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

• Order and compare integers on a number line.
• Convert between fractions, decimals, and percentages.
• Apply the laws of exponents to simplify expressions.
• Identify rational and irrational numbers.
• Solve problems using these concepts.

Lesson summary

This week, we delve deeper into the world of numbers by exploring integers, rational numbers, and exponents. These concepts are fundamental building blocks for more advanced mathematical concepts you'll encounter in higher grades. Understanding these topics will not only improve your mathematical skills but will also help you make informed decisions in your daily life. For example, understanding negative numbers helps you manage your bank account, comprehending rational numbers enables you to calculate fractions in recipes or divide resources fairly, and exponents become essential when calculating compound interest or understanding population growth.

Lesson notes

Integers: Integers are whole numbers (no fractions or decimals) and their negatives. They include numbers like -3, -2, -1, 0, 1, 2, 3, and so on. A number line is a useful tool for visualizing integers and understanding their order. Numbers to the right are greater than numbers to the left. Zero is an integer that is neither positive nor negative.

Example: Consider these integers: -5, 2, -1, 0,

4. On a number line, they would be arranged from left to right: -5, -1, 0, 2,

4. Therefore, -5 is the smallest, and 4 is the largest.

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 integers (since any integer n can be written as n/1), fractions (like 1/2, 3/4, -2/5), terminating decimals (like 0.25, which is 1/4), and repeating decimals (like 0.333..., which is 1/3).

Example 1: Converting a fraction to a decimal Convert 3/8 to a decimal. To do this, divide 3 by 8: 3 ÷ 8 = 0.375 Example 2: Converting a decimal to a fraction Convert 0.75 to a fraction. 0.75 = 75/

1

0

0. Simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 25. 75/100 = (75 ÷ 25) / (100 ÷ 25) = 3/4 Example 3: Converting a percentage to a fraction and decimal Convert 60% to a fraction and a decimal. 60% means 60 out of 100, so 60% = 60/

1

0

0. Simplify this fraction by dividing both numerator and denominator by their greatest common factor, which is 20. 60/100 = (60 ÷ 20) / (100 ÷ 20) = 3/5 To convert to a decimal, divide 60 by 100: 60 ÷ 100 = 0.6 Example 4: Identifying Rational and Irrational Numbers Explain why 2.5 is rational and √2 is irrational. 5 can be expressed as the fraction 5/2, so it meets the definition of a rational number. √2 cannot be expressed exactly as a fraction p/q, where p and q are integers. Its decimal representation is non-terminating and non-repeating.

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

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

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

Example: (5 2 ) 3 = 5 23 = 5 6 = 15625 Power of a product: (ab) n = a n b n (The power of a product is the product of the powers).

Example: (23) 2 = 2 2 3 2 = 4 * 9 = 36 Power of a quotient: (a/b) n = a n /b n (The power of a quotient is the quotient of the powers).

Example: (4/2) 3 = 4 3 / 2 3 = 64 / 8 = 8 Zero exponent: a 0 = 1 (Any non-zero number raised to the power of 0 is 1).

Example: 7 0 = 1 Negative exponent: a -n = 1/a n (A negative exponent means the reciprocal of the base raised to the positive exponent).

Example: 2 -3 = 1/2 3 = 1/8 Example 1: Simplifying an expression with exponents Simplify: 4 2 * 4 -1 Using the product of powers rule: 4 2 + (-1) = 4 1 = 4 Example 2: Simplifying an expression with exponents and division Simplify: (5 3 ) / 5 0 Using the quotient of powers rule: 5 3-0 = 5 3 = 125 Guided Practice (With Solutions)

Question 1: Arrange the following integers in ascending order: -8, 5, -2, 0, 3, -6 Solution: The integers in ascending order (smallest to largest) are: -8, -6, -2, 0, 3,

5. Explanation: We identify the most negative number as the smallest and progressively move towards the most positive number.

Question 2: Convert 0.8 to a fraction and then express it as a percentage.

Solution: 8 = 8/10 = 4/5 (simplified). To express as a percentage, multiply by 100: (4/5) * 100 = 80%.

Therefore, 0.8 = 80%.

Explanation: We first express the decimal as a fraction with a denominator of 10, then simplify. To convert a fraction to a percentage, we multiply by

1

0

0. Question 3: Simplify: (3 2 * 3 -1 ) 2 Solution: First, simplify inside the parentheses: 3 2 * 3 -1 = 3 2 + (-1) = 3 1 =

3. Now, raise the result to the power of 2: (3) 2 = 3 * 3 =

9. Explanation: We apply the product of powers rule inside the parentheses and then apply the power of a power rule.

Question 4: Identify which of the following numbers is a rational number and explain why: √4, √

5. Solution: √4 =

2. Since 2 can be expressed as the fraction 2/1, it is a rational number. √5 cannot be expressed exactly as a fraction and has a non-terminating, non-repeating decimal representation; therefore, it is irrational (not covered this week, but good to introduce concept).

Question 5: John has a bank balance of -R

5

0. He deposits R

1

2

0. What is his new bank balance?

Solution: His new balance is -R50 + R120 = R

7

0. Explanation: Adding a positive number to a negative number is equivalent to subtracting the absolute value of the negative number from the positive number.