Lesson Notes By Weeks and Term v5 - Grade 9

Lesson Video

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Performance objectives

• Solve linear equations with integers, brackets, and fractions.
• Solve linear inequalities and show them on a number line.
• Identify and find the general rule for quadratic number patterns.
• Apply these concepts to solve real-world problems.

Lesson summary

This week, we delve deeper into the fascinating world of equations, inequalities, and number patterns. These mathematical concepts are fundamental building blocks for problem-solving, critical thinking, and logical reasoning – skills highly valued in every aspect of life, from managing personal finances to understanding data in news reports. In South Africa, understanding equations and inequalities is crucial for interpreting socio-economic data (e.g., income distribution, unemployment rates) and making informed decisions. Recognising and extending number patterns is useful in fields like coding, design, and even appreciating the patterns in traditional South African art and crafts.

Lesson notes

2.1 Equations An equation is a mathematical statement that asserts the equality of two expressions. The goal when solving an equation is to find the value(s) of the unknown variable(s) that make the equation true. 2.1.1 Solving Linear Equations Linear equations contain a single variable raised to the power of

1. To solve them, we use inverse operations to isolate the variable on one side of the equation.

Example 1: Solve for x: 2x + 5 = 11 Subtract 5 from both sides: 2x + 5 - 5 = 11 - 5 2x = 6 Why: We subtract 5 to isolate the term with x.

Divide both sides by 2: 2x/2 = 6/2 x = 3 Why: We divide by 2 to isolate x.

Example 2: Solve for y: 3(y - 2) = y + 4 Expand the brackets: 3y - 6 = y + 4 Why: We use the distributive property to remove the brackets.

Subtract y from both sides: 3y - y - 6 = y - y + 4 2y - 6 = 4 Why: We group the y terms together.

Add 6 to both sides: 2y - 6 + 6 = 4 + 6 2y = 10 Why: We isolate the term with y.

Divide both sides by 2: 2y/2 = 10/2 y = 5 Why: We isolate y.

Example 3: Solve for m: m/2 + 1 = 4 Subtract 1 from both sides: m/2 + 1 - 1 = 4 - 1 m/2 = 3 Why: We isolate the term with m.

Multiply both sides by 2: (m/2) 2 = 3 2 m = 6 Why: We isolate m. 2.2 Inequalities An inequality is a mathematical statement that compares two expressions using symbols like (greater than), ≤ (less than or equal to), or ≥ (greater than or equal to). Solving an inequality involves finding the range of values for the variable that makes the inequality true. Important

Note: When multiplying or dividing both sides of an inequality by a negative number, you must reverse the inequality sign.

Example 1: Solve for x: x + 3 > 7 Subtract 3 from both sides: x + 3 - 3 > 7 - 3 x > 4 Why: We isolate x. The solution is all values of x greater than

4. Example 2: Solve for y: -2y ≤ 6 Divide both sides by -2 (and reverse the inequality sign): -2y/-2 ≥ 6/-2 y ≥ -3 Why: We divide by a negative number, so we reverse the inequality. The solution is all values of y greater than or equal to -3. 2.2.1 Representing Inequalities on a Number Line Open circle (o): Represents (the value is not included in the solution). Closed circle (•): Represents ≤ or ≥ (the value is included in the solution). For example, x > 4 is represented on a number line with an open circle at 4 and an arrow extending to the right. y ≥ -3 is represented on a number line with a closed circle at -3 and an arrow extending to the right. 2.3 Number Patterns A number pattern (or sequence) is an ordered list of numbers. Each number in the sequence is called a term. 2.3.1 Quadratic Number Patterns A quadratic number pattern is a sequence where the second difference between consecutive terms is constant. The general formula for a quadratic sequence is Tn = an² + bn + c, where a, b, and c are constants, and n represents the term number. Finding the general rule (Tn = an² + bn + c): Calculate the first difference: Find the difference between consecutive terms.

Calculate the second difference: Find the difference between consecutive first differences.

Find 'a': a = (second difference) / 2 Find 'b': Use the value of 'a' and two consecutive terms to form a system of equations and solve for 'b'.

Find 'c': Substitute the values of 'a' and 'b' and any term in the original sequence into the general formula to find 'c'.

Example: Consider the sequence: 3, 7, 13, 21, ...

First difference: 4, 6, 8 Second difference: 2, 2 'a': a = 2/2 = 1 'b': T1 = a(1)² + b(1) + c = 3 => 1 + b + c = 3 T2 = a(2)² + b(2) + c = 7 => 4 + 2b + c = 7 Subtracting the first equation from the second: 3 + b = 4 => b = 1 'c': 1 + 1 + c = 3 => c = 1 Therefore, the general rule is Tn = n² + n + 1 Guided Practice (With Solutions)

Question 1: Solve for x: 4x - 7 = 9 Solution: Add 7 to both sides: 4x - 7 + 7 = 9 + 7 => 4x = 16 Divide both sides by 4: 4x/4 = 16/4 => x = 4

Commentary: We isolated x by performing inverse operations in the correct order.

Question 2: Solve for y: -3(y + 1) > 6 Solution: Expand the brackets: -3y - 3 > 6 Add 3 to both sides: -3y - 3 + 3 > 6 + 3 => -3y > 9 Divide both sides by -3 (and reverse the inequality sign): -3y/-3 y 1 + b + c = 2 T2 = a(2)² + b(2) + c = 5 => 4 + 2b + c = 5 Subtracting the first from the second: 3 + b = 3 => b = 0 1 + 0 + c = 2 => c = 1 Therefore, Tn = n² + 1 The next term (T5) is 5² + 1 = 26 The term after that (T6) is 6² + 1 = 37

Commentary: We systematically found the coefficients of the quadratic rule. The next two terms were then easily found by substitution.