Exploring Pascal's Triangle
This activity can be done with or without a computer. Here are two possibilities:
Print a copy of the Pascal worksheet and color the cells using crayons, markers or colored pencils. Choose one color for the cells that contain a multiple of 3, a second color for cells that contain numbers that are one less than a multiple of 3, and a third color for cells that contain numbers that are two less than a multiple of 3.
In the example above, cells with multiples of 3 are colored red, cells with numbers one less than a multiple of 3 (2, 5, 8, etc.) are colored green, and cells with numbers two less than a multiple of 3 (1, 4, 7, etc.) are colored blue.
Use the ClarisWorks paint program to color the cells.
- Download a copy of the (compressed) Pascal worksheet. The file will automatically unpack with Stuffit Expander. If you don't have Stuffit Expander, you can download a copy from the Math Forum Software page.
- Open the file in ClarisWorks.
- Select your favorite color from the paint palette.
- Use the paint bucket tool to pour the chosen color into all cells that contain a multiple of 3.
- Select a second color from the paint palette and color all cells that contain a number that is one less than a multiple of 3.
- Select a third color from the paint palette and color all cells that contain a number that is two less than a multiple of 3.
Use either coloring technique with:
four colors to color the multiples of four, numbers one less than multiples of four, numbers two less than multiples of four, and numbers three less than multiples of four.
five colors to color the multiples of five, numbers one less than multiples of five, numbers two less than multiples of five, numbers three less than multiples of five, and numbers four less than multiples of five.
any number of colors.
The Mathematics in the Patterns
The divisibility rule for three requires students to add the digits in the number. If the sum of the digits is a multiple of three, then the number is divisible by three. When they use Pascal's triangle to practice this rule, students are less likely to reach for their calculators to check for divisibility.
Students will also begin to recognize patterns in numbers. For example, 924 is divisible by 3. Students may observe that 9 + 2 + 4 = 15, or they may realize that since 9 is already a multiple of 3 and the sums of multiples of 3 produce other multiples of 3, they need only note that 2 + 4 = 6 to show that 924 is a multiple of 3.
The divisibility rule for three can be used to determine which numbers are one less than a multiple of three. The number 200 is one less than a multiple of three: 2 + 0 + 0 = 2, which is one less than 3.
The rule for divisibility for three can be used to determine which numbers are two less than a multiple of three. The number 52 is two less than a multiple of three: 5 + 2 = 7, which is two less than 9.
The Commutative Property of Addition
This property, together with the structure of Pascal's Triangle, explains the symmetry that can be observed in the colors of each row of cells.
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 optical illusion of a cube in three dimensions.
For more ideas, see Patterns in Color (Advanced Level).
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