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.
• Simplify expressions involving exponents.
• Solve problems using the order of operations (BODMAS).
• Apply these concepts to real-life scenarios.

Lesson summary

This week, we delve into integers, rational numbers, and exponents. These concepts are fundamental building blocks for more advanced mathematics. Understanding them will empower you to solve complex problems, not only in the classroom but also in everyday life. Imagine budgeting your pocket money, understanding interest rates on a loan, or even calculating distances on a map - these all involve these mathematical tools. In the South African context, understanding financial literacy and making informed decisions requires a solid grasp of these foundational concepts.

Lesson notes

2.1 Integers: Integers are whole numbers (no fractions or decimals) that can be positive, negative, or zero.

Examples: -3, -2, -1, 0, 1, 2, 3...

Number Line: The number line is a visual representation of integers. Zero is in the middle, positive numbers extend to the right, and negative numbers extend to the left.

Adding Integers: Adding two positive integers: The sum is positive.

Example: 3 + 5 = 8 Adding two negative integers: The sum is negative.

Example: -3 + (-5) = -8 Adding a positive and a negative integer: Consider the absolute values (the number without the sign). Subtract the smaller absolute value from the larger absolute value. The sign of the answer is the same as the sign of the integer with the larger absolute value.

Example: -7 + 3 = -4 (Absolute values are 7 and 3. 7 - 3 =

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

Example: 5 + (-2) = 3 (Absolute values are 5 and 2. 5 - 2 =

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

Subtracting Integers: Subtracting an integer is the same as adding its opposite.

Example: 5 - 3 = 5 + (-3) = 2

Example: 2 - 5 = 2 + (-5) = -3

Example: 3 - (-2) = 3 + 2 = 5

Example: -4 - (-1) = -4 + 1 = -3 Multiplying and Dividing Integers: Positive x Positive = Positive.

Example: 2 x 3 = 6 Negative x Negative = Positive.

Example: -2 x -3 = 6 Positive x Negative = Negative.

Example: 2 x -3 = -6 Negative x Positive = Negative.

Example: -2 x 3 = -6 The rules for division are the same as for multiplication. 2.2 Rational Numbers: A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.

Examples: 1/2, -3/4, 5 (5/1), 0.75 (3/4), -2.5 (-5/2).

Fractions: A fraction represents a part of a whole. It has a numerator (top number) and a denominator (bottom number).

Decimals: Decimals are another way to represent rational numbers. Terminating decimals (e.g., 0.25) and repeating decimals (e.g., 0.333...) are rational.

Percentages: Percentages are fractions out of

1

0

0. To convert a fraction to a percentage, multiply by 100%. To convert a percentage to a fraction, divide by

1

0

0. Example: 1/4 = (1/4) 100% = 25%

Example: 75% = 75/100 = 3/4 Converting between Fractions, Decimals, and Percentages: Fraction to Decimal: Divide the numerator by the denominator.

Decimal to Fraction: Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., and simplify.

Decimal to Percentage: Multiply by 100%.

Percentage to Decimal: Divide by 100. 2.3 Exponents: An exponent tells us how many times to multiply a base number by itself. `a^n` means "a" multiplied by itself "n" times. "a" is the base, and "n" is the exponent.

Example: 2^3 = 2 x 2 x 2 = 8

Example: 5^2 = 5 x 5 = 25 Important Rules (for positive integer exponents): Any number raised to the power of 1 is itself: a^1 = a.

Example: 7^1 = 7 Any number (except 0) raised to the power of 0 is 1: a^0 = 1 (where a ≠ 0).

Example: 4^0 =

1. Product of Powers: When multiplying powers with the same base, add the exponents: a^m a^n = a^(m+n).

Example: 2^2 * 2^3 = 2^(2+3) = 2^5 = 32 Quotient of Powers: When dividing powers with the same base, subtract the exponents: a^m / a^n = a^(m-n).

Example: 3^5 / 3^2 = 3^(5-2) = 3^3 = 27 Power of a Power: When raising a power to another power, multiply the exponents: (a^m)^n = a^(mn).

Example: (2^3)^2 = 2^(3*2) = 2^6 = 64 2.4 Order of Operations (BODMAS/PEMDAS): BODMAS/PEMDAS is an acronym that helps us remember the order in which to perform operations in a mathematical expression: Brackets / Parentheses Orders / Exponents Division and Multiplication (from left to right) Addition and Subtraction (from left to right)

Example: 2 + 3 x 4 - 10 / 2 = 2 + 12 - 5 = 14 - 5 = 9 Another Example with exponents: (5 + 2)^2 - 3 x 4 = (7)^2 - 3 x 4 = 49 - 12 = 37 Guided Practice (With Solutions)

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

Solution: Rewrite the subtraction as addition of the opposite: -8 + 5 + 3 Add the integers from left to right: -8 + 5 = -3 Continue adding: -3 + 3 = 0 Therefore, -8 + 5 - (-3) = 0 Question 2: Convert 0.6 to a fraction in its simplest form.

Solution: Write 0.6 as a fraction with a denominator of 10: 6/10 Simplify the fraction by dividing both numerator and denominator by their greatest common factor (2): 6/10 = (6/2)/(10/2) = 3/5 Therefore, 0.6 = 3/5 Question 3: Simplify: 3^2 x 3^4 / 3^3 Solution: Apply the product of powers rule: 3^2 x 3^4 = 3^(2+4) = 3^6 Apply the quotient of powers rule: 3^6 / 3^3 = 3^(6-3) = 3^3 Calculate 3^3: 3 x 3 x 3 = 27 Therefore, 3^2 x 3^4 / 3^3 = 27 Question 4: Evaluate: 2 x (4 - 1)^2 + 6 / 3 Solution: Brackets: 4 - 1 = 3 Exponents: 3^2 = 9 Multiplication: 2 x 9 = 18 Division: 6 / 3 = 2 Addition: 18 + 2 = 20 Therefore, 2 x (4 - 1)^2 + 6 / 3 = 20 Independent Practice (Questions Only)

Calculate: -12 - (-5) + 7 Calculate: (-4) x 6 / (-2) Convert 3/8 to a decimal. Convert 65% to a fraction in its simplest form.