Interpreting Data, just for fun

Let's finish the week and the year with a bit of research. We will make more charts about the hot dog (subject of yesterday's posting on picture graphs).

Using a newly-announced Google tool, we can investigate the number of times the words hot dog appear in English literature. Yes, I know you are asking WHY? but hey, it's my blog! Why not?

In this case, I compared hot dog with hamburger and the frequency that they occurred (Y-axis)  in a random set extracted from a Google-scanned database of books published in English between 1908 and 2008 (X-axis). An entire century! What do we learn from this?

It looks like hot dogs were more popular than the hamburger (at least in literature) in the decade between 1920 and 1930. Then the hamburger took off and hasn't been caught since. It is interesting to see that the hamburger has dropped in frequency of mention in the last decade. That makes sense. Now let's add the taco. Being a specialty ethnic food, the taco ranks third in this comparison.

Next I thought of fried chicken, so I added it in to the comparison. Notice in the chart below that early in the last century,  fried chicken got plenty of treatment in literature. It was much more popular than the other foods until after WWII. Now it cruises along right next to hot dogs.

What food will it take to change this chart dramatically? Pizza! Sure enough, it changes the whole scale of the graph, because it appears at least 3-4 times more frequently than hamburger in literature.

That's enough for now. I have equipped you with the tools to waste your entire holidays if you like ... and after a short vacation, I will be back after the New Year in January.

PS = Chocolate? YES! Trumps over pizza ...

Displaying Data Discretely

When I propose to display data discretely, I mean to show it with picture graphs (a discrete unit of measure on the graph) versus line, pie or other kinds of charts. I didn't mean "Leave out the embarrassing facts!"

Picture graphs are the simplest forms of charts, and are closest to representing the reality of what you are displaying. They are easiest for kids to understand. For example, here's a chart showing hot dogs eaten at a picnic:

Notice that the kids who ate the hot dogs are shown. Once we understand the idea of a picture graph, we can add some additional information, so the viewer doesn't have to squint at the tiny picture to identify the kids:

If the names are added we don't really need to have their pictures, so we can simplify a bit. Here is the chart using only the names. Now it's getting more abstract (farther from reality).

I am going to add another level of complexity. I am asking you to imagine that each 1 of the hot dogs shown represents 2 hot dogs in reality. Andrew ate 4, Hannah had 2, etc. The hot dog pictures are not really meaning a single dog in a bun, but two units of hot-dogginess. We have to do math in our heads to calculate Devin's picnic intake of 5 hot dogs.

In order to show this, math gurus invented the legend, or explanatory note. We put a legend at the bottom of the chart indicating that one dog picture = 2 hot dogs. We could have made it 3, 5, 10 or whatever we wanted.

In preparation for another level of abstraction, I have now added grid lines to the chart and value labels at the bottom of each grid line. We still have the hot dogs (we are about to remove them).

Here's the final version of our chart. It's turned from a picture graph to a bar graph. No more drawing little hot dogs, no more pictures of kids, just a boring old chart.

Isn't math fun? I think I will run down to Costco and pick up a hot dog for lunch.

Displaying Data again

Yesterday I showed three ways to construct a pie chart, or circle graph. Today we can look at a line graph, which is one of the more complex ways we teach elementary school kids to display data.

Here's a sample chart or graph which comes from our fourth grade material.

The chart shows distance (across the bottom or X-axis) and elevation (on the vertical or Y-axis). The green line is drawn connecting various points that our rider noted during his training ride. These points are also described in a story that accompanies the chart.

For this problem, we asked students to identify the points labeled 1, 2 & 3 on the chart. These corresponded to things mentioned in the text. This process allows kids to become familiar with the concept of the chart without having to construct everything themselves. Then we asked a question based on the data. Here are the answers:

The next week we asked for a bit more. We described a bike race, with a narrative story, then asked them to plot the ride on a blank grid. They had to decide the units on the Y-axis, label both axes, label the graph, and plot points based on the text. Finally, they had to ask their own question based on the data, and provide the correct answer.

Here's the completed graph:

Is this rocket science? No. Is it difficult to do? Yes. Does it help to visualize this if you took the bike ride first? Yes. Math is a representation of the reality around us, but it is not totally independent of that reality.

Displaying Data

In Excel Math we teach kids how to plot points on a graph, and to make charts.

Believe it or not, this can be done without using Excel or another spreadsheet or without Powerpoint or another display program. Close that window! and work along with me.

We still can do it the old-fashioned way, by thinking and drawing. For example,

Here are the recent viewers in the top 4 countries from which this blog is viewed. 

USA 7447, UK 1199, Canada 874, Aus 436. 

I got these figures from the Visitors box in the left-hand margin of this screen.

Let's make a pie chart - that's a hard one. Here's how I would construct it:

1. Add all the viewer numbers 7447+1199+874+436=9956 or roughly 10,000

2. Calculate the percentage of viewers in each country. Because it's very close to an even number, we will just estimate. USA 75%, UK, 12%, Canada 9%, Aus 4% = 100%

3. We could draw this freehand, but let's do it the proper way and calculate the degrees of the pie that belong to each country. There are 360 degrees in a circle, so the USA gets 75% of 360, or 270 degrees, the UK gets 43 degrees, Canada 32 degrees and Aus gets the other 15 degrees of the circle.

4. We think up a title for the chart. How about  Percentage of Viewers in Top 4 Countries

5. We draw a circle and divide up the pie using a protractor or by eye.

6. Here's a hand-drawn version captured with my phone camera. This took me about 90 seconds to draw, including looking around in the drawer for my green highlighter.

7. Here's my high-quality version, created using Adobe Illustrator. This took me about 10 minutes to create:

8. Finally, here's a chart made using Microsoft Excel. It took me about 60 seconds to make it, including entering the data. Why do you think they invented the spreadsheet Chart Wizard!

That's partly because I already knew what I wanted to show. I didn't have to stop at the front screen (shown at the top above) and ponder the dozens of variations of charts. That could have added 5 minutes.

Of course by now, the numbers have changed. Rats!

Digging into the Data

In our Excel Math curriculum we help kids to study a problem, then re-evaluate data, re-present it in different ways, and learn more about it using their math skills. Let me give you an example.

I read an article today about increasing shoe production in Indonesia. Shoe makers there are getting some of the shoe business that used to be performed in China. But the article was all text - about various companies and countries. Apples and oranges really, too hard to compare. Things like "China share of output currently at 80 per cent, expected to fall to 70 per cent over the next two years", etc. I decided to make some graphs.

Looking at these two pie charts, you can readily see how the NIKE company has balanced their production across three countries, while Payless is heavily oriented towards China. They are in the process of "balancing" their production to reduce exposure to rising currencies.

In this case, my point is not about where your sport shoes are made, but how to visually depict the data so it's more informative.

Another quote in the article said that Indonesia is expected to produce 300 million pairs of shoes this year, with a value of approximately $2.5 billion US dollars. I wondered,  

How much is that per pair?

You can't do it easily in your head and it's not easy on my hand-held calculator either because there are too many zeros to display. We can solve this on paper, and drop zeros to simplify the task:

2,500,000,000 ÷ 300,000,000  =  2,5∅∅,∅∅∅,∅∅∅ ÷ 3∅∅,∅∅∅,∅∅∅ = 25 ÷ 3 = $8.33

or we can do it on a spreadsheet:

Here's how it looks on the calculator that pops up on my spreadsheet software. You can see that the average value (to "the country of Indonesia"; NOT the sales price) is about $8 per pair. This value represents some labor and some materials, and possibly some packaging.

The same article told me that one of the shoe companies sold 170,000,000 pairs of shoes resulting in revenues of $3.3 billion. Let's do a little math on the numbers.

3,300,000,000 ÷ 170,000,000  =  3,30∅,∅∅∅,∅∅∅ ÷ 17∅,∅∅∅,∅∅∅ = 330 ÷ 17 = $19.41

My math says about $20 per pair total revenue to the company. Presumably this includes shipping the shoes out to stores, inventory, sales expense, etc. Plus the $8 earned in Indonesia, China or Vietnam.

If we wanted to learn more about the economics and politics of the shoe industry it would take a lot more research. Today we can just go away knowing that those three countries make many sport shoes, and get about $8 a pair for their work.

Ring Them Bells, Part V

Let's finish our week of Bell Math.

I found lots of software related to bells. The first category was factory and school bell software, to ring bells to indicate work shifts or classes are over. Then there was some software for churches, so they could "ring bells" using a computer and speakers.Then I got complete side-tracked by a bunch of sites ranting about "Ma Bell" (AT&T) and its issues! And discussions on how to draw a bell curve using Microsoft Excel.

I took a short trip through Apple's App store, where I found a few bell programs.

• Virtual Cow Bell
• Jingle Jingle Bell
• Dinner Bell
• H.Bell
  
• Hand Bell
• Handbell
• Golden Handbell
• Dingaling
• Mobel

The reviews were generally negative on most of these apps. Why? Because they don't sound like bells. Not surprising, considering an iPhone has a tiny speaker.

There is some uniqueness to the sound of a bell that is very hard to quantify. Bell sounds are unlike many other sounds that can be reproduced by speakers or headphones. As yesterday's post indicated, some of the sound is generated in our ears, and not from the source (bell or speaker).

Finally, I found some nifty software called Abel, Mabel and Mobel. Abel runs on PCs and Mabel on Macs. Mobel is for iPhones. The creator of these programs gives all the profits to restoration of real bells.

Of course, if you really want to hear bells, go find some real ones! Most ringers welcome new folks. Be prepared to use your logic, timing, math and arms.

Or get your own! You can buy here. Use your math while shopping, and you will learn that an average church bell is 28 inches in diameter and weighs 400+ pounds. I found a source that shows typical weights for bronze bells, by bell diameter. Here are a few:

6 inch weighs 12 lbs.
12 inch weighs 35 lbs.
18 inch weighs 125 lbs.
24 inch weighs 320 lbs.
30 inch weighs 560 lbs.
36 inch weighs 980 lbs.
42 inch weighs 1620 lbs.
48 inch weighs 2300 lbs.
54 inch weighs 3200 lbs.
60 inch weighs 4500 lbs.

This data should be in a table.
So I made one.


Finally, my wife asked me to explain the poor grammar in my choice of the Bell Post titles. It's the title of a song I like, by Bob Dylan. I also found another recording by Liza Minelli that was completely different - but the same title. There appear to be at least 100 different songs available with Bells in the title!

We can end with Ring Them Bells' cryptic lyrics:

Ring Them Bells

Ring them bells, ye heathen
From the city that dreams
Ring them bells from the sanctuaries
’Cross the valleys and streams
For they’re deep and they’re wide
And the world’s on its side
And time is running backwards
And so is the bride

Ring them bells St. Peter
Where the four winds blow
Ring them bells with an iron hand
So the people will know
Oh it’s rush hour now
On the wheel and the plow
And the sun is going down
Upon the sacred cow

Ring them bells Sweet Martha
For the poor man’s son
Ring them bells so the world will know
That God is one
Oh the shepherd is asleep
Where the willows weep
And the mountains are filled
With lost sheep

Ring them bells for the blind and the deaf
Ring them bells for all of us who are left
Ring them bells for the chosen few 
Who will judge the many when the game is through
Ring them bells, for the time that flies
For the child that cries
When innocence dies

Ring them bells St. Catherine
From the top of the room
Ring them from the fortress
For the lilies that bloom
Oh the lines are long
And the fighting is strong
And they’re breaking down the distance
Between right and wrong

Copyright © 1989 by Special Rider Music

I'd like this set for of 6 bells for Christmas, just in case you are feeling generous:

Ring Them Bells, Part IV

We've been talking about bells this week in the blog. Today we get into a bit more math. We start with how do we display the sound of a ringing bell visually?

Here is a picture of the waveform generated by a ringing bell, from Bill Hibbert's doctoral dissertation website:

The splash is when the clapper strikes the bell, then the tail is the slowly decaying sound we hear afterward.

Click here for a sample of decaying sound:  Bell Ring

Now to tie this in with Excel Math, the chart below is the first line graph our students ever see, in fourth grade Lesson 80. Like most line graphs, it shows amplitude (change in temperature) on the vertical axis, and a period of time going from left to right on the horizontal axis. This way of displaying value vs time is used constantly, in all sorts of math problems.

Here we show the values as discrete points with straight lines connecting them, as opposed to the rounded curves in the chart above. Curves require many more individual points OR analog data (as opposed to digital or individual data points).

Here's another graph from Bill's dissertation, this time not showing a line plotted over time, but a series of peaks that represent a bell's loudness or amplitude at different frequencies (vibrations; cycles per second).

If I understand his theory, the ground-breaking contribution of this dissertation is concluding that bell sounds are assembled "in the ear of the hearer" (like beauty is "in the eye of the beholder"). Some of the "minor" vibrations combine to produce a set of tones that are not necessarily what we would expect.

One of his tests involved playing sounds from 25 bells for 30 test subjects who were strong and confident singers. After hearing the bell, each singer sang the sound he/she had heard. The research showed that people clearly heard significantly different tones from the same bells. Any set of bells that should have been "in tune" (according to the scientific equipment) was heard to be out of tune by the listeners.

The math behind these results overwhelmed me.

Ring Them Bells, Part III

This is the third in a series on Bells and why they are related to math we learn in elementary school.

MAKING A BELL
Most bells are made of bronze - an 80-20% combination of copper and tin. The bronze is melted and poured into various kinds of molds. The bell is removed from the mold and tuned. Bells may also be made from steel alloys, ceramic or even glass.

In case you ever have to inspect or describe a cast bell, these are the main parts:

Here's a short description of how bells are cast in Russia.

SOUNDS
Cast metal bells are made in shapes and sizes that produce the best tone. A bell's ringing frequency varies in relation to the square of its thickness, and inversely with its diameter. The sound bow (thickest part of a bell) is often one thirteenth its diameter.

A bell's sound is composed of various tones, including the nominal note (what the bell is named after), the hum, and the second, tierce and quint partials. After a bell is cast at the foundry, elaborate  sound measuring equipment may be used for initial tuning. The final tweaks are done by hand by experts.

DIFFERENT TYPES OF BELLS

I learned in my research that European bells are played like a piano or melodic instrument, while Russian bells are played like a drum or rhythmic instrument. For this and other reasons, Russian bells are not tuned in the same way as European bells, but have a more individual character to their sound.

Calling bells are used to call people to dinner, summon servants, and so on.

Warning bells are employed to sound an alarm or warn of the approach of danger (closing gates at a railway crossing, for example).

Symphonic bells are specially formed to complement the sound of a symphony, and to produce tones that harmonize well with other instruments.

Decorative bells include this cute little ceramic bell I spotted while walking through our offices today. Jingle bells and sleigh bells are primarily decorative rather than functional.

Here's a website that shows you all sorts of handbells.

If you like bell sounds, you could look for Brian Eno's album called Bell Studies for the Clock of the Long Now.

You can listen to the bells of Amersham, near where I used to live in England.

Or if you are really interested, you can go read the doctoral dissertation of Bill Hibbert, on bells. It's really fascinating.

This thesis quantifies how the pitch or strike note of a bell is determined by the frequencies of its partials. Bell pitch is more often generated in the listener's ear rather than being radiated out as a frequency from the bell. The exact pitch varies from the expected pitch by changes in the frequency of various partials. This can cause bells whose partials are tuned precisely (in theory), to sound out of tune to some people.

The pitch shifts were quantified at frequencies across the audible spectrum by experiments carried out on 30 subjects; these showed that partial amplitude does not significantly affect bell pitch. A simple model of pitch shift was devised from the test results which gave good agreement with the stretch tuning in a number of peals of bells. 

A comprehensive investigation was done on over 2,000 bells with from a range of dates, weights and foundries. An unexpected, straightforward relationship was found between the frequencies of the upper partials which generate virtual pitches, which seems to apply to all bronze and steel bells of Western shape. The relative frequencies of these partials are determined by the thickness of the bell's wall near the rim. This relationship between the partials has not been previously reported, and explains previous failed attempts by bellfounders to tune these partials independently.

If you only know elementary school math you can still follow most of the discussion!

Ring Them Bells, Part II

We're looking at and listening to bells this week. Here's my Paolo Soleri bell:



Surprisingly, my clip art collection, which we use as raw material for the math curriculum, has 418 different bell illustrations. Of course after scanning them all, I found the list includes bell peppers too. Here are a few types of bells:

Besides the large hanging bells, there are hand bells, cow bells, dinner bells (triangles), tubular bells (long pipes) and the desk counter bell (ding ding) and jingle bells. Not shown here are the doorbell, alarm bell, and the bell gong.

I have also omitted the following which we associate with the word bell:

• pepper
• hop
• jar
• -y dancer
• Alexander Graham
• helmets
• curve
• Tinker
• helicopter
• Salvation Army
• Packard
• diving
• Taco
• telephone
• & Howell
• sleigh

The bell works on math principles we'll talk about later. In the meantime, you can go this website and learn all about how bells and chimes are made, repaired and maintained. See if you can find out how Russian bells differ from European bells.

The art of making and ringing bells is called campanology. There is a field of mathematics called Braid Theory that addresses ringing sets of bells. (We don't teach that in Excel Math, sorry.)

See if you can define these two common sayings:  For whom the bell tolls   and   That rings a bell

Ring Them Bells, Part I

Time is a concept we discuss in math class. Students learn how to tell time, and to read a clock or watch. The word clock comes from a Latin word clocca. Related words that mean bell include the French cloche, Latin glocio, Saxon clugga and German glocke.

Why are bells related to clocks? Because before we had watches we could inspect, or towers clocks with hands that people could look up to see, we rang bells to inform people of the time.

Here's a video I made when on vacation in a tiny village in northeastern France, about 10 years ago. We were rudely awakened by this bell. It tolled for a long time to get everyone up out of bed.



My latest Watch and Clock Collector magazine has an article on tower bells. These are the varieties that were mentioned:

• Carillon
• Chimes
• Change Ringing Bells
• Peals
• Clock Chimes
• Tubular Bells
• Great Bell

I went to a website on Change Ringing and learned this is "a team sport; a highly coordinated musical performance; an antique art; a demanding exercise that involves a group of people ringing rhythmically a set of tuned bells through a  series of changing sequences that are determined by mathematical  principles and executed according to learned patterns."

Whew! I did see the word mathematics in there though, so I thought why not tackle Bells in my blog?

Here are some change ringers pulling on their ropes.



We'll look into bells a bit more in the next few days.

How do you measure a life? Part V

Today is the last blog in measuring a life. I'm starting off on a tangent, talking about gold...

Gold is a precious metal, right? A place to stash your wealth when the markets are down and banks are failing. Today the price of gold is $1389 per ounce.

Is gold a color too? (not really, but it seems sort of yellow). The word comes from the Latin word aurum, meaning shining dawn. It has the atomic number of 79. It's relatively heavy and soft. It doesn't corrode or tarnish easily. Gold is found as nuggets and in veins running through other rocks and minerals. Because it's inert in most cases, it can be consumed (eaten) without any ill effect.

According to Wikipedia, all the gold mined in human history could be formed into a cube about 20 meters on a side. That seems too little to me. Although because it is so heavy, that amounts to 5.3 billion ounces! Gold has some industrial uses because it is extremely conductive and reflective. Here's a McLaren F1 engine compartment hood, lined with gold leaf to reflect the engine's heat.

 Gold is mostly formed into jewelry or used in dentistry. It's also used in art. Here's a gilded (gold leaf coated) Russian icon:

Gold symbolizes power, strength, wealth, warmth, happiness, love, hope, optimism, perfection and the sun. In the Biblical book of Revelation, gold is used to describe Heaven. I'll finish this weeks math blog with this song's lyrics. I like to play it when someone I know leaves this earth:

City Of Gold


There is a city of gold
Far from the rat-race that eats at your soul
Far from the madness and the bars that hold
There is a city of gold.

There is a city of light
Raised up in heaven, and the streets are bright
Glory to God, not by deeds or by might
There is a city of light.

There is a city of love
Surrounded by stars and the power above
Far from this world and the stuff dreams are made of
There is a city, city of love.

There is a city of grace
You drink holy water in a sanctified place
No one's afraid to show their face
There is a city, a city of grace

There is a city of peace
Where all destruction will cease
When the mighty have fallen and there's no police
There is a city, a city of peace

There is a city of hope
Across the ravines by the green sunlit slope
All I need is an axe and a rope
To get to the city of hope.

I'm headed for the city of gold
Before it's too late, before it gets too cold
Before I'm too tired, before I'm too old
I'm headed for the city of gold

How do you measure a life? Part IV

Today, as part of this short series on math and our lives, I will list some of the items of an obituary that are math-related.

An obituary is a short review of a person's life, normally published in a newspaper or other public document. Writing them is a real art. In the past newspapers would keep obituary files or even pre-written statements for famous people and government figures. Nowadays they often done in a rush shortly after death, and/or strongly influenced by the family.

In this interview with Walter Cronkite, he talks about the art of the obituary.

If you write one, use a respectful tone of voice throughout, although it's permissible to include random outrageous or unexpected acts that the person might have performed.

An obituary should contain these items involving math:

• Birth and Death Dates (but don't expect the reader to any do calculations)
• Age at death
• Locations of living and traveling
• Schools attended and/or notable degrees held by the deceased person
• Employment including the number of years worked at major jobs
• Involvement in global events such as wars, scientific discoveries, sports, etc.
• Mention of wealth if a person was very rich or famously poor
• Name and/or number of siblings and descendents

Here's an entertaining math-packed obituary I found today:

Dr. B died in January in San Diego at 84. He was a physics professor for four decades, a director of the Kitt Peak Observatory, and ran the Palomar and Mt. Wilson telescopes. In 1957 in a pivotal discovery of 20th-century astrophysics, he proposed that everything around us is made of star dust. B was born in 1925 in England, half-way between Oxford and Stratford-on-Avon, the only child of a builder and a milliner. He married in 1948 and received his first PhD in 1951. The family was invited to UCSD in 1962 after his discovery involving quasars, radio galaxies and black hole gravity. B called a friend of his 3 times a week for 40 years to debate creation theories, and edited the Annual Review of Astronomy and Astrophysics for more than 30 years. In 1990 he published a paper arguing against the Big Bang theory, insisting instead on a steady-state universe with mini-bangs every 20 billion years. His wife, an astronomer in her own right, has survived him, along with one daughter and a grandson.

Here's a rich man's obituary, with lots of numbers:

Mr P died Monday in Sydney. He became the richest man in the country, turning his inheritance worth millions into an empire worth billions. He was 68 when he died and had been ill for decades. He controlled television networks and magazines, as well as many casinos. He was the first to cover cricket matches on satellite television, transforming the global cricket scene. P sold his TV network for $500 million, then bought it back 3 years later for $100 million. He suffered from polio and dyslexia as a child, and lost a kidney to cancer at the age of 40. He later had a heart attack in 1990 and a kidney transplant in 2000. A friend of 20 years gave him the kidney. Pwas reported to be the "rudest and most frightening man ever" but also known for his generosity; taking care of poor working families and supporting hospitals across the country. His only son James succeeded him in the family business in 2000.

How do you measure a life? Part III

If you are a careful and regular reader you will know that I started this thread on How Do You Measure A Life on Monday. A member of my family died yesterday and I failed to write my blog. I was totally consumed and could not write. Today I am ready.

Much of what happens when a person dies can't be explained or measured by math:

• how does the cat know we are sad?
• at what speed is the flood of thoughts and feelings and memories moving?
• how can the normal passage of time become warped? moving more slowly, then quickly
• what's the height of the wave of good feelings, encouragement, hope and prayers that wash over you?
• who's turned the volume control of love, remorse, guilt, forgiveness and joy up to 11?

Then we have the math items which come along too:

• the doors of the social security vault slam shut; zero more dollars will be coming out
• the fees of the attorneys begin to add up; cha-ching!
• the limits for mobile phone minutes and texts are soon exceeded
• the estate (a vague and slowly coalescing mass) will eventually be divided 4 ways
• another 2500 slowly-fading photographs are added to the "I'll scan them someday" pile

Those are relatively straight-forward math operations, even if we accomplish them in the midst of an elaborate dance dictated by tax rules and probate courts. Much more elaborate trains of reasoning (a major subject area of Excel Math) begin winding through the minds of those who knew the dearly-departed:

1. I want to go to the memorial service
2. I don't want to go, I might cry foolishly. 
3. I have nothing suitable to wear.
4. I can get a new outfit!
5. Aunt Mabel will be there, no way am I going
6. Cousin Bill will be there, of course I'll go
7. It's on a weekday, I'll take off work (Hooray!)
8. This will use some of my frequent flier miles - hmmm (Grandma vs spring break) hmmm
9. If I don't go someone will notice, then no one will come to my memorial service
10. She's my Grandma, I'm going to say goodbye, no matter what