Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

This page supports the lesson note with a companion video and a short classroom-ready summary.

For class groups and homework, share this lesson page so learners also get the summary, objectives, and full lesson context.

Performance objectives

• Perform calculations with percentages (discounts, markups).
• Apply ratios and rates to solve problems (sharing, currency exchange).
• Calculate simple and compound interest.
• Use a calculator effectively for complex calculations.
• Estimate answers and assess reasonableness of results.

Lesson summary

This week, we delve deeper into the fundamental building blocks of Mathematical Literacy: Numbers and Calculations. This topic is not just about crunching numbers; it's about empowering you to make informed decisions in your everyday life, from budgeting your pocket money to understanding loan repayments and interpreting statistics in the news. We will revise basic operations and extend our understanding to more complex calculations involving percentages, ratios, rates, and financial calculations relevant to the South African context. A strong grasp of these concepts is crucial for success in this subject and for navigating the financial and numerical aspects of the real world.

Lesson notes

2.1 Revision of Basic Operations: Before diving into more complex calculations, let's ensure we're comfortable with the basics: addition (+), subtraction (-), multiplication (x or *), and division (÷ or /). Remember the order of operations (BODMAS/PEMDAS): Brackets / Parentheses Orders / Exponents Division and Multiplication (from left to right) Addition and Subtraction (from left to right)

Example: 12 + 3 x (8 - 2) ÷ 2 = 12 + 3 x 6 ÷ 2 = 12 + 18 ÷ 2 = 12 + 9 = 21 2.2 Percentages: "Percent" means "out of one hundred." A percentage is a fraction with a denominator of

1

0

0. Converting a percentage to a decimal: Divide by 100 (e.g., 25% = 25/100 = 0.25)

Converting a decimal to a percentage: Multiply by 100 (e.g., 0.75 = 0.75 x 100 = 75%)

Finding a percentage of a quantity: Multiply the quantity by the percentage (as a decimal).

Example 1 (Discount): A pair of jeans costs R350, and there's a 20% discount. What is the discount amount, and what is the sale price?

Discount amount: 20% of R350 = 0.20 x R350 = R70 Sale price: R350 - R70 = R280 Example 2 (Percentage Increase): The price of bread increases from R15 to R

1

8. What is the percentage increase?

Increase in price: R18 - R15 = R3 Percentage increase: (Increase / Original Price) x 100 = (R3 / R15) x 100 = 0.2 x 100 = 20% 2.3 Ratios and Rates: Ratio: Compares two or more quantities of the same type.

Ratios can be written as a:b, a/b, or "a to b." Rate: Compares two quantities of different types. Rates often involve units of measurement (e.g., km/h, rands per liter).

Example 1 (Ratio): A recipe for vetkoek requires 3 cups of flour and 1 cup of water. What is the ratio of flour to water?

Ratio of flour to water: 3:1 Example 2 (Rate): A car travels 400 km in 5 hours. What is the average speed (rate)?

Average speed: 400 km / 5 hours = 80 km/h Example 3 (Currency Conversion): If the exchange rate is R15 per US dollar, how many US dollars can you buy with R300?

US dollars: R300 / R15/dollar = 20 dollars 2.4 Simple and Compound Interest: Simple Interest: Interest is calculated only on the principal amount.

Formula: Interest = Principal x Rate x Time (I = PRT)

Where: P = Principal, R = Interest Rate (as a decimal), T = Time (in years)

Compound Interest: Interest is calculated on the principal amount and on the accumulated interest from previous periods.

Formula: A = P(1 + R)^T Where: A = Amount (Principal + Interest), P = Principal, R = Interest Rate (as a decimal), T = Time (in years)

Example 1 (Simple Interest): You invest R1000 at a simple interest rate of 8% per year for 3 years. How much interest will you earn? Interest = R1000 x 0.08 x 3 = R240 Example 2 (Compound Interest): You invest R1000 at a compound interest rate of 8% per year for 3 years. How much will you have at the end of the period? A = R1000 (1 + 0.08)^3 = R1000 (1.08)^3 = R1000 x 1.259712 = R1259.71 (approximately) 2.5 Using a Calculator Effectively: Familiarize yourself with your calculator's functions: +, -, x, ÷, %, √, ^ (exponent), memory functions (M+, M-, MR, MC). Pay attention to the order of operations (BODMAS/PEMDAS). Use brackets to ensure calculations are performed in the correct order. Practice using the calculator for percentage calculations, square roots, and exponents. Be mindful of the number of decimal places required in your answer. 2.6 Estimation and Reasonableness: Before performing a calculation, estimate the answer to get a sense of what to expect. This helps you identify potential errors. After performing a calculation, check if the answer is reasonable in the given context. For instance, if you're calculating the price of an item after a discount, the sale price should be lower than the original price. Guided Practice (With Solutions)

Question 1: A shop sells a cellphone for R

2

5

0

0. They offer a 15% discount for students. How much will a student pay for the cellphone?

Solution: Discount amount: 15% of R2500 = 0.15 x R2500 = R375 Price for student: R2500 - R375 = R2125 Answer: A student will pay R2125 for the cellphone.

Commentary: We first calculated the discount amount by finding 15% of the original price. Then, we subtracted the discount from the original price to find the final price.

Question 2: A farm has 40 cows and 60 sheep. What is the ratio of cows to sheep, expressed in its simplest form?

Solution: Ratio of cows to sheep: 40:60 Simplify the ratio by dividing both sides by their greatest common factor (20): 40 ÷ 20 : 60 ÷ 20 = 2:3 Answer: The ratio of cows to sheep is 2:

3. Commentary: To simplify a ratio, divide both sides by the greatest common factor. This gives you the simplest whole-number ratio.

Question 3: A taxi charges R12 per kilometer. How much will it cost to travel 25 kilometers?

Solution: Cost: R12/km x 25 km = R300 Answer: It will cost R300 to travel 25 kilometers.

Commentary: This is a simple rate problem. We multiplied the rate (R12 per kilometer) by the distance (25 kilometers) to find the total cost.