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Working with a partner, study the following array of numbers. What patterns do you see in the arrangement of the numbers? Describe each pattern using words and symbols. Exploring Pascal's Triangle:
Patterns, Relations, Functions, and Algebra
As you look for patterns, try to answer the following questions:
- Can you predict the next row of numbers?
- Is there a pattern in the sums of the numbers in the rows?
- Do any numbers repeat?
- Can you find a pattern in the diagonal numbers?
Share your discoveries with the group.
See if you can find:
natural numbers
1, 2, 3, 4, ...
1, 5, 15, 35, 70, ...
2, 4, 8, 16, ...
1, 2, 5, 14, 42, ...
11, 121, 1331, 14641, ...
1, 1, 2, 3, 5, 8, ...
1, 3, 6, 10, ...
1, 4, 10, 20, ...
1, 6, 15, 28, ...
Interactive Number Sets: The Geometer's Sketchpad
If you have The Geometer's Sketchpad, you can download and experiment with an interactive version of Pascal's Triangle that will show some of the patterns (401K). You can also download sketches that illustrate Sierpinski's Triangle and Multiples of Threes (385K) and Hexagonal, Tetrahedral, and Pentatope Numbers (389K). If you don't have the Geometer's Sketchpad, you can download a demo version.
If you are using Netscape 3.0, Internet Explorer 3.0 or a later version of these browsers you can view an interactive version of Pascal's Triangle that uses java script. Please be patient; this page takes a few moments to load.
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