Introduction to Linear Programming Problems

Students retain linear programming concepts better when they move from abstract symbols to real contexts. Working in pairs, small groups, and whole class settings lets them experience how decision variables model problems like production planning, making the shift from theory to practice meaningful.

• Class 12
• Mathematics
• 4 activities
• 20 min–45 min

Learning Objectives

1. Formulate a mathematical model for a given real-world scenario involving resource allocation or production planning.
2. Identify and differentiate between the objective function and constraints in a linear programming problem.
3. Analyze the graphical representation of constraints to determine the feasible region for a linear programming problem.
4. Calculate the optimal value of the objective function at the corner points of the feasible region.

Think-Pair-Share

⏱ 30 min·Pairs

Pairs: Scenario Formulation

Pairs receive cards with real-life situations, such as a farmer maximising crop yield under irrigation limits. They define variables, write the objective function and three constraints. Pairs present one formulation to the class for feedback.

Prepare & details

Explain the purpose of linear programming in optimizing real-world situations.

Facilitation Tip: During Scenario Formulation, circulate and listen for students naming decision variables clearly, such as ‘number of shirts’ instead of vague terms like ‘items’.

Setup: Works in standard Indian classroom seating without moving furniture — students turn to the person beside or behind them for the pair phase. No rearrangement required. Suitable for fixed-bench government school classrooms and standard desk-and-chair CBSE and ICSE classrooms alike.

Materials: Printed or written TPS prompt card (one open-ended question per activity), Individual notebook or response slip for the think phase, Optional pair recording slip with 'We agree that...' and 'We disagree about...' boxes, Timer (mobile phone or board timer), Chalk or whiteboard space for capturing shared responses during the class share phase

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 45 min·Small Groups

Small Groups: Feasible Region Graphing

Groups plot two to three inequalities on graph paper, shade the feasible region, and mark vertices. They test sample objective functions at corners to find optimal points. Groups compare regions and discuss boundary effects.

Prepare & details

Differentiate between an objective function and a constraint in a linear programming problem.

Facilitation Tip: For Feasible Region Graphing, remind groups to label axes with units and scale them consistently to avoid distorted feasible regions.

Setup: Works in standard Indian classroom seating without moving furniture — students turn to the person beside or behind them for the pair phase. No rearrangement required. Suitable for fixed-bench government school classrooms and standard desk-and-chair CBSE and ICSE classrooms alike.

Materials: Printed or written TPS prompt card (one open-ended question per activity), Individual notebook or response slip for the think phase, Optional pair recording slip with 'We agree that...' and 'We disagree about...' boxes, Timer (mobile phone or board timer), Chalk or whiteboard space for capturing shared responses during the class share phase

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 25 min·Whole Class

Whole Class: Application Modelling

The class brainstorms Indian business scenarios like textile production. Teacher guides collective formulation on the board, graphing constraints together. Students vote on the optimal solution and justify choices.

Prepare & details

Construct a simple real-world problem that can be formulated as a linear programming problem.

Facilitation Tip: In Application Modelling, ask probing questions like ‘Why did you choose these constraints?’ to push students to explain their reasoning.

Setup: Works in standard Indian classroom seating without moving furniture — students turn to the person beside or behind them for the pair phase. No rearrangement required. Suitable for fixed-bench government school classrooms and standard desk-and-chair CBSE and ICSE classrooms alike.

Materials: Printed or written TPS prompt card (one open-ended question per activity), Individual notebook or response slip for the think phase, Optional pair recording slip with 'We agree that...' and 'We disagree about...' boxes, Timer (mobile phone or board timer), Chalk or whiteboard space for capturing shared responses during the class share phase

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Think-Pair-Share

⏱ 20 min·Individual

Individual: Corner Point Evaluation

Students receive a pre-graphed feasible region with an objective function. They list vertices, compute values, and identify the optimum. Share results in a quick gallery walk.

Prepare & details

Explain the purpose of linear programming in optimizing real-world situations.

Facilitation Tip: During Corner Point Evaluation, prompt students to double-check coordinates by substituting back into constraints to verify feasibility.

Setup: Works in standard Indian classroom seating without moving furniture — students turn to the person beside or behind them for the pair phase. No rearrangement required. Suitable for fixed-bench government school classrooms and standard desk-and-chair CBSE and ICSE classrooms alike.

Materials: Printed or written TPS prompt card (one open-ended question per activity), Individual notebook or response slip for the think phase, Optional pair recording slip with 'We agree that...' and 'We disagree about...' boxes, Timer (mobile phone or board timer), Chalk or whiteboard space for capturing shared responses during the class share phase

• Understand
• Apply
• Analyze
• Self-Awareness
• Relationship Skills

Teaching This Topic

Teach linear programming by starting with simple, relatable problems before moving to complex ones. Avoid rushing through graphing; students need time to internalise how inequalities define regions. Research shows that students grasp corner-point optimality more deeply when they physically plot constraints and test points themselves, rather than relying on teacher-drawn graphs.

What to Expect

Successful learning is visible when students can formulate scenarios into mathematical models, graph feasible regions accurately, and justify why corner points hold optimal solutions. They should connect graphical solutions to real-world decisions with confidence.

Watch Out for These Misconceptions

Common Misconception

During Scenario Formulation, watch for students assuming the objective must always be maximised.

What to Teach Instead

Have pairs create two versions of the same problem: one maximising profit and another minimising cost. Ask them to compare how the feasible region changes and where the optima shift, using the same constraints and objective values.

Common Misconception

During Feasible Region Graphing, watch for students treating all constraints as equalities.

What to Teach Instead

Provide graph paper and ask groups to shade the feasible region step-by-step, starting with one constraint at a time. Have them compare their shaded area with another group’s to spot differences in interpretation.

Common Misconception

During Corner Point Evaluation, watch for students believing non-linear objectives can be solved graphically.

What to Teach Instead

Give students a problem with a non-linear objective like Z = x² + y² and ask them to attempt graphing. When they realise the feasible region cannot be shaded uniformly, prompt them to identify why linear objectives are necessary for standard methods.

Assessment Ideas

Quick Check

After Scenario Formulation, collect each pair’s objective function and constraints. Check if they have correctly identified decision variables, written the objective in linear form, and expressed constraints as inequalities with proper units.

Discussion Prompt

During Application Modelling, ask students to explain why the optimal solution is always at a corner point, using a simple graphical example they plotted earlier. Listen for references to how constraints form a bounded polygon and how testing vertices ensures the best outcome.

Exit Ticket

After Corner Point Evaluation, collect students’ feasible region graphs and corner point coordinates. Assess whether they can list the points accurately and justify which one maximises the given objective function, such as Z = 2x + 3y.

Extensions & Scaffolding

• Challenge early finishers to modify the problem by adding a third product, forcing them to re-graph and re-evaluate the feasible region.
• Scaffolding for struggling students: Provide pre-drawn graphs with constraint lines already plotted, so they focus only on shading and identifying corner points.
• Deeper exploration: Ask students to research how linear programming is used in local industries like dairy farming or textile mills, and present one real-world case to the class.

Key Vocabulary

Objective Function	A mathematical expression, typically linear, representing the quantity to be maximized or minimized, such as profit or cost.
Constraints	Linear inequalities or equations that set limitations on the decision variables, reflecting resource availability or operational restrictions.
Decision Variables	The variables in a linear programming problem that represent the quantities to be determined, such as the number of units of a product to manufacture.
Feasible Region	The set of all possible solutions that satisfy all the constraints of a linear programming problem, represented graphically as a polygon.
Corner Points	The vertices of the feasible region, where the optimal solution to a linear programming problem is found.