# IGCSE Mathematics Foundation – Working with Numbers

## Scheme of work: IGCSE Foundation: Year 10: Term 1: Working with Numbers

#### Prerequisite Knowledge

1. Understanding Place Value
• Concept: Knowing the value of each digit in a number based on its position.
• Example: $\text{In the number 345, the digit 3 represents 300, the digit 4 represents 40, and the digit 5 represents 5}$
2. Basic Arithmetic Operations
• Concept: Understanding addition, subtraction, multiplication, and division.
• Example: $2 + 3 = 5, \quad 7 – 4 = 3, \quad 6 \times 2 = 12, \quad \frac{8}{2} = 4$

#### Success Criteria

1. Using Integers
• Performing Calculations: Students should be able to understand and use positive, negative, and zero integers in calculations.
• Example: $-3 + 5 = 2, \quad -7 – 4 = -11, \quad 6 – (-2) = 8$
• Describing Integers: Students should be able to describe practical situations using integers, such as temperatures.
• Example: $\text{Temperature: } -5^\circ C \text{ is 5 degrees below freezing point}$
2. Using Directed Numbers
• Performing Calculations: Students should be able to use directed numbers in practical situations, such as calculating profit and loss.
• Example: $\text{Profit of } \200 \text{ and a loss of } \50 \text{ results in a net gain of } \150$
• Describing Directed Numbers: Students should be able to describe situations using directed numbers, such as altitude.
• Example: $\text{Altitude: } +200 \text{ meters above sea level, } -50 \text{ meters below sea level}$
3. Using Brackets and Hierarchy of Operations
• Performing Calculations: Students should be able to use brackets and the hierarchy of operations to solve complex calculations.
• Example: $(2 + 3) \times 4 = 20, \quad 8 \div (2 + 2) = 2$
• Describing Hierarchy of Operations: Students should be able to describe the order of operations using BIDMAS/BODMAS rules.
• Example: $\text{BIDMAS/BODMAS: Brackets, Indices/Orders, Division and Multiplication, Addition and Subtraction}$
4. Identifying and Using Factors and Multiples
• Identifying Factors and Multiples: Students should be able to identify prime factors, common factors, and common multiples.
• Example: $\text{Factors of 12: } 1, 2, 3, 4, 6, 12 \quad \text{Multiples of 3: } 3, 6, 9, 12, 15, \ldots$
• Describing Factors and Multiples: Students should be able to use the terms ‘odd’, ‘even’, ‘prime numbers’, ‘factors’ and ‘multiples’ accurately.
• Example: $\text{Prime factors of 12: } 2, 2, 3 \quad \text{Common multiples of 4 and 5: } 20, 40, 60, \ldots$

#### Key Concepts

1. Understanding Place Value
• Concept: Knowing that the value of a digit depends on its position in the number.
• Example: $\text{In the number 345, the digit 3 represents 300, the digit 4 represents 40, and the digit 5 represents 5}$
2. Using Integers in Practical Situations
• Concept: Understanding how integers are used in real-life contexts such as temperatures and financial calculations.
• Example: $\text{Temperature: } -5^\circ C \text{ and } +10^\circ C$
3. Directed Numbers
• Concept: Knowing how to perform operations with positive and negative numbers.
• Example: $-3 + 5 = 2, \quad -7 – 4 = -11, \quad 6 – (-2) = 8$
4. Hierarchy of Operations
• Concept: Understanding the importance of the order of operations in calculations.
• Example: $(2 + 3) \times 4 = 20, \quad 8 \div (2 + 2) = 2$

#### Common Misconceptions

1. Using Integers
• Common Mistake: Students might incorrectly add or subtract integers, especially when dealing with negative numbers.
• Example: Calculating $$-3 + 5$$ as: $\text{Incorrect: } -3 + 5 = -8$ $\text{Correct: } -3 + 5 = 2$
2. Using Directed Numbers
• Common Mistake: Students might misunderstand the direction of the numbers, leading to incorrect calculations.
• Example: Calculating profit and loss incorrectly: $\text{Incorrect: } \text{Profit of } \200 \text{ and a loss of } \50 \text{ results in a net loss of } \150$ $\text{Correct: } \text{Profit of } \200 \text{ and a loss of } \50 \text{ results in a net gain of } \150$
3. Using Hierarchy of Operations
• Common Mistake: Students might ignore the order of operations, leading to incorrect results.
• Example: Calculating $$2 + 3 \times 4$$ as: $\text{Incorrect: } 2 + 3 \times 4 = 20$ $\text{Correct: } 2 + 3 \times 4 = 14$
• Common Mistake: Students might incorrectly use brackets, leading to incorrect results.
• Example: Calculating $$8 \div (2 + 2)$$ as: $\text{Incorrect: } 8 \div 2 + 2 = 6$ $\text{Correct: } 8 \div (2 + 2) = 2$
4. Identifying and Using Factors and Multiples
• Common Mistake: Students might incorrectly identify factors and multiples, leading to incorrect results.
• Example: Identifying the factors of 12 as: $\text{Incorrect: } 1, 2, 4, 12$ $\text{Correct: } 1, 2, 3, 4, 6, 12$
• Common Mistake: Students might confuse prime factors with regular factors.
• Example: Identifying the prime factors of 12 as: $\text{Incorrect: } 2, 3, 4$ $\text{Correct: } 2, 2, 3$

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