COSMO · STUDY + TEACHING DECK

Proof

September 6, 2025

A question worth sharing

Proof and Induction

From convincing examples to an argument that covers every case.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use base case, inductive hypothesis, inductive step 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

The first five odd numbers sum to 25. What should the first n odd numbers sum to, and what would count as proof?

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

Draw the sums as growing square borders.
Separate the pattern from the reason it continues.
Name what a proof must cover that examples cannot.

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.

base case

the first verified link

inductive hypothesis

the case temporarily assumed

inductive step

the rule that forces the next case

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

The core idea

Keep the idea. Explain the conditions.

P(1),\quad P(n)\Longrightarrow P(n+1)\quad\Longrightarrow\quad P(n)\text{ for all }n\ge1

The idea

From convincing examples to an argument that covers every case.

What to remember

Induction is a chain of implication, not a pattern detector.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Prove 1 + 3 + ... + (2n - 1) = $n^{2}$ .

Predict each next step

Base: n = 1 gives 1 = $1^{2}$ .
Assume the sum through n is $n^{2}$ .
Add the next odd number 2n + 1.
$n^{2}$ + 2n + 1 = (n + 1)^2.

Which step needs a reason, not just a calculation?

See the structure

Induction needs a first domino and a reliable push

The base starts the chain. Each step must reach the next case.

Read the representation

The base case starts the chain. The inductive step proves that any fallen domino forces the next one.

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.

Evidence

Checks selected examples.
Can reveal a pattern.
Can still miss a later failure.

Proof

Names the mechanism.
Covers every allowed case.
Explains why failure is impossible.

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

Reasoning clinic

Find the step that needs repair.

Investigate these concerns

Names the mechanism.
Covers every allowed case.
Explains why failure is impossible.

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

Repair a broken induction

Your shared task

Groups receive a proof missing either the base case, the hypothesis, or the link to n + 1.

Explore

4 min - locate the gap

Compare

5 min - repair the logic

Defend

4 min - explain why the gap mattered

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

The hypothesis is permission to use exactly one assumed statement.
The step must reach n + 1, not merely restate n.
A correct algebra line is not enough unless its logical role is named.

Which observation explains the mechanism?

Try → explain → check

Your turn to build the route

The problem

Prove 1 + 2 + ... + n = n(n + 1)/2.

Predict each next step

Verify n = 1.
Assume the formula for n.
Add n + 1 and combine fractions.
Factor to obtain (n + 1)(n + 2)/2.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

A missing base case makes a true step useless.

Build the argument

The rule 'if one domino falls, the next falls' starts nowhere by itself.
A chain can be perfectly linked while every domino remains standing.
The base case supplies the first true statement.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Find the flaw in the false proof that all horses have the same color. At what overlap step does induction fail?

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

Induction verifies recursive programs, sums, divisibility claims, inequalities, and structures built one stage at a time.

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 may the inductive hypothesis be used for?

Question 2

Why must the step end at n + 1?

Question 3

Give one true pattern that examples alone do not prove.

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

Induction is a chain of implication, not a pattern detector.

Annotate each worksheet proof with BASE, ASSUME, STEP, and CONCLUDE.

A good question belongs to everyone in the room.