# IGCSE Mathematics Foundation: Powers and Roots

Scheme of work: IGCSE Foundation: Year 10: Term 3: Powers and Roots

#### Prerequisite Knowledge

1. Understanding Prime Numbers
• Concept: Knowing that a prime number is a number greater than 1 that has no positive divisors other than 1 and itself.
• Example: $\text{Prime numbers include } 2, 3, 5, 7, 11, \text{ etc.}$
2. Finding Factor Pairs of an Integer
• Concept: Identifying pairs of numbers that, when multiplied together, equal a given integer.
• Example: $\text{Factor pairs of } 12 \text{ are } (1, 12), (2, 6), \text{ and } (3, 4).$
3. Understanding Lowest Common Multiple (LCM)
• Concept: Knowing how to find the lowest common multiple of two or more numbers.
• Example: $\text{LCM of } 4 \text{ and } 5 \text{ is } 20.$

#### Success Criteria

1. Identify Square Numbers and Cube Numbers
• Objective: Recognize and identify square numbers and cube numbers.
• Example: $\text{Square numbers: } 1, 4, 9, 16, 25, \text{ etc.}$ $\text{Cube numbers: } 1, 8, 27, 64, 125, \text{ etc.}$
2. Calculate Squares, Square Roots, Cubes, and Cube Roots
• Objective: Calculate the square, square root, cube, and cube root of a number.
• Example: $4^2 = 16, \quad \sqrt{16} = 4, \quad 2^3 = 8, \quad \sqrt[3]{8} = 2$
3. Use Index Notation and Index Laws
• Objective: Use index notation and apply index laws for multiplication and division of positive and negative integer powers, including zero.
• Example: $a^m \times a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad a^0 = 1$
4. Express Integers as a Product of Powers of Prime Factors
• Objective: Factorize integers into prime factors and express the result using powers.
• Example: $60 = 2^2 \times 3 \times 5$
5. Find Highest Common Factors (HCF) and Lowest Common Multiples (LCM)
• Objective: Calculate the highest common factor and lowest common multiple of two or more numbers.
• Example: $\text{HCF of } 12 \text{ and } 18 \text{ is } 6$ $\text{LCM of } 4 \text{ and } 5 \text{ is } 20$

#### Key Concepts

1. Identify Square Numbers and Cube Numbers
• Concept: Square numbers are the product of an integer multiplied by itself, and cube numbers are the product of an integer multiplied by itself twice.
• Example: $\text{Square numbers: } 1 = 1^2, 4 = 2^2, 9 = 3^2$ $\text{Cube numbers: } 1 = 1^3, 8 = 2^3, 27 = 3^3$
2. Calculate Squares, Square Roots, Cubes, and Cube Roots
• Concept: Squaring a number means multiplying the number by itself. Finding the square root is the inverse operation of squaring. Cubing a number means multiplying the number by itself twice. Finding the cube root is the inverse operation of cubing.
• Example: $4^2 = 16, \quad \sqrt{16} = 4, \quad 2^3 = 8, \quad \sqrt[3]{8} = 2$
3. Use Index Notation and Index Laws
• Concept: Index notation is a way to represent repeated multiplication of the same factor. Index laws are rules for simplifying expressions involving indices.
• Example: $a^m \times a^n = a^{m+n}, \quad \frac{a^m}{a^n} = a^{m-n}, \quad a^0 = 1$
4. Express Integers as a Product of Powers of Prime Factors
• Concept: Prime factorization involves expressing a number as the product of its prime factors, using exponents to indicate repeated factors.
• Example: $60 = 2^2 \times 3 \times 5$
5. Find Highest Common Factors (HCF) and Lowest Common Multiples (LCM)
• Concept: The highest common factor (HCF) is the largest number that divides two or more numbers without leaving a remainder. The lowest common multiple (LCM) is the smallest number that is a multiple of two or more numbers.
• Example: $\text{HCF of } 12 \text{ and } 18 \text{ is } 6$ $\text{LCM of } 4 \text{ and } 5 \text{ is } 20$

#### Common Misconceptions

1. Identify Square Numbers and Cube Numbers
• Common Mistake: Confusing square numbers with even numbers or cube numbers with multiples of three.
• Example: $\text{Incorrect: } 6 \text{ is a square number because it is even.}$ $\text{Correct: Square numbers are } 1, 4, 9, 16, \text{ etc.}$
2. Calculate Squares, Square Roots, Cubes, and Cube Roots
• Common Mistake: Incorrectly applying the operations for squaring, taking square roots, cubing, or taking cube roots.
• Example: $\text{Incorrect: } \sqrt{16} = 8 \text{ or } 2^3 = 6.$ $\text{Correct: } \sqrt{16} = 4 \text{ and } 2^3 = 8.$
3. Use Index Notation and Index Laws
• Common Mistake: Misapplying index laws, especially with negative or zero exponents.
• Example: $\text{Incorrect: } a^m \times a^n = a^{mn} \text{ or } a^0 = 0.$ $\text{Correct: } a^m \times a^n = a^{m+n} \text{ and } a^0 = 1.$
4. Express Integers as a Product of Powers of Prime Factors
• Common Mistake: Missing a prime factor or incorrectly using exponents when factorizing a number.
• Example: $\text{Incorrect: } 60 = 2 \times 30 \text{ or } 60 = 2 \times 2^2 \times 3 \times 5.$ $\text{Correct: } 60 = 2^2 \times 3 \times 5.$
5. Find Highest Common Factors (HCF) and Lowest Common Multiples (LCM)
• Common Mistake: Confusing the processes for finding HCF and LCM, or miscalculating the factors or multiples.
• Example: $\text{Incorrect: HCF of } 12 \text{ and } 18 \text{ is } 36 \text{ (which is actually the LCM).}$ $\text{Correct: HCF of } 12 \text{ and } 18 \text{ is } 6.$ $\text{Incorrect: LCM of } 4 \text{ and } 5 \text{ is } 1 \text{ (the HCF).}$ $\text{Correct: LCM of } 4 \text{ and } 5 \text{ is } 20.$

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