At a glance. One lesson of about 60 minutes. It assumes systems of equations from lesson 4 and regions from lesson 5. Students find the vertices of a feasible region, evaluate an objective at each, choose the best, handle ties and minimization, and say what a whole-number limit does.
Calculator: GDC optional. The lesson is by hand. The optional section shows GDC users lists for corner values and warns that tracing a graph is only approximate.
What students will learn
Know
- Variables, constraints, feasible region, objective function.
- An optimum of a linear objective on a bounded region is reached at a vertex.
- A tie along an edge happens when the objective line is parallel to it.
Understand
- The objective is a family of parallel lines; sliding one shows why a vertex is enough.
- A vertex must be inside the whole region, not just on two boundary lines.
- A model is a simplification: whole-number limits and missing constraints can change the decision.
Do
- Model a decision as inequalities and an objective.
- Find the vertices and evaluate the objective at each.
- State the decision in context and comment on a limitation.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: vertices, objective values and optimum (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students record the best corner for several objectives and describe how the slope of the line decides it.
- Criterion C. Communicating: a clear table of corners and values, ending in a sentence in context (worked example 1, faded example).
- Criterion D. Applying mathematics in real-life contexts: the tailor's workshop and the bakery, and the whole-number limit in question 7.
The one thing to do with the figure
Slide before you show. Keep the corner values hidden. Ask students to slide the value of P up until the line just leaves the region and to say which corner it left through. Only then press the button to show the corner values and compare.
Then set p = 2 and q = 1 and ask what is different: the line leaves along a whole edge, so two corners tie. Ask which other choices of p and q give a tie (p = q gives a tie on x + y = 9).
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: where is the best choice? Middle, edge or corner? Vote. |
| 12 min | Figure: slide the line, then the corner values; key idea; the warning card. |
| 10 min | Worked examples 1 to 3; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C, return to the corner values. |
| 18 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | x = 5, y = 4; yes; b ≤ 18. |
|---|---|
| Hook | At a corner (or along a whole edge). For the tailor, P = 3x + 2y is largest at (5, 4): 5 totes and 4 purses, 23 thousand francs. |
| Worked examples | 340 at (4, 6); minimum 19 at (5, 1); tie of 36 along an edge. |
| Faded example | 60 at (10, 0). |
| 1. Choosing the best vertex | B. The corner values are 0, 24, 17 and 4. |
| 2. Largest value at the corners | 25. At (5, 5). |
| 3. Writing a boundary | 20. 60 ÷ 3. |
| 4. Finding a vertex and the optimum | 44. At (4, 6). |
| 5. Minimizing | 42. At (2, 6). |
| 6. A tie along an edge | 2. 36 = 24 + 6k. |
| 7. A whole-number limit | 5. For example 3 chairs and 2 tables. |
| 8. What is guaranteed | D. A, B and C are guaranteed; whole numbers are not. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Chose the corner that looks largest. | Work out P at every corner in a table. |
| 18 or 16 in question 2 | Took the corner with the largest y or x. | Compute all four values before choosing. |
| 12 or 63 in question 3 | Used the y-intercept, or added. | Put y = 0 into the boundary equation. |
| 62, 32 or 30 in question 4 | Used an infeasible corner, or an axis corner. | Check each corner in every inequality. |
| 17 or 47 in question 5 | Used an infeasible corner, or the larger of two. | Test (2, 1) in x + y ≥ 8. |
| 12 or 6 in question 6 | Used the wrong pair of corners, or forgot a term. | Write P at both corners with k in it. |
| 6 or 5.6 in question 7 | Rounded up, or gave the real optimum. | Test whole-number pairs against both inequalities. |
| A, B or C in question 8 | Doubted a guaranteed statement. | Draw a region and slide a line across it. |
Questions worth asking
- Why is a vertex enough? Listen for: the objective lines are parallel, and the last one to touch the region meets it at a corner.
- When is the best answer not a single point? Listen for: the objective is parallel to an edge, so the whole edge ties.
- What could make a real decision differ from the model? Listen for: a missing constraint, a whole-number limit, or a changing price.
- Why check that a vertex is feasible? Listen for: two boundary lines always meet, but the point may break another constraint.
Differentiation
A way in for anyone stuck
- Give a four-column table: corner, x, y, value of P. Fill it row by row.
- Provide a checklist: variables, constraints, graph, corners, objective, decision in context.
- Colour each inequality's side differently so the region is where all colours overlap.
Stretch
- Find the values of k for which P = 3x + ky is largest at (0, 9) in the first figure.
- Explain why an unbounded region may have no maximum but still have a minimum.
Students learning in English as an additional language
“Constraint” is a limit. “Feasible” means possible: it satisfies every limit. “Objective” is the goal, the quantity you want to be largest or smallest. “Vertex” is a corner. “Optimum” means the best value.
| Word | Plain meaning | Example |
|---|---|---|
| constraint | a limit written as an inequality | x + y ≤ 30 |
| feasible region | all the points that satisfy every constraint | the shaded polygon |
| objective function | the quantity to be made largest or smallest | profit = 8x + 5y |
| vertex | a corner of the region, where two boundary lines meet | where x = 2 meets y = 3 |
| optimum | the best value of the objective | the largest profit |
| maximize | make as large as possible | the most money |
Sentence starters for explaining aloud or in writing:
- The constraints are … so the vertices are …
- The objective is largest at … where P = …
- In context, this means …
Two things not to say
“The middle must be the best, because it balances everything.” For a linear objective the best is never strictly inside.
“Just read it off the graph.” A traced corner is approximate; solve the two equations.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- EvaluateCorners (0, 0), (5, 0), (2, 4). Find the maximum of P = 3x + 2y.
15 at (5, 0). - WriteWrite “at most 12 hours” as an inequality in h.
h ≤ 12.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- MinimizeCorners (1, 6) and (4, 2). Find the minimum of C = 2x + 3y.
14 at (4, 2). - Earlier work (lesson 5)Give the corner of y = x + 1 and y = 7 − x.
(3, 4).
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningThe objective P = x + y is largest on a whole edge of a region. What can you say about that edge?
It lies on a line x + y = constant, with gradient −1. - ExtendA model gives an optimum of (3.5, 2). Explain what you check before using it for chairs and tables.
Whole numbers: test the nearby whole-number pairs against every constraint.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider or button, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? Squared paper and a ruler are enough: draw the region and slide a ruler across it. The With a GDC section is optional.