One of the more interesting projects of the last year was the wooden Quarto set I made for our middle schoolers to use during their study hall.

The game is quite an interesting one, as was the build. The pieces (rectangular and cylindrical prisms capped with solid or hollow tops) were fairly simple to make using the table-saw for the bodies and laser cutter for the tops. However, I wanted to make a box for the pieces and have the board with its 4×4 grid of circle serve as the top.

Cutting out the top and bottom of the box out of plywood (on the CNC machine) was easy enough, as was lasering on the grid, but the most fascinating part was making the sides of the box. The rounded corners on the top required rounded sides, so I used the laser to cut a living hinge on a piece of plywood and glued it to the base. Then I used some spacer pieces for the inside to hold up an inset that would hold the pieces.

**Citing this post**: **Urbano**, L., 2017. Quarto Set, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Last year, for an interim project, one of my more musically inclined students decide to build a Cajón. It’s a box shaped drum that you can sit on, with a snare inside. He worked up a simple design in Inkscape and I cut it out on the CNC machine at the Techshop. It turned out quite well, and he even built one that I could keep at school.

**Citing this post**: **Urbano**, L., 2017. Cajón, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Our first through third grade teachers requested these finger labyrinths. I asked Dr. Steurer to explain how they used them:

Sometimes you just need to be alone. Welcome to our 1-2-3 Classroom “Comfort Zone.” When a child needs a break, they may relax and take a time away from any feeling of pressure or being overwhelmed.

One activity found in the “Comfort Zone” is a finger labyrinth crafted by Dr. Urbano. Children are taught three easy steps for slowly tracing the beautifully designed wooden labyrinth.

Step One: Release – Pause and take a deep breath.Take a deep breath before you begin your finger walk to the center. This is the time for you to calm yourself and get focused. Let go of everything.

Step Two: Receive – Take in the center.The center is a place for you to gain calm and peace. You can stay in the center point as long as you need.

Step Three: Return – Slowly take the journey back.Move back out of the center point. Make the transition from the center back into your daily routine, ready and armed with the experience of peace and calm.

The “Comfort Zone” is one area in our classroom used to support our children in improving their abilities to pay attention, to calm down when they are upset and to make better decisions.

Being

Mindfulhelps with emotional regulation and cognitive focus.– Dr. Steurer

**Citing this post**: **Urbano**, L., 2017. Finger Labyrinths, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

A nice explanation, from the excellent Real Engineering channel, of the physics of GPS that explains how the satellites must adjust for the effects of special and general relativity.

**Citing this post**: **Urbano**, L., 2017. The Physics of GPS, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

After a lot of hours of experimentation I’ve finally settled on a workable method for generating large-scale 3d terrain.

**Data** from the NGDC’s Grid Extraction tool. The ETOPO1 (bedrock) option gives topography and bathymetry. You can select a rectangle from a map, but it can’t be too big and, which is quite annoying, you can’t cross the antimeridian.

The ETOPO1 data is downloaded as a GeoTIFF, which can be easily converted to a png (I use ImageMagick convert).

**Adjusting the color scale**. One interesting property of the data is that it uses a grayscale to represent the elevations that tops out at white at sea-level, then switches to black and starts from there for land (see the above image). While this makes it easy to see the land in the image, it needs to be adjusted to get a good heightmap for the 3d model. So I wrote a python script that uses matplotlib to read in the png image as an array and then I modify the values. I use it to output two images: one of the topography and one of just land and water that I’ll use as a mask later on.

The images I export using matplotlib as grayscale png’s, which can be opened in OpenSCAD using the surface command, and then saved as an stl file. Bigger image files are take longer. A 1000×1000 image will take a few minutes on my computer to save, however the stl file can be imported into 3d software to do with as you will.

Note: H.G. Deitz has a good summary of free tools for Converting Images Into OpenSCAD Models

**Citing this post**: **Urbano**, L., 2017. Generating 3d Terrain, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

One of my students with a TechShop membership wanted a bike silhouette for a wall hanging. He wanted it to be bigger than he could fit on the laser cutter, so I tried doing it on the CNC router. The problem was that to get the maximum detail we needed to use the smaller drill bits (0.125 inches in diameter), however, after breaking three bits (cheap ones from Harbor Freight) and trying both plywood and MDF, we gave up and just used the larger (0.25 inch) bit. Since the silhouette was fairly large (about 45 inches long), it worked out quite well.

**Citing this post**: **Urbano**, L., 2017. Bike Silhouette, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Ben Eater’s excellent series on building a computer from some basic components. It goes how things work from the transistors to latches and flip-flops to the architecture of the main circuits (clock, registers etc). The full playlist.

Other good resources include:

- Nand2Tetris : Building a computer from first principles (this one uses a hardware simulator however): See overview video.

**Citing this post**: **Urbano**, L., 2017. How to Build an 8-bit Computer on a Breadboard, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Managing cell phone usage at school is a tricky topic. We have some teachers who’d like to ban them outright, but we also have a growing number of parents who are expecting to be able to communicate with their kids–to organize pickups and carpooling during the day for example. The phones can be great for data-collection and documentation in classes, and a lot of my upper level math students prefer the Desmos app to using their graphical calculators.

Our current compromise is that middle schoolers have to leave their phones in the front office, where they can check them at lunch time or check them out if a teacher wants them to use them.

The high schoolers are allowed to keep their phones with them, but have to put them in a basket at the front of the classroom. Since they don’t like piling them into the basket, I experimented with the CNC machine to cut some plywood into a cell-phone shelf.

The shelf can hold about 30 phones, and I can easily see how many phones are on there from across the room, so I’d say this one worked out pretty well.

**Citing this post**: **Urbano**, L., 2017. Cell Phone Shelf, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

I recently discovered that, although they may look it, Go boards are not necessarily square. They’re slightly longer in one dimension so that the board looks more square to the players on both sides.

A student asked me to make one for him–he’d ordered a set recently and didn’t like the board it came with–so, I wrote a small python program to generate the Go grid, then lasered it onto a nice piece of sanded plywood.

It worked out quite well. Apparently the plywood makes just the right “thunk” sound when you put down the pieces.

The script to generate the grid.

*go_board_2.py*

from visual import * from svgInator_3 import * length = 424.2 #mm width = 454.5 #mm nLines = 19 dx = length/(nLines-1) dy = width/(nLines-1) print "Lenght = ", length print "dx = ", dx f = svgInator("go_board.svg") lineStyle = {"stroke": "#000", "stroke-width": "2pt",} #lines for i in range(nLines): x = i * dx y = i * dy #vertical f.line(pos=[vector(x,0), vector(x,width)], style=lineStyle) #horizontal f.line(pos=[vector(0,y), vector(length,y)], style=lineStyle) #circles grid_pos = [(3,3), (3,9), (3,15), (9,3), (9,9), (9,15), (15,3), (15,9), (15,15)] for i in grid_pos: (x, y) = (i[0]*dx, i[1]*dy) f.circle(pos=vector(x,y), radius=2.0, style={"stroke": "#000", "fill":"#000"}) #bounding box f.rect(dim=vector(length,width), style=lineStyle) f.close()

Now I just have to learn to play.

**Citing this post**: **Urbano**, L., 2017. Go Board, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Inspired by a video of a temporary bridge built out in the woods for mountain biking, my students wanted to try building a “natural” bridge with no fasteners–no screws, no nails–over a small ravine that feeds into our creek.

We found a couple large fallen logs to cut into two 10 foot lengths for the basic structural support for the bridge. These were dug into the ground to anchor them on either side of the ravine. We then chopped a couple more logs into 2 foot sections to go across the structural logs. The dense mud from the banks of the creek was then packed onto the top to hold it all together.

In the end, the bridge turned out to be pretty solid, and definitely usable.

**Citing this post**: **Urbano**, L., 2017. Our Natural Bridge, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

My crew from the Gaga Ball pit decided to make a trail through the woods and across the creek. So they built two short (12 ft long) bridges to cross the creek itself, and a third, “natural” log bridge to cross a small ravine that runs into the creek and cuts across the trail.

The short bridges were made of overlapping 2×4’s for structure (held together by 2.75 inch structural screws), with 24 inch long, 1×6 planks across the top.

The short bridges needed to be small and light enough to be moved when the creek rises, like it did today. I’ll attest that they can be moved, but not easily. They’re pretty heavy: it took a team of three or four middle schoolers to get it down to the creek, and it was hard going trying to drag it over to the side by myself this afternoon. Note to self: next time make sure the structural cross pieces are not at the very end of the bridge.

**Citing this post**: **Urbano**, L., 2017. Building Bridges (Literally), Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

To start with, we’ll look at the temperature at five points (call them nodes): three are in the wall and two at the edges (see Figure 1). We’ll say the temperature at node 1 which is next to the furnace is T_{1}, and the temperature at the outer edge of the wall is the 5th node (T_{5}) so:

- T
_{1}= 1560 K - T
_{5}= 300 K

Now we need to find the temperature at nodes 2, 3, and 4.

For each of the interior nodes, we can consider how things are at equilibrium. Heat is always moving through the wall, but as some point in time the heat flowing from the furnace into the wall will be equal to the heat flowing out of the wall, and for each node the heat flowing in will equal the heat flowing out.

The heat flow (Q) from one location to another is driven by the difference in temperature (ΔT): heat flows from high temperature to cooler temperatures (which makes ΔT negative). However, you also need to consider the distance the heat is traveling (Δx) since, given the same temperature drop, heat will flow faster if the distance is short. So, it’s best to consider the temperature gradient, which is how fast the temperature is changing over distance:

How fast heat flows is also mediated by the type of material (different materials have different thermal conductivities (K)), so our heat flow equation is given by the equation:

HEAT IN: Heat is flowing into Node 2 from Node 1, so:

HEAT OUT: Heat flows out to Node 3 so:

At equilibrium the heat into the node equals the heat out:

so:

Which we can simplify a lot because we’re assuming the thermal conductivity of the wall material is constant and we’ve (conveniently) made the spacing between our nodes (Δx) the same. Therefore:

And now we solve for temperature at T_{2} to get:

Finally, we can break the fraction into separate terms (we need to do this to make it easier to solve using matricies as you’ll see later) and start using decimals.

If we do the same for all the internal nodes we get:

You should be able to see here that the temperature at each node depends on the temperature of the nodes next to it, and we can’t directly solve this because we don’t know the temperatures of the interior nodes.

Let’s collect all of our information to get a system of equations:

- T
_{1}= 1560 K - T
_{2}= 0.5 T_{1}+ 0.5 T_{3} - T
_{3}= 0.5 T_{2}+ 0.5 T_{4} - T
_{4}= 0.5 T_{3}+ 0.5 T_{5} - T
_{5}= 300 K

Now to rewrite this as a matrix. We start by putting all terms with variables on the left and all the constants on the right.

- T
_{1}= 1560 K - -0.5 T
_{1}+ T_{2}– 0.5 T_{3}= 0 - -0.5 T
_{2}+ T_{3}– 0.5 T_{4}= 0 - -0.5 T
_{3}+ T_{4}– 0.5 T_{5}= 0 - T
_{5}= 300 K

Now to matrixize:

This you can solve on paper, and, if you’d like, check your answer using Alex Shine’s Gaussian Elimination solver, which gives step by step output (although it’s a little hard to follow). I used Alex’s solver to get the result:

Solution Variable 1 = 1560.0 Variable 2 = 1245.0 Variable 3 = 930.0 Variable 4 = 615.0 Variable 5 = 300.0

This problem is set up to be easy to check because the results should be linear (if you plot temperature versus distance through the wall you’ll get a straight line). It will also give you whole number results up to 10 nodes.

This procedure for setting up the matrix give the same basic equations no matter how many nodes you use, because as long as the distance between the nodes (Δx) and the thermal conductivity (K) are constant, the equation for each internal node (T_{i}) will be:

[math] T_i = \frac{T_{i-1} + T_{i+1}}{2}

and the matrix will continue to just have three terms along the diagonal.

If the wall is not uniform then the thermal conductivity coefficient does not just drop out of the equation, so you’ll have to pay attention to the conductivity going into and out of each node. Same goes with the node spacing.

Students could try:

- writing their own matrix solvers.
- setting up the system of equations in a spreadsheet program (they’ll need to change your program’s settings so that it uses its iterative solver).

**Citing this post**: **Urbano**, L., 2017. Solving 1D Heat Transfer Problems with Matricies, Retrieved October 22nd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.