It’s hard being an idealist in a world of utilitarians. – student C.J.
Author: Lensyl Urbano
The Farm School Blog
The interest in our efforts to establish a Farm School program has been tremendous. Parents have enthusiastically chipped in time and resources to get things started, and we’ve been able to recruit Dr. Sansone to manage the practical side of things on the farm. Indeed, things are going so well that I’ve started a new TFS Farm blog to help us keep track of what we’re doing and to help us coordinate our efforts.
Dr. Sansone’s degree is in veterinary medicine and he has helped out a lot in my previous endeavors (see chickens and rabbits) to incorporate farming as practical life while studying anatomy.
Intelligence Squared: The Killer Apps of Western Civilization
Naill Ferguson gives a provocative talk about his thesis that there are six “Killer Apps” that made western civilization so successful over the last five centuries.
The killer apps he suggests are:
- Political and economic competition
- The scientific method
- Property rights (linked to voting)
- Modern Medicine
- Consumer based economies
- Protestant work ethic
There’s a PBS series about it as well.
The Intelligence2 podcast, recommended by Mr. Schmidt, is a great resource for talks like this one. They have a nice archive.
Volumes of Rotation: The Disk Method: 3d with Javascript Three.js
Finally, relatively easy interactive 3d on the web. You can rotate and zoom into the scene. (Although it may not yet be compatible with all browsers it does work with Firefox at least).
This method uses the three.js Javascript library. Here I use it to show the volume of a rotated surface using the disk method. It’s almost identical to my calculus student’s project, except here I’m finding the volume between x=1 and x=3, using disks that are 0.5 units in height (Δx).
Since the volume of cylinder is:
where r is the radius of the cylinder.
We’re finding the volume created by a function that’s rotated around the x-axis. Using the function:
The radius of each cylinder is the value of the function for that x value, so you could write the radius as:
Therefore the volume of each disk is:
There are four disks and we use the function value at the far end of the disk to draw the disk so the total volume is:
Factoring out the π and the Δx gives:
Since Δx = 0.5, a = 1.0, and b = 3.0, we can define the number of disks as n = 4 then we can rewrite using summation formula:
reverting back to a and b gives the general equation:
where:
Fireplace
The most productive place to work varies for each student. Some need more rigorous structure–chair and desk; pen and paper–while others can get a lot done while lying in front of the fire. A key here, I think, is that the students have enough space. They’re working on different projects and they don’t need to collaborate, so a meter separation (the same distance they’re required to be apart during personal reflection time) gives everyone space to do their own thing.
Project Euler: Math/Programming Challenge
On the recommendation of Mr. Schmidt, two of my students have been quite fascinated over the last few days trying to solve problems on Project Euler. They’ve been working on them together to, I suspect, the detriment of some of their other classes, but as their math teacher I find it hard to object.
An example problem is something like this:
If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
They’ve been solving them numerically using Python. It’s been quite fascinating to see.
Finding Volumes Using the Disk Method
This VPython program was written by a student, Mr. Alex Shine, to demonstrate how to find the volume of a curve that’s rotated around the x-axis using the disk method in Calculus II.
The program finds volume for the curve:
between x = 0 and x = 3.
To change the curve, change the function R(x), and to set the upper and lower bounds change a and b respectively.
volume_disk_method.py by Alex Shine.
from visual import*
def R(x):
y = -(1.0/4.0)*x**2 + 4
return y
dx = 0.5
a = 0.0
b = 3.0
x_axis = curve(pos=[(-10,0,0),(10,0,0)])
y_axis = curve(pos=[(0,-10,0),(0,10,0)])
z_axis = curve(pos=[(0,0,-10),(0,0,10)])
line = curve(x=arange(0,3,.1))
line.color=color.cyan
line.radius = .1
line.y = -(1.0/4.0) * (line.x**2) + 4
#scene.background = color.white
for i in range(-10, 11):
curve(pos=[(-0.5,i),(0.5,i)])
curve(pos=[(i,-0.5),(i,0.5)])
VT = 0
for x in arange(a + dx,b + dx,dx):
V = pi * R(x)**2 * dx
disk = cylinder(pos=(x,0,0),radius=R(x),axis=(-dx,0,0), color = color.yellow)
VT = V + VT
print V
print "Volume =", VT
Electricity and Magnetism Experiments
Last week, my middle schoolers did a set of experiments on electricity and magnetism. They answered the questions:
- How does the voltage across each light bulb change as you add more and more bulbs to a parallel circuit?
- How does the voltage across each light bulb change as you add more and more bulbs to a series circuit?
- How does the number of coils of wire wrapped around a nail affect it’s magnetism (as measured by the number of paperclips it can pick up)?
- How does the amount of salt mixed into water affect its conductivity?
Each question is designed so that students have something to measure and will be able to use those measurements to make predictions. For example, once they’ve measured the voltage across four bulbs in series, they should be able to predict the voltage across the bulbs in a series of ten.
Some of the experiments, like the nail electromagnet, should have simple linear trends, with students choosing the advanced option having to find an equation to fit their data for the predictions. And I’ll challenge the students in Algebra II to find the equations for the inverse relationships–I’ve already asked their math teacher (Mr. Schmidt) to help them out if they need it.
This has also provided the opportunity for them to apply what they’ve just learned about drawing circuit diagrams (we use this set of symbols).
