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 rational numbers in decimal and fraction form.
• Simplify expressions using the laws of exponents.
• Solve simple problems using these concepts.

Lesson summary

This week, we delve deeper into integers, rational numbers, and how exponents work. Understanding these concepts is crucial not just for math class, but also for making smart financial decisions, understanding measurements in construction, and even predicting population growth. Think about managing your airtime balance (integers), dividing a pizza fairly amongst friends (rational numbers), or understanding how quickly bacteria can multiply and spread (exponents). These topics are fundamental building blocks for more advanced mathematical concepts you'll encounter in high school and beyond. A solid grasp here will make future math much easier!

Lesson notes

2.1 Integers: Integers are whole numbers (no fractions or decimals) and include positive numbers, negative numbers, and zero. We can represent them on a number line, with zero in the middle. Numbers to the right of zero are positive, and numbers to the left are negative.

Addition: Adding two positive integers: This is straightforward; just add the numbers as usual.

Example: 3 + 5 = 8 Adding two negative integers: Add the absolute values of the numbers and keep the negative sign.

Example: -4 + (-2) = -6 Adding a positive and a negative integer: Find the difference between their absolute values. The sign of the answer is the same as the sign of the number with the larger absolute value.

Example: -7 + 3 = -4 (because |-7| > |3| and -7 is negative)

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

Example: 5 - 8 = 5 + (-8) = -3 a - (-b) = a + b.

Example: 2 - (-6) = 2 + 6 = 8 Multiplication and Division: Positive x Positive = Positive.

Example: 4 x 2 = 8 Negative x Negative = Positive.

Example: -3 x -5 = 15 Positive x Negative = Negative.

Example: 6 x -2 = -12 Negative x Positive = Negative.

Example: -4 x 3 = -12 The same rules apply to division.

Example 1: A shopkeeper made a profit of R50 on Monday, a loss of R30 on Tuesday, a profit of R20 on Wednesday, and a loss of R40 on Thursday. What was their total profit/loss over these four days?

Solution: Profit = + Loss = - Total profit/loss = (+50) + (-30) + (+20) + (-40) = 50 - 30 + 20 - 40 = 0 Answer: The shopkeeper broke even (neither profit nor loss). 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 zero. This includes fractions, terminating decimals, and repeating decimals.

Fractions to Decimals: Divide the numerator by the denominator. For example, 1/4 = 0.

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5. Decimals to Fractions: Write the decimal as a fraction with a denominator of 10, 100, 1000, etc., and then simplify. For example, 0.75 = 75/100 = 3/

4. Operations with Rational Numbers: Follow the same rules as with integers, but remember to find common denominators when adding or subtracting fractions.

Example 2: Convert 3/8 to a decimal.

Solution: Divide 3 by 8: 3 ÷ 8 = 0.375 Example 3: Convert 0.6 to a fraction in its simplest form.

Solution: 0.6 = 6/10 = 3/5 2.3 Exponents: An exponent tells you how many times to multiply a base number by itself. For example, 2 3 (read as "2 to the power of 3" or "2 cubed") means 2 x 2 x 2 =

8. Base: The number being multiplied.

Exponent: The number that indicates how many times the base is multiplied by itself.

Laws of Exponents: We'll focus on two key rules this week: Product of powers: When multiplying powers with the same base, add the exponents. a m x a n = a m+n .

Example: 2 2 x 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 Example 4: Simplify 5 2 x 5 4 .

Solution: Using the product of powers rule, 5 2 x 5 4 = 5 2+4 = 5 6 = 15625 Example 5: Simplify 7 5 / 7 3 .

Solution: Using the quotient of powers rule, 7 5 / 7 3 = 7 5-3 = 7 2 = 49 Guided Practice (With Solutions)

Question 1: Calculate: -8 + 5 - (-2)

Solution: -8 + 5 - (-2) = -8 + 5 + 2 (Subtracting a negative is the same as adding) = -3 + 2 = -1

Commentary: We worked from left to right, applying the rules for adding integers. Remembering that subtracting a negative number is the same as adding the positive number is key.

Question 2: Convert 0.45 to a fraction in simplest form.

Solution: 0.45 = 45/100 = 9/20 (Dividing both numerator and denominator by 5)

Commentary: We first write the decimal as a fraction with a denominator of

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0. Then, we simplify the fraction by finding the greatest common factor (GCF) of the numerator and denominator (in this case, 5) and dividing both by the GC

F. Question 3: Simplify: 4 3 x 4 -1 Solution: 4 3 x 4 -1 = 4 3+(-1) (Product of powers rule: add exponents) = 4 2 = 16

Commentary: Even though the exponent is negative, the rule for multiplying exponents with the same base still applies. Remember that x -n = 1/x n Question 4: A student owes R25 for a school trip. They earn R40 washing cars and then spend R15 on snacks. What is their final financial position?

Solution: Start: -R25 Earn: +R40 Spend: -R15 Final Position: -25 + 40 - 15 = 15 - 15 = R0

Commentary: We represent owing money as a negative number and earning money as a positive number. Adding and subtracting these integers gives us the final financial position. Independent Practice (Questions Only)

Calculate: -12 - (-5) + 3 Calculate: (-6) x 4 ÷ (-2) Convert 7/20 to a decimal. Convert 0.85 to a fraction in simplest form.

Simplify: 2 4 x 2 -2 Simplify: 9 3 / 9 A thermometer reads -3°

C. The temperature rises by 7°C, then falls by 5°

C. What is the final temperature?

Simplify: (5/8) + (1/4) - (1/2) Thabo has a loan of R

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0. He pays back R250 and then takes out another loan of R100.