Tag: geometry

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

Transit

NWI Instruments transit.
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.
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.
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.
Students trying out the transit.

Geometry at the City Museum

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.

Seeing Functions at the City Museum

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.

Jupiter and Venus in Conjuction

Venus (brighter) and Jupiter.

Jupiter and Venus have been sitting near the western horizon, shining so brightly that even I have noticed them. Phil Plait explains with some back-of-the-envelope math, why Venus is brighter even though it’s smaller than Jupiter. It’s a nice example of how a little math can do a great job explaining how the world (and others) works.

Math & Art Contest

The Missouri Council of Teachers of Mathematics (MoCTM) has a Math & Art Contest that focuses on Geometry. It has fairly simple expectations, and it’s aimed at Middle School students and lower. The tie between the math and the art does not require much depth, but that’s probably appropriate for students who are still developing abstract thinking.

A tessellation. Image via Wikipedia.

I’m usually a bit cautious about the utility of contests. Their primary benefit is in the work that they motivate, not the reward (or hope of a reward) at the end; although, students do need to learn to win or lose with equanimity.