Lensyl Urbano – Page 27 – Montessori Muddle

Useful Games and Apps

Ellen Holderman has an excellent post about “24 Apps, Games, and Websites Teachers are Using in STEAM Classrooms” that are mostly free. (STEAM stands for Science, Technology, Engineering, Arts, and Mathematics).

They’re mostly aimed at a younger audience–middle school and below–, but there are things like Khan Academy that’s useful for all ages. Quite a number have to do with introducing coding.

↬ Ms. Lannert.

Why Math

It’s hard being an idealist in a world of utilitarians. – student C.J.

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.

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).

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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:

$V_{cylinder} = \pi r^2 h$

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:

$y = -\frac{x^2}{4}+4$

The radius of each cylinder is the value of the function for that x value, so you could write the radius as:

$R(x) = -\frac{x^2}{4}+4$

Therefore the volume of each disk is:

$V_{disk} = \pi R(x)^2 \Delta x$

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:

$V = \pi R(1.5)^2 \Delta x + \pi R(2.0)^2 \Delta x + \pi R(2.5)^2 \Delta x + \pi R(3.0)^2 \Delta x$

Factoring out the π and the Δx gives:

$V = \pi \Delta x \left(R(1.5)^2 + R(2.0)^2 + R(2.5)^2 + R(3.0)^2 \right)$

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:

$V = \pi \Delta x \sum\limits_{i=1}^n R(1.0+i \Delta x)^2$

reverting back to a and b gives the general equation:

$V = \pi \Delta x \sum\limits_{i=1}^n R(a+i \Delta x)^2$

where:
$n = \frac{b-a}{\Delta x}$

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

Mr. Alex Shine's program to calculate the volume of a curve rotated around the x-axis using the Disk Method in Calculus. — Student’s program to calculate the volume of a curve rotated around the x-axis using the Disk Method in Calculus.

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:

$y = -\frac{x^2}{4} + 4$

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