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Exploring Pascal's Triangle

Advanced Level

Patterns in Color




Choose any five colors.

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



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.)


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

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


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.






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?

See also Coloring Multiples (Intermediate Level).

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