Lesson Notes By Weeks and Term v5 - Grade 7

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.
• Evaluate algebraic expressions by substituting values.
• Solve simple one-step equations.
• Formulate equations from real-world scenarios.

Lesson summary

This week, we delve deeper into the exciting world of algebraic expressions and simple equations. Understanding algebra is like learning a secret code that unlocks many problem-solving possibilities, not just in mathematics, but also in everyday life. Imagine calculating the cost of airtime bundles with varying data amounts or figuring out how much pocket money you need to save each week to buy that new pair of takkies. Algebra helps us represent these situations using symbols and letters, making them easier to understand and solve. It's a foundational skill that will be crucial for your future studies in maths and science, and for navigating various real-world scenarios.

Lesson notes

2.1 What is an Algebraic Expression? An algebraic expression is a mathematical phrase that combines numbers, variables (letters representing unknown values), and operation symbols (+, -, ×, ÷).

Variable: A letter (like x, y, or a) representing an unknown number.

Constant: A number that stands alone (like 5, -3, or 0.7).

Coefficient: The number that multiplies a variable (like 3 in 3x, or -2 in -2y).

Examples: `2x + 3`: This has two terms: `2x` (coefficient 2, variable x) and `3` (a constant). `5y - 7z + 1`: This has three terms: `5y` (coefficient 5, variable y), `-7z` (coefficient -7, variable z), and `1` (a constant). `a + b - c`: This has three terms: `a`, `b`, and `-c`. (Remember `c` is the same as `1c`, so the coefficient is -1). `4`: This is just a constant term. 2.2 Simplifying Algebraic Expressions: Combining Like Terms Like terms are terms that have the same variable(s) raised to the same power. We can combine like terms by adding or subtracting their coefficients.

Example 1: Simplify `3x + 5x - 2x` All terms have the same variable, `x`, raised to the power of

1. They are like terms.

Combine the coefficients: 3 + 5 - 2 = 6 Therefore, `3x + 5x - 2x = 6x` Example 2: Simplify `4y + 2 - y + 6` Identify like terms: `4y` and `-y` are like terms, and `2` and `6` are like terms (constants). Combine `4y` and `-y`: 4y - y = 3y (Remember `-y` is the same as `-1y`) Combine `2` and `6`: 2 + 6 = 8 Therefore, `4y + 2 - y + 6 = 3y + 8` Example 3: Simplify `2a + 3b - a + 5b - 4` Identify like terms: `2a` and `-a` are like terms, and `3b` and `5b` are like terms. The `-4` is a constant and has no like terms. Combine `2a` and `-a`: 2a - a = a (Remember `a` is the same as `1a`) Combine `3b` and `5b`: 3b + 5b = 8b Therefore, `2a + 3b - a + 5b - 4 = a + 8b - 4` 2.3 Evaluating Algebraic Expressions To evaluate an algebraic expression, we substitute the given value(s) for the variable(s) and then simplify using the order of operations (BODMAS/PEMDAS: Brackets, Orders, Division and Multiplication, Addition and Subtraction).

Example 1: Evaluate `2x + 5` when `x = 3` Substitute `x = 3`: 2(3) + 5 Multiply: 2(3) = 6 Add: 6 + 5 = 11 Therefore, `2x + 5 = 11` when `x = 3` Example 2: Evaluate `3a - 2b` when `a = 4` and `b = 1` Substitute `a = 4` and `b = 1`: 3(4) - 2(1)

Multiply: 3(4) = 12 and 2(1) = 2 Subtract: 12 - 2 = 10 Therefore, `3a - 2b = 10` when `a = 4` and `b = 1` Example 3: Evaluate `(x + y) / 2` when `x = 7` and `y = 3` Substitute `x = 7` and `y = 3`: (7 + 3) / 2 Add inside the brackets: 7 + 3 = 10 Divide: 10 / 2 = 5 Therefore, `(x + y) / 2 = 5` when `x = 7` and `y = 3` 2.4 Solving Simple One-Step Equations An equation is a mathematical statement that two expressions are equal. Solving an equation means finding the value of the variable that makes the equation true. To solve a one-step equation, we perform the inverse operation on both sides of the equation to isolate the variable.

Addition Equation: If the equation involves addition, subtract the same number from both sides.

Subtraction Equation: If the equation involves subtraction, add the same number to both sides.

Multiplication Equation: If the equation involves multiplication, divide both sides by the same number.

Division Equation: If the equation involves division, multiply both sides by the same number.

Example 1: Solve `x + 5 = 12` To isolate `x`, subtract 5 from both sides: `x + 5 - 5 = 12 - 5` Simplify: `x = 7` Therefore, the solution is `x = 7` Example 2: Solve `y - 3 = 8` To isolate `y`, add 3 to both sides: `y - 3 + 3 = 8 + 3` Simplify: `y = 11` Therefore, the solution is `y = 11` Example 3: Solve `3a = 15` To isolate `a`, divide both sides by 3: `3a / 3 = 15 / 3` Simplify: `a = 5` Therefore, the solution is `a = 5` Example 4: Solve `b / 4 = 2` To isolate `b`, multiply both sides by 4: `(b / 4) 4 = 2 4` Simplify: `b = 8` Therefore, the solution is `b = 8` 2.5 Formulating Simple Algebraic Equations from Word Problems We can translate real-life scenarios into algebraic equations. The key is to identify the unknown quantity (our variable) and the relationships between the quantities.

Example 1: "A taxi fare costs R15 plus R8 per kilometre. If the total fare is R55, how many kilometres was the journey?" Let `k` represent the number of kilometres. The cost per kilometre is R8, so the cost for `k` kilometres is `8k`. The total fare is R15 (fixed cost) plus R8k (variable cost), which equals R

5

5. The equation is: `15 + 8k = 55` Example 2: "Sipho has twice as many marbles as Thandi. Together, they have 21 marbles. How many marbles does Thandi have?" Let `t` represent the number of marbles Thandi has. Sipho has twice as many marbles as Thandi, so Sipho has `2t` marbles. Together they have `t + 2t` marbles, which equals

2

1. The equation is: `t + 2t = 21` Guided Practice (With Solutions)

Question 1: Simplify the expression: `7x - 3 + 2x + 5` Solution: Identify like terms: `7x` and `2x` are like terms, and `-3` and `5` are like terms.