Math Forum: Exploring Pascal's Triangle

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Exploring Pascal's Triangle

Advanced Level

Patterns in Color

1.

2.
Choose any five colors.
Assign a different color
to each number and shade
each block on the color
chart accordingly.
(See closure, below.)

3.
Print a blank Pascal Triangle grid from the student worksheets page.
Color the top three hexagons color 1. (Using black for color 1 provides a nice outline.)

4.
To determine the color of the next row of cells, look at the last row:

if there is only one cell above a cell, make that cell color 1.
if there are two cells above a cell, use the chart to find the color to use.
if the two cells above are both color 1, look at row 1 of column 1 on the chart for the color to use. It is color 2.
if the two cells above are colors 1 and 2, look at row 1 of column 2: it tells you to use color 3.

5.

The first few rows would be colored like this:
When the grid has been completely colored, cut it out carefully along the edges.

6. Six identically colored triangles can be joined
to form a hexagon.
These constructions make great classroom or hall decorations. Looking at the center point gives the
illusion of a cube in three dimensions.

7.
By joining or slightly
overlapping twelve
colored triangles,
you can create
a star.

Property Connections

1.	Closure: A set S is closed under the operation * if whenever a and b are in S, *a b** is in S. For our coloring we defined a set of 5 colors and an operation defined by the chart. It was necessary that our set be closed under the given operation. Why?
2.	Identity Property: There is a number 0, called the additive identity, such that a + 0 = a and 0 + a = a. What is the identity for our coloring operation?
3.	Inverse Property: For every a, there is a number -a, called the additive inverse, or opposite, of a, such that a + (-a) = 0 and -a + a = 0 where 0 is the identity for the operation. Does each color in our set have an inverse? If yes, name them.
4.	Commutative Property: a + b = b + a Is our operation of coloring a commutative operation? Why or why not?
5.	How does our set of colors compare to mod 5 arithmetic?

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Math Forum * * * * 4 May 1998

1. 2.	Choose any five colors. Assign a different color to each number and shade each block on the color chart accordingly. (See closure, below.)

3.	Print a blank Pascal Triangle grid from the student worksheets page. Color the top three hexagons color 1. (Using black for color 1 provides a nice outline.)

4.	To determine the color of the next row of cells, look at the last row: if there is only one cell above a cell, make that cell color 1. if there are two cells above a cell, use the chart to find the color to use. if the two cells above are both color 1, look at row 1 of column 1 on the chart for the color to use. It is color 2. if the two cells above are colors 1 and 2, look at row 1 of column 2: it tells you to use color 3.

5.	The first few rows would be colored like this: When the grid has been completely colored, cut it out carefully along the edges.

6.	Six identically colored triangles can be joined to form a hexagon. These constructions make great classroom or hall decorations. Looking at the center point gives the illusion of a cube in three dimensions.