Lesson Notes By Weeks and Term v5 - Grade 8

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

• Define and represent integers and rational numbers on a number line.
• Perform all four basic operations with integers.
• Simplify expressions with positive exponents.
• Convert between fractions, decimals, and percentages.
• Solve simple real-world problems using these concepts.

Lesson summary

This week, we embark on a journey to solidify our understanding of integers, rational numbers, and exponents – foundational concepts that underpin much of higher-level mathematics. Understanding these concepts well is crucial for success not just in the classroom, but also in many real-world scenarios. For example, managing your monthly expenses involves working with integers (credits and debits), dividing a pizza fairly among friends utilizes rational numbers, and calculating interest on a savings account requires knowledge of exponents.

Lesson notes

a)

Integers: Integers are whole numbers (no fractions or decimals) and their negatives, including zero. They can be positive, negative, or zero. Think of them as points evenly spaced along a number line extending infinitely in both directions.

Examples: ..., -3, -2, -1, 0, 1, 2, 3, ... The number line is a visual representation of integers and their order. Numbers to the right are greater than numbers to the left. Zero is neither positive nor negative.

Arithmetic Operations with Integers: Addition: Adding two positive integers results in a positive integer. (e.g., 3 + 5 = 8) Adding two negative integers results in a negative integer. (e.g., -3 + (-5) = -8)

Adding a positive and a negative integer: Subtract the smaller absolute value from the larger absolute value. The result has the sign of the integer with the larger absolute value. (e.g., -7 + 4 = -3; 5 + (-2) = 3)

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

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

Example: -2 - (-6) = -2 + 6 = 4 Multiplication: Positive × Positive = Positive (e.g., 2 × 3 = 6) Negative × Negative = Positive (e.g., -2 × -3 = 6) Positive × Negative = Negative (e.g., 2 × -3 = -6) Negative × Positive = Negative (e.g., -2 × 3 = -6)

Division: The rules for division are the same as for multiplication regarding signs. Positive ÷ Positive = Positive (e.g., 6 ÷ 2 = 3) Negative ÷ Negative = Positive (e.g., -6 ÷ -2 = 3) Positive ÷ Negative = Negative (e.g., 6 ÷ -2 = -3) Negative ÷ Positive = Negative (e.g., -6 ÷ 2 = -3)

Example 1 (Integers): A Spaza shop owner in Soweto had a balance of R

2

0

0. He spent R350 on stock. What is his new balance?

Solution: His balance decreases by R

3

5

0. Therefore, his new balance is R200 - R350 = -R

1

5

0. He is now R150 overdrawn. b)

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 means they can be expressed as a ratio of two integers.

Examples: 1/2, -3/4, 5 (which is 5/1), 0.25 (which is 1/4), -1.75 (which is -7/4) Non-

Examples: √2, π (these are irrational numbers) Fractions, Decimals, and Percentages: Rational numbers can be written in three common forms: fractions, decimals, and percentages.

Fraction to Decimal: Divide the numerator by the denominator. (e.g., 1/4 = 0.25)

Decimal to Fraction: Write the decimal as a fraction with a denominator that is a power of 10 (10, 100, 1000, etc.), then simplify. (e.g., 0.75 = 75/100 = 3/4)

Fraction to Percentage: Multiply the fraction by 100%. (e.g., 1/2 = (1/2) 100% = 50%)

Percentage to Fraction: Divide the percentage by 100 and simplify. (e.g., 25% = 25/100 = 1/4)

Decimal to Percentage: Multiply the decimal by 100%. (e.g., 0.6 = 0.6 100% = 60%)

Percentage to Decimal: Divide the percentage by 100. (e.g., 80% = 80/100 = 0.8)

Example 2 (Rational Numbers): A survey in Durban showed that 3 out of 5 people prefer eThekwini Municipality's beaches. What percentage of people prefer these beaches?

Solution: The fraction of people who prefer the beaches is 3/

5. To convert this to a percentage, we multiply by 100%: (3/5) * 100% = 60%.

Therefore, 60% of people prefer eThekwini Municipality's beaches. c)

Exponents: An exponent indicates how many times a base number is multiplied by itself. For example, in the expression a n , 'a' is the base, and 'n' is the exponent. a n means 'a' multiplied by itself 'n' times. a n = a × a × a × ... × a (n times)

Examples: 2 3 = 2 × 2 × 2 = 8 5 2 = 5 × 5 = 25 10 4 = 10 × 10 × 10 × 10 = 10000 Example 3 (Exponents): Calculate the area of a square garden with sides of length 7 meters.

Solution: The area of a square is side × side, or side 2 . In this case, the area is 7 2 = 7 × 7 = 49 square meters. Guided Practice (With Solutions)

Question 1: Order the following integers from least to greatest: -5, 2, 0, -1,

7. Solution: The order from least to greatest is: -5, -1, 0, 2,

7. Remember, negative numbers are less than zero and positive numbers. The further a negative number is from zero, the smaller it is.

Question 2: Evaluate: -8 + 3 - (-2)

Solution: -8 + 3 - (-2) = -8 + 3 + 2 = -5 + 2 = -

3. First, we change subtraction to addition of the opposite. Then, we perform the addition from left to right.

Question 3: A recipe for vetkoek calls for 2/3 cup of flour. You want to make half the recipe. How much flour do you need? Express your answer as a fraction.

Solution: We need to find half of 2/3, which means (1/2) * (2/3) = 2/

6. We can simplify this fraction by dividing both numerator and denominator by 2, which gives us 1/

3. Therefore, you need 1/3 cup of flour.

Question 4: Calculate 3 4 .

Solution: 3 4 = 3 3 3 3 = 9 9 =

8

1. We multiply 3 by itself four times. Independent Practice (Questions Only) What is the opposite of -12?

Evaluate: 15 - 22 + 8 Calculate: -4 * 6 / (-3) Convert 0.85 to a fraction in simplest form. Express 45% as a decimal.

Evaluate: 5 3 Simplify: 2 2 + 3 2 - 1 4 A shop sells a T-shirt for R

8

0. They offer a 15% discount.