Lesson Notes By Weeks and Term v4 - SHS 1
NUMBER AND ALGEBRAIC PATTERNS
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NUMBER AND ALGEBRAIC PATTERNS
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Subject: Additional Mathematics
Class: SHS 1
Term: 1st Term
Week: 3
Grade code: 1.1.1.LI.5
Strand code: 1
Sub-strand code: 1
Content standard code: 1.1.1.CS.1
Indicator code: 1.1.1.LI.5
Theme: MODELLING WITH ALGEBRA
Subtheme: NUMBER AND ALGEBRAIC PATTERNS
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This lesson introduces a powerful and elegant tool from mathematics for expanding binomials raised to a power, such as $(x+y)^5$ or $(2a-3b)^4$. Multiplying these expressions out by hand is a long, tedious process prone to errors. We will discover a fascinating pattern of numbers, known as Pascal's Triangle, that provides a shortcut. This method is not just a mathematical trick; it forms the foundation for topics in probability (like predicting outcomes in games or genetics), finance (calculating compound interest), and higher-level mathematics.
A. What is a Binomial Expression? A binomial is simply an algebraic expression that contains two terms. *Examples:* $x + y$ $a - 3b$ $2p + 5$ $m - \frac{1}{2}n$
"Binomial expansion" is the process of multiplying a binomial by itself a certain number of times.
Let's try a few by hand to see the challenge: $(a+b)^1 = a+b$ $(a+b)^2 = (a+b)(a+b) = a^2 + ab + ba + b^2 = \boldsymbol{1}a^2 + \boldsymbol{2}ab + \boldsymbol{1}b^2$ $(a+b)^3 = (a+b)(a^2 + 2ab + b^2) = a^3 + 2a^2b + ab^2 + a^2b + 2ab^2 + b^3 = \boldsymbol{1}a^3 + \boldsymbol{3}a^2b + \boldsymbol{3}ab^2 + \boldsymbol{1}b^3$
Notice the numbers in bold (the coefficients): For $(a+b)^2$, they are 1, 2, 1. For $(a+b)^3$, they are 1, 3, 3, 1.