Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Perform calculations with multiple operations (BODMAS).
• Apply the order of operations correctly.
• Estimate answers and justify reasonableness.
• Solve problems with direct and inverse proportions.
• Convert between different units of measurement.

Lesson summary

This week, we're revisiting and extending our understanding of numbers and calculations. This isn't just about doing sums; it's about building a crucial foundation for making informed decisions in your daily life, especially in South Africa's diverse economic and social landscape. From budgeting your money to understanding loan repayments, from interpreting statistics in the news to calculating discounts at the shops, a solid grasp of numbers is essential for navigating the world effectively. This topic equips you with the skills to be a financially literate and empowered citizen.

Lesson notes

2.1 Order of Operations (BODMAS/PEMDAS) The order of operations is a set of rules that dictate the sequence in which calculations should be performed to arrive at the correct answer. In South Africa, BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) is commonly used.

However, PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) is equivalent.

Remember: Brackets/Parentheses: Perform operations within brackets or parentheses first, starting with the innermost set.

Orders/Exponents: Evaluate exponents (powers and roots).

Division and Multiplication: Perform division and multiplication from left to right.

Addition and Subtraction: Perform addition and subtraction from left to right.

Example 1: Calculate: `2 + 3 × (6 - 4) ÷ 2` Brackets: `6 - 4 = 2` Multiplication: `3 × 2 = 6` Division: `6 ÷ 2 = 3` Addition: `2 + 3 = 5` Therefore, `2 + 3 × (6 - 4) ÷ 2 = 5` 2.2 Rational and Irrational Numbers Rational Numbers: Numbers that can be expressed as a fraction p/q, where p and q are integers and q ≠

0. Examples: 2/3, -5, 0.75 (which is 3/4). Terminating and repeating decimals are rational.

Irrational Numbers: Numbers that cannot be expressed as a fraction p/q. Their decimal representations are non-terminating and non-repeating.

Examples: √2, π (pi).

Example 2: Identify which of the following numbers are rational and irrational: 3.14159, √9, √10, 0.333..., -7/8 3.14159: Rational (it's a terminating decimal, though it's an approximation of pi) √9 = 3: Rational (an integer) √10: Irrational (non-terminating, non-repeating decimal) 0.333... = 1/3: Rational (a repeating decimal) -7/8: Rational (a fraction) 2.3 Scientific Notation Scientific notation is a way of expressing very large or very small numbers compactly. A number in scientific notation is written as a × 10^n, where 1 ≤ |a| < 10 and n is an integer.

Example 3: Express 5,400,000 in scientific notation. Move the decimal point until there is only one non-zero digit to the left of the decimal point: 5.4 Count the number of places you moved the decimal point: 6 places. Since the original number was greater than 1, the exponent is positive: 5.4 × 10^6 Example 4: Express 0.0000038 in scientific notation. Move the decimal point until there is only one non-zero digit to the left of the decimal point: 3.8 Count the number of places you moved the decimal point: 6 places. Since the original number was less than 1, the exponent is negative: 3.8 × 10^-6 2.4 Ratios and Proportions Ratio: A comparison of two quantities, often expressed as a:b.

Proportion: An equation stating that two ratios are equal, a/b = c/d.

Direct Proportion: Two quantities are directly proportional if an increase in one quantity results in a proportional increase in the other, and vice versa. y = kx, where k is the constant of proportionality.

Inverse Proportion: Two quantities are inversely proportional if an increase in one quantity results in a proportional decrease in the other, and vice versa. y = k/x, where k is the constant of proportionality.

Example 5: Direct Proportion (Currency Exchange) The exchange rate between South African Rand (ZAR) and US Dollars (USD) is approximately 1 USD = 18 ZAR. How many ZAR would you get for 50 USD? Since the relationship is directly proportional (more USD means more ZAR), we can set up a proportion: 1 USD / 18 ZAR = 50 USD / x ZAR Cross-multiply: 1 x = 18 50 x = 900 ZA

R. Therefore, 50 USD is equal to 900 ZA

R. Example 6: Inverse Proportion (Work Rate) It takes 3 painters 8 hours to paint a house. How long will it take 6 painters, assuming they work at the same rate? This is an inverse proportion (more painters means less time). The total work remains constant. Work = Painters × Time 3 8 = 6 x 24 = 6x x = 4 hours.

Therefore, it will take 6 painters 4 hours to paint the house. 2.5 Unit Conversions Understanding unit conversions is critical for solving real-world problems. Common conversions in South Africa include: Length: mm, cm, m, km Mass: g, kg, tonnes Volume: mL, L Time: seconds, minutes, hours, days, weeks, months, years Example 7: Convert 3.5 kilometers to meters. We know that 1 km = 1000 m.

Therefore, 3.5 km = 3.5 * 1000 m = 3500 m. Guided Practice (With Solutions)

Question 1: Calculate: 15 + (8 - 2) × 4 ÷ 2 - √25 Solution: Brackets: 8 - 2 = 6 Square root: √25 = 5 Multiplication: 6 × 4 = 24 Division: 24 ÷ 2 = 12 Addition: 15 + 12 = 27 Subtraction: 27 - 5 = 22 Therefore, 15 + (8 - 2) × 4 ÷ 2 - √25 =

2

2. Commentary: This question tests the order of operations and includes a square root.

Question 2: Express 0.00072 in scientific notation and then multiply it by 4 × 10^

5. Solution: Scientific notation: 0.00072 = 7.2 × 10^-4 Multiplication: (7.2 × 10^-4) × (4 × 10^5) = (7.2 × 4) × (10^-4 × 10^5) 7.2 × 4 = 28.8 10^-4 × 10^5 = 10^(5-4) = 10^1 Therefore, the result is 28.8 × 10^1 = 288

Commentary: This question tests scientific notation and exponent rules.