The bright blue iridescence of the wings of this insect results from the way light refracts through the thin layered membranes of the wings.

When you look at the sunlight reflected off this black insect’s wings at just the right angle, they blaze bright blue. The phenomena is called iridescence, and results from the way different wavelengths of light refract through the wing membrane. Blue light is of just the right wavelength that the light reflected off the top of the membrane and the light that’s refracted through the membrane constructively interfere. The Natural Photonics program at the University of Exeter has an excellent page detailing the physics of iridescence in butterflies (Lepidoptera), and the history of the study of the subject.

Citing this post: Urbano, L., 2012. Iridescent Wings - The Physics, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Surface tension supports the water strider on the surface of the still water of the creek.

Down at the creek the water striders are out. They can stand, walk and jump on the surface of the water without penetrating the surface because of the force of surface tension that causes water molecules to stick together — it’s the same cohesive force that make water droplets stick to your skin. I got a decent set of photos to illustrate surface tension.

The water striders still create ripples on the surface of the water, even though they never break the surface.

The green canopy that over hangs the creek allows for some nice photographs.

Water strider in still water.

Citing this post: Urbano, L., 2012. Surface Tension, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Tool‘s Lateralus has the Fibonacci Sequence embedded into it. Ms. Wilson tells me that some of her Algebra II students were able to detect it out after listening to the song (a few times), but this video has the highlights where the sequence occurs.

↬ Ms. Wilson

Citing this post: Urbano, L., 2012. Music of the Fibonacci Sequence, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Gaussian Elimination Matrix Solver by Alex Shine (comments by me).

from visual import *

'''The coefficient matrix (m) can be any sized square matrix
   with an additional column for the solution'''
m = [ [1,-2,1,-4],[0,1,2,4],[2,3,-2,2]]

'''Convert the input matrix, m (which is a list) into an array'''
a = array(m,float)
print "Input matrix:"
print a
print

'''Get the shape of the matrix (number of rows and columns)'''
(r,c) = a.shape 

rs = (1-r)

'''Solve'''
for j in range(r):

    print "Column #:", j
    for i in range(r):
        if a[i,j] <> 0:
            a[i,:] = a[i,:] / a[i,j]
    print a
    print

    for i in range(rs+j,j):
        if a[i,j] <> 0:
            a[i,:] = a[j,:] - a[i,:]
    print a
    print

print "Solution"
for i in range (r):
    a[i,:] = a[i,:] / a[i,i]
    print "Variable", i, "=", a[i,-1]

print
print "Solution Matrix:"
print a

The code above solves the following system of equations:

  x - 2y +  z = -4
       y + 2z =  4
 2x + 3y - 2z =  2

Which can be written in matrix form as such:

You use the solver by taking the square matrix on the left hand side of the equation and combining it with the column on the hand side as an additional column:

This is entered into the program as the line:

m = [ [1,-2,1,-4],[0,1,2,4],[2,3,-2,2]]

When you run the above program you should get the results:

>>> ================================ RESTART ================================
>>>
Input matrix:
[[ 1. -2.  1. -4.]
 [ 0.  1.  2.  4.]
 [ 2.  3. -2.  2.]]

Column #: 0
[[ 1.  -2.   1.  -4. ]
 [ 0.   1.   2.   4. ]
 [ 1.   1.5 -1.   1. ]]

[[ 1.  -2.   1.  -4. ]
 [ 0.   1.   2.   4. ]
 [ 0.  -3.5  2.  -5. ]]

Column #: 1
[[-0.5         1.         -0.5         2.        ]
 [ 0.          1.          2.          4.        ]
 [-0.          1.         -0.57142857  1.42857143]]

[[ 0.5         0.          2.5         2.        ]
 [ 0.          1.          2.          4.        ]
 [ 0.          0.          2.57142857  2.57142857]]

Column #: 2
[[ 0.2  0.   1.   0.8]
 [ 0.   0.5  1.   2. ]
 [ 0.   0.   1.   1. ]]

[[-0.2  0.   0.   0.2]
 [ 0.  -0.5  0.  -1. ]
 [ 0.   0.   1.   1. ]]

Solution
Variable 0 = -1.0
Variable 1 = 2.0
Variable 2 = 1.0

Solution Matrix:
[[ 1. -0. -0. -1.]
 [-0.  1. -0.  2.]
 [ 0.  0.  1.  1.]]
>>>

Be aware that:

• The code is designed to take any size of matrix.
• The matrix you put in can not have any zeros on its diagonal, so some manipulation is often necessary before you can use the code.

Other notes:

• The negative zeros (-0) that show up especially in the solution matrix may not look pretty but do not affect the solution.
• The code imports the vpython module in the first line but what it really needs is the numpy module, which vpython imports, for the arrays.

The next step is to turn this into a function or a class that can be used in other codes, but it’s already proved useful. My calculus students compared their solutions for the coefficients of a quadratic equation that they had to solve for their carpet friction experiment, which was great because their first answers were wrong.

A calculus student uses the matrix solver. Mr. Shine is now trying to convert the solver into an iPhone app.

Citing this post: Urbano, L., 2012. Learning Matricies by Programming a Matrix Solver, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

The solar water heater in action.

Next year I want to try building an actual solar water heater, similar to the passive air heater my students built two years ago, with the tubing in a greenhouse box to see just how efficient we can make it.

Citing this post: Urbano, L., 2012. A Model Solar Water Heater, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Formative assessment happens during learning, usually in the classroom. Students do something, like an assignment, and get immediate feedback on what they did. A teacher walking around from student to student or group to group, following what the students are doing and helping students identify which concepts they’re not getting, is a typical example of formative assessment.

Authentic assessments are assignments that are or mimic real-world problems, and require students to apply the stuff they should have learned to solving them. I’m using projects like the draining of a bottle and carpet friction experiments to assess if my students truly understand why they do algebra and calculus, and are able to apply the techniques they’ve learned.

Caveat: It is important to note, however, that being able to solve real-world problems requires some abstract thinking skills that adolescents are still developing. Yet, even though a lot of the basic learning in middle and high school consists of ingesting the language of the different fields of study — they type of thing that is easy to test — a more useful assessment is likely to be one that requires students to use their new vocabulary in written assignments, such as project reports and essays.

Citing this post: Urbano, L., 2012. Formative and Authentic Assessment, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

… cramming—short-term memorizing—does not contribute to retention or transfer [my emphasis]. It may, however, yield positive short-term results as measured by exam scores.

– Jaffee, 2012: Stop Telling Students to Study for Exams in The Chronicle of Higher Education.

It’s getting close to the end of the academic year, and exams are coming up. David Jaffee advocates that we stop telling students to study for their exams; they should, instead, study for learning and understanding.

Jaffee especially piles on on Final Exams:

This dysfunctional system reaches its zenith with the cumulative “final” exam. We even go so far as to commemorate this sacred academic ritual by setting aside a specially designated “exam week” at the end of each term. This collective exercise in sadism encourages students to cram everything that they think they need to “know” (temporarily for the exam) into their brains, deprive themselves of sleep and leisure activities, complete (or more likely finally start) term papers, and memorize mounds of information. While this traditional exercise might prepare students for the inevitable bouts of unpleasantness they will face as working adults, its value as a learning process is dubious.

– Jaffee (2012).

The alternative to exams, Jaffee suggests, is formative and authentic assessment.

Citing this post: Urbano, L., 2012. Don't Study for Your Exams, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Crayfish camouflaged against the rocks in the creek.

One of my favorite things about the Fulton School campus is the little creek that runs along the boundary. It’s small, dynamic, and teeming with life.

The crayfish are out in force at the moment. Some of the high-schoolers collected one last fall and it survived the winter in our fish tank (also populated with fish from the creek).

They are quite fascinating to observe; wandering around the sandy bed as if they own the place; aggressive with their pincers occasionally; but then darting backward amazingly fast if they feel threatened.

The one in our tank has just molted a second time, so now we have two almost perfect exoskeletons sitting around the science lab.

Crayfish exoskeleton. From pincers to tail the skeleton is approximately 10cm long.

Citing this post: Urbano, L., 2012. Crayfish in the Creek, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

The area around the Sun is blue on Mars because the gasses in the thin atmosphere don't scatter much, but the Martian dust does (it scatters the red). Image via NASA.

The dust in Mars’ atmosphere scatters red, while the major gasses in Earth’s atmosphere (Nitrogen and Oxygen) scatter blue light. Longer wavelengths of light, like red, will bounce off (scatter) larger particles like dust, while shorter wavelengths, like blue light, will bounce of smaller particles, like the molecules of gas in the atmosphere. The phenomena is called Rayleigh scattering, and is different from the mechanism where different molecules absorb different wavelengths of light.

Ezra Block and Robert Krulwich go into details on NPR.

Blue sky in the upper right, but the dust scatters the red light.

Citing this post: Urbano, L., 2012. Why are Earth's Sunsets Red While Mars' are Blue?, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

For every action there is an equal and opposite reaction.

– Newton’s Third Law of Motion

I introduced my Middle Schoolers to the principles of Newton’s Laws of Motion last week.

The discussion started off with projectiles. If you’re floating in space — zero gravity — and throw something, like a basketball, away from you, you’ll be pushed off in the opposite direction. In fact, if you throw something that has the exact same mass as you do away from yourself, you’ll move off in the opposite direction with the exact same speed as the thing you threw.

Then I brought up rockets, and how they’re expelling gas to move them forward. I think it was the phrase, “expelling gas” that did it. The next question, which the student brought up somewhat circumspectly, sidling around the issue and the language, was (more or less), “So if you expel gas in space will you move off in the other direction?”

The simple answer, appropriate to that stage of the discussion, was, of course, “Yes.”

Which lead to to, “What about spitting?”

“Yes.”

“What about, you know, peeing?”

“Yes, except …”

At that point I thought it would be wise to rein it in a little, and make a further point about the whole action-reaction thing.

“You see, if you expel anything, wouldn’t it just be stuck in your spacesuit with you? Then you’re not really expelling it, it’s still attached to you, so you wouldn’t really move. What would be more useful would be to collect it in something like a spray can or a squeeze bottle. Then you can just squirt it out opposite the direction you want to go in to control your movement.”

This produced a moment of thoughtful silence as they figured out the logistics.

Notes

I thought this was a useful conversation to have. The students were interested and animated. And I believe it’ll be memorable too.

An artist's concept depicts the Deep Space 1 probe with its ion engine operating at full thrust. Image via NASA.

P.S.: I’d wanted to talk about ion drives, which operate on the same reaction principle, but are much cooler (after all they’re used in Star Trek). Instead of burning fuel to create the propulsive force ion drives generate an electric field that ejects charged particles; we’d been talking about ions and charged particles earlier in the day. However, I decided on the day that it would just complicate what was a new issue. I’ll probably bring it up this week though as we recurse through Newton’s Laws.

Citing this post: Urbano, L., 2012. Flatulence ... in Space, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

2. Some of your worst days lie ahead. Graduation is a happy day. But my job is to tell you that if you are going to do anything worthwhile, you will face periods of grinding self-doubt and failure.

– Wheelan, 2012: 10 Things Your Commencement Speaker Won’t Tell You in The Wall Street Journal

Charles Wheelan provides an excellent perspective on what should be important in a commencement address.

I particularly like this warning about the danger of working only for rewards:

8. Don’t model your life after a circus animal. Performing animals do tricks because their trainers throw them peanuts or small fish for doing so. You should aspire to do better.

– Wheelan, 2012: 10 Things Your Commencement Speaker Won’t Tell You in The Wall Street Journal

And this point on conservation and the real meaning of being conservative:

3. Don’t make the world worse. I know that I’m supposed to tell you to aspire to great things. But I’m going to lower the bar here: Just don’t use your prodigious talents to mess things up. Too many smart people are doing that already.

– Wheelan, 2012: 10 Things Your Commencement Speaker Won’t Tell You in The Wall Street Journal

↬ The Dish

Citing this post: Urbano, L., 2012. A Better Commencement Address, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

I also like how clear they make the fact that science is a processes of discovery, and what fascinates scientists is the unknown. Students do experiments all the time and if they don’t find what they expect — if it “doesn’t work” — they’re usually very disappointed. I try my best to let them know that this is really what science is about. When your experiment does not do what you want, and you’re confident you designed it right, then the real excitement, the new discoveries, begin.

Citing this post: Urbano, L., 2012. Searching for the Higgs Boson: How Science Really Works, Retrieved May 19th, 2012, from Montessori Muddle: http://MontessoriMuddle.org/ .
Attribution (Curator's Code ): Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.