Lesson Notes By Weeks and Term v5 - Grade 9

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

• Solve linear equations with one variable.
• Represent solutions to linear inequalities on a number line.
• Identify and extend quadratic number patterns.
• Apply algebra to solve word problems.
• Use number patterns to make predictions.

Lesson summary

This week, we delve into the fascinating world of equations, inequalities, and number patterns. These aren't just abstract mathematical concepts; they are fundamental tools for understanding and solving problems in our everyday lives. From managing your pocket money and understanding data bundles to predicting growth in a small business or designing a safe housing structure, a strong grasp of these concepts is crucial. In the South African context, understanding budgets, planning community projects, and analyzing social trends all rely on these mathematical skills.

Lesson notes

2.1 Equations An equation is a mathematical statement that two expressions are equal. It contains an equals sign (=). Our focus is on linear equations with one variable, meaning the highest power of the variable (usually 'x') is

1. Solving Equations: The goal is to isolate the variable on one side of the equation. We do this by performing the same operation on both sides to maintain the equality. Remember BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction) in reverse when unwinding the equation.

Equations with Brackets: First, expand the brackets using the distributive property (a(b+c) = ab + ac). Then, simplify and solve.

Equations with Fractions: Clear the fractions by multiplying both sides of the equation by the lowest common multiple (LCM) of the denominators.

Example 1 (Equations with Brackets): Solve for x: 2(x + 3) = 10 Expand the bracket: 2 x + 2 3 = 10 => 2x + 6 = 10 Subtract 6 from both sides: 2x + 6 - 6 = 10 - 6 => 2x = 4 Divide both sides by 2: 2x / 2 = 4 / 2 => x = 2 Therefore, x =

2. Example 2 (Equations with Fractions): Solve for y: (y/3) + 1 = 5 Subtract 1 from both sides: (y/3) + 1 - 1 = 5 - 1 => y/3 = 4 Multiply both sides by 3: (y/3) 3 = 4 3 => y = 12 Therefore, y = 12. 2.2 Inequalities An inequality is a mathematical statement that compares two expressions using inequality symbols: \ (greater than) ≤ (less than or equal to) ≥ (greater than or equal to)

Solving Inequalities: The process is very similar to solving equations, except when you multiply or divide both sides by a negative number, you must reverse the inequality sign.

Representing Solutions: Number Line: A visual representation where the solution set is shown as a shaded region. Open circles (o) indicate , while closed circles (•) indicate ≤ or ≥.

Interval Notation: A notation that uses brackets and parentheses to represent the solution set. `(` and `)` indicate , while `[` and `]` indicate ≤ or ≥. Infinity (∞) is always enclosed by a parenthesis.

Example 3 (Solving Inequalities): Solve and represent the solution on a number line and in interval notation: 3x - 2 > 7 Add 2 to both sides: 3x - 2 + 2 > 7 + 2 => 3x > 9 Divide both sides by 3: 3x / 3 > 9 / 3 => x > 3 Number Line Representation: Draw a number line. Place an open circle at 3 and shade the region to the right, indicating all values greater than

3. Interval Notation: (3, ∞) Example 4 (Solving Inequalities with Negative Numbers): Solve and represent the solution in interval notation: -2x + 4 ≤ 10 Subtract 4 from both sides: -2x + 4 - 4 ≤ 10 - 4 => -2x ≤ 6 Divide both sides by -2 (and reverse the inequality sign): -2x / -2 ≥ 6 / -2 => x ≥ -3 Interval Notation: [-3, ∞) 2.3 Number Patterns A number pattern is a sequence of numbers that follow a specific rule. We will focus on quadratic number patterns. These patterns have a constant second difference between the terms.

Finding the General Term (nth term): The general term of a quadratic sequence is given by: T n = an 2 + bn + c To find a, b, and c: Calculate the first difference (the difference between consecutive terms). Calculate the second difference (the difference between consecutive first differences). The second difference is constant. `2a = second difference` `3a + b = first difference of the first two terms of the original sequence` `a + b + c = the first term of the original sequence` Example 5 (Quadratic Number Patterns): Find the general term for the sequence: 2, 5, 10, 17, ...

First Difference: 3, 5, 7 Second Difference: 2, 2 (constant)

Find 'a': 2a = 2 => a = 1 Find 'b': 3a + b = 3 => 3(1) + b = 3 => b = 0 Find 'c': a + b + c = 2 => 1 + 0 + c = 2 => c = 1 Therefore, the general term is T n = n 2 + 1 Guided Practice (With Solutions)

Question 1: Solve for x: 3(x - 2) = x + 4 Solution: Expand the bracket: 3x - 6 = x + 4 Subtract x from both sides: 3x - x - 6 = x - x + 4 => 2x - 6 = 4 Add 6 to both sides: 2x - 6 + 6 = 4 + 6 => 2x = 10 Divide both sides by 2: 2x / 2 = 10 / 2 => x = 5 Therefore, x =

5. This question tests the ability to solve linear equations with brackets. The key is to expand the brackets correctly and then isolate the variable.

Question 2: Solve and represent the solution on a number line: 4x + 3 ≤ 11 Solution: Subtract 3 from both sides: 4x + 3 - 3 ≤ 11 - 3 => 4x ≤ 8 Divide both sides by 4: 4x / 4 ≤ 8 / 4 => x ≤ 2 Number Line Representation: Draw a number line. Place a closed circle at 2 and shade the region to the left, indicating all values less than or equal to

2. This question focuses on solving inequalities and representing the solution graphically. Remember to use a closed circle because the inequality includes "equal to." Question 3: Find the general term for the following quadratic sequence: 0, 3, 8, 15, … Solution: First Differences: 3, 5, 7 Second Difference: 2, 2 (constant)

Find 'a': 2a = 2 => a = 1 Find 'b': 3a + b = 3 => 3(1) + b = 3 => b = 0 Find 'c': a + b + c = 0 => 1 + 0 + c = 0 => c = -1 Therefore, T n = n 2 - 1.