Lesson Notes By Weeks and Term v5 - Grade 10
Revision – Week 9 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Revision – Week 9 focus
Download the Lessonotes Mobile South Africa app for faster lesson access on Android and iPhone.
Subject: Mathematics
Class: Grade 10
Term: Term 4
Week: 9
Theme: General lesson support
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.
This week's focus is revision of key topics covered so far this term, including simplifying algebraic expressions, solving linear equations and inequalities, and understanding number patterns. These concepts are foundational for success in mathematics and have direct relevance to everyday life. Whether you're budgeting your pocket money, calculating discounts at a clothing store, or understanding the logic behind cell phone data bundles, these mathematical skills are essential. A strong grasp of these concepts now will ensure a smoother transition to more complex topics in future years and is essential for subjects like Physics and Accounting.
2.1 Simplifying Algebraic Expressions Algebraic expressions involve variables (like x, y, a) and constants combined with operations (+, -, ×, ÷). Simplifying means writing the expression in its simplest form, usually by combining like terms and removing brackets. Remember to follow the order of operations: BODMAS/BIDMAS (Brackets, Orders/Indices, Division/Multiplication (from left to right), Addition/Subtraction (from left to right)).
Example 1: Simplify: 3(2x - 5) + 4x - (x + 2)
Step 1: Expand the brackets. Multiply the number outside the bracket by each term inside the bracket. 3(2x - 5) = 3 2x - 3 5 = 6x - 15 -(x + 2) = -1 x - 1 2 = -x - 2 Step 2: Rewrite the expression. 6x - 15 + 4x - x - 2 Step 3: Combine like terms. Like terms are terms with the same variable and exponent. (6x + 4x - x) + (-15 - 2) = 9x - 17 Therefore, 3(2x - 5) + 4x - (x + 2) simplifies to 9x - 17 Example 2: Simplify: (x^2 + 3x - 4) - (2x^2 - x + 1)
Step 1: Distribute the negative sign. The minus sign in front of the second set of parentheses means we multiply each term inside by -1. -(2x^2 - x + 1) = -2x^2 + x - 1 Step 2: Rewrite the expression. x^2 + 3x - 4 - 2x^2 + x - 1 Step 3: Combine like terms. (x^2 - 2x^2) + (3x + x) + (-4 - 1) = -x^2 + 4x - 5 Therefore, (x^2 + 3x - 4) - (2x^2 - x + 1) simplifies to -x^2 + 4x - 5 2.2 Solving Linear Equations and Inequalities Linear equations involve finding the value of a variable that makes the equation true. Linear inequalities involve finding a range of values for the variable. The goal is to isolate the variable on one side of the equation or inequality. Remember that when multiplying or dividing an inequality by a negative number, you must reverse the inequality sign.
Example 1: Solve for x: 2x + 5 = 11 Step 1: Subtract 5 from both sides. This isolates the term with 'x'. 2x + 5 - 5 = 11 - 5 2x = 6 Step 2: Divide both sides by
2. This isolates 'x'. 2x / 2 = 6 / 2 x = 3 Therefore, the solution to the equation 2x + 5 = 11 is x =
3. Example 2: Solve for x: 3x - 2 n ), which allows us to calculate any term in the sequence.
Arithmetic Sequence: A sequence where the difference between consecutive terms is constant (called the common difference, 'd'). T n = a + (n - 1)d, where 'a' is the first term and 'n' is the term number.
Geometric Sequence: A sequence where the ratio between consecutive terms is constant (called the common ratio, 'r'). T n = ar n-1 , where 'a' is the first term and 'n' is the term number.
Example 1: Find the general term and the 10th term of the arithmetic sequence: 2, 5, 8, 11, ...
Step 1: Identify the first term (a) and the common difference (d). a = 2 d = 5 - 2 = 3 Step 2: Write the general term formula. T n = a + (n - 1)d T n = 2 + (n - 1)3 T n = 2 + 3n - 3 T n = 3n - 1 Step 3: Calculate the 10th term (T 10 ). T 10 = 3(10) - 1 = 30 - 1 = 29 Therefore, the general term is T n = 3n - 1 and the 10th term is
2
9. Example 2: Find the general term and the 6th term of the geometric sequence: 3, 6, 12, 24, ...
Step 1: Identify the first term (a) and the common ratio (r). a = 3 r = 6 / 3 = 2 Step 2: Write the general term formula. T n = ar n-1 T n = 3 * 2 n-1 Step 3: Calculate the 6th term (T 6 ). T 6 = 3 2 6-1 = 3 2 5 = 3 * 32 = 96 Therefore, the general term is T n = 3 * 2 n-1 and the 6th term is 96. 2.4 Factorization Factorization is the reverse of expansion. It involves expressing an algebraic expression as a product of its factors. Common methods include taking out a common factor, difference of two squares, and factorizing trinomials.
Common Factor: Identify the greatest common factor (GCF) of all terms and factor it out.
Difference of Two Squares: a 2 - b 2 = (a + b)(a - b)
Trinomials: For a trinomial of the form ax 2 + bx + c, find two numbers that multiply to 'ac' and add up to 'b'.
Example 1: Factorize: 4x 2 + 8x Step 1: Find the GCF of 4x 2 and 8x. The GCF is 4x.
Step 2: Factor out the GCF. 4x 2 + 8x = 4x(x + 2)
Example 2: Factorize: x 2 - 9 Step 1: Recognize this as a difference of two squares. x 2 is a square, and 9 (3 2 ) is a square.
Step 2: Apply the formula a 2 - b 2 = (a + b)(a - b). x 2 - 9 = (x + 3)(x - 3)
Example 3: Factorize: x 2 + 5x + 6 Step 1: Find two numbers that multiply to 6 and add up to
5. The numbers are 2 and
3. Step 2: Rewrite the trinomial using these numbers. x 2 + 2x + 3x + 6 Step 3: Factor by grouping. x(x + 2) + 3(x + 2)
Step 4: Factor out the common factor (x + 2). (x + 2)(x + 3) 2.5 Solving Quadratic Equations by Factorization A quadratic equation has the form ax 2 + bx + c =
0. When possible, factorizing the quadratic expression allows us to easily find the solutions (roots).
Example 1: Solve: x 2 - 4x + 3 = 0 Step 1: Factorize the quadratic expression. Find two numbers that multiply to 3 and add up to -
4. The numbers are -1 and -3. x 2 - x - 3x + 3 = 0 x(x - 1) - 3(x - 1) = 0 (x - 1)(x - 3) = 0 Step 2: Set each factor equal to zero and solve for x.