Working with the thousand cube, hundred square, ten bar and unit cells in algebra.
I sent a couple of my algebra students down to the pre-Kindergarten classroom to burrow one of their Montessori works. They were having a little trouble adding polynomials, and the use of manipulatives really helped.
The basic idea is that when you add something like:
you can’t add a n3 term to a n2 or a n term. You only combine the terms with the same degree (and same variables). So the equation above becomes:
which simplifies to:
The kids actually enjoyed the chance to run downstairs to burrow the materials from their old pre-K teacher (and weren’t they quite good about returning the materials when they were done with them).
And it clarifies a lot of misconceptions when you can clearly see that that you just can’t add a thousand cube to a ten bar — it just doesn’t work.
There are some things in this world that we are willing to trade, things that we can put a dollar value on, but there are other things — call them sacred things — values and beliefs that just don’t register on any monetary scale. New research (summarized by Keim, 2012) emphasizes this intuitive understanding, by showing that different part of the brain are used to evaluate these two different types of things.
[W]hen people didn’t sell out their principles, it wasn’t because the price wasn’t right. It just seemed wrong. “There’s one bucket of things that are utilitarian, and another bucket of categorical things,” [neuroscientist Greg Berns] said. “If it’s a sacred value to you, then you can’t even conceive of it in a cost-benefit framework.”
Some of the biggest implications of this work has to do with economics. The traditional, rational view has been that people evaluate everything by comparing the costs versus the benefits. When economists take that rational view of human behavior into other fields, there is a strong sense of overreach (see Freakonomics).
The growing research into behavioral economics, on the other hand, is making a spirited effort grapple with the irrationality of human behavior, much of which probably stems from these two different value systems (sacred vs. cost/benefit). While it’s not exactly the same thing, Dan Ariely‘s books are a good, popular compilation of observations and anecdotes that highlight how people’s irrational behavior extends even into the marketplace.
Parabolic mirrors magnify by reflecting parallel rays of incoming light onto a single point. (Adapted from Wikimedia Commons User:Nargopolis).
We’re talking about light and sound waves in physics at the moment, and NPR’s Morning Edition just had a great article on how the enormous, ultra-precise, mirrors that are used in large telescopes are made.
Astronomical observatories tend to use mirrors instead of lenses in their telescopes, largely because if you make lenses too big they tend to sag in the middle, while you can support a mirror all across the back, and because you have to make a lens perfect all the way through for it to work correctly, but only have to make one perfect surface for a parabolic mirror.
ScienceClarified has a great summary of the history of the Hubble Space telescope, that includes all the trouble NASA went through trying to fix it when they realized it was not quite perfect.
Large parabolic mirrors are used for magnification in telescopes. (Image via Wikipedia).
In addition, it’s interesting to note that you can also make a parabolic surface on a liquid by spinning it, resulting in liquid telescope mirrors .
The Guardian asked six major newspapers from across Europe about their local stereotypes of the other countries: Great Britain, France, Italy, Spain, Poland, and Germany. Then they asked six “cultural commentators” from those countries to respond. It’s quite an interesting read.
Aerial robots are used to construct a tower. It’s pretty awesome, especially when you note that the robots don’t collide with each other, and plug themselves in when they realize they’re running out of power.
Moving the magnet through a wire coil creates an electric current in the wire. Animation captured from the VPython program: Magnetic Induction - Coils.
My students asked me this question the other day, and while slapping together an animation of electromagnetic induction I gave it some thought.
This program itself is really simple. It took about 15 minutes.
But that’s not counting the half hour I spent searching the web for an image I could use to illustrate magnetic induction and not finding one I could use.
Nor does it count the four hours I spent after I got the animation working to get the program to take screen captures automatically. Of course, I must admit that figuring out the screen captures would have gone a lot quicker if I’d not had to rebuild all my permissions on my hard drive (I’d recently reformatted it), and reinstall ImageMagick and gifsicle to take the screen captures and make animations.
Just in time for our physics test — on electromagnetism — the Sun has had a Coronal Mass Ejection of charged particles that is heading toward the Earth.
[The Coronal Mass Ejection] is moving at almost 1,400 miles per second, and could reach Earth’s magnetosphere – the magnetic envelope that surrounds Earth — as early as tomorrow, Jan 24 at 9 AM ET (plus or minus 7 hours). This has the potential to provide good auroral displays, possibly at lower latitudes than normal.
The Earth's magnetic field deflects charged particles around the planet, although some do get redirected down toward the poles, making the arouras. (Image from NASA's Spaceplace).
A Coronal Mass Ejection has about 100 billion tons of electrons, protons and other particles (NASA Cosmicopia, 2011), usually ionized, that would bombard the Earth and the atmosphere if we weren’t protected by the Earth’s magnetic field.
Most of the ions are deflected around the Earth but some get focused down toward the poles. At the poles, these ions hit nitrogen and oxygen molecules (that make up 98% of the atmosphere), exciting many of them. Excited atoms and molecules give off light. The light shows created are called the auroras.
Aroura australis, as seen from the International Space Station.
I like the second video they post because, at the end, there is a splatter of interference from all the charged particles affecting the detector.
Students enjoyed doing it, even though it was challenging making the coil just right so it would spin easily. They persisted and enjoyed that wonderful eureka moment when it actually worked.
Motor Speed
One group wanted to figure out how fast the armature was spinning. Because of small imperfections inherent to hand-made parts, we found that the armatures would bounce, ever so slightly, with each rotation. So the students recorded the sound using their laptop, and then counted rotations off the recorded sound wave. I think they came up with about 10 rotations per second.
I need to check if there’s a free phone app we can use that will show the sound waveform more efficiently — Pocket WavePad seems like it might work.
Accidental arc welding
At the end, some students wanted to figure out just how fast they could get the motor running. Disdaining the online instructions, they went for more power, hooking up all the batteries they could scavenge from the other groups before I had to make them stop. They did manage to weld the insulating varnish on the coil wire to the paperclip contact before the end.
All the sparks did lead to a discussion of how arc welding works, however, which I was able to tie into the maths of transformers; cheaper arc welders (like this one) take in 20 Amps at 120 Volts, and output 70 Amps at 22 Volts.
This group did not want to stop, so I gave them permission to pass on P.E. for that one day, with a note that said they were doing some “remedial” physics. They got a kick out of that.
