Mathematics · Class 12

Active learning ideas

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.

• CBSE Learning Outcomes
• NCERT: Linear Programming - Class 12

• 20–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• 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.

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’.

What to look for

Present students with a short word problem, for example: 'A tailor makes shirts and trousers. Each shirt requires 2 hours of cutting and 3 hours of stitching. Each trouser requires 3 hours of cutting and 2 hours of stitching. The tailor has 120 hours of cutting time and 110 hours of stitching time available. Formulate the objective function and constraints for maximizing profit, assuming a profit of Rs. 50 per shirt and Rs. 70 per trouser.'

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

Activity 02

• 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.

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.

What to look for

Ask students to explain in their own words why the optimal solution for a linear programming problem is always found at a corner point of the feasible region. Encourage them to use a simple graphical example to support their explanation.

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

Activity 03

• 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.

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.

What to look for

Provide students with a set of linear inequalities representing constraints. Ask them to: 1. Graph the inequalities to find the feasible region. 2. List the coordinates of the corner points of the feasible region. 3. Identify which corner point would maximize an objective function like Z = 2x + 3y.

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

Activity 04

• 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.

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.

What to look for

Present students with a short word problem, for example: 'A tailor makes shirts and trousers. Each shirt requires 2 hours of cutting and 3 hours of stitching. Each trouser requires 3 hours of cutting and 2 hours of stitching. The tailor has 120 hours of cutting time and 110 hours of stitching time available. Formulate the objective function and constraints for maximizing profit, assuming a profit of Rs. 50 per shirt and Rs. 70 per trouser.'

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

A few notes on teaching this unit

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.

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.

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Watch Out for These Misconceptions

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

  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.

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

  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.

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

  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.

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Methods used in this brief

• Think-Pair-Share