Math Web Sites

2.   

Powers of Three

               
I first ran across the fraction 1/243 in one of the Rama series of books by
Arthur C. Clarke. Now 243 is the fifth power of 3. The decimal equivalent forms
a very curious pattern as I describe in the notes. I then created an Excel
spreadsheet to calculate decimal equivalents of the reciprocals of different
powers of three to help analyze the patterns ( the notes explain how to create
this spreadsheet ). I then could extend the work to explore powers of other
repeating decimals. The only limitation is the size of your spreadsheet and your
imagination.

2.   

Powers of Three

               
I first ran across the fraction 1/243 in one of the Rama series of books by
Arthur C. Clarke. Now 243 is the fifth power of 3. The decimal equivalent forms
a very curious pattern as I describe in the notes. I then created an Excel
spreadsheet to calculate decimal equivalents of the reciprocals of different
powers of three to help analyze the patterns ( the notes explain how to create
this spreadsheet ). I then could extend the work to explore powers of other
repeating decimals. The only limitation is the size of your spreadsheet and your
imagination.

3.   
Graphing Project

               
As part of an analytic geometry I would often assign as a project the creation
of a picture, a picture constructed from the equations in the geometry section.
In some courses this mean only linear relations, or linear and quadratic
functions or, in senior courses, trig and conic sections functions. A typical
assignment is included. I would create an example ( to save space I've omitted
the picture for this example,- you're welcome to try and produce it from the
equations ). I would hand out the instructions, with the example, a blank table
and a piece of graph paper, and, of course, a due date . Any picture in good
taste was acceptable - geometric designs, copies of logos, whatever. The majority
of the marks were for the math but I always included a small part for creativity
and neatness. 

               
Much along the same lines, I once had senior students, once we finished the
section on ellipses, try to construct the Toyota symbol as a combination of
three ellipses ( best done on a computer ).

8.     
Sloping Letters

                       
This is the start of a simple grade 9 introduction to the difference between
positive and negative slopes ( also the slopes of horizontal and vertical lines
). It can easily be extended by having the students make up their own phrase (
always something in good taste ) and the corresponding description using slopes.

9.       Lotteries

                         
I used lotteries at the junior and senior levels. At the junior level it is
fairly easy to explain the basis of probability. Then I can use the
multiplication principle as an example of fraction multiplication. I've attached
a PowerPoint slide show I created to illustrate the chance of winning a lottery
- the last page is a worksheet the students would have in order to follow along with the
slide show. At the senior level I've created a variety of assignments ( I
changed the lotteries I used from year to year ) primarily based on calculating
the expected value of a set of lotteries.

• ODDS
• SLOT MACHINE
• CRAPS
• 
  THE EXPECTED VALUE OF A SEVEN-GAME PLAYOFF
  SERIES
• BACK TO THE TOP

11.       
Pi in a Can

                       
This is a simple grade 9 or 10 exercise that combines measurement skills ,
graphing skills and the concept of slope. You need a variety of different
diameter cans. You might assign this as a take-home project in which each
student had to find the different cans at home or you might have different cans
brought into the class ( once the activity is done, the cans could be donated to
the local food bank ). The students, working in groups of two, need to determine
as accurately as possible the diameter and circumference of each type of can. If
this is to done in class you would need rulers and string ( wrap the string
around the can to get the circumference, then lay it out against a ruler ). Each
group of students creates a table of circumference versus diameter, then plots a
graph, with diameter along the horizontal axis. The points should fit a linear
relation fairly closely. For analysis, depending on the students and the course, you
could do one or more of the following :

• leave it there, simply as
  an exercise in making a table and graphing
• calculate the slope,
  showing that it equals pi
• do curve fitting with the
  TI-83 or in Excel
• discuss direct variation
  as the line should go through (0,0)
• do error analysis,
  suggesting ways to improve the results ( i.e. averaging
  group results )

14.       Stylometry

                        
Stylometry is the statistical study of written material. It turns out that the
style of a particular author shows up in the distribution of word length, sentence length, in the repetition of certain words, etc. . There have been many books
written on the subject ( a good starting point would be

http://en.wikipedia.org/wiki/Stylometry ) . Stylometry is sometimes used to settle disputes of
authorship ( from the bible to Shakespeare to the courtroom ). As an exercise in
graphing and descriptive statistics I would have students read passages from
different texts ( or passages from the same text but in different chapters or
sections ) and do some or all of the following ( depending on time ) :

graph the distribution of
word length

graph the distribution of
sentence length

find average word length and
sentence length

compare graphs to help identify which passages are
from the same author

for an author who writes
under his or her own name but also under a pseudonym, determine if
there is a change in style

look at the different
methods used to calculate a readability level ( see

http://en.wikipedia.org/wiki/Readability )

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