# IGCSE Mathematics Foundation: Algebraic Manipulation

## Scheme of work: IGCSE Foundation: Year 10: Term 2: Algebraic Manipulation

#### Prerequisite Knowledge

1. Understanding Variables
• Concept: Knowing that letters can be used to represent unknown numbers.
• Example: $\text{If } x = 5, \text{ then } 2x = 2 \times 5 = 10.$
2. Generalizing Arithmetic Rules to Algebra
• Concept: Understanding that the rules of arithmetic (addition, subtraction, multiplication, and division) can be applied to algebraic expressions involving letters.
• Example: $\text{For any numbers } a \text{ and } b, \text{ } a + b = b + a \text{ (commutative property of addition).}$

#### Success Criteria

1. Evaluate Expressions by Substituting Numerical Values for Letters
• Objective: Substitute numerical values into algebraic expressions and evaluate the result.
• Example: $\text{Evaluate } 3a + 2b \text{ when } a = 2 \text{ and } b = 3.$ $3(2) + 2(3) = 6 + 6 = 12.$
2. Collect Like Terms
• Objective: Identify and combine like terms in an algebraic expression.
• Example: $2x + 3x = 5x.$
3. Multiply a Single Term over a Bracket
• Objective: Use the distributive property to multiply a single term over a bracket.
• Example: $3(a + b) = 3a + 3b.$
4. Take Out Common Factors
• Objective: Factor out the greatest common factor from terms in an algebraic expression.
• Example: $4x + 8 = 4(x + 2).$
5. Expand the Product of Two Simple Linear Expressions
• Objective: Use the distributive property to expand the product of two binomials.
• Example: $(x + 2)(x + 3) = x^2 + 5x + 6.$
6. Understand and Factorize Quadratic Expressions (limited to $$x^2 + bx + c$$)
• Objective: Recognize the general form of a quadratic expression and factorize simple quadratics.
• Example: $x^2 + 5x + 6 = (x + 2)(x + 3).$

#### Key Concepts

1. Evaluate Expressions by Substituting Numerical Values for Letters
• Concept: Understanding that variables represent numbers and substituting specific values into expressions allows for the evaluation of these expressions.
• Example: $\text{For } 3a + 2b, \text{ substitute } a = 2 \text{ and } b = 3 \text{ to get } 3(2) + 2(3) = 12.$
2. Collect Like Terms
• Concept: Recognizing and combining terms with the same variable and exponent to simplify algebraic expressions.
• Example: $2x + 3x \text{ can be simplified to } 5x \text{ because the terms are like terms.}$
3. Multiply a Single Term over a Bracket
• Concept: Applying the distributive property to expand expressions where a single term multiplies terms inside a bracket.
• Example: $3(a + b) = 3a + 3b.$
4. Take Out Common Factors
• Concept: Factoring out the greatest common factor (GCF) from each term in an expression to simplify the expression.
• Example: $4x + 8 = 4(x + 2) \text{ by factoring out the GCF of 4.}$
5. Expand the Product of Two Simple Linear Expressions
• Concept: Using the distributive property (FOIL method) to expand the product of two binomials.
• Example: $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6.$
6. Understand and Factorize Quadratic Expressions (limited to $$x^2 + bx + c$$)
• Concept: Recognizing the standard form of a quadratic expression and using factorization techniques to express it as a product of two binomials.
• Example: $x^2 + 5x + 6 = (x + 2)(x + 3).$

#### Common Misconceptions

1. Evaluate Expressions by Substituting Numerical Values for Letters
• Common Mistake: Students may forget to apply the correct order of operations (PEMDAS/BODMAS) when substituting values into expressions.
• Example: $\text{Incorrect: } 3a + 2b \text{ when } a = 2 \text{ and } b = 3 \text{ is evaluated as } 3(2) + 2(3) = 6 + 6 = 12.$ $\text{Correct: Apply the operations correctly: } 3(2) + 2(3) = 6 + 6 = 12.$
2. Collect Like Terms
• Common Mistake: Students may combine unlike terms or fail to combine all like terms.
• Example: $\text{Incorrect: } 2x + 3 + x = 5x + 3.$ $\text{Correct: Combine like terms correctly: } 2x + x = 3x, \text{ so the expression is } 3x + 3.$
3. Multiply a Single Term over a Bracket
• Common Mistake: Students may forget to multiply the single term by all terms inside the bracket.
• Example: $\text{Incorrect: } 3(a + b) = 3a + b.$ $\text{Correct: Distribute the term correctly: } 3(a + b) = 3a + 3b.$
4. Take Out Common Factors
• Common Mistake: Students may not factor out the greatest common factor from all terms in the expression.
• Example: $\text{Incorrect: } 4x + 8 = 2(2x + 4).$ $\text{Correct: Factor out the GCF: } 4x + 8 = 4(x + 2).$
5. Expand the Product of Two Simple Linear Expressions
• Common Mistake: Students may incorrectly apply the distributive property and miss terms when expanding.
• Example: $\text{Incorrect: } (x + 2)(x + 3) = x^2 + 6.$ $\text{Correct: Apply the distributive property correctly: } (x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6.$
6. Understand and Factorize Quadratic Expressions (limited to $$x^2 + bx + c$$)
• Common Mistake: Students may not correctly identify the factors of the quadratic expression or may forget to set the expression to zero.
• Example: $\text{Incorrect: } x^2 + 5x + 6 = (x + 1)(x + 6).$ $\text{Correct: Identify the correct factors: } x^2 + 5x + 6 = (x + 2)(x + 3).$

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