Grades 8-11
Discrete Geometry: Key Episodes
These are scripted scenarios based on what actually happened when this module was run at two schools in India (referred to as Ganga and Indus). They are written as teacher-ready dialogues so you can see what the conversations look like in practice, what went well, what went poorly, and what to do differently.
Student names are real first names from the implementation. The letter T refers to the teacher.
Episode 1: Posing the Question
Context
This is the very start of the module. The teacher has written the question on the board: "In a world with exactly 6 points, can every straight line be bisected?" Two schools had very different reactions. At Ganga, students were comfortable with the ambiguity and began exploring on their own. At Indus, students found the openness frustrating and needed more scaffolding.
The dialogue (Ganga)
T: Here is the question. "In a world with exactly 6 points, can every straight line be bisected?" What do you think?
Silence for about 30 seconds. Then:
Vivaan: This is meaningless.
T: What do you mean, meaningless?
Vivaan: You haven't told us what a straight line is. In a world with 6 points, what does "straight" even mean? There is nothing between the points.
T: Good. So what do you want to do about that?
Vivaan: We need to define it.
Karan: But how? In normal geometry a straight line is... well, it's straight. You can see it.
T: You can see it in Euclidean geometry because there are infinitely many points. Here there are only 6. So how will you decide what counts as a straight line?
Vivaan: Maybe the shortest path?
T: Shortest path between what?
Vivaan: Between two of the 6 points. If the shortest way to get from one point to another is through certain other points, that is the straight line.
T: Let us try that. But first we need a world to work in.
What the teacher did well
- Let the silence sit for 30 seconds. The temptation to fill the silence with hints is strong, but Vivaan's "this is meaningless" was more valuable than anything the teacher could have said.
- Responded to "meaningless" with curiosity rather than correction. Vivaan was not being dismissive — he was making a precise observation about the question being ill-formed.
- Did not provide the definition of straight line. Instead, guided Vivaan toward proposing one himself.
What the teacher could have done better
- The teacher moved to "let us try that" quite quickly after Vivaan proposed "shortest path." It would have been worth asking the class: "Does everyone agree that shortest path is a good definition of straight line? Can anyone think of a reason it might not work?" This would have opened space for alternative definitions and made the choice more visible.
What to do differently
At Indus, the same question produced confusion rather than productive inquiry. Students did not know where to start. The teacher had to provide the teleportation analogy and the graph representation much more quickly. The lesson: if students are not comfortable with open-ended mathematical exploration, scaffold more heavily at the start. Pose the question, then immediately introduce the teleportation analogy and draw the simple world, and then ask what "straight line" should mean. You can always remove scaffolding later; adding it after students have already become frustrated is harder.
Episode 2: Two Definitions of Bisection
Context
This episode happened at Ganga, partway through Session 1. Students had been working with the simple world (chain A-B-C-D-E-F) and had agreed that "straight line" means "shortest path." The teacher now asked about bisection. Two students — call them Student A and Student B — gave different answers about whether the line A-B could be bisected.
The dialogue
T: So, we have all these straight lines. The question asks whether every straight line can be bisected. What does bisect mean?
Student A: Cut it in half. Divide it into two equal parts.
T: Good. Can the line from A to B be bisected?
Student A: No. There is no point between A and B. The line has length 1 — you cannot split 1 into two equal whole numbers.
Student B: Yes it can. You just cut the edge in half.
Student A: You cannot cut the edge. There is nothing between A and B. You said so yourself — it is like teleportation.
Student B: But in normal geometry you can bisect any line. You just find the midpoint. The midpoint of A-B is halfway along the edge.
T: Interesting. You are both making sense, but you seem to mean different things by "bisect." Student A, what does bisect mean for you?
Student A: There has to be a point — one of the 6 points — that is equally far from both endpoints.
T: And Student B?
Student B: You split the line into two equal halves. The split can happen between two points.
Student C: Oh. They are using different definitions. Student A wants the midpoint to be an actual point in the world. Student B is okay with the midpoint being in between.
T: Exactly. Let us give these names. Student A's version: A-Bisection — bisecting at a point. Student B's version: B-Bisection — bisecting between points. Now, under A-Bisection, can the line A-B be bisected?
Multiple students: No.
T: Under B-Bisection?
Multiple students: Yes.
T: So whether this line "can be bisected" depends on which definition we choose. Same line, same world, different definitions, different answers.
What the teacher did well
- Immediately identified that the two students were using different definitions rather than making a factual error. This is the core skill of this module.
- Named the two definitions (A-Bisection, B-Bisection) to make them trackable. Without names, students would lose track of which definition they were using.
- Let Student C articulate the insight ("they are using different definitions") rather than stating it himself. Student C's observation was more powerful coming from a peer.
What the teacher could have done better
- The naming convention (A-Bisection, B-Bisection) happened to match the students' initials, which was convenient but potentially confusing. In a different class, use more descriptive names like "point-bisection" and "edge-bisection."
What to do differently
This episode is the emotional centre of Session 1. The moment students realise that the disagreement is not about facts but about definitions is the moment the module clicks. If it does not happen naturally (two students genuinely disagreeing), you can manufacture it by asking: "I think A-B can be bisected. Who agrees? Who disagrees?" Then ask both sides to explain. The disagreement will surface the definitional difference.
Episode 3: Circles in the Simple World
Context
This episode happened at Ganga, in Session 2. The class had moved on from bisection to circles. The teacher asked students to borrow the circle definition from Euclidean geometry: a circle is the set of all points at a fixed distance from a centre. Students were working with the simple world (chain A-B-C-D-E-F).
The dialogue
T: In Euclidean geometry, what is a circle?
Karan: A round shape. All points at the same distance from the centre.
T: Good. Let us use that. In our simple world, what is the circle centred at A with radius 1?
Karan: B. Just B.
T: Just one point?
Karan: Yes. B is the only point at distance 1 from A.
Vivaan: So AB is a circle.
T: What do you mean?
Vivaan: The set A, B is a circle. A is the centre and B is on the circle.
T: Is A on the circle?
Vivaan: A is the centre.
T: Right, but in Euclidean geometry, is the centre of a circle part of the circle?
Karan: No. The centre is in the middle. The circle is the points around it.
Vivaan: Then the circle is just B. A circle with one point.
T: Is that strange?
Multiple students: Yes.
Vivaan: In normal geometry circles have infinite points.
T: Why?
Vivaan: Because there are infinitely many directions you can go from the centre. Here there is only one direction from A — towards B.
T: Exactly. In Euclidean geometry, you can go in any direction from the centre. In our world, from A you can only go towards B. So the "circle" has only one point on it.
Later, working on the circle centred at C with radius 2:
Karan: The points at distance 2 from C are... A and E.
T: Good. So the circle centred at C, radius 2, is A and E.
Karan: But A is to the left and E is to the right. That does not look like a circle.
T: What does a circle look like?
Karan: Round. These points are on opposite sides.
T: In Euclidean geometry, a circle looks round because there are points at the same distance in every direction. Here, "round" does not apply. What matters is the definition: equidistant from the centre. A and E are both at distance 2 from C. They satisfy the definition.
Karan: So a circle can be just two points on a line?
T: In this world, yes.
What the teacher did well
- Pushed back on Vivaan's "AB is a circle" to clarify whether the centre is part of the circle. This small clarification prevented a persistent confusion.
- When Karan said "that does not look like a circle," the teacher did not dismiss the concern but asked what a circle looks like, then explained why the visual intuition does not transfer.
- Connected the strangeness (one-point circles) to the structural reason (limited directions from each point).
What the teacher could have done better
- Karan's remark "these points are on opposite sides" was an opportunity to discuss what "direction" means in a discrete world. In a chain, there are only two directions from any interior point (left and right). The teacher could have asked: "How many directions can you go from C?" This would have set up a useful comparison with the necklace world, where there are still only two directions but they wrap around.
- The teacher could have asked students to predict what would happen in other worlds before computing. "Do you think circles will be different in a necklace world?" would have made the transition to Session 2's necklace exploration more purposeful.
What to do differently
Some students found the idea of a one-point circle genuinely unsettling. If this happens, lean into it: "Is a single point really a circle? What properties of circles does it have? What properties does it lack?" This can lead to a productive discussion about which properties of circles are definitional (equidistant from centre) and which are accidental consequences of Euclidean geometry (being round, having infinitely many points).
Episode 4: "That Shouldn't Be a Triangle"
Context
This episode happened at Indus, in Session 3. Students had been working with the necklace world and had found NC-triangles like A, C, E. The teacher then pointed out that in the simple world (chain), the triple A, B, C also satisfies the triangle definition — three points with three shortest paths forming a closed figure. But all three points are on the same line.
The dialogue
T: Let us look at the simple world again. Take the points A, B, C. What is the shortest path from A to B?
Tarini: A-B. Length 1.
T: From B to C?
Tarini: B-C. Length 1.
T: From A to C?
Tarini: A-B-C. Length 2.
T: So we have three points, three straight lines between them — AB, BC, and AC. Is this a triangle?
Tarini: No.
T: Why not? We said a triangle is three points with three straight-line sides forming a closed shape. These three paths share endpoints and form a closed figure: A to B to C and back to A.
Devika: But they are all on the same line. A, B, C are all on the chain.
T: So?
Devika: A triangle has to have area. This has no area. It is flat.
T: In Euclidean geometry, what do we call three points that are all on the same line?
Tanya: Collinear.
T: And do collinear points form a triangle in Euclidean geometry?
Tanya: No. We usually say the points of a triangle must be non-collinear.
T: Exactly. So in Euclidean geometry, we add an extra condition to the definition: the three points must be non-collinear. We did not add that condition to our definition. Should we?
Devika: Yes. Otherwise everything is a triangle.
T: Not everything. But many things you would not want to call triangles would count. So let us distinguish: a C-triangle is a "triangle" where all three points lie on a single straight line. An NC-triangle is one where they do not. Do NC-triangles exist in the simple world?
Students think.
Uday: No. In the simple world, every pair of points has a unique shortest path, and all shortest paths go along the chain. So every triple of points is collinear.
T: Good. What about the necklace world?
Tarini: Yes. A, C, E is an NC-triangle. The shortest path from A to C does not go through E.
T: So the simple world has no NC-triangles and the necklace world does. Why?
Devika: Because the necklace has a cycle. You can go around. In the chain, you can only go back and forth.
What the teacher did well
- Let students articulate why collinear triangles feel wrong before providing the resolution. Devika's "it has no area" and Tanya's recall of the non-collinear condition both contributed to the insight.
- Connected the issue back to the Euclidean definition, showing that even in Euclidean geometry we exclude collinear triangles — students just never noticed because textbooks state it as part of the definition.
- Introduced the C-triangle / NC-triangle terminology to make the distinction precise and trackable.
What the teacher could have done better
- Devika's observation that "the necklace has a cycle" was the key structural insight — NC-triangles require the world to have cycles. The teacher accepted this without pushing further. It would have been worth asking: "Is that always true? Can you have an NC-triangle in any world that has a cycle? Can you find a world with a cycle but no NC-triangles?" This would have pushed students toward a deeper understanding of the relationship between world structure and geometric properties.
- Uday's reasoning ("every pair of points has a unique shortest path, and all shortest paths go along the chain") was a valid proof that the simple world has no NC-triangles. The teacher could have named this as a proof and written it on the board, connecting to the assumption digging skills from the Triangle Theory Building module.
What to do differently
The "that shouldn't be a triangle" moment is the discrete geometry version of the definitional debates in the Podgon Game. If students have done the Podgon module, reference it explicitly: "Remember how in Podgon, the same word could pick out different shapes depending on the definition? The same thing is happening here. Whether collinear points count as a triangle depends on your definition."
If students have not done the Podgon module, spend more time on this moment. It is one of the most valuable learning opportunities in the entire module: the realisation that a formal definition can include cases that violate your intuition, and that the response is to refine the definition, not to reject the formalism.