Lensyl Urbano – Page 27 – Montessori Muddle

Glass Bending

Students bend glass tubing for their steam distillation apparatus. — Bending the glass tubing is fairly straightforward, but looks awfully sciency.

Our steam distillation apparatus for extracting lavender oil started off fairly simply — a steamer connected to a glass tubing running under the cold water tap to a collection flask — and evolved from there. One of the final tweaks we attempted, was to make a coil in the glass tubing so the steam would have a longer transit through the ice-water bath to enhance condensation.

We heated the glass using a small butane burner until it became pliable, then bent the tube into shape around a piece of wet wood. Using the wood was not as effective as we’d hoped because the glass tubing is fairly thin and cools down quickly when in contact with the water.

You also have to be very careful when bending the tubing to make sure you don’t pull on it. Pulling stretches the glass, making the walls thinner, making it more likely to break. My students discovered this the hard way.

The Bees are Coming

Students use sandpaper to prepare the outsides of donated hives for painting.

Ms. Mertz and Mr. Dittrich attended a beekeeping class last month. They were quite excited about what they learned, and it fits right into our Farm School initiative. They were able to secure a couple donated hives, which they’re busily preparing because queens are on the way.

Students also scraped the insides of the hives to remove old comb material.

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).

</p> <p>Your browser does not support iframes.</p> <p>

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.