Looking for Patterns

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Submitted By buddahmonk
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Looking for Patterns
Blaise Pascal was a French mathematician who lived during the 17th century. He is credited with creating a mechanical calculator, and he influenced the development of probability theory, economics, programming, and geometry. A pattern of numbers in the form of a triangle was named after him, although the pattern was known by other, ancient civilizations. The triangle demonstrates the inherent order of numbers and math. The exercises below challenge you to find and define patterns. Often being a good problem solver relies on one’s ability to recognize patterns and changes in patterns. •

How many triangles can you find in this picture? 27

Develop a rule to describe the number of circles in the figures below. Find the number of circles in the 20th figure, and indicate the reasoning.

Figure 1 Figure Number Number of Circles 1 1

Figure 2 2 6 3 11

Figure 3 Figure 4 4 16 5 21 6 26 7 31

...Figure 20 8 36 20 96

What is the general rule to describe how many circles are in each figure? • Find the relationship between the pyramid steps needed to build rows 6 through 10. Ro w# 0 1 2 3 4 5 6 7 8 9 10 1 1 1 1 0 9 4 5

1 1 1 1 1 1 1 1 8 3 6 12 0 7 28 8 4 21 0 6 2 1 56 12 6 25 2 5 15 35 70 12 6 21 0 4 10 20 35 56 8 4 12 0 3 6 10 15 2 1 28 3 6 4 5 2 3 4 5 6 7 8 9 1 0 1 1 1 1 1 1 1 1 1 1

3a. Find as many relationships between the numbers in the pyramid as you can, and describe them. • • • Both sides are symmetrical You can get each number by adding the two numbers above it. Each horizontal row doubles when added together from the previous row.

3b. Let the row number represent the toss of a coin. List the possible outcomes. For four coin tosses: • How many possible outcomes are there? 16 • Is this the sum of any of the rows? If yes, which? Yes, the second and fourth rows. • How many outcomes give you exactly two tails? 6 • What…...

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