Math 151

Fall 03

Project Ideas

 

            Feel free to suggest something not on this list. 

 

1.  Use Fractalina to create images that look like Pascal’s triangle colored according to various modulos.   Initial Exploration: try to create a “chaos game” whose attractor looks like Pascal’s triangle mod 3 (i.e. with zeroes white and 1’s and 2’s black).

 

2.  Learn why the Pascal’s triangle modulo patterns happen as they do.  I have a reading on this  you’ll learn about numbers in various bases.  Very interesting.  Initial exploration: figure out how many odds and evens there are in each row of pascal’s triangle.  Look for patterns.  Can you predict the number of odds/evens in row 100?  (If you know about base 2 notation, compare the base 2 notation of the row number with the number of odds/evens in the row; remember first row is 0).

 

3.  Explore numerical patterns in Pascal’s Triangle (not reduced some modulo).  Initial exploration:  Look at the sums of numbers in each row.  What pattern do you notice?  Can you justify it?

 

4.  Iteration/Chaos.  Explore these topics in more depth.  Initial Exploration: Go to http://math.bu.edu/DYSYS/applets/Iteration.html and choose the equation cx(1-x).  First, explain what choice of r and X will convert the equation at the bottom of page 99 in the Devlin article to the form cx(1-x).  Play with iterating several values of the function and see if you can make better sense of pages 98-99 of the Devlin article.

 

5.  Mandelbrot/Julia Sets:  Initial Exploration:  Spend a few minutes zooming into the Mandelbrot set at http://math.bu.edu/DYSYS/applets/JuliaIteration.html just to get an idea of the pretty pictures.  What do you notice about the associated Julia sets?  To begin understanding some of the mathematics, go to http://scholar.hw.ac.uk/site/maths/topic4.asp?outline=  for an introduction to /review of complex numbers.  Note that we have activity books and also a video on this topic.

 

6.  Explore the powers of various integers in various mods.  Initial exploration:  start with mods 2, 3, 4, and 5.  In mod 4, for example, find the first, second, third, fourth, fifth, sixth etc. powers of 0, 1, 2, and 3.  Note that you can use your mod 4 multiplication table (for example, 3 x 3 =1 so 3³ = (3 x 3) x 3 = 1 x 3=3.  Look for patterns.  When are the powers 0?  1? Which numbers are “perfect squares?”

 

7.  Exploring figures analogous to cubes in various dimensions.  We can think of a cube as being formed by taking two squares and joining corresponding vertices with an edge.  Similarly, we can think of a square as being formed by taking two segments and joining corresponding endpoints, or a segment as being formed from joining two points.  Initial exploration:  Make a table of the number of vertices, sides, edges, and faces for 0, 1, 2, and 3 dimensional analogies to cubes (i.e. a point, a segment, a square, and a cube).  Look for patterns.  Imagine how you’d expect your patterns to continue into four dimensions.

 

8.  Billiard balls.  Suppose we have a rectangular billiard table, with pockets at the corners, like the one below.  We start hitting a ball from the lower left corner, at a 45 degree angle, and the ball always bounces at the same angle.  Using the shape of the table, can you determine into what pocket the ball will land?  Initial exploration:  Try this problem with ten different rectangles and look for patterns.

 

 

 

       

 

 

 

 

 

 

9.  Sketchpad investigation: area of inscribed triangles.  In the figure below, the points A’, B’, and C’ are 1/4 of the way from C, A, and B respectively (use dilation in sketchpad to create such triangles, e.g. C’ is a dilation of point C with ratio 1/4 about the point B).  Explore the ratio of the area of ABC to the area of A’B’C’, as the dilation factors change.  Initial exploration:  try dilation ratios of 1/2, 1/3, 1/4, 1/5, and 1/6.  Look for patterns.

 

 

 

10.    Tower of Hanoi Variation:  The rules of the puzzle are essentially the same: There are three pegs and disks are transferred between pegs one at a time. At no time may a bigger disk be placed on top of a smaller one. The difference is that now for every size there are two disks: one green and one red. Also, there are now two towers of disks of alternating colors. The goal of the puzzle is to make the towers the same color. The biggest disks at the bottom of the towers are assumed to swap positions.  Initial exploration:  Find the minimum number of moves with six disks: you can try the applet at http://www.cut-the-knot.org/recurrence/BiColorHanoi.shtml.  

 

Copyright 2005, Debra K. Borkovitz.  You may copy or edit this material for non-profit, educational use only.

 

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