#### 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 February 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 February 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 February 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 February 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 February 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 February 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

#### October 31, 2012

The slide on the third floor of the City Museum. A co-ordinate system is overlayed, and points showing the curve of the slide are selected.

Elegant curves.

I asked my students to take pictures of the curves they found while on our field trip to the scrap metal playground that is the City Museum. The plan is to see if we can determine what functions best fit the curves. To do so, we need to transfer the curves from the images to a co-ordinate system. Since I’m primarily interested in what type of functions might best fit the data, the scale of the co-ordinates does not matter that much.

Feet, inches, meters, centimeters, pixels, or any other units can be used. In fact, I use a purely arbitrary set of coordinates in the image above. All I require is that the grid be evenly spaced (although the vertical and horizontal spacing don’t have to be the same, it’s more straightforward if they are).

Now we take a set of points that lie on our shape and try to match them to some sort of curve using a spreadsheet, and, if we’re able, least squares regression.

There were lots of shapes to choose from.

There were lots of shapes to choose from, including the nice sinusoid in the background.

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

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

Posted in Algebra, immersion, MathematicsNo Comments » - Tags: algebra, field trips, geometry, immersion, math, pre-Calculus, St. Louis field trips

#### June 9, 2012

The Royal Observatory, Greenwich, explains:

Includes a nice animation of the Doppler effect; explains red shifting of distance galaxies; and how we can use geometry (parallax) to determine the distance to nearby stars.

↬ The Dish

**Citing this post**: **Urbano**, L., 2012. How do we know how big the Universe is?, Retrieved February 23rd, 2017, from *Montessori Muddle*: http://MontessoriMuddle.org/ .

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

Posted in Mathematics, Natural WorldNo Comments » - Tags: geometry, math, shorts, space science