COSMO · STUDY + TEACHING DECK

Applied Math

May 24, 2026

A question worth sharing

Applied Optimization

Choices, constraints, objectives, and why 'best' depends on the model.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use decision variable, constraint, objective to name what you notice.

Convince

Build an argument a partner can check.

The goal is a method someone else can use.

The opener

Start with a question.

Opening problem

Which route is best: shortest distance, least time, lowest cost, or fewest risky turns?

Think · 2 min

Sketch, test a small case, or make a first guess.

Pair · 3 min

Compare methods. Find one thing you agree on and one question.

Be ready to share your route, even if it is unfinished.

Compare before checking

Put two attempts in conversation.

Explain

Tell a partner how you started. Let them finish before responding.

Question

Which step feels certain? Which step still needs evidence?

Build

Combine one useful idea from each attempt into a new route.

Can you explain your partner’s method in your own words?

Board conversation

Notice before naming.

Look for structure

List decisions the traveler controls.
Separate hard constraints from preferences.
Name one objective before comparing routes.

Make a conjecture

We think ___ because ___. A case that might break this is ___.

What changed? What stayed the same?

Useful language

Now the language has a job.

decision variable

a choice the model controls

constraint

a choice that is not allowed

objective

the quantity being minimized or maximized

Use one of these words to explain something you already noticed.

The core idea

Keep the idea. Explain the conditions.

\max(8x+5y)\quad\text{subject to }3x+2y\le12,\quad x,y\in\mathbb Z_{\ge0}

The idea

Choices, constraints, objectives, and why 'best' depends on the model.

What to remember

Optimization begins by defining best, not by calculating it.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Snack X costs $3 and has 8 protein units. Snack Y costs$ 2 and has 5. With $12, which nonnegative integer quantities maximize protein?

Predict each next step

Let x and y count snacks X and Y.
The budget gives 3x + 2y ≤ 12.
Maximize the objective 8x + 5y.
For x = 0, 1, 2, 3, 4, the best totals are 30, 28, 31, 29, 32. Four X snacks win.

Which step needs a reason, not just a calculation?

See the structure

The feasible region contains every allowed choice

Illustrative feasible region: x ≥ 0, y ≥ 0, x + 2y ≤ 8.

Read the representation

Optimization searches only the choices that satisfy every constraint, then compares the objective among them.

Look together

Point to the part that explains the relationship. Change one input and predict what changes.

What can you see here that was harder to see in words?

Two approaches

Compare the reasoning.

Local improvement

Makes the next step better.
Can be fast and intuitive.
May become trapped away from the global best.

Global optimum

Beats every feasible alternative.
Needs a proof or exhaustive structure.
Can change when the objective changes.

Which route can you justify? What would you test first?

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

Beats every feasible alternative.
Needs a proof or exhaustive structure.
Can change when the objective changes.

Repair together

Name the missing condition. Try a small counterexample. Rewrite one step so it is justified.

A useful correction explains why the original step fails.

Work together

Optimize a school event plan

Your shared task

Teams allocate a fixed budget across food, materials, and transport, then defend both constraints and objective.

Explore

5 min - define variables

Compare

6 min - choose an objective

Defend

5 min - compare two models

Everyone should be able to explain the group’s final claim.

Share → connect

What did the room discover?

Bring your evidence

Share a conjecture, a useful example, or a route that changed.

Connections to investigate

A model cannot optimize a value it never measured.
Changing the objective can change the winner without changing the data.
Human priorities enter when objectives and constraints are chosen.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Why can nearest-next-stop fail to produce the shortest full route?

Predict each next step

The closest step is only locally best.
It may leave distant stops for an expensive ending.
A counterexample disproves the greedy rule.
A successful greedy method needs a structural proof.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

An optimum is only as meaningful as its objective and constraints.

Build the argument

The same feasible choices can be scored by cost, time, or fairness.
Different scores rank those choices differently.
Therefore 'best' is not defined until the model states what best means.

What does each step rely on? Where are the conditions used?

Stretch together

Take the idea somewhere new.

Transfer challenge

Create two objective functions for the same schedule that choose different winners.

Choose a route

Name the objects, the conditions, and a relationship that might help. Start with one test case.

What transfers from the earlier example? What is different?

Peer review

Check the argument together.

Restate

Explain the other person’s route before evaluating it.

Test

Try a small case or boundary case. Does every step still make sense?

Revise

Fix an assumption, add a reason, or find a second method.

Where did your thinking change?

Beyond this problem

Where might this idea help?

A connection

Routing, scheduling, resource allocation, engineering, public policy, and machine learning all optimize a chosen quantity under limits.

Discuss the model

What objects or quantities are represented?
Which relationship from today is preserved?
What assumption would you check before trusting it?

Invent one more situation with the same structure.

Exit ticket

Leave with an explanation.

Question 1

What is the difference between a constraint and an objective?

Question 2

Why can a greedy choice fail?

Question 3

Name one important factor a simple route model might omit.

Choose one. Write an answer with a reason, then compare with a partner.

Reflect together

Make your thinking visible.

I used to think…

Name a claim or approach that changed.

Now I can explain…

Give the reason, not only the result.

I still wonder…

Write a question worth bringing to the next circle.

Share one sentence. Ask one follow-up question.

The idea to keep

Optimization begins by defining best, not by calculating it.

On the worksheet, underline choices, box constraints, and circle the objective before solving.

A good question belongs to everyone in the room.