Saturday, December 24, 2011

Closest packing of Christmas cookies?

I was cutting out some cookies the other day, and noticed that my method wasn’t really that efficient.

DSC04577

So I started using this pattern, and challenge you to find an even more efficient way to pack stars in a plane.

DSC04578

Tuesday, December 20, 2011

A quick fractal Christmas Tree in Illustrator

Step 1: Make a thing. 
Fullscreen capture 12162011 113812 PM
I went with a leaf shape.

Step 2: Effect > Distort & Transform > Transform.  Make sure to preview, then make some copies of that leaf and scale ‘em down.  Add a reflect X, scale dimensions down to 90%, and put some Move Vertical, and with some tweaking you should get a fern-looking thing.
Fullscreen capture 12162011 113912 PM
Why does this remind me of the geometric series elves?

Step 3: Object > Expand Appearance so that the effect becomes materialized.  Select all, right-click and group that sucker.
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Step 4: Now for the fractal part: iterate!  I rotated my tree-looking thing 90ยบ clockwise so it resembled the initial leaf.  Then I repeated the transformation to make a bunch of these tree things with a scale factor of 90%. 
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With 50 copies of a shape made of 50 parts, you’ll notice your computer start to slow down a bit.  I decided to stop while I was ahead… not a perfect fractal, but good enough for your Christmas card.

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Friday, December 16, 2011

Geometric series in Adobe Illustrator

There’s a nice feature in Adobe Illustrator that makes it easy to visualize exponential change.  Here’s a little example to show your students how

\displaystyle\sum_{i=1}^\infty (\frac12)^i = 1

Start by drawing something… in this case, an elf.  Say he has a height of one unit.  Select your drawing, right-click, and group it.

Fullscreen capture 12162011 112748 PM

Now hit up the Effects > Distort & Transform > Transform

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Here’s where it gets cool.  First click Preview, then put a number in the Copies box (I used 100 to approximate infinity).  This will create a bunch of copies of the original, but since they’re all the same size, you can’t see them.  Change the horizontal and vertical scales to 50%, and space out the elves by increasing the Move>Horizontal slider.  Drop the vertical slider so that the elves are on the same baseline.

Fullscreen capture 12162011 113025 PM

You can now see that each elf is half the height of the previous.  This is a geometric sequence.  To find the value of the series, we need to add the heights of the elves, which we can do by adjusting the horizontal and vertical sliders to stack the elves on top of each other (make sure to have students make a prediction first).

Fullscreen capture 12162011 113246 PM

All that’s left is to see how tall that stack of elves on top of the first one is.  Remember that the big elf is one unit, so let’s use him as the measuring stick.  Copy and paste the big elf, then turn off his transformation effect and drag him up to measure the series.

Fullscreen capture 12162011 113332 PM

Students should see that the sum of the heights of all the little elves is equal to 1 big elf, which is to say 1/2 + 1/4 + 1/8 + … = 1.

Sunday, November 20, 2011

How tall is Yertle the Turtle?



That ego-maniacal Yertle, proclaiming himself king of all that he sees, calls for a stack of 500 turtles to sit upon.
And now Yertle the Turtle was perched up so high,
he could see forty miles from his throne in the sky



So tell me: how tall is each turtle?  Assume the earth is a sphere with radius 4000 mi, that Yertle and the reset of the turtles are a uniform height, and that Yertle's eyes are on the top of his head.  The first correct solution (posted as a comment) will earn a cool drawing of the solver on top of Yertle.

Tuesday, November 8, 2011

Flip the Sign

Katie Cunningham, a NBCT friend of mine at an alternative school in Washington helped her students make a music video to aid their solving of inequalities. Warning: you may have it stuck in your head for the rest of the day.

Saturday, November 5, 2011

Million-dollar division problem

When pressed, I'll reluctantly admit to be a sucker for reality shows.  While catching up on The Amazing Race, I came across this scenario.

Brian and Ericka are presented with a choice: Construct 12 hookahs or weigh out $500,000 worth of gold using the current price per ounce.  They choose the latter (wouldn't you?).  They find themselves in a room with a lot of gold, a scale, and a screen telling them the up-to-the-minute price: $934.75 per ounce.



The first thing they need to figure out is what to do with the numbers $500,000 and $934.75.  Fortunately they are able to answer this (another team had a harder time), but now they're stuck with a long division problem, something with which former Miss America doesn't feel confident: "My American education has dumbed me down to use a calculator for everything."  Plus, the rate just changed: $941.25/oz.



Now, take a moment to appreciate that a long-division problem is standing between them and their goal of winning $1 million.  Ericka had a point: how many students are graduating high school with competency in arithmetic?  I'm guilty of allowing too much calculator use in my classroom as well, having uttered justifications like, "It lets students stay focused on the higher-order concepts..."  This may be true and appropriate at times, but my students would have benefited from more arithmetic thrown into the mix.



Have your students work against the clock, giving them a new divisor every so often.



Poor Brian and Ericka eventually gave up - is that what your students would do?

Thursday, October 20, 2011

Play dough math

I've been having fun with play dough lately, making stuff like this with my kids:

I think it would be fun to do in the math classroom with younger students, as questions like this pop up:
You need to make a 30-second video, with 24 frames per second, and each image takes up 4 frames - how many images do you need?
Having the dinosaur not walk like a spaz could be done by taking into account the length of time for each step and the rate the images are being played back. I didn't do this, as you may have noticed.
You could show the application of quadratic functions by having students try to make a projectile look realistic in claymation. Maybe I've spent too long around the play dough.

Friday, October 14, 2011

Free math poster for your classroom!

I noticed this little guy on a wall as I was walking along the other day, and thought about the difference math education makes in perceiving the snail.



A snail builds its shell at a rate proportional to its size (see Nature's Numbers), resulting in exponential growth. An understanding of the exponential function is so important in today's world (see examples of its application) that I was inspired to make a poster. Feel free to print it for your classroom, and let me know what you think!

Click here to download the 24"x36" 300dpi version (12mb).


Creative Commons License
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported License.

Wednesday, October 5, 2011

Flash Math Art

Without an Internet connection in our new apartment, I was inspired to be creative.  Here are a few experiments with Flash, using math to make art.

1. Paths are generated by altering the 2nd-derivative of the x and y components.  It's not as tricky as it sounds.  A common way to simulate gravity (which has a constant acceleration, or second derivative) is to adjust position as follows.
x = x + dx;   //just increasing x by the amount dx in each time step.
dx = dx + g; //just increasing dx (velocity) by a constant amount (acceleration) in each time step.

In the case here, the amount g changes sinusoidally (for no particular reason), making the paths wiggle around and do weird stuff. There's some more stuff in there (I forget why they sometimes just go straight), but remember - but there's no randomness in how the paths are made (they are deterministic). Click to start:

Click to stop the first one so it doesn't slow you down.
2. Similar idea as the first one, but this time not as smooth. The value g changes as the remainder when the time is divided by some value (modulus).

3. This time the x and y derivatives make zig-zag graphs, so the x and y positions are smooth curves (almost like sinusoids). I call these guys Lissajou Worms:

4. The last one got me thinking about Lissajou curves (x and y are sine functions of time), so I started with this:

Click to stop, and again to restart with new conditions. Watch the x position go all the way left, then all the way right, and the y position go all the way up and all the way down. So we have two functions to control position: x(t) = sin(b*t + c) and y(t) = sin(d*t), where b, c, and d are random above.  When one of the periods is a multiple of the other one, you get a static shape like one of these.



We'll come back to this in a minute.

5. What would make these shapes more interesting?  Both x and y have a constant amplitude, so maybe we should mix that up a bit.  I added another component to the x function - now x(t) = f(t) + g(t), where f and g are both sine waves.  Watch for y continuing to go all the way up and down, while x does more complicated stuff:

6. What would it look like if y were the sum of two waves as well?  Well, it looks like craziness.

7. To tone down the craziness, I made sure the periods coincided better.  Here, if
x(t) = f(t) + g(t)
y(t) = j(t) + k(t)

then the period of f is a multiple of j's period (or vice versa), and g and k have the same period.  That way we get a periodic curve of some kind:

Click twice to get a new design. I hope you enjoyed your dose of math art!

Friday, September 30, 2011

In the free box

My family and I recently moved.  We had a "free box" out in front of our house, where we would throw miscellaneous junk that we didn't want to try to sell.  Almost everything in the free box was eventually taken, but one straggler remained: a handmade screen print of one of Ramanujan's formulas for the value of pi.  What is the world coming to?

Sunday, September 25, 2011

Cylindrical perspective



My favorite Escher print is called "House of Stairs," which shows a tiled room inhabited by silly creatures called wentelteefje.

The image makes us of cylindrical perspective, the effect you would get if you projected the world onto a cylinder with your eye in the middle. I noticed this in a panoramic picture of my son and me, taken by a friend.

(The Escher-inspired t-shirt is pure coincidence).

The book M.C. Escher: Visions of Symmetry gives nice explanation of the effect (among the many other mathematical aspects of the art). Upper-level students may be able to generate a "proof" that parallel lines appear as sinusoids, while other students might be able to use sinusoidal templates to draw images of their own. Here's a video of someone doing just that...

Tuesday, August 30, 2011

Which is worse: curriculum or instruction?

As the US continues to fall behind other industrialized nations in math education, there is no shortage of frantic finger-pointing to help us out. But isn't there a simple solution? Japan is scoring higher on the TIMSS assessment, so perhaps we would do well to nationalize education, get students in class 20% longer, and somehow reconfigure our popular culture to value and encourage academic excellence. If our students' test results directly determined their quality of life, they'd probably study harder.

Looking over my friend Dr. Wai's response to a recent opinion piece by Garfunkel and Mumford in the New York Times, I wonder to what extent the traditional math sequence could be tweaked (rather than overhauled) to achieve the financial, statistical, and programming competencies suggested by the authors of the NYT piece. If we agree the goal is "quantitative literacy" (as has been called "numeracy" in the past), or competence in tackling real-life quantitative application problems, then all we have to do is decide on the best way to get students there. Simple.

As the pendulum swings back from integrated to traditional math sequence in my own district, I find myself feeling underwhelmed. Aren't there some meaningful changes that could be made rather than rearranging the furniture again? For a start, it would be nice if schools would teach the same stuff. I don't know how many millions of dollars are spent on new and different textbooks every few years (which are then to be criticized), but I'd guess it's more than ideal. Or how much money goes into "alignment" between textbooks and state assessments. Why not have the people doing the assessing also make the books? And while we're at it, why not have those books be public domain? Then take the money saved and double teacher salaries.

We're a long way from a national (or even state) curriculum, but efforts are being made. I take issue with Garfunkel and Mumford's contention that the Common Core State Standards Initiative are too abstract and "simply not the best way to prepare a vast majority of high school students for life". The beauty of math is in its ability (through abstraction) to apply to seemingly disparate areas.

For example, the authors contend that replacing the "algebra" course with "finance" could work because students would still encounter the exponential function (I'm paraphrasing rather crudely). But would they appreciate that the exponential function also models radioactive decay, or the cooling of a corpse, or the growth of the fennel plant I just found in my yard?



Incidentally, the first Common Core standard to address exponentials is "F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions"- this strikes me as entirely reasonable, and conducive to any number of instructional methods.

I don't know of a high school instructor who is "teaching 'pure' math, with no context," which Garfunkel and Mumford set up as a straw man. Engaging applications play a huge role in good teaching, but the abstraction of mathematics is what makes it functional. Haven't we progressed from the days of the Rhind Papyrus?

But then, we're not just teaching math; we're teaching students.

Aside from the immense effects that culture and economics have on math achievement, I doubt the solution lies in what we teach so much as how we teach. The TIMSS 1999 Video Study found that we spend twice as long reviewing material as Japanese teachers. Students formulate procedures in 44% of Japanese lessons but only 1% of US lessons. In a recent NCTM publication, Smith and Stein remind us that "...curriculum provides a set of instructional possibilities; what actually happens in the classroom depends on the teacher's view of what students need to know and do and her capacity to support the enactment of curricular tasks that are most likely to achieve those competencies."

What's that line about best-laid plans?

Thursday, August 18, 2011

Numberism

At the Portland Saturday Market the other day I happened across an interesting stall.


I poked my head in and found Sienna Morris, creator of images created with numbers...quite literally. Just as the pointillists used small daubs of paint to create coherent pictures, Morris uses numbers and calls the technique "numberism". She explained that the numbers she chooses to construct her drawings are significant; for example, in the piece called "A Cello", each string of the cello is drawn using the frequency for its particular pitch. The body of the cello contains the speed at which sound propagates through the particular type of wood. Even the 12th root of 2, which makes our modern musical scale possible, occurs in the image. Pretty cool!





Monday, July 25, 2011

The Notes

To keep myself organized, I designed a notebook to use with each of my 4 classes. There was room for planning as well as class notes (I found I wasn't using my separate planning book very much), learning targets and justification, and graph paper. Each week I would scan the notes for the class web site, which parents and absent students appreciated.

At the end of the year one student asked if he could keep The Notes. Why not?

A few colleagues (not just in math) also used the notebook this last year and offered improvement suggestions. I made 18-week and 36-week versions:
[caption id="attachment_1116" align="alignnone" width="229" caption="36-week"][/caption]
[caption id="attachment_1117" align="alignnone" width="229" caption="18-week"][/caption]

Friday, July 8, 2011

Transformation Tai Chi

Here's how I teach transformations of functions.  The students get into it and start adding their own embellishments after they've gotten comfortable with the concepts.

Monday, June 27, 2011

Unwinding "The Biggest Ball of Twine in Minnesota" Pt. 1

Weird Al's awesome song, "The Biggest Ball of Twine in Minnesota," prompted me to visit Darwin, MN when I was in the area. The ball is not just the biggest in Minnesota, but the world!  If we're talking sisal twine.  Made by one man.  (Turns out there are a lot of big twine balls).  The song always leaves me wondering whence the narrator originated. Here we'll unravel the lyrics and use math to help us understand.

Let's start at the beginning .

Well, I had two weeks of vacation time coming
After working all year down at Big Roy's Heating And Plumbing

Artistic license appears to have been taken, as all online references to this establishment on are related to the song's lyrics.  We can come back to this later.

Oh, we couldn't wait to get there
So we drove straight through for three whole days and nights

Maybe we can estimate an average speed, but how "straight" were they going?  Let's use some arbitrary paths to get an idea of the radius we should consider.



Tuesday, June 21, 2011

The Story of Maths

Here's a cool series to show in class if you can fit it in (or just to watch yourself).  The first two episodes are appropriate for a wide range of skill levels, visually engaging, and emphasize the multicultural history of math.  The next two give an overview of the development of math within the last 400 years or so, as the narrator visits sites of historical significance.  The Newton-Leibniz piece would be great to show at the outset of a calculus class, or when discussing the notation for derivatives.  The final disc explores prime numbers and the Riemann hypothesis, which holds special significance to the narrator, Marcus du Sautoy.

Thursday, June 16, 2011

Fractal Magic Square

As I was watching The Story of Math with my class after finals, I thought I'd try a little experiment with the magic square shown in the video.



Each row, column, and diagonal sums to 15. Incidentally, if you add an amount n to each value, the square retains its magic and the sum changes to 15+3n.

But what if each unit square were comprised of nine smaller squares, like a sudoku board?  Could you follow the same overall pattern to generate a fractal magic square?

I made this one by adding successive multiples of nine to each mini-square, following the same "path" as in the original:

Saturday, May 28, 2011

Quad-core graphing calculator

Writing programs to generate fractals is fun, but the rendering takes too long. Why not distribute the window range across the classroom (in my case, 4 students on "senior skip day") to have each student generate a little piece of the whole?



This process will only hint at the beauty of the set:



(Generated with Fractal Explorer)

Thursday, May 26, 2011

Angry Birds teach calculus!

The angry yellow birds shoot off on [nearly] linear tangents to their parabolic flight paths. You know what that means - another Angry Birds math lesson! This time it's for calculus, or even precalculus, to motivate the concept of derivative. Start with this:



Here are my files - look for the PDFs to get started.

Friday, May 20, 2011

Angry Birds teach parabolas!

Update: Please check out the "Angry Birds" button in the top right!

I tried out the popular game Angry Birds the other day, and began thinking, "What excuse can I have to play this more?"

Using it in a lesson, perhaps? An Angry Birds Math Lesson!

To motivate the study of quadratic functions (usually an Algebra I topic), let's start with this video:



Next, students should realize we need some measurements:



Solve a system using substitution and voila - we know where the bird will land.  This last part will take some guidance, so here's a handout.


You may want to import the basic image into Geogebra (click the little arrow on the Slider Tool to show a drop-down menu) and have your students play with it first.



Don't watch the results until you've tried it out!

In this package you'll find:

  • Both HD videos

  • Handout

  • Images (with and without parabola)

  • Worked solution

Let me know if you students like having Angry Birds teach math!

Wednesday, May 18, 2011

Cheap polyhedra!

I found a vinyl-looking tube out of the gutter and brought it to school.  Using a pushpin and some bamboo skewers, a student put together some cool models for the classroom.  Thu pushpin was uncomfortable under prolonged use, so I made a hold-puncher with a nail and piece of wood. I came across a bag of 1000 thin plastic straws at Goodwill, and another student connected them with pipe cleaners to make some more sweet polyhedra.





 

Wednesday, May 11, 2011

Free pi posters!

Instead of going to bed on time last night I made this printable pi poster for the classroom!  This is just the first few digits, so you probably won't find Your Life in Pi, but it has other uses.

Click for the 4-page pdf.



Here's a single-page version of the above.

Here's a 2-page long-format:

Wednesday, April 20, 2011

Race to the Tree

A colleague's student teacher and I were trying to think up a way to hook our Math 1 students into wanting to solve a system.  An idea came, and we quickly whipped up this video:



(Clearer videos can be downloaded)
It took a few days before the students were confident enough to solve the problem, but during that time they kept asking about the outcome of the race.  The math works out nicely if you use two data points when both runners are in the scene.  Students can solve graphically, but the points are close enough that a range of answers will show up - hence the need for substitution or elimination.

My setup

Sunday, April 10, 2011

Brainventories

As I looked at the deficits that my current calculus students have, I worked with a colleague to compile a list of skills that our pre-calc students must master by the end of the year.  These skills form the "Brainventory" which is evaluated weekly, with each skill tested with 2 to 8 questions.  If a student scores poorly, he or she simply retakes that portion on subsequent Brainventories, and I overwrite lower grades with higher grades as students strive toward mastery.

For example, here is a student who has mastered factoring and logarithm (as far as I am concerned, anyway), but needs to improve his understanding of exponents.


This process has several benefits:

Students are highly aware of their own progress, and which skills need improvement

Assessment is both formative and summative - students use the Brainventories to improve and gauge their current understanding, and grades reflect that current level of understanding at all times.

There isn't so much "brain dumping" after a topic is assessed.

Assessment is differentiated, meeting each student's current needs.

It can be adapted to reflect concept mastery

 

Monday, March 28, 2011

Alphabet

Awhile back I made an alphabet book for my son, in which I tried to arranged the letters to highlight their symmetries.  At almost-three, he's just about ready to appreciate it...








Tuesday, March 22, 2011

Pentanonic awesomeness

I came across this simple but addictive music maker on M0ar, with no credit given:



It seemed that any possible pattern sounded good. I was reminded of this: