COSMO · STUDY + TEACHING DECK

Applied Math

July 26, 2026

A question worth sharing

Mathematical Modeling

Turn a situation into variables, assumptions, relationships, and a testable conclusion.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use variable, assumption, validation 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

A taxi charges $3 plus$ 2 per mile. Write and interpret a cost model.

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

Try a small case and record every step.
Separate what changes from what stays fixed.
State the pattern in a sentence before naming the theorem.

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.

variable

quantity allowed to change

assumption

simplification declared explicitly

validation

comparison between model and evidence

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

The core idea

Keep the idea. Explain the conditions.

V(t)=100-4t,\qquad 0\le t\le25

The idea

Turn a situation into variables, assumptions, relationships, and a testable conclusion.

What to remember

A model is an argument made from variables and assumptions, not a claim that context disappeared.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Model water draining at a constant 4 liters per minute from 100 liters.

Predict each next step

Let t be minutes and V be liters.
Start at V(0)=100.
Subtract 4 liters per minute.
V(t)=100-4t for 0≤t≤25.

Which step needs a reason, not just a calculation?

See the structure

The structure behind mathematical modeling

V(t)=100-4t,\qquad 0\le t\le25

Constant drainage is credible while water remains; the model includes its domain.

Read the representation

Turn a situation into variables, assumptions, relationships, and a testable conclusion.

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.

Structural method

Name the objects and conditions.
Use a reusable relationship.
Explain why the method must work.

Brute force

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

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

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

May help with a small test case.
Can hide the general reason.
Becomes unreliable when the scale grows.

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

Investigate mathematical modeling

Your shared task

Groups model the same situation with different assumptions and compare what each model can answer.

Explore

4 min - test cases

Compare

5 min - form a rule

Defend

4 min - defend the rule

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 is an argument made from variables and assumptions, not a claim that context disappeared.
A complete solution names the condition that makes the method valid.
A counterexample is useful when it exposes a missing condition.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Compare linear and exponential growth for an account increasing by $100 yearly versus 5 percent yearly.

Predict each next step

Linear model: $A_{n}$ = $A_{0}$ +100n.
Exponential model: $B_{n}$ = $B_{0}$ (1.05)^n.
The linear change is constant.
The exponential change scales with current size.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A useful model states its assumptions and the range where they are credible.

Build the argument

Every model omits details.
An assumption explains what was simplified.
A domain prevents extrapolation beyond the situation the relationships describe.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Build a model for cooling coffee and name one measurement that would test it.

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

Modeling supports planning, science, policy analysis, engineering, and everyday estimation.

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

A taxi charges $3 plus$ 2 per mile. Write and interpret a cost model.

Question 2

Explain why this claim is true: A useful model states its assumptions and the range where they are credible.

Question 3

Build a model for cooling coffee and name one measurement that would test it.

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

A model is an argument made from variables and assumptions, not a claim that context disappeared.

Use the matching practice sets to move from direct examples to proof and transfer.

A good question belongs to everyone in the room.