cold! I must be exhausted from all the hard work my heart is doing.

Some people are quite comfortable with the idea of talking about blood. It's not my favorite subject, as I tend to faint at the sight or mention of needles. Blood tests are a grueling experience and I avoid them whenever possible. Why hasn't the medical profession found another way to test people? 

The only time I tried to donate blood, back when I was in college, I was able to eke out about a cupful in 45 minutes - all the other donors had lain down, given blood, eaten a donut, drank some orange juice and were on their way in 15 minutes ... that's me, below:

I lived in the UK for a year during the period of time when "mad cow disease" was discovered, so I have been banned for life as a blood donor to the Red Cross. That saves me from appearing selfish or cowardly if asked to give blood ...

In some parts of the world, cattle can get an infectious, fatal brain disease called Mad Cow Disease. In these locations, humans have gotten a new disease called variant Creutzfeld-Jacob Disease (vCJD) which is also a fatal brain disease. Scientists believe that vCJD is Mad Cow Disease that has somehow transferred to humans. Evidence from a small number of case reports involving patients and laboratory animals suggest that vCJD can be transmitted through transfusion. There is no test for vCJD to protect the blood supply. This means that blood programs must take special precautions to keep vCJD out by avoiding collections from those who have been living where this disease is found.

I think I better go lie down. I'm feeling queasy!

Are you an Excel Math fan?

In the English language a single word can have many different meanings. Today we look at the FAN - a powered device for moving air.

 

FANS and MATH go together, and that's not just a bunch of hot air!

These fans are sitting around inside the Excel Math company offices:

We have floor fans, heater fans, fans with sanitizers, rotating and oscillating fans, etc. Some have vertical shafts around which the blades are set (the ones on either side of the photo) or horizontal shafts (the 2nd and 4th heater-fans). The middle fan with the blue lights is intended to kill germs.

The three middle fans are axial - meaning the blades are attached to a rotating shaft and air flow is in line with the shaft. The tall ones are centrifugal fans. On a centrifugal fan the air does not go in the direction of the rotating shaft.  Their blades are arranged in one of these ways:

I found lots more fans outside the company offices, in our warehouse:

How do we use math to understand fans?

Special units have been created to describe the air moved by fans. Because air can vary in many ways (temperature, density, humidity, particulates, etc.), people in the fan business have defined a set of standard air characteristics:

Standard air is clean, dry air with a density of 0.075 pounds mass per cubic foot (1.2 kg/m³), at barometric pressure of 29.92 inches of mercury (101.325 kPa) at sea level, at 70°F (21°C).

Using standard air allows companies to develop fans that compete on equal terms to move the maximum amount of air with the minimum of noise and power. Sorry, you can't design a fan with only elementary math, you have to be able to understand formulas like Specific Fan Power (how many kilowatts of electricity it takes to move some air):

If you want to know how loud some moving air is, you should understand sones, which describe the noise of moving air. (See my blog on horns and whistles if you like noise!) Here's a sones formula:

Of course, people who don't know math can still move air, and they can do it without electricity ...

If you were thinking about a different kind of FAN - educators, parents and kids who like Excel Math, that's a great use of the word FAN too.

These FANs happily follow and promote the activities of another person, or team, or company. Luckily for us, many educators and kids are fans of Excel Math:

Excel Math fans like to share their thoughts about our curriculum. Go here to read what they say or go to our Facebook page.

More Housecleaning, on a rainy day

Here's where we left off yesterday - trying to clean my computer, repair my crashing iPhoto software and fix up permissions on my damaged drive.

Capacity : 232.6 GB (249,715,376,128 Bytes)

Available : 121.5 GB (130,462,179,328 Bytes)

Used : 111.1 GB (119,253,196,800 Bytes)

Number of Files : 868,750

Number of Folders : 203,911

I'm happy to report the iPhoto crashing has been repaired, and the damaged drive has been fixed too. Now we just need to run a capacity report and see where we have ended up! Do I have more clean space?

Capacity : 232.6 GB (249,715,376,128 Bytes)

Available : 116.3 GB (124,861,341,696 Bytes)

Used : 116.3 GB (124,854,034,432 Bytes)

Number of Files : 916,206

Number of Folders : 205,894

Alas, this looks like more data rather than less! How did that happen? It's like going to a garage sale in your neighborhood - you take things over to sell and come back with more than you took ...

I can tell you what happened. I have an external drive and I moved a lot of things over there, thus cleaning the drive inside the computer. But my iPhoto database (27 GB and 50,000 photos) was on that external drive. In order to rescue the data, I had to move all the photos to my internal drive and create a new database.

I decided I don't want to move the photos back to the external drive (just for the blog clean-up) because it's slower and would take me more than an hour.

Now a question! Did you notice in the statistical report above that I have used almost exactly half of the total space on this drive?

Available : 116.3 GB (124,861,341,696 Bytes)

      Used : 116.3 GB (124,854,034,432 Bytes)

Both say 116.3 GB but when you look at the detailed figures, there's very slightly more available than was used. Do you think I could create a file exactly the right size to split the difference?? Let's see:

 124,861,341,696
-124,854,034,432

we can lose the 1248 at the front to simplify things a little bit

 61,341,696

-54,034,432

   7,307,264

So I need to create some thing(s) that use 7,307,264 bytes. How do I do that? I looked for a file of the right size and copied it. That was too big, so I took it off (emptied the trash) and tried a few others. I got to this point:

Available : 116.3 GB (124,857,974,784 Bytes)

        Used : 116.3 GB (124,857,401,344 Bytes)

 974,784
-401,344
 573,440

Now I need only 573,440 bytes. A little more trial and error, and I came up with this:

Available : 116.3 GB (124,857,679,872 Bytes)

      Used : 116.3 GB (124,857,696,256 Bytes) 

That's even closer to the middle. Nearly perfect, in fact, when you think in computer terms.

 96,256
-79,872
 16,384

I can't seem to get any closer than 16k (16 x 1024 = 16,384) because that's a sector on my hard drive and even the smallest file takes up that amount of room. And if I keep fooling around, caches and buffers fill up, emails sneak in, and my count gets wacky.

So this is how you use your garage sale and elementary math skills to entertain yourself on a rainy day!

Housecleaning

I'm back to my old iMac machine. It has even more stuff on it than the new iMac computer I reported about yesterday.  Here is a comparison of the two:

Old Computer


Capacity : 232.6 GB (249,715,376,128 Bytes)

Available : 94.7 GB (101,656,604,672 Bytes)

Used : 137.9 GB (148,058,771,456 Bytes)

Number of Files : 991,850

Number of Folders : 248,160

New computer

Capacity :           232.6 GB (249,715,376,128 Bytes)

Free Space :  197.0 GB (211,498,024,960 Bytes)

Used :                   35.6 GB (38,217,351,168 Bytes)

Number of Files :  711,468

Number of Folders :  184,232

It looks like these are about 100 GB more on the old computer than on the new one. That seems reasonable as I have the same software on each machine, but fewer songs, images and documents on the newer one.

I've already cleaned all the obvious stuff - the caches, buffers, trash cans, etc. Let's see if I can save any more space... here are a few thousand files in my PROJECTABLE product - those can all go to my back-up drive that sits offline most of the time, as can my backup of this blog, some marketing files, etc. etc.

With that stuff all moved off and lots of duplicate photos deleted, I found 166,000 items in my TRASH folder. Gulp! Empty it and now here's what I see on the report:

Capacity : 232.6 GB (249,715,376,128 Bytes)

Available : 121.5 GB (130,462,179,328 Bytes)

Used : 111.1 GB (119,253,196,800 Bytes)

Number of Files : 868,750

Number of Folders : 203,911

OK, now it appears I saved lots more space. Do  I really have 166,000 fewer things here? Let's do the math

    Files + folders in the original report  (991,850 + 248,160 = 1,240,010 items
    Files + folders in the second report  (868,750 + 203,911 = 1,072,661 items
    Files dirty computer 1,240,010 - files clean computer 1,072,661 = 167,349 fewer items on computer

Here's a screen shot showing you how hard I am working to do this blog for you

Actually that is not the right photo, that is only the screen as I am working on Blogger. Let's back up:

Now you can see how hard I am working!

Good thing you cannot see my face, as I am horrified to find out there are 18,194 discarded photo images in my iPhoto trash can! Every time I check this, iPhoto crashes.

I'll report back again tomorrow...

1458 Items in a folder - is that too many?

Managing and cleaning up our computers is a drag, isn't it?

It's also a commonly-neglected task. Because I manage 6 computers, plus 2 mobile phones, my house-cleaning chores might be worse than most people's chores.

I did a quick survey this afternoon and discovered these numbers in various corners of my multiple Macs:

• 1458 items in my Blog Artwork folder
• 1118 Messages in InBox (this is my back-up, mail-catching machine)
•   843 Messages in SentBox
• 5173 Items, 15.7 days, 25.61 GB in my iTunes

I wanted to learn more about this machine, but when I chose the "explore this disc" option I got an error. Oops! Time for that deferred maintenance... need to repair the permissions on my computer.

I can't wait because I also got a message from Google that the blog image uploading will be down for two hours starting at 5 pm today - why? - they are doing maintenance!

Now I have moved and I am working on one of my other Macs. It says things like this:

Capacity :          232.6 GB (249,715,376,128 Bytes)

Free Space : 197.0 GB (211,498,024,960 Bytes)

Used :                  35.6 GB (38,217,351,168 Bytes)

Number of Files : 711,468

Number of Folders : 184,232

Doesn't it seem amazing that there can be

Seven hundred eleven thousand, four hundred sixty-eight files?
Spread across one hundred eighty-four thousand, two hundred thirty-two folders?

Simple division in my head suggests that's only about 4 files in each folder. Maybe I don't want to have 1458 illustrations in my Blog Art folder.

Tomorrow I'll clean it up and report back!  (math never sleeps)

Can it be both 28% and 46% smaller at the same time?

Yesterday Steve Jobs talked about the new iPad and why its 10-inch screen is so much better than the 7-inch tablets being talked-up by a couple other companies. He said it looks better, it's easier for your fingers, etc. And he stated that the smaller screen is barely more than half the size of the larger one.

Let's talk about diagonal screen measurements (the bright magenta line) and why a small increase is really a big increase. In this case we really need a picture to save 1000 words of description, so here you go:

The first screen above at the left is the 3.5-inch screen of an iPhone. The second image is a (never-to-be-built-by-Apple) 7-inch tablet, and the third image shows the 9.7-inch iPad. The diagonal measures are shown below the images on the screens.

The 7-inch screen's diagonal measure is only 72% of the largest screen - meaning it's 28% smaller (100-72=28). It's just about 3/4 the size.

The iPhone's diagonal measure is only 36% of the largest screen - a bit more than 1/3 the size.

This bright picture shows you a comparison of the surface area on these three screens. Did you notice the yellow rectangles in the first image? I took those and duplicated them on this picture.

The smallest one is 13% of surface area of the largest one. The yellow tint on that small screen is 13% of the intensity of the largest screen (almost looks white, doesn't it?). It's only 1/7th the size of the large one.

The rectangle in the middle represents the area of our 7-inch screen, which is 54% of the surface area of the large screen and for good measure, it's 54% of the intensity of yellow. It's just over 1/2 the size of the large one! (Just like Steve said ... )

So the large screen's surface area is 100%, the intermediate one 54%, and the smallest 13%, while the diagonal measures are 100%, 72% and 36%? Are you thinking, How can that be?

Let me just say, If you are confused by how that same 7-inch middle screen can be 72% when measured diagonally, and 54% when measured by area, welcome to elementary school math!

Three Hundred seems like a lot

Since this blog is about math, let's look at a number today.

300

Three hundred equals three sets of tens. Or ten sets of thirty. It's a perfect score in bowling. A movie title.

All of these items have the number 300 in their file name in my clip art files - the Coptic symbol in the middle means 300.

Three Hundred has been the model name of many nice cars:

A High-Performance Chrysler (old style, when I was a kid!)

 A regular Chrysler (new style, when I am an old guy!)

 A few different Mercedes-Benzes; especially this beautiful Gull-wing model

 A Lexus with an IS at the front of its name

A Nissan with a Z at the end of its name

OK, I'm starting to exhaust your patience, already as I have almost exhausted myself. Why?  Today's post is my 300th in slightly more than 400 days.

In that time we have had roughly 12,000 unique visitors from 136 countries. A new country every 3 days! And 40 new people a blog, or 30 new people a day.

If it's your first visit, I write to remind you how grown-ups (like us) use the math we learned in school. This gives me a break from writing Excel Math curriculum. If you're in the USA and have a child or grandchild or local classroom that needs some help with their math, check out our website!

Thanks for stopping by!

What is Your 5+4 Number?

I just came back from racing to the post office to mail a letter which needs to be delivered by Monday. Have you ever done that? Despite emails and Internet, telephones and texting, we still sometimes need to rush out on a special trip to make sure our mail gets into the hands of the Post Office.

Let's see if we can find some math in this, ok?

It's a very traumatic time right now - for me and my post office, because a few months ago we learned the Postal Service is trying to sell the building and the land our post office sits upon. I know it's happening to lots of you too, because they are considering closing nearly 10% of the 35,000 post offices in the country.

A site I just checked estimates that there have been possibly 200,000 post offices in the USA over the years. So only about 1/6 of them are in operation today.

Our post office is not a small place - it was formerly the main distribution center for all of San Diego. It processes passports.  It's open til midnight every year on April 15th so late tax returns can be postmarked before the deadline. In fact, TV crews always show up and call it "The Procrastinator's Post Office" as they show us the long lines of cars.

They have copy machines, ATM machines, computerized automated postal processing machines - drop in your packages, feed in some cash, and off they go! In addition it has a special Philatelic area where you can buy collectible stamps and other items.

It's what's called Brutalist Architecture - bluff structures of concrete with few windows and harsh surface treatments. The architect was Ward Wyatt Deems (his site had this photo). You couldn't call the structure itself a friendly place. I've been using it for nearly 40 years and never feel very comfortable inside. Strangely - I learned when researching his site that his firm also designed two other San Diego buildings where I worked - Industrial Indemnity and Cordura Publications/Mitchell International.

But you might like it ... in fact if you have a couple million $$ you could buy it!

This postal site is 26.64 acres! That's 1,160,438 square feet. The main building itself is 436,000 square feet and there's a 22,000 square foot vehicle maintenance garage as well.

Some portions are open until at least 11 pm, and we can get to post office boxes until 1 am. It used to be open all night, much to the delight of homeless people. That's been stopped now, but it's still the place to go for special inquiries and services.

That's about enough of the facts and figures from me. If you want more, go directly to the PO!

Oh - the meaning of the title of this blog? ZIP plus 4 of course. At Ansmar Publishers we are 92064-7116.

What is division?

What is division? Besides being one of the 4 fundamental arithmetic operations?

As we try to define it we'll use this sample equation:

X ÷ Y = Z r

These are the component parts of a division problem:

X is the dividend - the amount being divided up
Y is the divisor - the unit doing the dividing
Z is the quotient - the result of having done this division
r is the remainder if the problem does not come out evenly
Y and Z are factors of X if the division problem comes out evenly (without a remainder)
X is evenly divisible if the division problem comes out without a remainder

Here are some sample problems, before we go on with more definitions:

  9 ÷ 3 = 3 or nine divided by three equals three; three and three are factors of nine
10 ÷ 3 = 3 r1  or ten divided by three equals three with a remainder of one
10 ÷ 2 = 5 or ten divided by two equals five; two and five are factors of ten

We can call division repeated subtraction, so Z is the number of times can I remove Y from X before I get to zero.

1. 10 - 2 = 8
2.   8 - 2 = 6
3.   6 - 2 = 4
4.   4 - 2 = 2
5.   2 - 2 = 0  ZERO

The subtraction was repeated 5 times, so Z =5.

We can say division is determining how many times one quantity goes into another number, so Z is the number of times that we can put Y into X before X is "full".

1.   0 + 2 = 2
2.   2 + 2 = 4
3.   4 + 2 = 6
4.   6 + 2 = 8
5.   8 + 2 = 10 FULL

The filling was repeated 5 times, so Z =5.

We can call division sharing where Z is the number of people who can share X evenly. Each share contains Y pieces. 

In this case 5 people can share 10 evenly if each share is 2 pieces.

Here's a division problem that might amuse you - artwork was adapted from the Arif & Ali blog.

There are lots of other ways to describe division, but this is enough for now.

Dividing A Summit by one-half

I read recently in a book about teaching elementary mathematics that division is the hardest of the four basic arithmetic operations. I agree.

The author went on to say that fractions were among the most difficult concepts in elementary math. I'll concede that for the sake of argument. Therefore, division by fractions is the pinnacle; the summit; the Mt Whitney of elementary arithmetic. What does she mean?

(Here's the top of Mount Whitney, just for fun.)

How about an easy one first. 3/2 ÷ 1/2 = ?

3 halves divided by 1 half. How many times does 1 of the halves go into 3 halves? Three. Right?

Another approach I like is to convert 3/2 into a decimal number (1.5) and 1/2 into a decimal number (.5)
now we can see  how many times .5 goes into 1.5?    Three. Right? Are you with me?

Now let's do a harder problem.

37/11 ÷ 1/2 = ?

Does the idea of multiplying 37/11 ÷ 1/2 scare you?  I used prime numbers just to challenge (irritate) myself and you, but it's not really harder to use primes than any other numbers. You just can't simplify.

Let's see - this means if a whole is divided into 11 parts, we have 37 of those parts. We want to divide that sum by 1/2, which is the same as multiplying by 2.

How do I know? That's the reciprocal property of division. Dividing by 1/2 = multiplying by 2. Or to state it more generally, dividing by a fraction is the same as multiplying by its reciprocal.

If we multiply by 2 we now have 74 of those 11th parts. We can simplify this a bit - that's the same as 6 wholes (6 x 11 = 66) plus (74 - 66 = 8) another eight 11ths left over. That's 6 and 8/11.

This is relatively straightforward if the numerator (top number) is a one. Let's try a more complicated version of this problem.

37/11 ÷ 7/2 = 

Ok, that's an ugly combination of numbers but it's not difficult.

If we invert (reciprocal) the fraction we're dividing by, we have this multiplication problem:

Thirty-seven 11ths times two 7ths which is equal to seventy-four  77ths, which looks to be just a smidgeon under one whole.

Can we then convert it into a decimal number to see how much less than one whole we really have?

Sure, just divide 74 by 77 and you have .96 which is just a tiny bit less than one whole item.

Let's try doing this by converting to decimals like the first problem. I'm going to use my calculator now but I'll show how it happens:

37 ÷ 11 = 3.3636...  and  7 ÷ 2 =  3.5 so now we are dividing 3.36 by 3.5 or multiplying 3.36 by 1/3.5

Both are the same as writing it 3.36/3.50.

It's still a tiny bit less than one whole item. So our answers match.

Now that you remember how to divide by a fraction, have we lopped off the summit of difficulty?

Cross Section to the other side

This post is on the concept of the cross section. If you've forgotten your elementary school geometry, this material is for you!

First - the definition: a cross section is the intersection of a plane surface with that of a three-dimensional geometric shape (or alternatively the intersection of a line with a 2-D object).

When you slice a block of cheese, a loaf of bread or even a hot dog, you create a cross section from a geometric shape. For example, here's a cube.

Here's our cube intersected by a reddish plane. The bright red surface is the cross section. With this diagonal cut we get a triangle.

Here's another way you can do a cross section through a cube. This cut gives us a square shape cross section. If we did a cut at a different angle, could we get a rectangle?

I think we can and we have here, but this is about the limit of my 3-D conceptualization and drawing skills. 

Now let's try cutting a section off of a cylinder that's laying on its side. We get a circular shape if we cut at 90% to the main axis of the cylinder. If we cut at an angle, we'd get an oval (ellipse).

Now let's cut up a sphere. Looks like an olive, doesn't it?

After all this virtual practice, I think it's time to go cut up some real food now - it's time for lunch.  

If you want to play with some online cross-sectioning tools, you can go to the Learner.Org website and chop up some geometric shapes. 

The middle of the road

I've proposed several times in this blog that we consider math to be a language of counting, measurement, shapes and calculation; a language with precise definitions and specialized terms.

I'd like to adjust my definition to include the concept of comparison. Math provides us with the means to compare and evaluate the world around us.

While there are subtleties which require us to use a wide range of comparison terms, the simplest comparisons could be narrowed down to two choices.

• This object is the same as that one.
• This object is not the same.

A much more satisfactory comparison can be made using three terms. In essence, we are still using the single comparison as well, but we are leaving that comparison unspoken.

• This object is the same as that one
• This object is better than that one (therefore it is not the same)
• This object is not as nice as that one (thus it too is not the same)

Kindergarten Excel Math starts off not with numbers, but with comparison words that a 5-year-old person could use. The table below contains a variety of these sets of words used by English-speaking people of all ages. I invite you to think of more terms:

Minus	Equals	Plus
less than	equal to	greater than
lower	same height	higher
below	same height	above
shorter	same height	taller
shorter	same length	longer
narrower	same width	wider
under	middle	over
bottom	middle	top
left	middle	right
thinner	same thickness	thicker
inside	on the line	outside
off	midway	on
before	during	after
fewer	same	more
fewest	same	most
least	same	most
different	similar	identical
front	middle	back
forward	not moving	backward
in front of	aligned with	behind
younger	same age	older

This concludes my list of comparison terms. But before I leave, here's a gift for those who insist on numbers in every math blog. Your assignment? Describe the relative positions of these three numbers using at least 3 sets of comparative terms from the table above: