Lesson Notes By Weeks and Term v5 - Grade 8

Lesson Video

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Performance objectives

• Simplify expressions involving addition, subtraction, and multiplication.
• Solve linear equations in one variable using inverse operations.
• Formulate simple algebraic equations from word problems.
• Expand simple algebraic expressions using the distributive property.

Lesson summary

This week, we delve deeper into the fascinating world of algebraic expressions and equations. Algebra is a fundamental building block in mathematics, allowing us to represent unknown quantities and solve problems using symbols and rules. Mastering algebraic expressions and equations is crucial for success in higher-level mathematics and provides a powerful tool for problem-solving in various real-life scenarios.

Think about it: when you're calculating the best deal at the grocery store (comparing prices per kilogram), you're implicitly using algebra! When someone is building a new structure or working with money, they are using algebra!

Lesson notes

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

Variable: A symbol (usually a letter like x, y, or n) that represents an unknown quantity.

Constant: A fixed number (e.g., 5, -3, 1/2).

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

Like Terms: Terms that have the same variable(s) raised to the same power(s). Only like terms can be added or subtracted. For example, 3x and -5x are like terms, but 3x and 3x 2 are not.

Simplifying Algebraic Expressions: This involves combining like terms to write the expression in its simplest form.

Example 1: Simplify 4x + 2y - x + 5y Identify like terms: (4x and -x), (2y and 5y)

Combine like terms: (4x - x) + (2y + 5y) = 3x + 7y Example 2: Simplify 2(a + 3) - 5 Apply the distributive property: 2 a + 2 * 3 - 5 Simplify: 2a + 6 - 5 Combine like terms: 2a + 1 2.2 Algebraic Equations: An algebraic equation is a statement that two algebraic expressions are equal. It contains an equals sign (=). The goal when solving an equation is to find the value(s) of the variable(s) that make the equation true.

Solving Linear Equations in One Variable: A linear equation in one variable is an equation that can be written in the form ax + b = c, where a, b, and c are constants and x is the variable. The key to solving linear equations is to isolate the variable on one side of the equation using inverse operations. Inverse operations "undo" each other (e.g., addition and subtraction, multiplication and division).

Example 3: Solve the equation 3x + 5 = 14 Subtract 5 from both sides: 3x + 5 - 5 = 14 - 5 3x = 9 Divide both sides by 3: 3x/3 = 9/3 x = 3 Why does this work? By performing the same operation on both sides of the equation, we maintain the equality. We are essentially "balancing" the equation.

Example 4: Solve the equation 2(x - 1) = 8 Distribute the 2: 2x - 2 = 8 Add 2 to both sides: 2x - 2 + 2 = 8 + 2 2x = 10 Divide both sides by 2: 2x/2 = 10/2 x = 5 2.3 The Distributive Property: The distributive property states that a( b + c) = a b + a c. This means you multiply the term outside the parentheses by each term inside the parentheses.

Example 5: Expand 3(x + 2)

Apply the distributive property: 3 x + 3 * 2 Simplify: 3x + 6 Example 6: Expand -2(y - 4)

Apply the distributive property: -2 y + (-2) * (-4)

Simplify: -2y + 8 (Remember that a negative times a negative is a positive) 2.4 Translating Word Problems into Algebraic Equations: This is a crucial skill! Look for key words that indicate mathematical operations: Sum, plus, added to, more than: Addition (+) Difference, minus, subtracted from, less than: Subtraction (-) Product, multiplied by, times: Multiplication () Quotient, divided by, per: Division (/) Is, equals, results in: Equals (=)

Example 7: "Thabo has x marbles. Sipho has 5 more marbles than Thabo. Together they have 23 marbles. Write an equation to represent this situation." Thabo's marbles: x Sipho's marbles: x + 5 Total marbles: x + (x + 5) = 23 Guided Practice (With Solutions)

Question 1: Simplify the expression: 5a - 3b + 2a + b Solution: Identify like terms: (5a and 2a), (-3b and b)

Combine like terms: (5a + 2a) + (-3b + b) = 7a - 2b

Commentary: Remember to pay attention to the signs (+ or -) in front of each term when combining like terms. -3b + b is the same as -3b + 1b, which equals -2b.

Question 2: Solve the equation: x/4 - 2 = 3 Solution: Add 2 to both sides: x/4 - 2 + 2 = 3 + 2 x/4 = 5 Multiply both sides by 4: (x/4) 4 = 5 4 x = 20

Commentary: We use the inverse operations in the reverse order of the operations performed on x. Since x was divided by 4 and then 2 was subtracted, we first add 2 and then multiply by

4. Question 3: Expand the expression: -3(2p - 1)

Solution: Apply the distributive property: -3 2p + (-3) (-1)

Simplify: -6p + 3

Commentary: Be careful with the signs! A negative multiplied by a negative results in a positive.

Question 4: Formulate an equation from the following: "A tuck shop sells sweets for R2 each. John buys x sweets and spends R

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4. Write an equation to represent this." Solution: The cost of x sweets is 2 x, which is written as 2x. Since John spends R14, the equation is 2x =

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4. Commentary: This demonstrates translating a real-life situation into an algebraic equation. Independent Practice (Questions Only)

Simplify: 7y - 2x + 3x - 4y Simplify: 5(a + 2) - 3a Solve: 2x + 7 = 15 Solve: y/3 + 1 = 6 Solve: 4(z - 2) = 12 Expand: 2(3m + 4)

Expand: -5(2n - 3) Nomusa buys airtime worth x Rand. She then spends half of it. She has R15 left. Write an equation to represent this situation. The length of a rectangle is twice its width. If the width is w, and the perimeter is 30 cm, write an equation to represent this.

Simplify: (6x + 4) + (2x -1)