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

• Simplify algebraic expressions by combining like terms.
• Solve linear equations using inverse operations.
• Formulate simple equations from word problems.
• Substitute values into expressions.
• Use the distributive property to simplify expressions.

Lesson summary

Algebraic expressions and equations form the foundation for more advanced mathematics and are crucial for problem-solving in many real-life situations. Understanding these concepts allows us to represent unknown quantities, make predictions, and solve problems involving relationships between different variables. In the South African context, from budgeting household expenses to understanding cellular data plans, algebraic thinking helps us make informed decisions. This week, we will focus on simplifying algebraic expressions, solving simple equations, and applying these skills to real-world problems.

Lesson notes

2.1 Algebraic Expressions An algebraic expression is a combination of variables (letters representing unknown values), constants (numbers), and mathematical operations (+, -, ×, ÷).

Variable: A letter (e.g., x, y, a, b) that represents an unknown value.

Constant: A number that has a fixed value (e.g., 5, -3, 0.25).

Coefficient: The number that multiplies a variable (e.g., in 3x, 3 is the coefficient of x).

Like Terms: Terms that have the same variable raised to the same power (e.g., 3x and -5x are like terms; 2y² and 7y² are like terms). 3x and 3x² are not like terms. Only like terms can be combined. Simplifying Algebraic Expressions Simplifying an algebraic expression means writing it in its shortest, most efficient form. This involves combining like terms.

Example 1: Simplify 3x + 2y - x + 5y Step 1: Identify like terms: 3x and -x are like terms; 2y and 5y are like terms.

Step 2: Combine like terms: (3x - x) + (2y + 5y)

Step 3: Perform the operations: 2x + 7y Therefore, the simplified expression is 2x + 7y.

Example 2: Simplify 5a - 3b + 2a - b + 4 Step 1: Identify like terms: 5a and 2a; -3b and -b Step 2: Combine like terms: (5a + 2a) + (-3b - b) + 4 Step 3: Perform the operations: 7a - 4b + 4 Therefore, the simplified expression is 7a - 4b + 4. 2.2 Algebraic Equations An algebraic equation is a statement that two algebraic expressions are equal. It always contains an equals sign (=). The goal is to find the value of the variable that makes the equation true. This is called "solving the equation." Solving Linear Equations A linear equation is an equation where the highest power of the variable is

1. We solve linear equations by using inverse operations to isolate the variable on one side of the equation. The key principle is to do the same operation to both sides of the equation to maintain the balance.

Inverse Operations: Addition and subtraction are inverse operations. Multiplication and division are inverse operations.

Example 1: Solve for x: x + 5 = 12 Step 1: To isolate x, we need to undo the addition of

5. The inverse operation is subtraction. Subtract 5 from both sides of the equation. x + 5 - 5 = 12 - 5 Step 2: Simplify: x = 7 Therefore, the solution is x =

7. Example 2: Solve for y: 3y = 15 Step 1: To isolate y, we need to undo the multiplication by

3. The inverse operation is division. Divide both sides of the equation by 3. 3y / 3 = 15 / 3 Step 2: Simplify: y = 5 Therefore, the solution is y =

5. Example 3: Solve for z: z - 4 = 9 Step 1: To isolate z, we need to undo the subtraction of

4. The inverse operation is addition. Add 4 to both sides of the equation. z - 4 + 4 = 9 + 4 Step 2: Simplify: z = 13 Therefore, the solution is z =

1

3. Example 4: Solve for m: m/2 = 6 Step 1: To isolate m, we need to undo the division by

2. The inverse operation is multiplication. Multiply both sides of the equation by 2. (m/2) 2 = 6 2 Step 2: Simplify: m = 12 Therefore, the solution is m = 12. 2.3 The Distributive Property The distributive property states that a(b + c) = ab + ac. This means we multiply the term outside the parentheses by each term inside the parentheses.

Example 1: Simplify 2(x + 3)

Step 1: Apply the distributive property: 2 x + 2 * 3 Step 2: Simplify: 2x + 6 Therefore, the simplified expression is 2x +

6. Example 2: Simplify -3(y - 4)

Step 1: Apply the distributive property: -3 y + (-3) * (-4) (

Note: a negative times a negative is a positive)

Step 2: Simplify: -3y + 12 Therefore, the simplified expression is -3y + 12. 2.4 Forming Equations from Word Problems To form an equation from a word problem, we need to identify the unknown quantity (represented by a variable) and translate the words into mathematical symbols.

Example: Thando has x marbles. Sipho has 5 more marbles than Thando. Sipho has 12 marbles. How many marbles does Thando have?

Step 1: Identify the unknown: The number of marbles Thando has, which is x.

Step 2: Translate the words into an equation: Sipho has 5 more marbles than Thando: x + 5 Sipho has 12 marbles: x + 5 = 12 Step 3: Solve the equation (as shown in section 2.2): x = 7 Step 4: Answer the question. Thando has 7 marbles. 2.5 Substitution Substitution involves replacing a variable in an algebraic expression with a given numerical value and then evaluating the expression.

Example: Evaluate the expression 3x + 2y if x = 4 and y = -

1. Step 1: Substitute the values: 3(4) + 2(-1)

Step 2: Perform the multiplication: 12 - 2 Step 3: Perform the subtraction: 10 Therefore, the value of the expression is

1

0. Guided Practice (With Solutions)

Question 1: Simplify the expression: 7a - 2b + 4a + 5b - 3 Solution: Step 1: Identify like terms: 7a and 4a; -2b and 5b.

Step 2: Group the like terms: (7a + 4a) + (-2b + 5b) - 3 Step 3: Combine like terms: 11a + 3b - 3 Therefore, the simplified expression is 11a + 3b -

3. Commentary: Remember to pay attention to the signs (+ or -) in front of each term.