COSMO · STUDY + TEACHING DECK

Geometry

January 4, 2026

A question worth sharing

Transformations and Symmetry

Classify motions by what they preserve and use composition to reveal symmetry structure.

ThinkCompareExplain

Today’s route

A question becomes an explanation.

Try

Enter through a problem. A partial attempt counts.

Connect

Use isometry, symmetry, composition 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

How many rotational symmetries does a regular hexagon have?

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.

isometry

distance-preserving transformation

symmetry

transformation leaving an object unchanged

composition

performing transformations in sequence

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

The core idea

Keep the idea. Explain the conditions.

R_{\beta}\circ R_{\alpha}=\operatorname{Rot}_{2(\beta-\alpha)}

The idea

Classify motions by what they preserve and use composition to reveal symmetry structure.

What to remember

A symmetry is identified by the structure it preserves.

What information does this method keep?

Worked example

One route, one decision at a time

The problem

Reflect point (3,-2) across the y-axis.

Predict each next step

The y-coordinate stays fixed.
The x-coordinate changes sign.
The image is (-3,-2).
Distances to the y-axis are preserved.

Which step needs a reason, not just a calculation?

See the structure

The structure behind transformations and symmetry

R_{\beta}\circ R_{\alpha}=\operatorname{Rot}_{2(\beta-\alpha)}

Reflections in lines through one point, at angles alpha then beta, compose into a rotation.

Read the representation

Classify motions by what they preserve and use composition to reveal symmetry structure.

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 transformations and symmetry

Your shared task

Students compose transparent reflection and rotation cards and record which motions commute.

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 symmetry is identified by the structure it preserves.
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

Reflect first across x=0, then across x=2. Describe the combined transformation.

Predict each next step

Reflecting across x=0 sends x to -x.
Reflecting that across x=2 sends -x to 4+x.
The net map is x to x+4.
The composition is a translation 4 units to the right.

Which step needs a reason, not just a calculation?

Convince a skeptic

Turn a pattern into a proof.

The claim

The composition of two reflections across intersecting lines is a rotation through twice the angle between them.

Build the argument

Track a ray making angles with both mirrors.
Each reflection doubles its angular displacement from the mirror.
The total change is twice the angle between the lines.

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

Stretch together

Take the idea somewhere new.

Transfer challenge

Describe the full symmetry group of a non-square rectangle.

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

Symmetry organizes molecules, tilings, art, coding, and conservation laws.

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

How many rotational symmetries does a regular hexagon have?

Question 2

Explain why this claim is true: The composition of two reflections across intersecting lines is a rotation through twice the angle between them.

Question 3

Describe the full symmetry group of a non-square rectangle.

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 symmetry is identified by the structure it preserves.

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

A good question belongs to everyone in the room.