Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Apply BODMAS with integers and rational numbers.
• Convert between fractions, decimals, and percentages.
• Simplify expressions with exponents.
• Solve problems with squares and cubes.
• Differentiate between rational and irrational numbers.

Lesson summary

This week, we delve deeper into the world of numbers, focusing on integers, rational numbers, and how to work with exponents. Understanding these concepts is crucial not just for success in mathematics, but also for making informed decisions in everyday life. From managing your pocket money to understanding how interest rates affect savings, these foundational skills are essential.

Furthermore, grasping exponents lays the groundwork for more advanced concepts in algebra and science, which will be vital in future studies and careers. Thinking about a career in engineering, finance, or even game development? These number skills are your starting point!

Lesson notes

2.1 Integers: Integers are whole numbers (no fractions or decimals) and their negatives, including zero.

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

Addition and Subtraction: Remember to consider the sign of the numbers. Adding a negative number is the same as subtracting. Subtracting a negative number is the same as adding.

Multiplication and Division: Positive x Positive = Positive Negative x Negative = Positive Positive x Negative = Negative Negative x Positive = Negative The same rules apply to division. 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: Proper fractions (e.g., 1/2), improper fractions (e.g., 5/3), and mixed numbers (e.g., 1 2/3).

Terminating Decimals: Decimals that end (e.g., 0.25, 3.14). These can always be written as fractions (0.25 = 1/4).

Repeating Decimals: Decimals that have a repeating pattern (e.g., 0.333..., 1.666...). These can also be written as fractions (0.333... = 1/3, 1.666... = 5/3).

Integers: All integers are rational numbers because they can be written as a fraction with a denominator of 1 (e.g., 5 = 5/1).

Example 1: Converting a repeating decimal to a fraction Convert 0.777... to a fraction. Let x = 0.777... Then 10x = 7.777...

Subtract x from 10x: 10x - x = 7.777... - 0.777... 9x = 7 x = 7/9 Example 2: Converting a mixed number to a decimal Convert 2 1/4 to a decimal. First, convert the mixed number to an improper fraction: 2 1/4 = (2 * 4 + 1) / 4 = 9/4 Then, divide the numerator by the denominator: 9 ÷ 4 = 2.25 2.3 Exponents: An exponent indicates how many times a base number is multiplied by itself. For example, a n means a multiplied by itself n times. a is the base. n is the exponent (or power).

Key Rules (for whole number exponents): a 0 = 1 (any non-zero number raised to the power of 0 is 1). a 1 = a (any number raised to the power of 1 is itself). a m a n = a m+n (When multiplying powers with the same base, add the exponents). a m / a n = a m-n (When dividing powers with the same base, subtract the exponents). (a m ) n = a mn (When raising a power to another power, multiply the exponents).

Example 3: Simplifying exponents Simplify: 2 3 * 2 2 Using the rule a m a n = a m+n , we have: 2 3 2 2 = 2 3+2 = 2 5 = 2 2 2 2 * 2 = 32 Example 4: Simplifying exponents with division Simplify: 5 4 / 5 2 Using the rule a m / a n = a m-n , we have: 5 4 / 5 2 = 5 4-2 = 5 2 = 5 * 5 = 25 2.4 Squares and Cubes: Square: A number multiplied by itself (raised to the power of 2). For example, 3 2 = 3 3 = 9. 9 is the square of

3. Cube: A number multiplied by itself twice (raised to the power of 3). For example, 2 3 = 2 2 * 2 = 8. 8 is the cube of

2. Example 5: Calculating the square and cube of a rational number Find the square and cube of 1/

2. Square: (1/2) 2 = (1/2) * (1/2) = 1/4 Cube: (1/2) 3 = (1/2) (1/2) (1/2) = 1/8 2.5 Irrational Numbers: Irrational numbers are numbers that cannot be expressed as a fraction p/q, where p and q are integers. These are non-terminating, non-repeating decimals.

Examples: π (pi), √2, √

3. Important

Note: Understanding the difference between rational and irrational numbers is key. If you can't write it as a fraction of two integers, it's irrational! Guided Practice (With Solutions)

Question 1: Evaluate: -3 + 5 * 2 - 8 ÷ 4 Solution: Following BODMAS/PEMDAS: Division: 8 ÷ 4 = 2 Multiplication: 5 * 2 = 10 Addition and Subtraction (from left to right): -3 + 10 - 2 = 7 - 2 = 5 Answer: 5

Commentary: This question tests the order of operations. Make sure to perform multiplication and division before addition and subtraction.

Question 2: Convert 1.4 to a fraction in its simplest form.

Solution: Write 1.4 as a fraction: 1.4 = 14/10 Simplify the fraction by dividing both numerator and denominator by their greatest common factor (GCF), which is 2: 14/10 = (14 ÷ 2) / (10 ÷ 2) = 7/5 Answer: 7/5

Commentary: This question tests your ability to convert decimals to fractions and simplify fractions.

Question 3: Simplify: (3 2 * 3 3 ) / 3 4 Solution: Simplify the numerator using the rule a m a n = a m+n : 3 2 * 3 3 = 3 2+3 = 3 5 Simplify the expression using the rule a m / a n = a m-n : 3 5 / 3 4 = 3 5-4 = 3 1 = 3 Answer: 3

Commentary: This question tests your understanding of exponent rules. Remember to simplify the numerator before dividing.

Question 4: Determine if the following number is rational or irrational: √16 Solution: √16 =

4. Since 4 can be written as 4/1 (a fraction of two integers), √16 is a rational number.

Answer: Rational

Commentary: Many square roots are irrational, but perfect square roots are rational. Recognizing perfect squares is key. Independent Practice (Questions Only)

Evaluate: 12 - 4 * (6 + (-3)) ÷ 2 Convert 2 3/5 to a decimal.

Simplify: (5 5 ) / (5 * 5 2 )

Find the value of x: x = 4 3 - 2 5 Determine if the following number is rational or irrational: √7 Simplify: (-2) 3 + (1/2) 2 A square piece of land has a side length of 7 meters.