Lesson Notes By Weeks and Term v5 - Grade 7

Lesson Video

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Performance objectives

• Identify variables, constants, and coefficients.
• Simplify expressions by combining like terms.
• Evaluate expressions by substituting values.
• Solve simple one-step equations.
• Translate word problems into equations.

Lesson summary

Algebra is a fundamental building block in mathematics. Understanding algebraic expressions and simple equations is crucial for success in higher-level mathematics and for solving everyday problems. In South Africa, understanding these concepts can help with managing personal finances, understanding business transactions, and even analyzing statistics related to social and economic issues. For instance, calculating cellphone data usage or figuring out how much money to save each month relies on understanding algebraic principles. This week, we will focus on understanding the basics of algebraic expressions and simple equations.

Lesson notes

What is Algebra? Algebra is a branch of mathematics that uses symbols, usually letters, to represent unknown quantities or values. These symbols are called variables.

Algebraic Expressions: An algebraic expression is a combination of variables, constants, and mathematical operations (addition, subtraction, multiplication, division).

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

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

Coefficient: The number that is multiplied by a variable (e.g., in the term 3x, 3 is the coefficient).

Like Terms: Terms that have the same variable raised to the same power. For example, 3x and 5x are like terms, but 3x and 3x² are not.

Example: The expression 5x + 2y – 3 has: Variables: x and y Constants: -3 Coefficients: 5 (for x) and 2 (for y)

Terms: 5x, 2y, and -3 Simplifying Algebraic Expressions: Simplifying an algebraic expression means writing it in its shortest and simplest form. We can only combine like terms.

How to Simplify: Identify like terms. Combine the coefficients of the like terms. Remember to pay attention to the signs (+ or -) in front of each term.

Example 1: Simplify 3x + 5x – 2x All terms are like terms (they all have the variable x).

Combine the coefficients: 3 + 5 – 2 = 6 Simplified expression: 6x Example 2: Simplify 4a + 2b – a + 7b Identify like terms: 4a and –a are like terms. 2b and 7b are like terms.

Combine like terms: 4a – a = 3a (Remember that –a is the same as –1a) 2b + 7b = 9b Simplified expression: 3a + 9b Example 3: Simplify 7y + 3 – 2y + 5 Identify like terms: 7y and -2y are like terms. 3 and 5 are like terms.

Combine like terms: 7y - 2y = 5y 3 + 5 = 8 Simplified expression: 5y + 8 Evaluating Algebraic Expressions: Evaluating an algebraic expression means finding its value by substituting given values for the variables.

How to Evaluate: Substitute the given value for each variable in the expression. Perform the operations according to the order of operations (BODMAS/PEMDAS - Brackets, Orders, Division/Multiplication, Addition/Subtraction).

Example 1: Evaluate 2x + 3 if x = 4 Substitute x = 4: 2(4) + 3 Multiply: 8 + 3 Add: 11 The value of the expression is

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1. Example 2: Evaluate 5a – b if a = 2 and b = 1 Substitute a = 2 and b = 1: 5(2) – 1 Multiply: 10 – 1 Subtract: 9 The value of the expression is

9. Simple Equations: An equation is a mathematical statement that shows that two expressions are equal. It always contains an equals sign (=). A simple equation usually involves one variable and a few operations.

Solving Simple Equations: Solving an equation means finding the value of the variable that makes the equation true. We do this by isolating the variable on one side of the equation. To isolate the variable, we use inverse operations. Inverse operations "undo" each other. The inverse operation of addition is subtraction. The inverse operation of subtraction is addition. The inverse operation of multiplication is division. The inverse operation of division is multiplication.

Important Rule: Whatever you do to one side of the equation, you must do to the other side to keep the equation balanced.

Example 1: Solve x + 5 = 12 To isolate x, we need to get rid of the +

5. The inverse operation of +5 is –

5. Subtract 5 from both sides of the equation: x + 5 – 5 = 12 – 5 Simplify: x = 7 Therefore, the solution is x =

7. Example 2: Solve y – 3 = 8 To isolate y, we need to get rid of the –

3. The inverse operation of –3 is +

3. Add 3 to both sides of the equation: y – 3 + 3 = 8 + 3 Simplify: y = 11 Therefore, the solution is y =

1

1. Example 3: Solve 2z = 10 2z means 2 multiplied by z. To isolate z, we need to get rid of the multiplication by

2. The inverse operation of multiplication by 2 is division by

2. Divide both sides of the equation by 2: 2z / 2 = 10 / 2 Simplify: z = 5 Therefore, the solution is z =

5. Example 4: Solve p / 4 = 3 p / 4 means p divided by

4. To isolate p, we need to get rid of the division by

4. The inverse operation of division by 4 is multiplication by

4. Multiply both sides of the equation by 4: (p / 4) 4 = 3 4 Simplify: p = 12 Therefore, the solution is p =

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2. Guided Practice (With Solutions)

Question 1: Simplify the expression: 6a – 2a + 3b – b Solution: Identify like terms: 6a and -2a are like terms; 3b and -b are like terms.

Combine like terms: 6a – 2a = 4a 3b – b = 2b (Remember that –b is the same as –1b)

Simplified expression: 4a + 2b

Commentary: The key here is recognizing like terms. Learners often make mistakes by trying to combine unlike terms. Emphasize that only terms with the same variable can be combined.

Question 2: Evaluate the expression 3x – 2y if x = 5 and y =

2. Solution: Substitute x = 5 and y = 2: 3(5) – 2(2)

Multiply: 15 – 4 Subtract: 11 The value of the expression is

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1. Commentary: This question tests the ability to substitute values correctly and follow the order of operations. Encourage learners to write down each step clearly to avoid errors.