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SEQUENCES

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Photo courtesy of Chibijosh

On Monday, I introduced you to Elizabeth Perkins, an up-and-coming glass artist in Seattle.  (She also happens to be one of my former students, but that is mere coincidence. I take no credit whatsoever for her success and talent.)  In her interview, she mentioned that she depends on the Fibonacci sequence to develop some of her annealing programs, or processes for cooling the glass so that is remains structurally sound.

But what the heck is a Fibonacci sequence?

Well, it’s a pretty cool list of numbers. And it’s also really, really easy to figure out. See for yourself:

0, 1, 1, 2, 3, 5, 8, 13, 21, ?

What’s the next number?

I’ll give you a chance to think about it.

Need a hint? Pick any number in the list (except for the first 0 and first 1), and look at the two numbers before it.

Get it yet?  (The correct answer is 34.)

The Fibonacci sequence is generated by adding the last two numbers together to get the next number.  Take a look:

0 + 1 = 1

1 + 1 = 2

1 + 2 = 3

2 + 3 = 5

3 + 5 = 8

5 + 8 = 13

8 + 13 = 21

13 + 21 = 34

Now that you know this rule, you could conceivably add numbers to this sequence until you got bored or exhausted (which ever comes first).

The fellow who discovered this sequence was, you guessed it, Fibonacci — an Italian mathematician and philosopher who was reportedly born in 1175 AD.  But to be honest, his sequence is not the greatest contribution Fibonacci (or Leonardo de Pisa) gave to humankind.  In fact, he is the father of our decimal system.  Yep, the fact that you can count the $5.23 you have in your wallet is due to a guy whose real name we don’t even know for sure.

But I digress.

The Fibonacci sequence isn’t just an easy and cool math fact.  It’s cool — and really, really important — because it shows up everywhere.  Here are just a few examples:

If you count the petals of various species of daisies, you’ll get one of the Fibonacci numbers.

The length of the bones in your wrist and hand are a Fibonacci sequence.

The spiral of a pineapple is arranged in Fibonacci numbers.

Branches of a tree grow in a Fibonacci sequence (one branch, two branches, three branches, five branches, and so on, moving up the height of the tree).

The gender of bees in reproduction mirrors the Fibonacci sequence.

Photo courtesy of Mr. Velocipede

And then there’s art.  Art loves the Fibonacci sequence.  Since the Greeks formalized what is beautiful in architecture and paintings, this little list of numbers has been front and center in a variety of artistic fields.

For example, this seven plate print is gorgeous and also represents something called the golden spiral.  The sides of each square (starting in the center with the smallest squares) correlate to the numbers in the Fibonacci sequence.  So, the smallest square has side length of 1 unit, the next largest is 2 units, the next is 3 units, the next is 5 units, etc.

Cool huh?

It gets better.  Remember the lady with the mysterious smile?  Leonardo da Vinci was fascinated by mathematics, and some folks have noticed that his lovely lady’s facial characteristics follow the path of the Fibonacci sequence.

Image courtesy of www.shoshone.k12.id.us

Do you see how the squares line up with the base of her eyes and  bottom of her chin, and surround her nose perfectly?

So there you have it.  What we see as beautiful could very well be because of mathematical wonders like Fibonacci’s sequence.  And as Beth the glass blower shows, this magical little list of numbers is useful in the science of making art as well.

Earlier this year, I posted a really, really cool video about the Fibonacci sequence in nature. Check it out here.Save

It’s the perennial question from students of all ages: “When will I use this stuff?” So when tutor, Ryan faced this query (probably for the upteenth time), he took to the streets to find the answer.  What he found is in the video below:

And of course I have some thoughts — for teachers and students.

It is absolutely true that series (that’s what the funny looking E — an uppercase sigma — means in this problem) are not the stuff of ordinary folks in non-science fields.  But they’re not as difficult as they seem.  It’s the notation that’s confusing.

Skip this part, if you don’t really want to do any algebra today.

A series is just the sum of a sequence (or list) of numbers.  That’s it.  Nothing more, nothing less.  So when you have

sum{n=1}{7}{3n-1}

you’re simply saying, “Find the sum of the first 7 values of 3n-1, where the first value of n is 1.” In other words: 2 + 5 + 8 + 11 + 14 + 17 + 20 = 77.

Now back to my opinions.

Okay, so I don’t need to know what a series is in order to visit the grocery store or get a good deal on a car or even figure out how much I earned this year over last year.  But here’s what I wish some of those folks who were interviewed for this video had been able to say:

“That funny-looking E is a Greek letter, right?”

“Doesn’t this have to do with adding things together?”

“Hey, I dated a girl from {Sigma}{Sigma}{Sigma} once!”

And second, this tutor did pick a humdinger of a problem to focus on.  Series (and their brothers, sequences) are not the main focus of any mathematics course.  But honestly, they wouldn’t be taught if they weren’t useful somewhere.  And boy-howdy are they useful!

So, here are a few ways that real people do use series in their real jobs (courtesy of Algebra Lab and Montana State University:

1.  Architecture:  “An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on and has 30 rows of seats. How many seats are in the theatre?”

2.  Business: “A company is offering a job with a salary of $30,000 for the first year and a 5% raise each year after that. If that 5% raise continues every year, find the amount of money you would earn in a 40-year career.”

3. Investment Analysis: “A person invests $800 at the beginning of each year in a superannuation fund.  Compound interest is paid at 10% per annum on the investment. The first $800 was invested at the beginning of 1988 and the last is to be invested at the beginning 2017. Calculate the total amount at the beginning of 2018.”

4. Physics: “The nucleii of a radioactive isotope decay randomly. What is the total number of nucleii after a given period of time?”

And this brings me to some additional news of the week.  Sol Garfunkel (Consortium for Mathematics and Its Applications) and David Mumford (emeritus professor of mathematics at Brown) made a bit of a splash on Wednesday, with an editorial in the New York Times: How to Fix Our Math Education.

Their proposal is that we teach tons of math that applies to everyday life — and focus on those applications. (Yay!) And we ditch “highly conceptual” math for folks who won’t need it for their jobs. (Boo!)

Hopefully, you’ve already identified the problem: How do we know if a kid won’t decide to go into physics or engineering or high school math education? Hell, how do we even attempt to lure them into these fields, if they don’t see the math at all?  (And by the way, physics, engineering and applied mathematics were recently identified as the top-paying degrees in the U.S.)

Look, I empathize with the student who isn’t interested in what any of the Greek letters mean in math class.  And I think it’s true that most folks won’t use these skills at all after high school.  (It is worth mentioning that everyone depends on series in their daily lives–they just don’t see the math.) But my response to the kid who asks, “What’s this good for?” is to tell him where it can be applied.

And if he says he won’t be going into any of those fields, I would say, “Suck it up, cupcake, because you’re too darned young to know for sure.”

Please share your thoughts in the comments section.  Do you agree that these concepts should be taught in high school, even though most kids won’t use them in their everyday lives? How do you think we should encourage more students to go into science, technology, engineering and math (STEM) fields?