Lesson Note on Algebraic Fractions: Equations and Substitution for SSS 2

Class: Senior Secondary School 2 (SS2 / SSS 2)
Term: Second Term
Week: 4
Age: 16 years
Duration: 45 minutes
Subject: General Mathematics
Curriculum Theme: Algebraic Expressions
Previous Lesson: Circle Geometry: Tangent to a Circle.
Topic: ALGEBRAIC FRACTIONS (II)
Subject Matter: Equation involving fractions, substitution in fractions, simultaneous equations involving fractions, finding the value of unknown to make a fraction undefined.

Specific Objectives

By the end of the lesson, pupils should be able to:

Cognitive Domain:

• Solve equations involving fractions.
• Perform substitution in given algebraic fractions.
• Solve simultaneous equations involving fractions.
• Determine the value(s) of an unknown that make a fraction undefined.

Affective Domain:

• Appreciate the importance of algebraic fractions in solving mathematical problems.
• Develop a systematic approach to solving complex algebraic fraction problems.

Psychomotor Domain:

• Accurately manipulate algebraic expressions to simplify and solve equations.
• Present solutions clearly and logically for various algebraic fraction problems.

Social Domain:

• Collaborate effectively with peers to solve challenging algebraic fraction problems.

Reference Materials

The following resources were used in planning this lesson:

• 9 Years Basic Education Curriculum
• State Unified Scheme of Work
• New General Mathematics for Senior Secondary Schools 2
• https://www.mathsisfun.com/algebra/fractions-equations.html
• https://www.dummies.com/education/math/algebra/how-to-solve-simultaneous-equations-with-fractions/

Instructional Materials

The teacher will teach this lesson with the aid of:

• Oranges
• Apples
• Ruler
• Sticks
• Whiteboard/Blackboard
• Markers/Chalk
• Textbooks

Rationale for the Lesson

Understanding algebraic fractions helps pupils to solve real-world problems that involve relationships between quantities. It is an important foundation for higher mathematics and various scientific fields, enabling pupils to handle more complex mathematical concepts.

Prerequisite/Previous Knowledge

Pupils should have prior knowledge of basic algebraic operations, simplification of algebraic fractions, finding the Least Common Multiple (LCM), and solving linear equations.

Lesson Content/Board Summary

ALGEBRAIC FRACTIONS (II)

1. Equations Involving Fractions

Equations involving fractions are algebraic equations where one or more terms are algebraic fractions. To solve these equations, the first step is usually to clear the denominators.

Steps to solve equations involving fractions:

• Find the Least Common Multiple (LCM) of all the denominators in the equation.
• Multiply every term in the equation by the LCM to clear the denominators.
• Simplify the resulting equation (it should now be free of fractions).
• Solve the simplified equation for the unknown variable using standard algebraic methods.

Example 1: Solve for x in the equation: (x/2) + (x/3) = 5

Solution:

1. Step 1: Find the LCM of the denominators (2 and 3). The LCM is 6.
2. Step 2: Multiply every term by the LCM (6):
   6 * (x/2) + 6 * (x/3) = 6 * 5
3. Step 3: Simplify the equation:
   3x + 2x = 30
5x = 30
4. Step 4: Solve for x:
   x = 30 / 5
x = 6

Example 2: Solve for y in the equation: (2y - 1)/3 = (y + 2)/4

Solution:

1. Step 1: Find the LCM of the denominators (3 and 4). The LCM is 12.
2. Step 2: Multiply every term by the LCM (12):
   12 * ((2y - 1)/3) = 12 * ((y + 2)/4)
3. Step 3: Simplify the equation:
   4(2y - 1) = 3(y + 2)
8y - 4 = 3y + 6
4. Step 4: Collect like terms:
   8y - 3y = 6 + 4
5y = 10
5. Step 5: Solve for y:
   y = 10 / 5
y = 2

2. Substitution in Fractions

Substitution in fractions involves replacing variables in an algebraic fraction with given numerical values or other algebraic expressions, then simplifying the resulting expression.

Example 1: If x = 2 and y = 3, evaluate the fraction (x + y) / (x - y).

Solution:

1. Step 1: Substitute the given values of x and y into the fraction:
   (2 + 3) / (2 - 3)
2. Step 2: Simplify the numerator and the denominator:
   5 / (-1)
3. Step 3: Perform the division:
   -5

Example 2: If a = -1 and b = 4, evaluate (a^2 + b) / (2a - b).

Solution:

1. Step 1: Substitute a = -1 and b = 4 into the expression:
   ((-1)^2 + 4) / (2(-1) - 4)
2. Step 2: Simplify the numerator and the denominator:
   Numerator: 1 + 4 = 5
   Denominator: -2 - 4 = -6
3. Step 3: The result is:
   5 / -6 or -5/6

3. Simultaneous Equations Involving Fractions

Simultaneous equations involving fractions are a set of two or more equations with multiple variables, where at least one equation contains fractions. To solve them, first clear the fractions from each equation, then solve the resulting linear simultaneous equations using methods like elimination or substitution.

Example: Solve the following simultaneous equations:

1) x/2 + y/3 = 1
2) x - y/2 = 4

Solution:

1. Step 1: Clear fractions from Equation 1.
   Multiply Equation 1 by the LCM of 2 and 3 (which is 6):
   6 * (x/2) + 6 * (y/3) = 6 * 1
3x + 2y = 6 (This is Equation 3)
2. Step 2: Clear fractions from Equation 2.
   Multiply Equation 2 by the LCM of 1 and 2 (which is 2):
   2 * x - 2 * (y/2) = 2 * 4
2x - y = 8 (This is Equation 4)
3. Step 3: Now solve the new simultaneous equations (Equations 3 and 4):
   3) 3x + 2y = 6
   4) 2x - y = 8
4. Step 4: Use the elimination method. Multiply Equation 4 by 2 to make the coefficients of y equal:
   2 * (2x - y) = 2 * 8
4x - 2y = 16 (This is Equation 5)
5. Step 5: Add Equation 3 and Equation 5 to eliminate y:
   (3x + 2y) + (4x - 2y) = 6 + 16
7x = 22
x = 22/7
6. Step 6: Substitute the value of x into Equation 4 to find y:
   2(22/7) - y = 8
44/7 - y = 8
-y = 8 - 44/7
-y = (56 - 44)/7
-y = 12/7
y = -12/7

Therefore, x = 22/7 and y = -12/7.

4. Finding the Value of Unknown to Make a Fraction Undefined

A fraction is undefined when its denominator is equal to zero. This is because division by zero is not mathematically permissible.

To find the value(s) of the unknown that make a fraction undefined:

• Set the denominator of the fraction equal to zero.
• Solve the resulting equation for the unknown variable.

Example 1: For what value of x is the fraction 1 / (x - 3) undefined?

Solution:

1. Step 1: Set the denominator equal to zero:
   x - 3 = 0
2. Step 2: Solve for x:
   x = 3

Therefore, the fraction is undefined when x = 3.

Example 2: Find the value(s) of y for which the fraction (y + 1) / (y^2 - 4) is undefined.

Solution:

1. Step 1: Set the denominator equal to zero:
   y^2 - 4 = 0
2. Step 2: Solve for y (using factorization or square root method):
   y^2 = 4
y = ±√4
y = 2 or y = -2

Therefore, the fraction is undefined when y = 2 or y = -2.

Example 3: For what value(s) of z is the fraction (2z - 5) / (z^2 + 5z + 6) undefined?

Solution:

1. Step 1: Set the denominator equal to zero:
   z^2 + 5z + 6 = 0
2. Step 2: Factorize the quadratic equation:
   (z + 2)(z + 3) = 0
3. Step 3: Solve for z:
   z + 2 = 0 or z + 3 = 0
z = -2 or z = -3

Therefore, the fraction is undefined when z = -2 or z = -3.

Teaching Methods/Instructional Techniques

Discussion, Lecture, Demonstration, Question and Answer, Visual Aids

Instructional Procedures

Step 1: Introduction

Time: 5 minutes
Teaching Skill: Set Induction
Teacher’s Activity: The teacher reviews the previous lesson on simplification of algebraic fractions and introduces the new topic, “Algebraic Fractions (II)”, by stating that pupils will now learn to solve equations involving fractions, perform substitution, solve simultaneous equations with fractions, and identify values that make a fraction undefined.
Pupils’ Activity: Pupils respond to questions about the previous lesson and listen attentively to the introduction of the new topic.
Learning Point: Pupils recall previous knowledge and understand the scope of the new lesson.

Step 2: Equations Involving Fractions

Time: 10 minutes
Teaching Skill: Explanation/Demonstration
Teacher’s Activity: The teacher explains the steps involved in solving equations that contain fractions, emphasizing finding the LCM of denominators and multiplying through. The teacher demonstrates with clear step-by-step examples on the board, guiding pupils through each stage.
Pupils’ Activity: Pupils pay close attention, copy the examples, and ask questions for clarification. They attempt to follow the procedures.
Learning Point: Pupils learn how to systematically solve algebraic equations that involve fractions.

Step 3: Substitution in Fractions

Time: 5 minutes
Teaching Skill: Explanation/Practice
Teacher’s Activity: The teacher explains what substitution means in the context of algebraic fractions. The teacher provides examples, showing how to replace variables with given values and simplify the resulting fraction.
Pupils’ Activity: Pupils listen, copy the examples, and practice by attempting similar problems provided by the teacher.
Learning Point: Pupils develop the skill of substituting numerical values into algebraic fractions and simplifying them.

Step 4: Simultaneous Equations Involving Fractions

Time: 10 minutes
Teaching Skill: Demonstration/Guided Practice
Teacher’s Activity: The teacher explains how to approach simultaneous equations that include fractions. The teacher demonstrates by clearing the denominators in each equation first, then solving the resulting linear simultaneous equations using either the elimination or substitution method. The teacher guides pupils through a detailed example.
Pupils’ Activity: Pupils observe the demonstration, take notes, and actively participate by suggesting steps or asking questions.
Learning Point: Pupils learn to solve systems of equations where fractions are present.

Step 5: Finding the Value of Unknown to Make a Fraction Undefined

Time: 5 minutes
Teaching Skill: Concept Introduction/Problem Solving
Teacher’s Activity: The teacher introduces the concept of an “undefined” fraction, explaining that it occurs when the denominator is zero. The teacher demonstrates how to find the value(s) that make a fraction undefined by setting the denominator to zero and solving the resulting equation, including examples with quadratic denominators.
Pupils’ Activity: Pupils grasp the concept of an undefined fraction, copy the examples, and practice identifying such values.
Learning Point: Pupils understand what makes a fraction undefined and how to determine the specific values of the unknown that cause this.

Step 6: Evaluation/Review

Time: 5 minutes
Teaching Skill: Questioning/Assessment
Teacher’s Activity: The teacher evaluates the learning by asking the following questions:

1. Define what makes an algebraic fraction undefined.
2. Solve for x in the equation: (x/4) - (x/6) = 1.
3. If a = 5 and b = -2, evaluate (2a + b) / (a - b).
4. Find the value(s) of y for which the fraction (y + 5) / (y^2 - 9) is undefined.

Pupils’ Activity: Pupils answer orally and in writing.

Learning Point: Pupils demonstrate understanding of the lesson.

Step 7: Conclusion

Time: 5 minutes
Teaching Skill: Summarization/Assignment
Teacher’s Activity: The teacher summarizes the main points of the lesson, reiterating the methods for solving equations with fractions, performing substitution, solving simultaneous equations with fractions, and finding undefined values. The teacher then assigns relevant homework from the textbook.
Pupils’ Activity: Pupils listen to the summary and copy down the assigned homework.
Learning Point: Pupils reinforce their understanding of the key concepts covered in the lesson.

Lesson Keywords

• Algebraic fractions – Fractions that contain variables in the numerator, denominator, or both.
• Equation – A mathematical statement that shows two expressions are equal.
• Substitution – Replacing a variable with a specific value or expression.
• Simultaneous equations – A set of equations with multiple variables that are solved together to find a unique solution for each variable.
• Undefined – A mathematical expression that does not have a meaningful value, typically occurring when division by zero is attempted.
• Denominator – The bottom part of a fraction, indicating the number of equal parts into which the whole is divided.
• Unknown – A variable in an equation whose value is to be determined.

Differentiation

For pupils who grasp concepts quickly, the teacher will provide more complex problems involving fractions with quadratic denominators or more intricate simultaneous equations. For pupils who require additional support, the teacher will offer simpler, step-by-step examples, provide individual attention, and encourage peer tutoring. Visual aids and repeated demonstrations will be used to cater to diverse learning styles.

Note for teachers using this lesson plan

Ensure that pupils have a strong foundation in basic fraction operations and solving linear equations before introducing this lesson. Emphasize the importance of finding the LCM correctly and multiplying all terms by it to clear denominators. Encourage pupils to show all their working steps clearly, especially when dealing with multiple variables or complex expressions, to avoid errors.