Lesson Notes By Weeks and Term v5 - Grade 11

Lesson Video

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Performance objectives

• Perform calculations with whole numbers, fractions, decimals, percentages, ratios, and rates.
• Solve problems involving financial calculations like interest and discounts.
• Estimate and check the reasonableness of answers.
• Apply the order of operations (BODMAS) correctly.
• Convert between different units of measurement.

Lesson summary

This week, we will be revising and extending our understanding of numbers and calculations with numbers. This is a crucial foundation for Mathematical Literacy, as it underpins almost every calculation and problem we will encounter throughout the year. Being confident with basic arithmetic, percentages, ratios, and rates allows us to make informed decisions about our personal finances, understand statistical information in the news, and participate effectively in our communities and the broader South African economy.

Lesson notes

2.1 Whole Numbers, Integers, Fractions, and Decimals Whole Numbers: These are non-negative numbers without any fractional or decimal parts (0, 1, 2, 3...).

Integers: These include whole numbers and their negative counterparts (...-3, -2, -1, 0, 1, 2, 3...).

Fractions: Represent parts of a whole. A fraction has a numerator (top number) and a denominator (bottom number).

Proper fractions:* Numerator is less than the denominator (e.g., 1/2).

Improper fractions:* Numerator is greater than or equal to the denominator (e.g., 5/3).

Mixed numbers:* A whole number and a proper fraction (e.g., 1 2/3).

Decimals: Represent numbers using a base-10 system, with a decimal point separating the whole number part from the fractional part (e.g., 3.14).

Converting between Fractions and Decimals: To convert a fraction to a decimal, divide the numerator by the denominator. To convert a decimal to a fraction, write the decimal as a fraction with a denominator that is a power of 10 (e.g., 0.75 = 75/100 = 3/4). 2.2 Percentages Definition: Percentage means "out of one hundred" or "per hundred." It's a way of expressing a number as a fraction of

1

0

0. The symbol for percentage is %. Converting Percentages to Decimals and Fractions: To convert a percentage to a decimal, divide by 100 (e.g., 25% = 25/100 = 0.25). To convert a percentage to a fraction, write the percentage as a fraction with a denominator of 100 and simplify if possible (e.g., 25% = 25/100 = 1/4).

Calculating Percentages of Quantities: To find x% of y, multiply y by x/100 (e.g., 15% of 200 = (15/100) 200 = 30). 2.3 Ratios and Rates Ratio: A ratio compares two or more quantities of the same kind.

It can be written in the form a : b, a to b, or a/ b. Ratios should be simplified where possible.

Rate: A rate compares two quantities of different kinds. It includes units of measurement. Common examples are speed (km/h) and price per item (R/item). 2.4 Financial Calculations: Simple and Compound Interest, Discounts, Inflation Simple Interest: Interest calculated only on the principal amount.

Formula: I = P r t, where I = interest, P = principal, r = interest rate (as a decimal), and t = time (in years).

Total Amount: A = P + I Compound Interest: Interest calculated on the principal amount and any accumulated interest.

Formula: A = P(1 + r)^n, where A = total amount, P = principal, r = interest rate (as a decimal) per compounding period, and n = number of compounding periods.

Discounts: A reduction in the original price of an item. Discount Amount = Original Price Discount Rate (as a decimal) Sale Price = Original Price - Discount Amount Inflation: The rate at which the general level of prices for goods and services is rising, and subsequently, purchasing power is falling. Future Price = Current Price (1 + Inflation Rate)^Number of Years 2.5 Order of Operations (BODMAS/PEMDAS)

BODMAS/PEMDAS: A mnemonic to remember the order of operations: Brackets / Parentheses Orders / Exponents Division and Multiplication (from left to right) Addition and Subtraction (from left to right) 2.6 Unit Conversions Understanding and converting units is crucial in many real-world applications. Know the conversion factors between common units (e.g., 1 km = 1000 m, 1 Rand = 100 cents).

Worked example

Example 1: Calculating a Percentage Discount

A pair of jeans originally costs R

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5

0. It is on sale with a 20% discount. Calculate the sale price.

Step 1: Calculate the discount amount.

Discount = 20/100 R350 = 0.20 * R350 = R70

Step 2: Calculate the sale price.

Sale Price = R350 - R70 = R280

Example 2: Simple Interest Calculation

You invest R2000 in a savings account that pays 5% simple interest per year. How much interest will you earn after 3 years? What will be the total amount in your account?

Step 1: Calculate the interest earned.

I = P r t = R2000 0.05 * 3 = R300

Step 2: Calculate the total amount.

A = P + I = R2000 + R300 = R2300