Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Add, subtract, multiply, and divide integers.
• Convert between fractions, decimals, and percentages.
• Evaluate expressions with exponents.
• Apply the order of operations (BODMAS).
• Solve simple real-world problems.

Lesson summary

This week, we delve deeper into the world of integers, rational numbers, and exponents. Understanding these concepts is crucial because they form the building blocks for more advanced mathematical topics you'll encounter later. Integers and rational numbers are used daily, from calculating your spaza shop budget to understanding cricket scores. Exponents are essential for understanding growth, decay, and even computer calculations. In a country like South Africa, where financial literacy and understanding data are increasingly important, mastering these foundational concepts is essential for informed decision-making and success in various fields.

Lesson notes

2.1 Integers Integers are whole numbers (not fractions) that can be positive, negative, or zero. The set of integers includes {... -3, -2, -1, 0, 1, 2, 3...}.

Adding Integers: If the integers have the same sign, add their absolute values and keep the sign.

Example: -3 + (-5) = -8 (Both are negative, add 3 and 5 to get 8, keep the negative sign).

Example: 4 + 7 = 11 (Both are positive, add 4 and 7 to get 11, keep the positive sign). If the integers have different signs, subtract the smaller absolute value from the larger absolute value and keep the sign of the integer with the larger absolute value.

Example: -7 + 3 = -4 (Absolute values are 7 and 3. 7 is larger, so subtract 3 from 7 to get

4. Since -7 has the larger absolute value, the answer is -4).

Example: 5 + (-2) = 3 (Absolute values are 5 and 2. 5 is larger, so subtract 2 from 5 to get

3. Since 5 has the larger absolute value, the answer is 3).

Subtracting Integers: Subtracting an integer is the same as adding its opposite (additive inverse). a - b = a + (-b)

Example: 5 - 8 = 5 + (-8) = -3

Example: -3 - (-2) = -3 + 2 = -1 Multiplying and Dividing Integers: If the integers have the same sign, the result is positive.

Example: -4 -3 = 12

Example: 10 / 2 = 5 If the integers have different signs, the result is negative.

Example: 6 -2 = -12

Example: -15 / 3 = -5 Example (South African Context): Imagine Sipho has R50 in his bank account. He then spends R

8

0. What is his new balance?

Solution: 50 - 80 = 50 + (-80) = -

3

0. Sipho now has a balance of -R30 (he is overdrawn). 2.2 Rational Numbers Rational numbers are numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. This includes fractions, decimals that terminate or repeat, and integers. Converting between Fractions, Decimals, and Percentages: Fraction to Decimal: Divide the numerator by the denominator.

Example: 1/4 = 1 ÷ 4 = 0.25 Decimal to Fraction: Write the decimal as a fraction with a power of 10 in the denominator and simplify.

Example: 0.75 = 75/100 = 3/4 Fraction to Percentage: Multiply the fraction by 100%.

Example: 1/2 = (1/2) 100% = 50% Percentage to Fraction: Divide the percentage by 100 and simplify.

Example: 25% = 25/100 = 1/4 Decimal to Percentage: Multiply the decimal by 100%.

Example: 0.6 = 0.6 100% = 60% Percentage to Decimal: Divide the percentage by

1

0

0. Example: 80% = 80/100 = 0.8 Example (South African Context): A shirt costs R150, and there is a 20% discount. What is the discount amount in Rand?

Solution: 20% as a decimal is 20/100 = 0.

2. Discount amount = 0.2 * R150 = R30. 2.3 Exponents An exponent indicates how many times a base number is multiplied by itself. For example, in the expression 2 3 , 2 is the base and 3 is the exponent. 2 3 means 2 2 2 =

8. Basic Terminology: Base: The number being multiplied.

Exponent (or Power): The number indicating how many times the base is multiplied by itself.

Examples: 3 2 = 3 3 = 9 (3 squared) 5 3 = 5 5 * 5 = 125 (5 cubed) 2 4 = 2 2 2 2 = 16 (2 to the power of 4)

Example (South African Context): A certain bacteria population doubles every hour. If you start with 5 bacteria, how many will you have after 3 hours?

Solution: After 1 hour: 5 2 =

1

0. After 2 hours: 10 2 =

2

0. After 3 hours: 20 * 2 =

4

0. Alternatively: 5 2 3 = 5 8 =

4

0. You will have 40 bacteria. 2.4 Order of Operations (BODMAS/PEMDAS) When evaluating expressions with multiple operations, we must follow a specific order: Brackets / Parentheses Orders / Exponents Division and Multiplication (from left to right) Addition and Subtraction (from left to right)

Example: 2 + 3 * 4 - 10 / 2 = 2 + 12 - 5 = 14 - 5 = 9 Example (South African Context): A spaza shop sells airtime for R12 per voucher. They buy the vouchers for R10 each. If they sell 5 vouchers and then have to pay R3 for transport to the wholesaler, what is their profit?

Solution: (12 - 10) 5 - 3 = 2 5 - 3 = 10 - 3 = R

7. Their profit is R

7. Guided Practice (With Solutions)

Question 1: Evaluate: -8 + 5 - (-3)

Solution: -8 + 5 - (-3) = -8 + 5 + 3 (Subtracting a negative is the same as adding) = -3 + 3 (Add -8 and 5) = 0 Question 2: Convert 3/8 to a decimal and a percentage.

Solution: Decimal: 3 ÷ 8 = 0.375 Percentage: (3/8) 100% = 37.5% Question 3: Simplify: 2 3 + 3 2 - 4 1 Solution: 2 3 + 3 2 - 4 1 = (2 2 2) + (3 * 3) - 4 = 8 + 9 - 4 = 17 - 4 = 13 Question 4: Evaluate: 15 ÷ 3 + 2 * (6 - 4) 2 Solution: 15 ÷ 3 + 2 (6 - 4) 2 = 15 ÷ 3 + 2 (2) 2 (Brackets first) = 15 ÷ 3 + 2 * 4 (Exponents next) = 5 + 8 (Division and Multiplication from left to right) = 13 (Addition last) Independent Practice (Questions Only)

Calculate: -12 - (-5) + 7 What is -6 multiplied by 4, divided by -3? Convert 7/20 to a decimal and a percentage. Express 0.45 as a simplified fraction.

Evaluate: 4 2 - 2 3 + 1 5 Simplify: 2 * (5 + 3) - 16 ÷ 4 Calculate: ( -3 + 7 ) * 2 - 10 A shop sells chips for R8 per packet. They buy them for R

5. If they sell 12 packets and have expenses of R10, what is their profit?