#### March 29, 2017

Students playing Go.

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.

Go board in use.

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 July 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Makerspace, TechShopNo Comments » - Tags: board games, geometry, makerspace, programming, TechShop

#### March 1, 2017

Chessboard under regular (day) light.

I used the computer controlled (CNC) Shopbot machine at the Techshop to drill out 64 square pockets in the shape of a chessboard. One of my students (Kathryn) designed and printed the pieces as part of an extra credit project for her Geometry class.

The pockets were then filled with a clear eqoxy to give a liquid effect. However, I mixed in two colors of pigmented powder to make the checkerboard. The powder was uv reactive so it fluoresces under black (ultra-violet) light.

Under a black (ultra violet) light bulb.

The powder also glows in the dark.

Glowing in the dark.

**Citing this post**: **Urbano**, L., 2017. Liquid Chessboard, Retrieved July 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in 3d Print, Mathematics, TechShopNo Comments » - Tags: Art, geometry, TechShop

#### August 29, 2016

Our PE teacher just started up an archery program at school (first classes today actually), and she shared this video with me that seems like it might be useful in geometry class. Specifically, it uses circle packing to estimate the relative difficulty for an archer to hit different sized circular targets.

**Citing this post**: **Urbano**, L., 2016. Some Geometry in Archery, Retrieved July 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in MathematicsNo Comments » - Tags: archery, geometry

#### August 27, 2015

Toothpick shape sequences.

*Using a sequence of connected shapes to introduce algebra and graphing to pre-Algebra students.*

Make a geometric shape–a square perhaps–out of toothpicks. Count the sides–4 for a square. Now add another square, attached to the first. You should now have 7 toothpicks. Keep adding shapes in a line and counting toothpicks. Now you can:

- make a table of shapes versus toothpicks,
- write the sequence as an algebraic expression
- graph the number of shapes versus the number of toothpicks (it should be a straight line),
- figure out that the increment of the sequence–3 for a square–is the slope of the line.
- show that the intercept of the line is when there are zero shapes.

Then I had my students set up a spreadsheet where they could enter the number of shapes and it would give the number of toothpicks needed. Writing a small program to do the same works is the next step.

**Citing this post**: **Urbano**, L., 2015. Toothpick Shapes' Sequences, Retrieved July 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Algebra, MathematicsNo Comments » - Tags: geometry, math, pre-algebra

#### August 27, 2014

Creating the outline of a guitar.

One of the assigned tasks from last summer’s guitar building workshop was to create a few modules for use in class. I worked on an assignment that has students calculate the volume of a guitar body using trapezoidal approximation methods that can be a bridge between pre-calculus and calculus.

The first draft of this module is here: **volume-activity-v01.pdf** (the LaTeX file is volume-activity-v01.tex.zip ). It has made contact with the ~~enemy~~ students and the results have so far been very good.

A method for finding the area of a guitar body by fitting trapezoids.

There were two things that I need to add for next time:

**How to find the area of a trapezoid**: I should have some more detail about how I came up with the formula for calculating the area of each trapezoid (see the figure above). I multiply the average of the heights of the two sides of the trapezoid by the width of the base to get the area. Students tend to want to find the area of the lower rectangle, then add the area of the upper triangle. Their method gives the same answer for area, but results in a more complicated equation that takes more effort to generalize.
**Have them also find the slope of a tangent line to the outline of the guitar at a certain point**. This assignment is intended to lead students up to the concept of limits with the idea that if you make the trapezoids thinner you’ll get less error in your calculation of the total area. So, as the width of the trapezoid approaches zero, you should get the exact area (with no error). The seemed to get that fairly well, however, when I get into the calculus, I actually first use limits to show them how to find derivatives of functions before I talk about finding areas under curves. As a result, I did ask the students to find the slope at a point on their guitar outline (I randomly chose a point from their outlines), and was very glad I did so. This should be included in the module.

Students drawing trapezoids to fit the outline of the guitar, and calculating their areas.

Finally, in addition, I also showed them how to quickly calculate the trapezoid areas once they’d entered the coordinates of each point on their graphs into Excel. I did not test them on this afterward, so I’m not sure how much of it they absorbed.

**Citing this post**: **Urbano**, L., 2014. Introducing Limits (Calculus) with a Guitar, Retrieved July 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in CalculusNo Comments » - Tags: calculus, geometry, guitar building, pre-Calculus

#### February 25, 2014

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:

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

**Citing this post**: **Urbano**, L., 2014. Finding Volumes Using the Disk Method, Retrieved July 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Mathematics, ProgrammingNo Comments » - Tags: calculus, geometry, volumes, vpython

#### November 13, 2013

NWI Instruments transit.

This spring I was nominated by my head of school for a small, Teacher of Distinction award offered by the Independent Schools of St. Louis (ISSL). I proposed to get a survey transit that our students could use to map ecological change on campus. My outdoor group has been slowly cutting down the invasive Bradford pear saplings on the slope and I’ve been curious to see if mapping their locations would help us better understand where they’re coming from.

Measuring the distance down to the creek.

The transit would also be a great tool for math. Geometry, algebra, and pre-calculus classes could all benefit because surveying can require quite a bit of geometry and trigonometry.

View through the transit. The middle mark on the reticule allows you to measure elevation change, while the upper and lower marks are used to measure distance. There’s a 100:1 conversion from the distance between the upper and lower marks and the distance from the transit to the measuring rod.

So, I’ve started training a few of my outdoor group in making the measurements. They’ve spent a few weeks learning how to use the transit; we only meet once a week so it goes slowly. However, we’ll start trying to put marks on paper at our next class.

Students trying out the transit.

**Citing this post**: **Urbano**, L., 2013. Transit, Retrieved July 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in Biology, Environmental Science, Mathematics, Natural WorldNo Comments » - Tags: campus ecology, ecology, geometry, outdoors, potential projects, student projects, transit

#### November 6, 2012

Ms. Wilson believes that the City Museum makes a great field trip for her geometry class. I think she has a point.

I had my pre-Calculus students take pictures of curves at the City Museum. Ms. Wilson’s geometry students had to photograph shapes and angles instead. Then they had to put together a slideshow of what they found, which, from what I heard, went very well.

**Citing this post**: **Urbano**, L., 2012. Geometry at the City Museum, Retrieved July 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

**Attribution (Curator's Code )**: Via: ᔥ Montessori Muddle; Hat tip: ↬ Montessori Muddle.

Posted in immersion, MathematicsNo Comments » - Tags: geometry, immersion, math