Competition: Math Writing

Would you like your math writing in a book? You might want to check out Plus Magazine's competition. Deadline is January 15th.

Online Course for Middle School Teachers

If you're reading me and not Christopher, check this out. He's teaching an online course for free, during the two-week span from March 17 to 30. If you're a middle school teacher and have the hour a day to commit to it, I bet it will be loads of fun.

Museum of Mathematics opening this Saturday in NYC!

I guess I wouldn't go on Saturday even if I lived nearby - openings are too crowded for me. I think I'd go on Monday. But alas, I live across the country, and don't know when I can get to NYC.

If you're nearby, I hope you'll go, and tell me what you think.

Here's a puzzle from the museum:

• Will the red point always be the product of the green points, for any line crossing the parabola?
• Does it work for other parabolas?

The Numberplay article sets this in a delightful Alice in Math Wonderland context.

How Big Is Infinity? Even Bigger Than the Internet

... which is so big it feels infinite sometimes. I've been following hundreds of math-related blogs for the past few years. How delightful to discover a whole niche of math-related goodies I'd never dreamed of.

Have you heard of 3-D printers? They're a basic part of the plot of Makers, by Cory Doctorow, a sci-fi novel I read a few weeks ago. (I bought the UK edition at my local independent bookstore in California. You can download the book for free at Cory's website.) Those 3-D printers are already real. (I think I knew that, but I haven't seen one in action yet.)

Christian Perfect, over at Aperiodical, has just posted a roundup of cool mathy stuff being made on 3-D printers. You can get designs at Thingiverse if you have access to a 3-D printer (which cost between $700 and thousands of dollars), and you can buy stuff at Shapeways. Christian reported the prices in pounds, so I had to visit the site to see if things were affordable in dollars. Yep. If you'd like to give math-inspired earrings as a gift, you can do it. Or weird dice. Or beautiful little sculptures. (My math-loving niece is getting this one.)

I have to wonder whether Cory had Shapeways in mind as he wrote Makers. This line from Shapeways' webpage sounds like something right out of the book:

"Shapeways is a spin-out of the lifestyle incubator of Royal Philips Electronics. Investors include Lux Capital and Union Square Ventures in New York and Index Ventures in London."

The San Francisco Chronicle has an article on 3-D printers that might help you get oriented. I just spent an hour or so looking for articles and watching youtube videos. I didn't find much that's well-written or exciting to watch, but I have a better basic understanding now of what they are.

After I was almost done ordering those earrings, I saw that they'd ship after the holidays. Apparently some things will ship faster. It depends on the materials. Also, if you do buy earrings, make sure you're getting two; some are priced per earring.

Repost: Holiday Logic Puzzles

This is reposted for your puzzling pleasure from my original post 3 years ago.

The first puzzle shouldn't be too hard. The second ... is tough. It's by Mike Shenk, who has a site called puzzability, and is interviewed here. 

I wish you all peace and joy during these winter holidays.

Holiday Logic 
by Sue VanHattum

1. The Green girl’s favorite Christmas tradition is singing carols.
2. The Brown boy celebrates Kwanzaa indoors.
3. DJ and Jordon joined their friend in her candle lighting ceremony.
4. Layla and Amani joined their friend for his annual walk through the woods.
5. The Gold girl came to Jordon’s house to join his family in their feast.
6. The Fox family celebrate the Yuletide, and Amani comes to their party.
7. Amani couldn’t make it to the Gold family’s Hanukkah celebration.

Each child celebrates just one holiday with one special activity as a tradition in their family, though they do join in the fun with their friends this year. Your mission: Decide who celebrates each holiday, and what they do to celebrate.

Oh Deer! A logic problem by Mike Shenk
(first published in Games Magazine, December 1992)

Twas the night before Christmas, and at the North Pole
The last-minute planning was taking its toll.
As Santa was hastily making a scheme
For the placement of deer in his sleigh-pulling team,
The good Mrs. Claus was crocheting bright bows
To be worn by these reindeer (four bucks and four does).

The ribbons were colored in eight festive hues:
One ocher, one rose, one cerise, one chartreuse,
One maroon, one magenta, one white, and one blue.
(These ribbons helped Santa keep track of who's who.)
The deer pulled the toy-laden sleigh in four rows,
Arranged so no row held two bucks or does.

The order of pullers was changed year by year,
For Santa was thoroughly fair with his deer.
He summoned the elves and instructed them thus:
"Let's hitch up the reindeer with minimum fuss.
The bow on the buck behind Dasher is white,
While Blitzen, a doe, sees cerise to her right.

The blue bow is nearer my sleigh than is Dancer,
But nearer the front of my team than is Prancer.
The doe in chartreuse gets a front-of-team honor,
But not on the same side as Cupid or Donner.
Now Comet stands two spots ahead of the rose.
And three deer of four on the right side are does.

The cerise bow is worn two in back of maroon,
One of which is beside the bright ocher festoon.
Oh-Cupid's in front of a buck, by the way.
Well, that's how they line up for pulling my sleigh.
I trust that you elves, being clever, now know
Each reindeer's position and color of bow."

In no time each colorful ribbon was tied
And the team was hitched up for the transglobal ride.
Can you ascertain where each member fits in?
Who's Comet? Who's Cupid" Where's Donner? And Blitzen?
Who's Dasher? Who's Dancer" Where's Vixen? And Prancer?
With logical thought, you'll determine the answer
And write down the color and place for each deer.
Happy Christmas to all, and to all much good cheer!

Math Girls 2 and Differentiating

Last year, I posted a review of Math Girls, by Hiroshi Yuki. I just discovered that Math Girls 2: Fermat's Last Theorem is coming out on December 12th. The first two chapters are available online, and look just as good as the first book. I love Pythagorean Triples, and that's the topic of the second chapter.

I also wanted to mention the cool math teacher in the book. I like how he offers his best students some personal challenges.

Mr. Muraki was our math teacher. He had taken a liking to us, and would regularly slip us index cards with all sorts of interesting math problems. They rarely had anything to do with our classwork, which made for a refreshing change of pace. We always looked forward to what he would come up with next.

I like this. I think I'll try to do it next semester.



I'll write a more complete review once I've managed to read the whole thing.

Good Calculus Textbook?

My department will be looking for a new calculus textbook over the next few months. We used to use Stewart, but there was some discontent, and we switched in part due to the high price. We've been using Briggs for about two years, and are very unhappy with it. So we want to switch again.

I asked a month and a half ago for a good discrete math text, and Josh suggested Discrete Math With Ducks. It looks fun, and was inexpensive compared to what we had been using. I'm excited that my request for suggestions panned out. (Thanks, Josh!) I'll have more to say about that next semester when I start teaching from it.

That was a decision I got to make on my own. The calculus textbook will be a group decision. The rest of the department will want a more conventional textbook than what I might want. I'm willing to work with whatever textbook we use, but I'm dreaming now of writing my own. (That will take a few years...)

What we didn't like in Briggs:

• The exercises often jumped too quickly to very hard problems
• There's nothing on centroids (until multivariable)

Hmm, I know there's more - I'll have to add to that list next week after our department meeting. I'd like to bring suggestions to the meeting, though. Have any of you used a calculus textbook that you love? Do any of you know of a complete textbook (for Calc I, II, and III, ie going through multivariable calculus) that's under $100?

My department doesn't seem interested in open source textbooks, and the two I used this semester weren't impressive enough for me to want to push it. I love the projects in Boelkins, but that only works if you want to teach through projects. The Guichard made some odd choices. I think any open source book will have more of its own personality than the commercial books. That could be fine, but I haven't seen one yet that will cover all the bases for us.

Ideas?

Centroid (Center of Mass)

This is a topic covered in Calculus II. The textbook explanation is inadequate, and I found nothing good online. So I wrote my own explanation. I now understand it better than I ever did before. (Not surprising, huh? If you're a student, this is an important principle of learning. After you think you understand something, try to write an explanation of it and see how much deeper your understanding can get.)

I'd love to improve this, so please let me know where it's unclear.


Imagine a thin sheet of metal cut in an artistic shape. Is there always a spot where you could hold it balanced on your finger? If there is, can we find that spot? We’ll assume the metal has uniform density. This allows us to treat area as equivalent to weight.

The 1-dimensional Case
To think about this, we first imagine a teeter-totter. We know that two people of the same weight must sit the same distance from the fulcrum (balance point) to balance. We also know that a heavier person must move inward if they want to balance with a lighter person. Experiments show that the weights times the distances from the fulcrum must be equal on the two sides for the teeter-totter to balance. (I wonder if there’s a thought experiment we could do that would convince us this must be true, without the actual experimental evidence.)

Example 1: I weigh 170 pounds and sit 5 feet from the center. My son weighs 75 pounds and sits in front of me, 4 feet from the center. Weights times distances = 170*5 + 75*4 = 1150 feet-pounds. We need someone who weighs 230 pounds to sit 5 feet from the center on the other side. We could write this as:  w₁*d₁ + w₂*d₂ = w₃*d₃

If we change our perspective to a number line below the teeter-totter, with 0 at the fulcrum, then the values on the left will be negative. We won’t have the same equality – we’ll have

 w₁*p₁ + w₂*p₂ = -(w₃*p₃), 

where each d (for distance, always positive) was replaced with a p (for position). This becomes

 w₁*p₁ + w₂*p₂ + w₃*p₃ = 0, 

given that the fulcrum is at 0. But suppose we don’t know where the fulcrum is? Let’s just put our 0 at the left end, and let the former proper place for the 0 - at the fulcrum - be f. Then the equation becomes

 w₁*(p₁ - f) + w₂*(p₂ - f) + w₃*(p₃ - f) = 0, 

or  w₁*p₁ + w₂*p₂ + w₃*p₃ =  w₁*f + w₂*f + w₃*f, 

or w₁*p₁ + w₂*p₂ + w₃*p₃ =  f(w₁ + w₂ + w₃). 

Let W = the sum of all the weights, then we have

f = (w₁*p₁ + w₂*p₂ + w₃*p₃)/W,

which of course extends from 3 weights and positions to n weights and positions.

On to 2 Dimensions
If we use areas instead of weights, we can look for the fulcrum of the x-values (written as an x with a bar over it) and the fulcrum of the y-values (written as a y with a bar over it). For a finite number of small areas, we would get (the same as above)

xbar = (a₁*x₁ + a₂*x₂ + ... + a_n*x_n)/A. 

If we imagine a shape formed by the area under a function f (where f has positive y values), between x=a and x=b, sliced into infinitely many vertical strips, with the area of each vertical strip given by height times width = f(x)^. Δ x, then taking the limit as Δ x goes to 0 gives us

For the y value, we need to notice that the vertical center of each vertical strip is at 1/2 *f(x), and we use this instead of x for the position. So we get

 or

Many times the area we're interested in will not be touching the x-axis, and so we need area between a top function, f(x), and a bottom function, g(x). The height of each slice will be (f (x)- g(x)), making the area  (f(x) - g(x))^. Δ x. The vertical center is now given by averaging f and g. We get:

 

 or

Now if only I could describe this bird shape with functions...

Top Ten Fun Math Books

I wrote about my favorite math books for the Nerdy Book Club site. It's up on their site now.

Have I left out anyone's favorite?

A *Math* Petition?! Yep.

Did you know that the U.S. Federal Government has a website called We the People, where you can post petitions?

This may be the first math-related petition I've ever 'signed'. 

Implement a Policy for Declassifying Discoveries by NSA Mathematicians

The NSA is the largest employer of mathematicians in the United States. Currently, the discoveries of those mathematicians in their official areas of research, being deemed potentially critical to national security, are indiscriminately classified for an indefinite period, with limited circumstances for declassification.
It is requested the White House press the NSA for an expiration policy for the classification status of non-applied discoveries and instituting an expiration for gag order patents in the interest of furthering American academia and industry advancement and in the interest of crediting the discoveries of our nation's talented NSA employees.

Sign here.

Calculus: Anti-derivatives and Area Under a Curve

The textbook we use (Briggs), and I think most of the textbooks I've used in the past (including Stewart and Thomas), introduce anti-derivatives before area under a curve. So they show students the notation for indefinite integrals before showing them the notation for definite integrals. I think this is a BIG mistake.

Here's what happens...

∫ f(x) dx (aka indefinite integral) means find all functions F(x) so that F'(x)=f(x). 

(Why does it use that funny symbol? Why does it have that dx part at the end? Hard to explain without referencing a connection that hasn't been made yet, isn't it?)

And then we start thinking about areas under curves and use a notation that's almost the same.

∫₁² f(x) dx (aka definite integral) means find the area which is between f(x) and the x-axis (area below the axis counts as negative), and between x=1 and x=2.

Seeing almost identical notation and names, we're going to assume that these two act the same in some ways. Students are going to expect anti-derivatives, even though it's area we're talking about here. So it's not much surprise when the Fundamental Theorem of Calculus tells us that to find area we can use anti-derivatives.

Wait! That should be a surprise. It's kind of amazing, isn't it? Derivatives give us slope. Why would going backwards in that process give us area?! Seems to me that's a big one we need to meditate on for a while.

This semester I knew I wanted to connect the new ideas with the position, velocity, and acceleration problems, so I introduced anti-derivatives first. And, I showed the indefinite integral symbol. Oops! I shouldn't have. If I had held off, I believe the meaning of the definite integral would have taken hold better in my student's minds.

Until this semester, I've followed the textbook pretty closely, so my way around this problem has been to introduce the 'Area Function' without using this notation. I found this idea/project in a book put out by the MAA.  I've revised it a lot over the years, but the original author, Charles Jones (of Grinnell College) still deserves credit for getting me started in this direction. (I wish I could figure out how to thank him personally, but he doesn't seem to be at Grinnell College these days, and google gives me lots of people with that name.)

I've put a pdf of the project here. If you'd like my Word file, just email me (mathanthologyeditor on gmail).

We've started that project, and it's going well enough, but I realized that if I hadn't introduced the indefinite integral, we'd be better off. Next semester I'll get that right.

Tomorrow we wrap up the project, and I clarify the implications of the Fundamental Theorem. Cool stuff!

Monk Climbing Mountain Puzzle

Have you seen this puzzle?

A monk climbs from the base of a mountain to its top on the one narrow path up and down, sleeps in a hut at the top, and then descends again to her monastery the next day. She leaves at about 6am on both days, and arrives around 6pm on both days. She stops for a break whenever she feels like it.

Will there be a time of day where she’s at the same spot on both days?

Factor Diagrams

A while back I reviewed You Can Count on Monsters, a delightful book showing monsters built from the prime factorizations of each number 2 through 100. (1 is in the book, but is sad, since it can't be made from primes.)

There are now lots of other takes on this idea. Brent, at The Math Less Traveled, made some gorgeous factor diagrams back in early October. When he posted them, many of his readers took them as inspiration to do more. One made a factor tango. Brent was then inspired to  improve on his own diagrams.  Here's a partial picture of what's he done:

He says he'll be making posters and t-shirts. I think this would make a great poster for a math classroom. Maybe I'll get copies for some of my colleagues.

Proving the Pythagorean Theorem

In a right triangle, where the lengths of the legs are given by a and b, and the length of the hypotenuse is given by c, a²+b²=c². 

We use this so much in math, I have no idea where I first saw it. And it's so simple that I never had trouble remembering it. (The quadratic formula, on the other hand, did not make it into my memory banks until after I had started teaching college. For the first few courses I taught, I had to have it written at the top of my notes.) So I've known and used the Pythagorean Theorem for longer than I can remember.

It comes up in beginning algebra, and for years I showed students how to use it to solve ridiculously artificial algebra problems, never once addressing the issue of proof. This seems terribly wrong to me now. Perhaps about 15 years ago, I realized I'd been 'teaching' this to students for about a decade without even knowing its proof. I tried to come up with a proof on my own and had no idea how to start. Since this was before google became a verb (or even a word), I had to search for a book that would show it. I eventually found it in a high school geometry textbook. Luckily it showed a visually simple proof that stuck with me. (There are hundreds of proofs, many of them hard to follow.)

One of the reasons Pythagoras is held in high esteem by mathematicians is his proof of this idea. It had been used long before Pythagoras and the Greeks, most famously by the Egyptians. Egyptian 'rope-pullers' surveyed the land and helped build the pyramids, using a taut circle of rope with 12 equally-spaced knots to create a 3-4-5 triangle (since 3²+4²=5² this is a right triangle, which is pretty important for building and surveying). But the first evidence we have that it was proven comes from Pythagoras. Ever since the Greeks, proof has been the basis of all mathematics. To do math without understanding why something is true really makes no sense.

Pam Sorooshian is a homeschooler who trusts kids' natural instinct for learning. So she unschooled her kids (who are now grown and doing very well). That means she never required them to learn something they weren't interested in, and never pushed her own interests on them. She has a story in Playing With Math* that really stuck with me. In a talk to other unschooling parents, she said:

Relax and let them develop conceptual understanding slowly, over time. Don't encourage them to memorize anything - the problem is that once people memorize a technique or a 'fact', they have the feeling that they 'know it' and they stop questioning it or wondering about it. Learning is stunted.

It took me decades to wonder about how we know that a²+b²=c². Now I feel that one of my main jobs as a math teacher is to get students to wonder. But my own math education left me with lots of 'knowledge' that has nothing to do with true understanding. (I wonder what else I have yet to question...) And beginning algebra students are still using textbooks that 'give' the Pythagorean Theorem with no justification. No wonder my Calc II students last year didn't know the difference between an example and a proof.

Just this morning I came across an even simpler proof of the Pythagorean Theorem than the one I have liked best over the past 10 to 15 years. I was amazed that I hadn't seen it before. (Maybe I did see it, but wasn't ready to appreciate it.)


My old favorite goes like this:

• Draw a square. 
• Put a dot on one side (not at the middle). 
• Put dots at the same place on each of the other 3 sides. 
• Connect them. 
• You now have a tilted square inside the bigger square, along with 4 triangles. At this point, you can proceed algebraically or visually.

Algebraic version:

• big square = small tilted square + 4 triangles
• (a+b)² = c² + 4*1/2*ab
• a²+2ab+b² = c² + 2ab
• a²+b² = c²



www.cut-the-knot.org/pythag

Visual version:

• Move the triangles around.
• What was c² is now a²+b².
• Also check out Vi Hart's video showing a paper-folding proof (with a bit of ripping). It's pretty similar to this one.

To me, that seemed as simple as it gets. Until I saw this:

from The Step to Rationality, by R. N. Shepard

This is an even more visual proof, although it might take a few geometric remarks to make it clear. In any right triangle, the two acute (less than 90 degrees) angles add up to 90 degrees. Is that enough to see that the original triangle, triangle A, and triangle B are all similar? (Similar means they have exactly the same shape, though they may be different sizes.) Which makes the 'houses with asymmetrical roofs' also all similar. Since the big 'house' has an 'attic' equal in size to the two other 'attics', its 'room' must also be equal in area to the two other 'rooms'.^# Wow!

Added note (6-9-13): I've been asked to clarify why the big house must be equal in size to the two smaller ones added together. Since all three houses are similar (exact same shape, different sizes), the size of the room is some given multiple of the size of the attic. More properly, area(square) = k*area(triangle), where k is the same for all three figures. The square attached to triangle A (whose area we will say is also A) has area kA, similarly for the square attached to triangle B. kA+kB=k(A+B), which is the area of the square attached to the triangle labeled A+B. But kA = a², and kB = b². So k(A+B) = a²+b². And it also equals c², giving us what we sought, a²+b² = c².

I stumbled on the article in which this appeared (The Step to Rationality, by R. N. Shepard) while searching on 'thought experiment weight times distance must equal to balance'. I'm working on a handout for my Calc II students to explain centroid (since the Briggs textbook leaves this topic out). I was wondering if we need experimental evidence to show that the two sides of a teeter-totter will balance only when the weights times distances from the fulcrum are equal on the two sides. I thought maybe we could come up with a thought experiment that would convince us it must be true. I wasn't having any bright ideas, and turned to google. It hasn't solved my centroid question yet, but I love what I discovered.

I think that, even though this proof is simpler in terms of steps, it's a bit harder to see conceptually. So I may stick with my old favorite when explaining to students. Or maybe there's a way to test out which one is more helpful for a deep understanding of both the notion of proof and this theorem in particular.

What do you think? 





____________
*Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers is a collection of great writing about math education (often outside the classroom), from over 30 authors. I've been working on for 4 years now; it will be available within 2 to 4 months. 
^#I got this language (of houses, attics, and rooms) from a similar description of this proof which I found on Cut-the-Knot.

Maths and Stats by Email (CSIRO)

About two and a half years ago, I signed up for this online newsletter. I have enjoyed most of the issues quite a bit. There's no link to head you over to the latest issue. They don't do it that way. (I have no idea why not.) You pretty much have to sign up for it if you want it.

This month's issue is typical. A simple topic, explained well for young people:

Take a sheet of A4 paper and measure its sides. A4 is 210 millimetres wide and 297 millimetres long. It’s probably the most common size of paper and it’s used in most countries. However, A4 side lengths aren’t simple numbers like 200 or 300 millimetres. So why don’t we use something easier to measure?

If you take a sheet of paper and cut it halfway down the longer side, you end up with two new pieces of paper. These pieces of paper each have half the area of the original sheet, but they are the same proportions as the original sheet! There’s only one type of rectangle that has this ability. Because these half sheets have the same proportions as A4, they also have a name – A5.  If you cut an A5 sheet in half, you get two pieces of A6 paper, with the same proportions as A5 and A4. All these paper sizes are part of a set called the A series.

This pattern also works if you want to go bigger instead of smaller.  If you take two sheets of A4 paper and stick the long sides together, you’ll end up with a sheet of paper that has the same proportions as A4, but is twice as big. This size is called A3. You can use the same process to make A3 sheets into A2, and even A2 sheets into A1 paper.

So why is A4 paper called A4? A4 is half an A3, or one quarter of A2, but more importantly, it’s one sixteenth of A0. A0 has an area of one square metre (but it isn’t a square), and every other paper size in the A series is based on A0. We use A4 for writing on because it is a lot more convenient than trying to write on a square metre sheet of paper!

After the introductory article, they always have a 'try this' activity. This month's is on tangrams (and the activity relates tangrams to the paper sizes described).

Let me know if you decide to sign up.

Seeking Math-Poets for a Reading (JMM, 1-11-13)

JoAnne Growney just posted this on her blog (Intersections - Poetry with Mathematics):

Call for Readers:
     The Journal of Humanistic Mathematics will host a reading of poetry-with-mathematics at the annual Joint Mathematics Meetings (JMM) on Friday, January 11, 5-6:30 PM in Room 1B, Upper Level, San Diego Convention Center.  If you wish to attend the reading and participate, please send,  by December 1, 2012 (via e-mail, to Gizem Karaali (gizem.karaali@pomona.edu)) up to 3 poems that involve mathematics (in content or structure, or both) -- no more than 3 pages -- and a 25 word bio.

There's more at her post, check it out.

I'll be co-hosting this event, and would love to meet you there. (I hope to read a poem or two of my own. I'd better get my submission in.)

Good Text for Discrete Math?

I'll be teaching Discrete Math next semester, for the first time ever. (Kind of surprising I waited this long, since many of the topics are dear to my heart and the whole course looks like playing with math to me.)

We've been using Discrete Mathematics and its Applications, by Kenneth Rosen. Cheapest used I see on Amazon is $92. My colleague likes a book by Washburn, Marlowe and Ryan, but it was published in 2000, and I don't see any newer editions online. It looks like it's probably out of print.

I have a few days to pick a textbook. Does anyone know a good one? Are there any open source texts? (I see one they use at UCSD.)


Edited to include the goodies:

• Exeter has a Discrete Math problem set. I'll probably use many of their problems as supplementary material.
• Discrete Math with Ducks, by sarah-marie belcastro, is about $60, and looks pretty good. I've chosen this for my course.
• John Golden pointed out another interesting book, Applied Combinatorics, by Keller and Trotter. Mitch Keller just changed it to a Creative Commons copyright. Thanks, Mitch!

Kindle Book: Let's Play Math!, by Denise Gaskins

I love Denise's blog, and now I'm loving her book. She wants people to understand that this is a beta version, which means that it will be even better when her official first edition comes out. She hopes to have it out in print next year.

But if you don't mind reading electronic books, it's only $3.99 for the Kindle version now. (You don't need a Kindle; you can read these online or on your computer.) Once she gets it perfected, I'm sure the price will go up.

Here's a peek at part of her chapter on home-made manipulatives:

Or put the number line at a slant, like a hill on which you can run up and down. In many applications of math and physics, “zero” is an arbitrary location, based on whatever makes our problem easy to solve.

In this case, you might invent a peasant who lives in a small hut at 0, partway up the hill, and dreams of becoming a knight. At the top of the hill is a castle, and at the bottom there is a cave where a terrible dragon lives. Take turns making up story problem adventures.

I don't know when I'll have time to savor the rest of this book, but I'm definitely looking forward to it. It'll be even nicer to have the print copy in my hands one day. I'll be waiting...

Math Stories: Special Triangles

Two weeks ago I was introducing the special triangles (with angles of 45-45-90 and 30-60-90) in my pre-calc class, and on a whim, I asked them to make up stories with those two triangles as characters. Only about a dozen students did the assignment, but what they created was really fun to read. A few students have given me permission to share their stories here. I'm sharing those 3 stories below. I may get to share a few more later.

 This post is my birthday present to myself. Anyone else want to share some math stories?

Different Views of Shapes, by Aldrich Pablo

Once upon a time, there was a square name Geo. Geo was a hard working square who worked in the slaughterhouse. Geo loved his work. He loved his work because his wife, Tri, an equilateral triangle, also worked at the slaughterhouse alongside of him.  Like any other day, both Geo and Tri cut fellow square and triangular shapes vertically and diagonally, making sure they cut them into 2 different types of triangles, a 45®-45®-90® triangle, derived from the squares, and a 30®-60®-90® triangle, derived from the equilateral triangles.

One day however, Geo got accidentally pushed into the cutting machines by his enemy, Bre, the circle. With the affectionate love Tri had for Geo, Tri jumped in and tried to save Geo. But it was too late. Both Geo and Tri have been sliced diagonally and vertically. Geo became a 45®-45®-90® triangle, and Tri became a 30®-60®-90® triangle.

Even though Geo and Tri got sliced in half, they both wanted to do the same to Bre. With all the mixed emotions both triangles have, they were able to learn something new about each other. Geo, the 45®-45®-90® triangle, learned that he still has the same leg lengths, but half of the original shape. Tri, the 30®-60®-90® triangle, learned that she is just half of the original shape, forming different angles.

With previous shapes that have been sliced by the workers, Geo and Tri were also able to understand that with their new shapes, multiple of the same shapes together form a circle. Bre was horrified. Once Geo and Tri were able to understand their new shapes, they too, pushed Bre into the slaughter machines. Because of a malfunction when Bre went into the machines, Bre got stuck, and the machine exploded.

When the machine exploded, both Geo and Tri saw him fly into the air. When both triangles went to go look for him however, Bre was nowhere to be found. They believe that Bre became the sun, and he was never to be seen again. So every time when Geo and Tri go to work at the factory, they will always remember Bre, as they look towards the sun.

The End.

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The Story of the Special Right Triangles, by Miranda Barron

            Once upon a time, there were two special triangle cousins, one named Mr. 45, full name 45-45-90 Triangle, and the other named Ms. 30, short for 30-60-90 Triangle. These two cousins both inherited their family’s 90° angle trait yet they each were very different and special in their own way. Mr. 45 had two legs that were equal lengths, meaning he also had the equal corresponding angles. If Mr. 45 could walk he’d walk like a human. Hard to believe, isn’t it?  Well Ms. 30 wasn’t so lucky as to have equal lengths. She had to constantly ask for help from her cousin. It was a hard life for her, but she never let it bring her down.

            The reason she learned to live with it was when she had kids. Each kid had different sized legs just like her. They even had the same proportion. She was always able to find the sizes of pant legs for her kids without having to measure each child’s leg, since it made them feel self-conscious. Each child, like her, had a proportion that had to do with one leg being x inches, then the other leg was x√3 inches, to go with the rest of their body that was 2x inches. It was the magical method to go with their cursed life.

            On the other hand, Mr. 45 and his kids weren’t so unique. He still found a way to make his children seem special with their normal equal length legs. He found that each of his kids were proportioned like him. Each of their legs would measure x inches to go with the rest of their body that was x√2.

            This seems like a weird story but it shows how unique and special you can be when you’re different. Mr. 45’s kids never got their pants personally made, one reason being their dad refused to sew, while Ms. 30’s kids got to have personally made pants from their loving and caring mother. Being special is amazing and that’s how things should be for everyone and everything. :D

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A Special Triangle, by Ayesha Saleem

Isis is a triangle and her friend, Trinity, is a triangle who is also a conjoined twin. She is conjoined with her brother and together they are an equilateral triangle, which has three congruent sides and angles. About two weeks ago, Trinity and her brother underwent a rigorous surgery to get separated from one another. They had to stay in the hospital for a little over a week so that the doctors could keep an eye on their recovery. The siblings were allowed to go home yesterday, and today Isis and Trinity went to the park to hang out. Isis wanted to see how her best friend was and how she looks now that she is not connected to her brother by one side.

            They met at a local park where they used to go to a lot when they were younger. Isis was so shocked at how differently Trinity looked. She was so happy that she has been recovering well and that she is happy with the surgery. Trinity brought along a measuring tape and a protractor. She wanted Isis to help her measure her sides and angles since she didn’t know what their measurements were anymore.

            Isis wanted to start by measuring her sides, so they started with Trinity’s base. They measured in across the bottom to be one foot long. Then they measured her hypotenuse, which was two feet long. Now they had to measure her height. They measured it and it came out to be a weird, decimal number. So, to be more exact, they decided to use the formula, a²+b²=c², to find the last length. They found that Trinity’s height was √3 feet. They found this by doing the following:

a²+b²=c²

a²+1²=2²

a²+1-1=4-1

√a²=√3

a=√3

            Next, they moved on to measuring her angles. They stared with the angle made between her height and base. They used the protractor and measured that it was a 90^o angle, or a right angle. Then they measured her bottom right angle and found it to be 60^o. Since Isis couldn’t reach up to the last angle at the top of Trinity, they decided to do it mathematically. They already knew that all triangles have three angles that will always add up to a total of 180^o. They found that the last angle is 30^o by doing the following:

 

90+60+x=180

150-150+x=180-150

x=30

            After doing all the measuring, they discovered that Trinity is now a Special, 30-60-90, Triangle. Trinity was so surprised; she didn’t think that she deserved to be a Special Triangle! Isis congratulated her and she was happy for her best friend. Trinity suggested that they measure Isis and maybe that she is also a Special Triangle, but Isis said she already knew that she was an Isosceles Triangle, meaning she had to equal sides and two equal angles. Isis was fine with being an Isosceles Triangle and Trinity was happy to find out that she was a Special, 30-60-90, Triangle.

 

Apps: Dragon Box

I don't have an ipad or an ipod touch myself, so I don't know much about apps yet. But lots of the folks on Living Math Forum are talking about how much fun the Dragon Box App is, and how much algebra their kids are learning from it.

Meet the New Bloggers (week 4)

The last installment... I hope you've subscribed to your favorites. Introducing Kyle, Maggie, Erin, Jillian, Kate, Nate, and (once again!) Algebrainiac.



Kyle Harlow (@KBHarlow), blogging at War and Piecewise Functions, wrote Just Some Cell Phone Photos From Denver. His summary: Spent last weekend in Denver, CO and went to the Broncos game.  Here are some pictures from my trip.

The hotel we stayed at had a restaurant called Pi Kitchen + Bar.  Its menu was a circle, and happy hour was from 3:14p to 6:28p.

Maggie Acree (@pitoinfinity8), blogging at pitoinfinity, wrote PreRequisite Knowledge/Rev. Her summary: This post is about reviewing concepts and what teachers do for reviewing. I used to spend a six weeks or so reviewing concepts from previous years, but it really did little to no good, so I have a solution I have found that has worked well for me.

No matter what, I am done reviewing concepts for the first six weeks.

What Maggie calls 'bell ringers' I call 'warmups'. I like her idea of using warmups to do the necessary review of concepts we wish our students already had down.



Erin Goddard (@ErinYBaker), blogging at Math Lessons on the Loose!!, wrote Thanks Blogger Community . Her summary: I related to mathemagicalmolly's blog. Only a teacher knows what a teacher's night sleep is like.

I always saw the real benefit of taking [from other blogs], but I learned the true benefit of reading, relating, learning, and also giving back hopefully as much as others have given me.

How nervous are you at the beginning of the school year?



Jillian Paulen (@jlpaulen), blogging at Laplace Transforms for Life, wrote My Math Autobiography (a week late). Her summary: I’ve always assigned a “Math Autobiography” to my Geometry students and I’ve really enjoyed reading them. But I’ve never written about myself! So here’s my (long) story.

My love for math has only grown since I started teaching, and I hope I can continue for a long, long time.

Her math ed course seemed too fluffy. I hear that. I wonder if there's a way to draw in the math talent in the math ed courses.



Kate (@fourkatie), blogging at Axis of Reflection, wrote Grade/Age Equivalents are NOT numbers!. Her summary: For my final post for the new blogger initiation I opted to write about what was on my mind. Today that was the use of age and grade equivalents by a special educator in a report. I stepped on my soap box to rant about why age/grade equivalents are NOT numbers and therefore should not be treated like they are.

Because they are not real numbers, so you can't do math with them like they are real numbers.

Do any one or two summary numbers really tell you anything important about a student?



Nate Gildersleeve, blogging at Hard Enough Problems, wrote Visual Multiplication. His summary: This post talks about a visual multiplication lesson I did, and what my rationale was for it.

It is that I want to use this as a way to practice and learn several things: the idea that there are multiple valid ways of doing something; if those ways do the same thing they are connected in some way; and by talking about these connections we gain a deeper understanding of whatever we're covering.

I'd like to hear more about the connections involved in doing the same thing two different ways.



Algebrainiac (@algebrainiac1), blogging at Algebrainiac, wrote Open House/Curriculum Night. AB's summary: I posted my plans for the set up of my room for 8th grade Open House, which was a new format for us this year. I included links to handouts and files I prepared.

The main difference I have seen so far is that the Open House format seems to require more front end work to prepare, but I don’t imagine I will leave as tired and drained as in past years.

I hope it went well. AB sure put in the preparation!



Maybe by next year, we'll have enough teachers at each level that we can split off. I'd love to review a half dozen new college math teachers' blogs.





_____ 
The round up of week four is at these blogs:: Julie, Fawn, Anne, Megan, Bowman, Sam, Lisa, John, Shelli, Tina, and Kate. And a roundup sorted by grade level taught (for those who responded to Julie's survey) is here.

Get Ready: September 25 is Math Storytelling Day

Maria Droujkova made it up in 2009, as a birthday present to herself. September 25th is my birthday too, and I loved the idea. So now I'd love to get lots of presents - math stories you all write. Here's my post on Math Storytelling Day 2 years ago. (I must have been too busy last year?)

If you'd like some inspiration, read The Man Who Counted, The Number Devil,  or Math Curse.

Poems are good, too.

In Seattle? Fun opportunity for Elementary Teachers

I love Dan's blog (Math for Love), and have met him in person. If you get a chance to do this free Math Teacher Circle, I think you'll love it.

Thanks for Reading Math Mama Writes ...

... I hit 100,000 page views late last night.

Planning Calculus: What Is My 2nd Unit?

A few weeks ago, just before the semester started, I had unit 2 as calculating the derivative, and unit 3 as using the derivative, much like Math Boelkins does. It seems like a logical division. But now I'm thinking it's not a good way to frame things pedagogically. We would spend way too much time coming up with 'rules' and learning to use them. That would get the students so stuck on the procedural that I'd never be able to get them to go back to trying to make sense out of these new ideas.

I want to introduce the 'rules' in the context of real problems. I don't have the facility with equipment that Shawn Cornally does. If I could spend a year as an intern in his classroom, I think I'd take lots of his ideas back home with me. But every time I think about using lab equipment, I get nervous. (We used a lab thermometer in pre-calc to measure the temperature of a hot cup of coffee during our murder mystery. And we built inclinometers with a protractor, tape, a straw, thread and a weight. That's about the extent of 'lab equipment' for me so far.)

To rethink this, I started with the sections of our textbook (Briggs Calculus) that I'm supposed to 'cover':

3.2: rules (constant, sum, power, constant multiple, e^x!)
3.3 product rule
3.4 trig function derivatives

3.5: derivative as rate of change

3.6: chain rule
3.7 implicit
3.8 derivative of log and exp fcns
3.9 derivative of inverse trig fcns

3.10 related rates

4.1 maxima and minima

4.2 what derivatives tell us

4.3 graphing

4.4 optimization

4.5 linear approximation and differentials
4.6 mean value theorem

[3.1 was part of the first unit, and the last two sections of chapter 4 will be part of the last unit.]

I played around with separating the 'rules' and the 'applications', and it seemed like there were only 4 applications (highlighted, with 4.1 to 4.3 as one). I'm not sure why the text splits things up the way it does. Graphing could come earlier, and seems like a simple way to begin seeing what derivatives can do for you.

In the first week of class I had the students read some history from Morris Kline's Calculus: An Intuitive and Physical Approach (pages 1 to 6). He gives 4 major problems that needed calculus (no wonder two people invented it at once):

1. Motion:  planetary & projectile,
2. Tangents to curves: for projectiles (will it hit head on?) & for lenses (telescopes and microscopes),
3. Optimizing (best angle to shoot a cannon, when will a planet be closest and farthest from sun?), and
4. Lengths of curves, areas, volumes, also center of gravity (for example, the volume of the earth, which is an oblate spheroid; these topics are mostly delayed to calc II).

I'm wondering if any of these historical applications are useful to bring into the mix.

What I have for now is:

App1: Rate of change & Graphs
Rate of change:
3.5 derivative as rate of change, velocity, rate of growth, cost (we mostly did this and this can be review)

In pre-calc, we factored to find x-intercepts of polynomials. What we couldn’t find was the maxes and mins, which are often pretty important.
3.2: rules (constant, sum, power, constant multiple, e^x! skip e^x for now)

4.1: maxima and minima
4.2: what derivatives tell us
4.3: graphing


App2: Sound!
3.4 trig function derivatives
3.3 product and quotient rule  (Shawn C uses 1/x * sin x to model guitar string)


App3: Differential Equations & Growth, Related Rates & Optimization
Infection (logistic application by Bowman D)
e^x
3.6: chain rule
3.7: implicit
3.10: related rates
3.8 derivative of log and exp fcns,
3.9 derivative of inverse trig
4.4: optimization

I still need a home for:

4.5 linear approximation and differentials
4.6 mean value theorem

This would make 3 units instead of two. That might be fine. I have to decide today, and get a unit sheet made by this evening. I keep wishing I could teach for a week and prep for a week. (Dream on!) But as hectic as this is, I'm having fun working on getting it right. And I know it will go more smoothly next semester.