Algebra 2
Name: ______________________________
Lamar Academy
Date: __________________ Period: ______
Pascal’s triangle investigation
This task will be assessed using Criterion B (investigating patterns) and Criterion D (reflection in
mathematics).
Maximum mark: 50
Part 1 – A brief historical background of Pascal’s triangle and the Chinese triangle
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Define Pascal’s triangle and discuss it’s number pattern
Provide a brief historical background of Blaise Pascal
Define the Chinese triangle and it’s relationship to Pascal’s triangle
Discuss uses and applications of both Pascal’s triangle and the Chinese triangle
Part 2 – Exploring number patterns in Pascal’s triangle
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As you look for number patterns, try to answer the following questions:
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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?
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Find as many of the following number patterns in Pascal’s triangle. Be sure to
demonstrate and explain how each pattern is obtained.
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natural numbers
powers of 2
powers of 11
triangular numbers
hexagonal numbers
pentatope numbers
Fibonacci numbers
Catalan numbers
1, 2, 3, 4, …
2, 4, 8, 16, …
11, 121, 1331, 14641, …
1, 3, 6, 10, … (why are these numbers called triangular?)
1, 6, 15, 28, …(why are these numbers called hexagonal?)
1, 5, 15, 35, …(why are these numbers called pentatope?)
1, 1, 2, 3, 5, 8, … (who was Fibonacci?)
1, 2, 5, 14, 42, … (who was Catalan?)
Suggest other number patterns that can be observed from using Pascal’s triangle. Be sure
to demonstrate and explain how each pattern is obtained.
Part 3 – Relations between algebra and Pascal’s triangle
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Expand (a  b) 0 , (a  b)1 , (a  b) 2 and (a  b) 3 .
Use your answers to make a conjecture about the relationship between Pascal’s triangle
and the multiplication of binomials.
Test your conjecture by expanding (a  b) 4 with multiplication and by using Pascal’s
triangle.
Generalize your conjecture for finding the expansion of (a  b) n .
Discuss the scope and limitations of your general statement.
Part 4 – Relations between coin flipping and Pascal’s triangle
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List all possible outcomes when flipping one coin, two coins, and three coins.
Use your answers to make a conjecture about the relationship between Pascal’s triangle
and flipping coins.
Test your conjecture by listing the possible outcomes when flipping 4 coins and by using
Pascal’s triangle.
Generalize your conjecture for listing the outcomes when flipping n coins.
Discuss the scope and limitations of your general statement.
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Part 5 – Picture patterns in coloring Pascal’s triangle
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Use one color to shade all odd numbers in Pascal’s triangle. The pattern produced is
called the Sierpinski triangle.
Briefly discuss some of the characteristics of the Sierpinski triangle.
Using four different colors,
use one color to shade all numbers that are multiples of 4
use another color to shade all numbers that are one less than the multiples of 4
use a third color to shade all numbers that are two less than the multiples of 4
use a fourth color to shade all numbers that are three less than the multiples of 4.
Comment on the pattern created by using four colors to shade the numbers in Pascal’s
triangle.
Use a similar strategy to shade the numbers in Pascal’s triangle using three (or five)
colors.
Comment on the pattern created.
You must include a list of the resources used on this task by providing a bibliography page.
You may find the two Pascal’s triangle worksheets that follow helpful as you work through your
investigation.
Read through the assessment criterion descriptions to make sure you are meeting the
requirements of this task.
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