Lesson Notes By Weeks and Term v4 - SHS 2

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Identify proper and improper rational functions.
• Resolve functions with linear factors in the denominator.
• Resolve functions with irreducible quadratic factors.
• Resolve improper rational functions.

Lesson summary

Welcome, learners! In our previous work with algebra, we learned how to combine several fractions into a single fraction. Today, we are going to learn the reverse process, called decomposition into partial fractions. Imagine you have a complex system, like the flow of traffic at the Kwame Nkrumah Circle interchange or how different factors affect the price of cocoa. Sometimes, to understand the whole system, you need to break it down into its simpler, individual parts. Partial fractions are a mathematical tool that allows us to do just that for complex algebraic fractions (rational functions).

Lesson notes

This lesson is divided into three main parts: a quick revision, and then two new, more advanced techniques. Part 1: Foundational Concepts (Revision)

A rational function is a fraction where both the numerator and the denominator are polynomials, like `(2x + 1) / (x² - 4)`. Proper Rational Function: The degree (highest power of x) of the numerator is less than the degree of the denominator. Example: `(x + 5) / (x² + 3x + 2)`. Degree of numerator is 1, degree of denominator is 2. (1 2).

Revision Case 1: Denominator with Distinct Linear Factors If a denominator has factors like `(x-a)(x-b)`, the form of the partial fraction is: `P(x) / ((x-a)(x-b)) = A/(x-a) + B/(x-b)`

Revision Case 2: Denominator with Repeated Linear Factors If a denominator has a factor like `(x-a)²`, you must include a term for each power of the factor: `P(x) / (x-a)² = A/(x-a) + B/(x-a)²` Part 2: New Concept - Irreducible Quadratic Factors

Evaluation guide

• Resolve rational functions into partial fractions, specifically quadratic denominators and
• improper rational functions.
• Activity1: Working with Partial Fractions Collaborative Learning, Experiential Learning, Whole class discussion, Talk for Learning
• Approaches and Problem -based Learning