Course Content
Algebraic Expressions
Introduction Algebra is a part of mathematics where letters are used together with numbers and operation signs. For example, 2x + 3y, ½a² + ¾bc, and 2x² − 3x + 2 are called algebraic expressions. They are made up of numbers, letters, and signs such as +, −, ×, and ÷. In algebra, letters are used to stand for numbers. Some letters can have different values depending on the question. These letters are called variables. For example, in ⅓πr²h, the letters r and h are variables because their values can change. Some values do not change. These are called constants. In the expression ⅓πr²h, ⅓ and π are constants because their values remain fixed. An algebraic expression is divided into smaller parts called terms. Terms are separated by plus (+) or minus (−) signs. For example, in the expression: 3xy² + 2xz³ − 4y + 5 the terms are 3xy², 2xz³, −4y, and +5. A number that appears alone, without any letter attached to it, is called the constant term. In 3xy² + 2xz³ − 4y + 5, the constant term is 5. The number written before a letter or a group of letters is called the coefficient. For example, in 2x − 3x² + 4y³, the coefficient of x is 2, the coefficient of x² is −3, and the coefficient of y³ is 4. Some terms have the same letters raised to the same powers. These are called like terms. For example, 4x²y and 7x²y are like terms because both contain x²y. However, 5x²y and 9xy² are not like terms because the powers of x and y are different. Important Points to Remember In algebra, the multiplication sign × is usually not written. For example, instead of writing a × b, we simply write ab. This also helps us avoid confusing the multiplication sign × with the letter x. In the same way, a ÷ b can be written as a/b. When writing algebraic terms, the number normally comes first, followed by the letters in alphabetical order. For example, 6x²yz and 9abc³ are written in the correct order. The plus (+) or minus (−) sign belongs to the term that comes immediately after it. For example, in 3xy² + 2xz³ − 4y + 5, the term involving 4y is −4y, not just 4y.
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Introduction of Series and Algebra with integer Operations
27:34
Simplifications of Like term Collecting like terms
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Sum, difference and multiplications of Algebraic Expressions
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Lesson 4: Introduction to Factorization
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Lesson 5: Factorization by Grouping content and Introduction to difference two squares
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Lesson 6 Difference of Two Squares constants and intro to Quadratic Trinomials
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GENERAL MATHEMATICS (WAEC FOCUS)

Topic Area: Number, Algebra, and Sequences

Level: Senior Secondary School / WASSCE Candidates

1. Topic Overview

This topic introduces learners to the basic ideas of series, algebra, and integer operations. It is designed to help students understand how numbers follow patterns, how letters can be used to represent numbers, and how positive and negative numbers are added, subtracted, multiplied, and divided.

In WASSCE General Mathematics, questions on series, algebra, and integers are very common. Learners are expected to recognize number patterns, simplify algebraic expressions, solve basic algebraic problems, and correctly handle operations involving positive and negative numbers.

This topic provides a foundation for more advanced areas of mathematics such as arithmetic progression, geometric progression, linear equations, quadratic equations, graphs, and word problems.

2. General Learning Objectives

By the end of this topic, learners should be able to:

  1. Explain the meaning of a series.

  2. Identify simple number patterns.

  3. Differentiate between a sequence and a series.

  4. Understand the meaning of algebra.

  5. Identify variables, constants, coefficients, and terms.

  6. Perform addition, subtraction, multiplication, and division of integers.

  7. Apply integer rules in algebraic expressions.

  8. Simplify simple algebraic expressions involving integers.

  9. Solve basic WASSCE-style questions involving series and algebra.

  10. Avoid common mistakes when working with negative numbers.

Part A: Introduction to Series

3. Meaning of Sequence and Series

A sequence is a list of numbers arranged in a particular order or pattern.

Example:

2, 4, 6, 8, 10

This is a sequence because the numbers follow a pattern. Each number increases by 2.

A series is the sum of the terms of a sequence.

Example:

2 + 4 + 6 + 8 + 10

This is a series because the terms are being added together.

Important Difference

A sequence is a list of numbers.

A series is the sum of the numbers in the sequence.

Example:

Sequence: 3, 6, 9, 12
Series: 3 + 6 + 9 + 12

4. Types of Simple Number Patterns

4.1 Increasing Pattern

An increasing pattern is a pattern where the numbers become larger.

Example:

5, 10, 15, 20, 25

The numbers increase by 5 each time.

4.2 Decreasing Pattern

A decreasing pattern is a pattern where the numbers become smaller.

Example:

50, 45, 40, 35, 30

The numbers decrease by 5 each time.

4.3 Multiplication Pattern

In this pattern, each term is multiplied by the same number.

Example:

2, 4, 8, 16, 32

Each term is multiplied by 2.

4.4 Division Pattern

In this pattern, each term is divided by the same number.

Example:

64, 32, 16, 8, 4

Each term is divided by 2.

5. Worked Examples on Series and Number Patterns

Example 1

Find the next two terms in the sequence:

4, 8, 12, 16, __, __

Solution

The numbers increase by 4 each time.

4 + 4 = 8
8 + 4 = 12
12 + 4 = 16
16 + 4 = 20
20 + 4 = 24

Answer

The next two terms are:

20, 24

Example 2

Find the next term:

3, 6, 12, 24, __

Solution

Each term is multiplied by 2.

3 × 2 = 6
6 × 2 = 12
12 × 2 = 24
24 × 2 = 48

Answer

The next term is:

48

Example 3

Find the sum of the series:

2 + 4 + 6 + 8 + 10

Solution

Add all the terms together.

2 + 4 = 6
6 + 6 = 12
12 + 8 = 20
20 + 10 = 30

Answer

The sum of the series is:

30

Part B: Introduction to Algebra

6. Meaning of Algebra

Algebra is a branch of mathematics where letters are used to represent numbers.

In algebra, letters such as x, y, a, b, and n may stand for unknown numbers.

Example:

x + 5 = 12

Here, x represents an unknown number.

To solve the problem, we find the value of x.

Since 7 + 5 = 12, then:

x = 7

7. Basic Terms Used in Algebra

7.1 Variable

A variable is a letter that represents an unknown number.

Examples:

x, y, a, b, n

In the expression:

3x + 5

x is the variable.

7.2 Constant

A constant is a number whose value does not change.

Example:

3x + 5

The number 5 is the constant.

7.3 Coefficient

A coefficient is the number multiplying a variable.

Example:

7x

The coefficient of x is 7.

Example:

-4y

The coefficient of y is -4.

7.4 Term

A term is a single part of an algebraic expression separated by plus or minus signs.

Example:

3x + 5y – 7

The terms are:

3x, 5y, and -7

7.5 Algebraic Expression

An algebraic expression is a mathematical phrase that contains numbers, letters, and operation signs.

Examples:

2x + 3
5a – 4
3m + 2n – 7

8. Like and Unlike Terms

Like Terms

Like terms are terms that have the same variable.

Examples:

3x and 5x are like terms.

2a and -7a are like terms.

Unlike Terms

Unlike terms have different variables.

Examples:

3x and 4y are unlike terms.

5a and 6b are unlike terms.

9. Simplifying Algebraic Expressions

To simplify algebraic expressions, collect like terms together.

Example 1

Simplify:

3x + 5x

Solution

Both terms are like terms because they contain x.

3x + 5x = 8x

Answer

8x

Example 2

Simplify:

7a – 2a + 4

Solution

Collect the like terms:

7a – 2a = 5a

So:

7a – 2a + 4 = 5a + 4

Answer

5a + 4

Part C: Integer Operations

10. Meaning of Integers

Integers are whole numbers that can be positive, negative, or zero.

Examples:

-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5

Positive integers are numbers greater than zero.

Examples:

1, 2, 3, 4, 5

Negative integers are numbers less than zero.

Examples:

-1, -2, -3, -4, -5

Zero is also an integer, but it is neither positive nor negative.

11. Addition of Integers

Rule 1: Positive + Positive = Positive

Example:

5 + 3 = 8

Rule 2: Negative + Negative = Negative

Example:

-4 + -6 = -10

This can also be written as:

-4 – 6 = -10

Rule 3: Positive + Negative

When adding a positive and a negative number, subtract the smaller number from the bigger number and take the sign of the bigger number.

Example:

8 + -3 = 5

Because 8 is bigger than 3, the answer is positive.

Example:

-9 + 4 = -5

Because 9 is bigger than 4 and the bigger number is negative, the answer is negative.

12. Subtraction of Integers

When subtracting integers, change the subtraction sign to addition and change the sign of the number after it.

Example 1

Simplify:

7 – 3

Solution

7 – 3 = 4

Answer

4

Example 2

Simplify:

7 – -3

Solution

Subtracting a negative number becomes addition.

7 – -3 = 7 + 3

7 + 3 = 10

Answer

10

Example 3

Simplify:

-5 – 4

Solution

-5 – 4 = -9

Answer

-9

13. Multiplication of Integers

Important Rules

Positive × Positive = Positive
Negative × Negative = Positive
Positive × Negative = Negative
Negative × Positive = Negative

Example 1

4 × 3 = 12

Example 2

-4 × -3 = 12

Example 3

-4 × 3 = -12

Example 4

4 × -3 = -12

14. Division of Integers

The rules for division are the same as multiplication.

Positive ÷ Positive = Positive
Negative ÷ Negative = Positive
Positive ÷ Negative = Negative
Negative ÷ Positive = Negative

Example 1

12 ÷ 3 = 4

Example 2

-12 ÷ -3 = 4

Example 3

-12 ÷ 3 = -4

Example 4

12 ÷ -3 = -4

Part D: Algebra with Integer Operations

15. Applying Integers in Algebra

In algebra, integer operations are used when simplifying expressions, substituting values, and solving equations.

Example 1

Simplify:

-3x + 7x

Solution

The terms are like terms.

-3x + 7x = 4x

Answer

4x

Example 2

Simplify:

5a – 9a

Solution

5a – 9a = -4a

Answer

-4a

Example 3

Simplify:

-2y – 6y

Solution

Both terms are negative.

-2y – 6y = -8y

Answer

-8y

16. Substitution with Integers

Substitution means replacing a letter with a number.

Example 1

If x = 4, find the value of:

3x + 2

Solution

Replace x with 4.

3x + 2 = 3(4) + 2

3 × 4 = 12

12 + 2 = 14

Answer

14

Example 2

If x = -3, find the value of:

2x + 5

Solution

Replace x with -3.

2x + 5 = 2(-3) + 5

2 × -3 = -6

-6 + 5 = -1

Answer

-1

Example 3

If a = -2 and b = 5, find the value of:

3a + 2b

Solution

Substitute the values:

3a + 2b = 3(-2) + 2(5)

3 × -2 = -6

2 × 5 = 10

So:

-6 + 10 = 4

Answer

4

17. Common Mistakes Students Make

Mistake 1: Confusing a Sequence with a Series

A sequence is a list of numbers.

A series is the sum of the numbers.

Mistake 2: Ignoring Negative Signs

Example:

-3 + 7 is not -10.

Correct answer:

-3 + 7 = 4

Mistake 3: Multiplying Negative Numbers Incorrectly

Remember:

Negative × Negative = Positive

Example:

-4 × -5 = 20

Mistake 4: Combining Unlike Terms

Example:

3x + 4y cannot become 7xy.

They are unlike terms and cannot be added together.

Mistake 5: Wrong Substitution

If x = -2, then 3x means:

3 × -2 = -6

It is not 3 – 2.

18. WASSCE Examination Tips

  1. Always read the question carefully.

  2. Look for the pattern before solving sequence questions.

  3. Show your working clearly in theory questions.

  4. Be careful with negative signs.

  5. Do not add unlike terms.

  6. Use brackets when substituting negative numbers.

  7. Check your final answer.

  8. Practice past WASSCE questions regularly.

19. Class Activities

Activity 1: Identify the Pattern

Find the next two terms:

  1. 2, 4, 6, 8, __, __

  2. 5, 10, 20, 40, __, __

  3. 30, 25, 20, 15, __, __

  4. 1, 4, 9, 16, __, __

Activity 2: Simplify the Expressions

  1. 4x + 3x

  2. 8a – 5a

  3. -2y + 9y

  4. -6m – 3m

  5. 7p + 2q

Activity 3: Integer Operations

Simplify:

  1. 7 + -3

  2. -5 + -4

  3. 9 – -2

  4. -6 – 5

  5. -4 × 3

  6. -8 × -2

  7. 15 ÷ -3

  8. -20 ÷ -5

20. Practice Questions

Answer all questions.

  1. State the difference between a sequence and a series.

  2. Find the next term in the sequence: 6, 12, 18, 24, __

  3. Find the sum of the series: 3 + 6 + 9 + 12

  4. Simplify: 5x + 2x

  5. Simplify: 9a – 13a

  6. Simplify: -4y + 10y

  7. Simplify: -7m – 6m

  8. If x = -4, find the value of 3x + 8.

  9. If a = 5 and b = -2, find the value of 2a + 3b.

  10. Simplify: -6 × -4

  11. Simplify: 24 ÷ -6

  12. Find the next two terms: 2, 6, 18, 54, __, __

21. Assignment

Answer the following questions and show all working.

  1. Explain the meaning of algebra.

  2. What is a variable? Give two examples.

  3. What is the coefficient of x in 8x?

  4. Simplify: 12x – 5x + 3x

  5. Simplify: -3a + 9a – 4a

  6. If x = -5, find the value of 4x + 7.

  7. If p = -2 and q = 6, find the value of 5p + 2q.

  8. Find the next three terms in the sequence: 4, 8, 12, 16, __, __, __

  9. Find the sum of the series: 5 + 10 + 15 + 20 + 25

  10. Simplify: -9 – -4

22. Lesson Summary

In this topic, learners were introduced to series, algebra, and integer operations. A sequence was explained as a list of numbers arranged in a pattern, while a series was explained as the sum of the terms of a sequence. Learners also studied the basic meaning of algebra and learned how letters can be used to represent unknown numbers.

The topic also covered integers, including positive numbers, negative numbers, and zero. Learners practiced how to add, subtract, multiply, and divide integers. Finally, they learned how integer operations are used in algebra when simplifying expressions and substituting values.

This topic is very important because it prepares learners for more advanced topics such as arithmetic progression, geometric progression, equations, inequalities, and algebraic word problems.

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