Additional Math Pages & Resources

Friday, December 17, 2010

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 ...

Thursday, December 16, 2010

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.


Wednesday, December 15, 2010

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.

Tuesday, December 14, 2010

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!

Monday, December 13, 2010

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.

Friday, December 10, 2010

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:


Thursday, December 9, 2010

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.

Wednesday, December 8, 2010

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!

Tuesday, December 7, 2010

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

Monday, December 6, 2010

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.

Thursday, December 2, 2010

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.

Wednesday, December 1, 2010

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

Tuesday, November 30, 2010

How do you measure a life? Part II

Yesterday I asked if we can use elementary math to describe a person's life. I haven't come to a conclusion or answer to the question. But I learned some people insist math and science are attempts to express new things we didn't know before, in a way that everyone will understand. In contrast, poetry will express what we already know, in new ways that some will understand.

Definition:
Poetry consists of an awareness of experience, expressed through meaning, sound, and rhythmic language to provoke an emotional response.

Mathematics is a science dealing with numbers, shapes, structures and change, and the relationships between these concepts.

So I wondered if a body of mathematical poetry might exist. Could it help us measure a life in more than just days lived or money accumulated?

Today I must admit that the bulk of math poetry does not. I've found plenty intended as education or entertainment, not expression of deep meaning of a person's existence. I saw memory aides - rhyming statements of how a math process works. I found plenty of problems presented to kids in a sing-song way, or limericks which pose a question to be answered or a puzzle to be solved. Here's an example:

Take five times which plus half of what,
And make the square of what you've got.
Divide by one-and-thirty square,
To get just four -- that's right, it's there.

This is not what I am looking for. After some searching, I did find tons (well, not really tons in a math sense) of serious discussion on math and how it might help us find meaning in the universe. Here's a sample:

Logic (thus math) starts with people labeling the existence of any object or phenomenon 
(God, rock, flower, etc.) by a symbol we call 1, or yes, or +, or dot, or true.  
The lack or absence of existence of an object is labeled as 0, or no, or -, or dash, or false
The symbols are used to describe objects around us as either existing or absent...

That discussion went on for many pages and my eyelids closed with a flutter! Then I found this:

If poetry is the love of carefully-chosen words and crafted phrases to convey image and idea;
  if a mathematician channels a love of pattern, quantity, and structure into carefully-chosen words and crafted phrases;
the intersection of their realms should be non-trivial.

Now we're talking! That statement led to this poem entitled An Equation for my children, by Wilmer Mills

It may be esoteric and perverse
That I consult Pythagoras to hear
A music tuning in the universe.
My interest in his math of star and sphere
Has triggered theorems too far-fetched to solve. 
They don't add up. 
 
But if I rack and toil
More in ether than mortal coil,
It is to comprehend how you revolve,
By formulas of orbit, ellipse, and ring.

Dear son and daughter, if I seem to range
It is to chart the numbers spiraling
Between my life and yours until the strange
And seamless beauty of equations click
Solutions for the heart's arithmetic.

Monday, November 29, 2010

How do you measure a life? Part I

Can math measure a life?

We can measure a lifespan.

How long have you lived? In my case, I can say "I'm 59 years old".

That makes sense, but it seems unreasonable; it seems nonsensical to say "I'm 21,600 days old, or 518,400 hours, or 31 million minutes, or 1,866 million seconds old".

Measuring life like this is meaningless. The larger the total of tiny units, the less the measuring means. Just as it makes sense to say "I drive 15 miles to work" but not to say "I drive 950,000 inches to work".

But regardless of units, the lifespan doesn't capture what the life was about. 

Measuring a life with math units doesn't capture its meaning, only the external evidence, such as "She earned a PhD and 94 graduate units" or "His retirement account has more than 1.5 million dollars" or "They had 4 children, 16 grandchildren and 44 great-grandchildren".

I saw this statement recently, In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. In math you try to prove something that's never been proven - and tell people how you did it. But in poetry, it's the exact opposite. With poetry you want to tell people (whether they understand it or not) what we all experience.

Can we describe a life? with mathematical poems?

We'll have to look at this more tomorrow. In the meantime, try searching on mathematical poetry.

Read some. Write one.

Wednesday, November 24, 2010

The Math Tree Seedlings

Math is a language, and the ability to use it fluently is valuable. From a parent's point of view, having great math skills should result in long-term benefits for their children. As shown here:
I wish that I could promise you that money will grow on trees if you are skilled in math. But I can't promise that. I can however promise that you will amaze and astound your friends throughout life if you are confident and fluent in math. You might impress your boss, and you'll certainly be able to manage your finances.

Excel Math is like a friend with a green thumb. We simply help to plant the math; someone else nurtures and waters the seedlings as they begin to take root.

My sister is a reading specialist with many years of experience and a PhD. She always encourages parents, aunts, uncles and cousins to read to and with children. I publish math curriculum and would also like to offer my own bit of advice:

Take some time over the holidays to talk about math with your kids. You can do this while shopping for a  Christmas tree, or while doing the dishes.

How many plates did we use; how much does this pot hold; how long a nap should we take on the couch; which football games will we watch; how many floats in the parade?

How far did they fly to come see us?  How many shopping days til Christmas? What good things can we do to help those less fortunate than we are - and how do you measure the value of good works?

Water those math seedlings that have been planted this fall!

Tuesday, November 23, 2010

The Math Family Tree and its Branches

It's Thanksgiving week and lots of kids are out of school and heading for grandma's house for turkey dinner. Time to use our imaginations and consider math for the holidays.

Here's a Mathematics tree, looking for other members in his family. Let's say he represents math. What other branches of his family might show up for the big dinner on Thursday? He knows he has a cousin named Geometry (gee, I'm a tree).

One amateur mathematician suggested that these are the definitive various branches of math:

1. Foundations
    -Logic & Model Theory
    -Computability Theory & Recursion Theory
    -Set Theory
    -Category Theory

2. Algebra
    -Group Theory -> Symmetry
    -Ring Theory -> Polynomials
    -Field Theory
    -Module Theory -> Linear Algebra
    -Galois Theory -> The Theory of Equations
    -Number Theory
    -Combinatorics
    -Algebraic Geometry

3. Mathematical Analysis
    -Real Analysis & Measure Theory -> Calculus
    -Complex Analysis
    -Tensor & Vector Analysis
    -Differential & Integral Equations
    -Numerical Analysis
    -Functional Analysis & The Theory of Functions

4. Geometry & Topology
    -Euclidean Geometry
    -Non-Euclidean Geometry
    -Absolute Geometry
    -Metric Geometry
    -Projective Geometry
    -Affine Geometry
    -Discrete Geometry & Graph Theory
    -Differential Geometry
    -Point-Set or General Topology
    -Algebraic Topology

4. Applied Mathematics
    -Probability Theory
    -Statistics
    -Computer Science
    -Mathematical Physics
    -Game Theory
    -Systems & Control Theory


Ok, this is a list compiled by an expert. And there are lots of big and confusing words in the list. But did you notice that he can't count?  Or doesn't know number sequence? We teach this in Kindergarten and First Grade.

Group 1, 2, 3, 4 and 4.

But that makes sense, because he left arithmetic off the list, which is what we teach in early grades of elementary school. We could do some of this list-making ourselves, I think. Here's what I come up with if I strip off all of Mr. Tree's leaves. We see these branches:


Most of these relatives will have to be introduced to you by someone else who knows them better than I do.

Monday, November 22, 2010

Backbreaking Math

I have a case study to examine today. Can we solve the problems with the elementary school math we teach to kids in our Excel Math curriculum? (Yes!)

Chris is building a planting bed in his back yard. It will keep soil from washing down from the neighbor's yard (a bit higher), and give him a place to grow vegetables. Take a look:

He bought these metal baskets called gabions, and will fill those with soil. How does he know how much soil to buy? Can he buy soil in bags from the garden center, or does he need a truck?

Chris has to fill these 8 containers:

5 large gabions = 1m x 1m x .5m = .5 cubic meters x 5 = 2.5 cubic meters
3 small gabions = 1m x .5m x .5m = .25 cubic meters x 3 = .75 cubic meters

He needs 3.25 about cubic meters of soil.

We'll estimate that Chris needs 3.5 cubic meters of soil since some will leak out the holes and wash away, or be lost in the grass. However, he has to buy soil by the cubic yard (plastic bags) or ton (determined by weighing the delivery truck), so now we need to convert from cubic meters to yards or tons.

A cubic yard of soil weighs 2000-3000 pounds or 1.0-1.5 tons.

A cubic meter = 1.3 cubic yards; a cubic yard = .76 cubic meters

3.25 cubic meters x 1.3 = 4.225 cubic yards

So we need about 4.25-4.5 cubic yards of soil.

You need 54 40-lb. bags of soil per yard, so Chris could buy 4.25 x 54 = 230 bags of soil from the garden center
OR
4.25 x 1.0 to 1.5 tons of soil = 4.25-6.35 tons delivered by a truck

After looking at the weight and his available vehicles, Chris didn't bother with any more math, such as calculating the number of trips it would take to pick up 230 bags.


He called the soil delivery man instead. Here's the the pile of dirt which is just arriving in his back yard:

Friday, November 19, 2010

Arithmetical Words, Part V

We have come to the end of the arithmetic alphabet with R-Z today. Most of the time this blog deals with using math in real life, but without knowing math words, it's hard to communicate precisely.

We tend to say square or rectangular when we are talking about 3D objects, even when we mean rectangular prism. I was talking with my friend Chris, who said he was putting square gabions in his back yard. I knew the word gabion meant a big wire cage, but I didn't know what shape they might be. I learned gabions can be purchased as rectangular prisms, cylinders or cubes. You put them in place, then fill them with heavy material to form earthen walls.

If we want to be extra-precise, gabions aren't solid figures until they unfolded and filled. Here are some pictures of his empty ones, waiting for 2 tons of dirt. We'll do the math calculations on the dirt in another blog:



Now, on to the definitions:
  • Radius straight line from the center point of a circle to any point on the circle
  • Ray line with one endpoint
  • Rectangular Prism solid figure with 8 vertices, 12 straight edges and a total of 6 rectangular flat faces, one of which is the base
  • Rectangular Pyramid solid figure with 5 vertices, 8 straight edges, 4 triangular flat faces and 1 rectangular base
  • Reflection movement of a figure over a line that results in a mirror image; a Flip
  • Remainder amount left over when one number is divided by another
  • Right Angle angle that measures exactly 90 degrees
  • Roman Numerals number system created by ancient Romans that uses letters rather than numerals; not based on place value
  • Rotate to move or turn a figure around a point; a Turn
  • Round (1) a circular or cylindrical shape
  • Round (2) replacing an exact number with an approximate number that is more convenient to use
  • Scalene Triangle a triangle where all three sides are of different lengths
  • Similar Figures figures having the same proportions but not the same size
  • Slide when a figure moves without changing its appearance, see translation
  • Sphere 3D solid figure where all points on the surface are equidistant from the center
  • Square 2D parallelogram with 4 congruent sides and 4 congruent angles
  • Square Pyramid 3D solid figure with a total of 5 vertices, 8 straight edges, 4 triangular faces and a square base
  • Square units (1) group of squares with sides one unit in length, which are laid on top of an object to measure its area
  • Square units (2) a unit of distance when multiplied by itself becomes a measure of area 
  • Surface area the sum of the areas of all the faces of a three-dimensional figure
  • Three-Dimensional Figures 3D geometric objects with length, width and height; they are “solid”
  • Translation when a figure moves without changing its appearance, see slide.
  • Triangular Prism 3D figure with 6 vertices, 6 straight edges, 3 rectangular flat faces and 2 triangular flat faces
  • Triangular Pyramid 3D figure with 4 vertices, 6 straight edges, and 4 triangular flat faces
  • Turn to make a figure revolve around a point, Rotate
  • Two-Dimensional Figures 2D geometric objects with only length and width; they are “flat”
  • Vertex point where at least two straight lines meet (flat figures) or three straight edges meet (solid figures); plural is vertices
  • Volume measurement of the amount of space occupied by material; expressed in cubic units
  • Week period of time consisting of 7 days; not related to solar or lunar activity
  • Weight measurement that describes how heavy an object is; due to earth's gravity
  • Yard unit of length equalling 3 feet; 36 inches; .9 meters
  • Year time period of 365-366 days or 12 months; based on rotation of Earth around the sun
  • Zero Property: Addition any number added to zero has itself as the sum
  • Zero Property: Multiplication any number multiplied by zero has a product of zero

Thursday, November 18, 2010

Arithmetical Words, Part IV

The phrase "Mind your P's and Q's" comes to my mind today, as we get closer to the end of the alphabet and arithmetic words. Many different origins have been suggested for this slang phrase, but it seems to have always been used to encourage people to be careful.

Today we will look (carefully) at the arithmetic words beginning with P and Q
  • Parallel Lines lines that never cross, no matter how far they extend; always the same distance apart from each other
  • Parallelogram quadrilateral whose opposite sides are parallel and congruent
  • Pattern regularly repeated arrangement of letters, numbers, shapes, etc.
  • Pentagon polygon with exactly five sides
  • Percent ratio that compares a number to 100 using the % symbol
  • Percent Pie Graph a circle graph where the sum of the percentages in each section equals 100 percent
  • Perimeter distance around a closed figure
  • Permutation one of several possible orders for a series of events or items
  • Perpendicular Lines intersecting lines that form “square corners” or right angles (90 degrees) where they cross
  • Pi (π) the ratio of the circumference of a circle to its diameter; approximately equal to 3.14, or 22/7
  • Plane Figure has only length and width; Two-Dimensional or 2D Figure
  • post meridiem label for time from noon up to, but not including, midnight
  • Polygon plane figure made up of 3 or more straight lines
  • Positive Number a number greater than zero
  • Prime Factor a factor that is also a prime number
  • Prime Number a number that has itself and one as its only factors
  • Probability the likelihood that a future event will occur; expressed as a value between 0 and 1, with 0 being impossible and 1 being certain
  • Product name of the result obtained by multiplying two or more numbers together
  • Property of One any number multiplied by one has itself as the product
  • Pythagorean Theorem For any right triangle, the area of a square constructed along the triangle’s longest side (the hypotenuse) is equal to the sum of the area of squares built along the other two sides
  • Quadrilateral a polygon with 4 sides
  • Quart standard unit of measure for volume; 32 ounces; slightly more than a liter
  • Quotient number resulting from a dividend being divided by a divisor; solution to a division problem

Tuesday, November 16, 2010

Arithmetical words, Part III

This is day three of the Great Arithmetic Glossary Series, where I am trying to create the most concise definitions possible for elementary math words and concepts.

These definitions are estimates; I am rounding the meanings slightly in order to save space and words. It would be possible to carry out the definitions to many significant digits!
  • Leap Year 366-day year created by adding an extra day (29th) in February; helps compensates for uneven motion of the Earth around the sun 
  • Least Common Factor smallest factor of two or more numbers
  • Least to Greatest arrangement of numbers from lowest value to highest value
  • Length distance along a figure’s longest side from one end point to the other
  • Less Than number of smaller, or lesser, value than another number; symbol <
  • Likely Event event with a probability of greater than 0.5 but not certain or equal to 1.0
  • Line 2D straight path extending infinitely in both directions without any endpoints
  • Line Graph diagram where plotted data points form a line that (usually) shows change over time
  • Line of Symmetry line that divides a figure so each portion is a mirror image of the other
  • Line Segment portion of a line that has two endpoints
  • Lowest Common Multiple the multiple of two or more numbers that has the least value
  • Mean description of a set of values calculated by adding the values and dividing their sum by the number of items in the set; Average
  • Median description of a set of values obtained by putting the values in order from least to greatest and selecting the middle value (for an odd number of items) or by calculating the mean of the two middle values (for an even number of items)
  • Mode description of a set of values obtained by selecting the value within the set that occurs most frequently; a set may have more than one mode
  • Month unit of time containing 28-31 days; 1/12th of a year; based on lunar motion around the Earth
  • Multiple product of two whole numbers
  • Multiplicand factor being multiplied in a multiplication problem
  • Multiplier factor by which the multiplicand is multiplied in a multiplication problem
  • Negative Number number less than zero
  • Numerator portion of a fraction that is written above the line; it represents parts of a whole
  • Obtuse Angle angle that measures more than 90 degrees and less than 180 degrees
  • Octagon polygon with exactly eight sides
  • Odd Number number that cannot be divided into two equal groups; ends in 1, 3, 5, 7 or 9
  • Open Figure begins and ends at two different points
  • Order of Operations rules used to determine the sequence of performing addition, subtraction, multiplication and division in an equation
  • Order of Symmetry number of different positions to which a figure can be rotated to match itself exactly
  • Ordered Pair pair of numbers used to locate a point on a coordinate grid; horizontal (x-coordinate) is given first and vertical (y-coordinate) is next
  • Ordinal Number whole number that indicates sequential position: first, second, third, etc
  • Origin intersection of x- and y-axes on a coordinate grid; designated as a point (0, 0)
  • Outlier value in a set that is an extreme deviation from the mean value
Stay tuned for more math words tomorrow!

Arithmetical Words, Part II

I decided to devote this week's blogs to the terms we use for elementary arithmetic - in an attempt to define them clearly in as few words as possible. Read Arithmetical Words, Part I.

I should mention that some of these definitions are unique to US English. The meanings may vary in other English-speaking countries around the world. Math symbols also vary - for example, the function of the decimal "point" (as we call it) is performed by a "comma" in other countries. For a fun diversion you could visit Jenny Eather's Maths Dictionary for Kids (she is based in Australia)

You are welcome to download an illustrated, expanded version of this glossary in English or Spanish from the Excel Math website.

Our next group begins with the letters D through K
  • Day unit of time containing 24 hours; equal to one rotation of the Earth
  • Decimal (1) symbol used to separate whole numbers from smaller parts of a whole
  • Decimal (2) symbol used to separate dollar amounts from cents
  • Decimal (3) word that refers to 10
  • Decimal (4) math system based on 10 different digits
  • Decimal Number number with a decimal point; not a whole number
  • Deductive Reasoning logical process; begins with evidence and draws a conclusion; used to solve word problems
  • Denominator portion of a fraction written below the line; the total number of parts into which a whole number is divided
  • Density (1) ratio of the weight or mass of a material to its volume
  • Density (2) the number of individual items in a given space or region
  • Diagonal line segment completely inside a polygon that connects two of its non-adjoining vertices
  • Diameter line segment passing through the center of a circle, ending at either side of the circle
  • Distributive Property: Multiplication multiplying the sum of several addends gives the same result as individually multiplying the addends first, then adding the products (A + B) x C = (A x C) + (B x C)
  • Dividend quantity to be divided; beginning number from which repeated subtractions are taken
  • Divisor quantity by which a dividend is divided; amount repeatedly subtracted from a dividend
  • Edge line segment where 2 faces on a 3D figure meet; flat or curved
  • Empty Set a set that contains no items; not the same as a set containing a zero
  • Equally Likely multiple events with the same probability of occurring; probability value of 0.5
  • Equation number statement that includes an equal symbol =
  • Equilateral Triangle plane or 2D figure whose 3 sides are equal in length
  • Equivalent Fractions fractions with the same value expressed using different numbers; 4/8 is equivalent to 2/4 and 1/2
  • Estimate to quickly calculate a number that is tolerably close to the exact answer
  • Even Number number that can be divided into two equal groups; all even numbers end in 0, 2, 4, 6 or 8
  • Exterior Angle an angle on the outside of 2 parallel lines that are intersected by another line
  • Face 2D polygon (plane figure) that forms one side of a 3D figure
  • Fact Family related addition/subtraction or multiplication/division facts involving the same set of numbers
  • Factor number that divides evenly into another number.
  • Factorial product generated by multiplying a number and every positive number less than the number
  • Fahrenheit (F) temperature scale with 180 units between the freezing point (32) and the boiling point (212) of water
  • Flip change in location of a figure that results in a mirror image of the original figure; Reflection
  • Formula a mathematical statement or rule used in calculations
  • Greater Than number of higher, or larger, value than another number; symbol >
  • Greatest Common Factor the largest factor of two or more numbers
  • Greatest to Least arrangement of numbers from highest value to lowest value
  • Height vertical dimension of a 2D or 3D figure
  • Hexagon polygon with exactly six sides
  • Histogram graph where the labels for the bars are numerical intervals; used to compare data
  • Impossible Event event that will not happen; probability value of 0.0
  • Improper Fraction fraction where the numerator is greater than or equal to the denominator
  • Inequality number statement that compares two unequal expressions
  • Integer a whole number and its opposite (-2,-1,0,1,2)
  • Intercept point where a line or curve meets the x- or y-axis on a grid
  • Interest 1) fee charged by a lender to a borrower for use of money
  • Interest 2) fee a bank pays to its depositors; often a percentage of the deposit, calculated over a period of time (5% per month)
  • Interior Angle angle on the inside of two parallel lines that are intersected by another line
  • Intersecting Lines lines that cross at some point
  • Intersection of Sets a group of values or items that are common to all the sets being evaluated
  • Isosceles Triangle triangle having only 2 sides of equal length
NOTE - there are no J or K arithmetic words in our glossary!

Monday, November 15, 2010

Arithmetical words, Part I

Words, words, words, I'm so sick of words. 
I get words all day through; First from him, now from you!

This refrain from My Fair Lady has stuck in my mind for decades. Of course, as a book editor, I have to look at words all day, if not hear them. And not just any old words, but math words.

Can we consider some of A, Bs and Cs of arithmetic today? These are words anyone who studied arithmetic will have learned (although we may have forgotten a few). I'll try to present these complex words along with clear, brief definitions. But first, how about arithmetic itself:

Arithmetic study of quantities resulting from combining and separating integers, decimals and fractions;

Arithmetic branch of mathematics concerned with calculating numbers using the operations of addition, subtraction, multiplication and division in a defined order

Go ahead, test yourself and see how many you know:
  • Acute Angle angle that measures less than 90 degree
  • Adjacent Angles angles that are next to each other
  • Adjoining Sides sides that meet to form the angles of a figure
  • Alternate Exterior Angles outside angles formed when a line intersects two other parallel lines
  • Alternate Interior Angles inside angles formed when a line intersects two other parallel lines
  • ante meridiem title for time from midnight up to, but not including, noon
  • Angle two rays or line segments that intersect or have the same endpoint
  • Angle Bisector line or line segment dividing an angle into 2 congruent angles
  • Arc continuous section of a circle’s circumference 
  • Area the size of an enclosed surface, measured in square units
  • Area of a Parallelogram  = base x height; expressed in square units
  • Area of a Rectangle area = length x width; expressed in square units
  • Area of a Triangle area = 1/2 x (base x height); expressed in square units
  • Associative Property: Addition sum stays the same if grouping of addends changes
  • Associative Property: Multiplication product stays the same if grouping of factors changes
  • Average single number that describes a set of values; could be the mean, median or mode
  • Bar Graph chart where bars represent numbers and display data, such as quantities
  • Base a polygon’s side or a solid figure’s face; usually the bottom, after which the figure is named
  • Bilateral Symmetry a figure that when folded along a line of symmetry forms two halves that are mirror images
  • Celsius metric temperature scale with 100 points between freezing and boiling points of water
  • Center point in a circle an equal distance from any point on its circumference
  • Central Tendency  numerical average; center of a set of values; Mean, median or mode
  • Certain an event that will definitely happen; has a probability of 1
  • Chord line segment connecting two points on a circle’s circumference 
  • Circle closed curve with all points equidistant from a fixed point in the center
  • Circular Base a special side of a cone or cylinder that forms a closed curve.
  • Circumference the perimeter of a circle
  • Closed Figure a figure that begins and ends at one point
  • Combination a possible set of events or items
  • Commutative Property: Addition  sum remains the same if order of addends changes
  • Commutative Property: Multiplication product remains the same if order of factors changes
  • Complementary Angles two angles whose sum is 90 degrees
  • Composite Number number with more than two factors
  • Concave Polygon polygon with four or more sides having at least one internal angle greater than 180°
  • Cone 3D figure with 1 vertex, 1 curved edge, 1 circular base and 1 curved surface
  • Congruent Figures figures with identical angles and sides of equal lengths; of the same shape and size
  • Convex Polygon figure whose internal angles are all less than 180°
  • Coordinate Grid area where multiple points may be located by their horizontal and vertical distance from the origin
  • Coordinate Point location on a coordinate grid described by 2 numbers (2, -3)
  • Cost Per Unit money needed to buy a measured quantity ($1.25 per pound or dozen)
  • Cube 3D figure with 8 vertices, 12 congruent straight edges and 6 congruent square flat faces
  • Curved Edge curved line segment formed where a curved surface meets a circular base
  • Cylinder 3D figure with 2 curved edges, 2 circular bases and 1 curved surface