Mathematics · Class 8 · The Language of Algebra · Term 1

Solving Equations with Variables on Both Sides

Students will solve linear equations where the variable appears on both sides of the equality.

• CBSE Learning Outcomes
• CBSE: Linear Equations in One Variable - Class 8

About This Topic

Factorisation is the process of breaking down an algebraic expression into its simplest building blocks, or factors. It is essentially the reverse of expansion. In Class 8, students learn to factorise by taking out common factors, grouping terms, and using the identities they mastered in the previous topic. This skill is foundational for solving quadratic equations and simplifying complex rational expressions in higher grades.

In the Indian context, factorisation is often taught as a series of steps, but it is actually a form of mathematical 'detective work'. Students must look for clues, like the number of terms or the presence of perfect squares, to decide which method to use. This topic benefits from collaborative investigations where students can debate which strategy is most efficient for a given expression. Students grasp this concept faster through structured discussion and peer explanation.

Key Questions

1. Analyze the strategic steps required to solve equations with variables on both sides.
2. Compare different approaches to moving variables to one side of an equation.
3. Explain how to verify the solution of a linear equation.

Learning Objectives

• Calculate the value of the variable that satisfies linear equations with variables on both sides.
• Compare the efficiency of different algebraic steps for isolating the variable.
• Explain the rationale behind moving terms across the equals sign in an equation.
• Verify the solution of a linear equation by substituting the value back into the original equation.

Before You Start

Solving Linear Equations in One Variable (Single Side)

Why: Students must be proficient in isolating a variable when it appears on only one side of the equation before tackling variables on both sides.

Operations on Algebraic Expressions

Why: Understanding how to add, subtract, multiply, and divide terms with variables is essential for manipulating equations.

Key Vocabulary

Variable	A symbol, usually a letter like 'x' or 'y', that represents an unknown value in an equation.
Coefficient	The numerical factor that multiplies a variable in an algebraic term. For example, in 3x, the coefficient is 3.
Constant Term	A term in an algebraic expression that does not contain variables; its value remains fixed.
Equality	The state of being equal. In an equation, the expression on the left side of the equals sign has the same value as the expression on the right side.

Watch Out for These Misconceptions

Common Misconception

Students often stop factorising after taking out one common factor, even if the expression can be simplified further.

What to Teach Instead

Use the 'Irreducible Factor' checklist. In a station rotation, have students check if their final factors can be broken down more. Peer review of each other's 'final' answers helps them spot remaining common terms.

Common Misconception

Incorrectly applying identities, such as using (a-b)² for a² - b².

What to Teach Instead

During the 'Sorting Expressions' activity, have students highlight the signs. Discussing as a group why a minus sign between two squares requires a different identity than a trinomial helps clarify the distinction.

Active Learning Ideas

Stations Rotation: Factorisation Challenge

Set up four stations: Common Factors, Grouping, Identity 1 & 2, and Identity 3. Groups spend 8 minutes at each station solving problems specific to that technique, moving only when they have verified their answers.

45 min·Small Groups

Peer Teaching: The 'Reverse' Game

One student expands an expression (like (x+3)(x+2)) and gives the result to their partner. The partner must then factorise it back to the original form, explaining each step of their logic aloud.

25 min·Pairs

Inquiry Circle: Sorting Expressions

Give groups 20 different algebraic expressions on cards. They must sort them into categories based on the best factorisation method to use, justifying their choices to the rest of the class.

30 min·Small Groups

Real-World Connections

• Budgeting for events: A school organiser might set up an equation like 500 + 15x = 800 + 10x to compare the total cost of two different party packages, where 'x' is the number of guests. They need to solve for 'x' to find the number of guests for which the costs are equal.
• Comparing mobile plans: A consumer might compare two mobile service plans. Plan A costs ₹500 per month plus ₹0.20 per minute, while Plan B costs ₹800 per month plus ₹0.10 per minute. They can set up an equation like 500 + 0.20m = 800 + 0.10m to determine the number of minutes 'm' at which both plans cost the same.

Assessment Ideas

Quick Check

Present students with the equation 7x - 5 = 3x + 11. Ask them to write down the first step they would take to get all the 'x' terms on one side and explain why that step is logical.

Exit Ticket

Give students the equation 4y + 9 = 2y - 3. Ask them to solve for 'y' and then write one sentence explaining how they would check their answer to ensure it is correct.

Discussion Prompt

Pose the question: 'Is it always better to move the variable with the smaller coefficient first when solving equations with variables on both sides? Why or why not?' Facilitate a class discussion where students justify their reasoning.

Frequently Asked Questions

What is the first step in any factorisation problem?

The first step should always be to look for a common factor in all terms. If there is one, factor it out immediately. This often makes the remaining expression much easier to handle using other methods like identities.

When should I use the grouping method?

Grouping is typically used when an expression has four terms and there is no single factor common to all of them. By grouping them into two pairs, you can often find a common binomial factor.

How is factorisation useful in real life?

Factorisation is used in fields like cryptography, computer science, and engineering to simplify complex formulas. In daily life, it helps in mental math and understanding how different variables interact in a system.

How can active learning help students understand factorisation?

Active learning, like the 'Sorting Expressions' activity, encourages students to look for patterns rather than just following a recipe. When students have to justify why they chose a specific method, they are engaging in higher-order thinking. Peer teaching also helps, as explaining the 'reverse' process of expansion forces a student to understand the deep connection between the two, making factorisation feel less like a chore and more like a logic puzzle.