COSMO · STUDY + TEACHING DECK

Computer Science & AI

July 10, 2026

A question worth sharing

Distance and Cosine Similarity

Compare vectors by absolute distance, direction, and the assumptions built into each metric.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use L2 distance, cosine similarity, normalization 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

Vectors (1,1) and (2,2) point in the same direction. Are they identical under every comparison?

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.

L2 distance

straight-line Euclidean distance between vectors

cosine similarity

normalized dot product measuring directional alignment

normalization

rescaling a vector to unit length

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

The core idea

Keep the idea. Explain the conditions.

\operatorname{cos}(u,v)=\frac{u\cdot v}{\|u\|_2\|v\|_2},\qquad u,v\ne0

The idea

Compare vectors by absolute distance, direction, and the assumptions built into each metric.

What to remember

A similarity score is meaningful only after the geometry and scale assumptions are named.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Find the L2 distance between u=(1,2) and v=(4,6).

Predict each next step

Subtract coordinates to get v-u=(3,4).
Square the differences: 9 and 16.
Add to get 25.
Take the square root: the distance is 5.

Which step needs a reason, not just a calculation?

See the structure

The structure behind distance and cosine similarity

\operatorname{cos}(u,v)=\frac{u\cdot v}{\|u\|_2\|v\|_2},\qquad u,v\ne0

Cosine compares direction for nonzero vectors; Euclidean distance also responds to magnitude.

Read the representation

Compare vectors by absolute distance, direction, and the assumptions built into each metric.

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 distance and cosine similarity

Your shared task

Students rank the same vector collection with L2 distance and cosine similarity, then explain every disagreement.

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 similarity score is meaningful only after the geometry and scale assumptions are named.
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

Find the cosine similarity between a=(1,0) and b=(1,1).

Predict each next step

The dot product is 1.
The lengths are 1 and $\sqrt{2}$ .
Divide to obtain 1/ $\sqrt{2}$ .
The similarity is about 0.707.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

Cosine similarity ignores overall scale, while L2 distance responds to both scale and direction.

Build the argument

Multiplying a vector by a positive constant preserves its direction.
The normalized dot product therefore stays the same.
Its Euclidean distance from the original usually changes.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Give two vectors with cosine similarity 1 but L2 distance greater than 10.

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

Metric choice changes search results, clusters, recommendations, and which embedding appears strongest.

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

Vectors (1,1) and (2,2) point in the same direction. Are they identical under every comparison?

Question 2

Explain why this claim is true: Cosine similarity ignores overall scale, while L2 distance responds to both scale and direction.

Question 3

Give two vectors with cosine similarity 1 but L2 distance greater than 10.

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 similarity score is meaningful only after the geometry and scale assumptions are named.

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

A good question belongs to everyone in the room.