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.
August 29, 2016
August 27, 2015
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.
August 27, 2014
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.
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.
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.
February 25, 2014
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
November 13, 2013
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.
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.
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.
November 6, 2012
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.
October 31, 2012
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.