Linear Programming Consolidation

Lesson 10 of 10

In this lesson:

• Apply the five-stage method to two assessment problems
• Self-diagnose which stage is your weakest link
• Build a one-sentence targeted revision plan

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Apply all five stages to a mixed-context assessment problem
2. Self-diagnose which stage of the method produced your errors
3. Prioritise one revision target with evidence

Hook: Ten Lessons, One Method

At which stage is your weakest link?

LP-01 through LP-09 taught one method in five stages:

1. Read → 2. Form → 3. Graph → 4. Corners → 5. Evaluate

Plus the LP-08 realism layer: integer check after stage 5.

Five-Stage Recall: Skills and Errors

• Stage 1 Read: identify variables, constraints, direction
• Stage 2 Form: write inequalities — include non-negativity
• Stage 3 Graph: shade feasible region using test point
• Stage 4 Corners: inspect axes; calculate intersections
• Stage 5 Evaluate: table every corner; state direction

Self-Rate Each Stage Before the Assessment

Rate your confidence 1–5 for each stage:

• Stage 1 Reading: ___
• Stage 2 Forming: ___
• Stage 3 Graphing: ___
• Stage 4 Corners: ___
• Stage 5 Evaluating: ___

Check-In: Match Error to Stage

Which stage does each error belong to?

• "Shaded the wrong side of the boundary line" → Stage ___
• "Missed the non-negativity constraint" → Stage ___
• "Reported the largest value in a minimization problem" → Stage ___

Name the stage and the revision lesson for each.

Assessment Format: Show All Five Stages

For each problem, show explicitly:

1. Direction statement ("I am maximizing/minimizing...")
2. System of inequalities
3. Sketch of feasible region with corners labelled
4. Evaluation table with all corners
5. Interpreted sentence with units

Missing any stage is a marks penalty — each stage carries independent credit.

Problem 1: Farmer Maximizes Profit

A farmer plants maize ( ha) and beans ( ha).

• (labour days)
• (fertiliser bags)
• , ; variables continuous

Objective: maximize (thousands UGX)

State direction. Work all 5 stages.

Problem 1: Stages 1–3 and Corners

Corners: , , ,

Interior corner : solve and .

Problem 1: Stage 5 and Answer

at each corner:

• (0, 0):
• (4, 0):
• (3, 2): 900 ← maximum
• (0, 4):

"Plant 3 ha maize and 2 ha beans for a maximum profit of 900,000 UGX."

Problem 2: School Desk-and-Chair Order

A school orders desks () and chairs ().

• (at least 12 items)
• (supplier limit)
• (supplier limit)
• , ; variables integer

Minimize (thousands UGX)

Problem 2: All Five Stages

Corners: , ,

• : 460
• :
• :

Problem 2: Integer Check and Interpretation

Algebraic minimum: at .

Diagnostic question: Are x and y integer-required? Yes.

Is already integer? Yes — no search needed.

"Order 2 desks and 10 chairs for a minimum cost of 460,000 UGX."

Check-In: Identify the Stage Error

Problem 1 error: A student reports the optimum as with .

Which stage produced this error? What revision lesson applies?

Map Every Error to a Revision Stage

• Confused objective with constraint → Stage 1
• Missed non-negativity → Stage 2
• Shaded wrong side → Stage 3
• Counted corner outside region → Stage 4
• Incomplete table or wrong direction → Stage 5
• Fractional answer for integer variable → Stage 5, LP-08

Worked: Classify a Sample Error Set

Four errors; two stages emerge as targets:

• Missing y ≥ 0 → Stage 2, LP-02
• Interior corner not verified → Stage 4, LP-04
• Evaluation table incomplete → Stage 5, LP-05
• No units in answer → Stage 5, LP-05

Revise LP-04 then LP-05.

Write Your Self-Targeted Revision Plan

Write one sentence:

"My weakest stage is ___, and I'll revisit lesson ___ to strengthen it."

Compare with your pre-assessment self-rating.

Did your errors confirm your prediction, or surprise you?

Check-In: Compare Self-Rating with Evidence

Look at your pre-assessment self-rating from Slide 5.

Did your errors confirm your lowest-rated stage?

Write: "My prediction was ___ (accurate / inaccurate) because ___."

Variant: What If the Region Were Unbounded?

Problem 1 with no upper bounds — would a maximum still exist?

Apply the LP-07 diagnostic rule:

1. Is the region unbounded? Yes (remove the ≤ constraints)
2. Does grow in the unbounded direction? Yes

No maximum would exist — state this explicitly.

Three Common Habits Worth Unlearning

Watch out:

• Equal-time review — re-practising strong stages wastes time; target the weakest
• Stop at a number — the answer is always the interpreted sentence, not a value
• Skip the diagnostic question — ask "integer?" before writing every LP answer

Check-In: Write Your Revision Plan

Write your one-sentence revision plan:

"My weakest stage is ___, and I'll revisit lesson ___ to strengthen it."

Use evidence from the assessment — not from your feeling.

The Five-Stage Method Never Changed

Lessons 1–9 applied the same five stages to:

LP-01: recognition · LP-02: forming · LP-03: graphing · LP-04: corners · LP-05: evaluating · LP-06: minimize · LP-07: unbounded · LP-08: integers · LP-09: design

Nine contexts. One method. Zero new stages after LP-05.

Key Takeaway: One Stage, One Target

✓ Five stages — same method from LP-05 onward

✓ Error classification maps every mistake to a stage

✓ Targeted revision: one stage, one lesson, one plan

✓ The interpreted sentence with units is always the answer

The Five-Stage Method Works Everywhere

LP will appear in:

• The term assessment
• Higher-level optimization problems in S5–S6
• Real-world applications: agriculture, transport, production planning

The five-stage method works on all of them.