Algebra: Euler’s dice
View Sequence overviewA polyhedron is a 3D object with flat polygonal faces that is fully enclosed.
Prisms and pyramids are two types of polyhedra, each with their own defining properties.
Algorithms can be written as a sequence of steps and decisions, and tested against examples.
Whole class
Euler’s dice Slides
Each group
Sticky notes
Task
Display Slide 3 of Euler’s dice Slides which shows a collection of dice. If possible, have physical examples available to pass around.
Ask: What do you notice about these dice? What do they all have in common?
If students comment on numbers, colours or size, acknowledge those observations and ask: What about the shape itself? What do they all have in common geometrically?
Students should notice that the dice have flat faces, straight edges and no curves.
Tell students:
A game designer is creating a board game called Fate and Fortune, a game built entirely around luck, superstition, and the roll of a die. They have one problem: they need a set of custom dice that use a set of numbers considered lucky or unlucky across different cultures. They’ve put out a call to designers.
But before anyone starts building, there’s a mathematical question that needs answering: can you actually make a die with any number of faces? Are some numbers impossible?
Ask students to write down their initial conjecture.
Display Slide 4 of Euler’s dice Slides, which shows two sorted groups of shapes: polyhedra on one side, non-polyhedra on the other. Point out that the singular is “polyhedron” while the plural is “polyhedra”.
Ask: What do you notice? What do all the shapes on this side have in common? What about the other side?
Give students a minute to look in silence, then ask them to write down their own definition independently. Tell students: Write a definition that describes every shape on this side, and excludes every shape on the other side. Don’t worry about getting it perfect.
Think: Students write their definitions individually.
Pair: Students share their definition with a partner and discuss. Are they saying the same thing or something different? Together, they agree on one definition to bring to the class.
Share: Invite pairs to share their definition. Write each definition on the board exactly as the student said it. Avoid paraphrasing or tidying the language.
Keep collecting until you have three or four genuinely different definitions on the board. Ask: Look at all of these. Which definition handles every shape we’ve seen so far without letting anything slip through to the wrong side?
Give students a moment to revise their own definition individually considering what they have heard, then work toward a class consensus: Can we agree on one definition the whole class is happy with? This will be our working definition but we haven’t tested it yet.
It doesn’t matter (in fact, it is useful) if the definition agreed upon is not entirely accurate or correct. Gaps and edge cases to be considered later give the class something to return to and refine.
Students may define polyhedra as:
- shapes that look like a box.
- shapes with flat sides.
- shapes with no curves.
- 3D objects made of flat pieces stuck together.
- objects with flat faces and straight edges.
The following example shows how a teacher might respond when a student offers a definition.
When a student says Polyhedra are shapes with flat sides:
| Instead of this... | Try this... | Why? |
| Tidying the language, paraphrasing, or confirming the definition as correct. | Write the definition on the board exactly as said. Ask: Does anyone have something different? | The class, not the teacher, should evaluate the definition. Paraphrasing puts the teacher’s words on the board instead of the student’s. |
| Telling students which definition is best. | Ask: Is there anything you would borrow from one definition to improve another? | Students evaluating each other’s definitions is more powerful than the teacher deciding. It also means the final definition genuinely belongs to the class. |
| Accepting the first reasonable definition and moving on. | Ask: Can anyone find a shape that our definition would accidentally include on the wrong side? | Every time a test case slips through or gets wrongly excluded, students have found a gap that needs closing. This cycle of testing and patching is how a rough description becomes a precise definition. |
Building definitions and Philosophy for Children (P4C)
The definition-building activity in this task draws on a practice common in Philosophy for Children (P4C), a structured approach to classroom inquiry where students construct knowledge together through dialogue and reasoning rather than receiving it from a teacher.
In P4C, forming and testing definitions is a core activity. Rather than being told what a polyhedron is, students are presented with examples, invited to construct a shared definition, and then challenged to refine it when tricky cases arise. The generated definition belongs to the class, which is precisely why students are invested in defending and revising it.
A few things that make this work well in practice:
- Treat all contributions as worth examining, even incorrect ones. A wrong definition is more useful than no definition because it gives the class something to test.
- Resist the urge to correct students immediately. Let the test cases do that work instead.
- When students disagree, don’t rush to resolve it. The disagreement is where the learning is.
- The accepted mathematical definition at the end should feel like a confirmation of what the class already worked out, not a correction of it.
For teachers who want to explore P4C further, Philip Cam’s Thinking Together: Philosophical Inquiry for the Classroom and 20 Thinking Tools are both practical and accessible starting points.
The definition-building activity in this task draws on a practice common in Philosophy for Children (P4C), a structured approach to classroom inquiry where students construct knowledge together through dialogue and reasoning rather than receiving it from a teacher.
In P4C, forming and testing definitions is a core activity. Rather than being told what a polyhedron is, students are presented with examples, invited to construct a shared definition, and then challenged to refine it when tricky cases arise. The generated definition belongs to the class, which is precisely why students are invested in defending and revising it.
A few things that make this work well in practice:
- Treat all contributions as worth examining, even incorrect ones. A wrong definition is more useful than no definition because it gives the class something to test.
- Resist the urge to correct students immediately. Let the test cases do that work instead.
- When students disagree, don’t rush to resolve it. The disagreement is where the learning is.
- The accepted mathematical definition at the end should feel like a confirmation of what the class already worked out, not a correction of it.
For teachers who want to explore P4C further, Philip Cam’s Thinking Together: Philosophical Inquiry for the Classroom and 20 Thinking Tools are both practical and accessible starting points.
Think-pair-share
Think-pair-share gives every student time to formulate their own idea before being asked to share it publicly. This is particularly valuable in definition-building activities, where students who are less confident may otherwise wait to hear what others say before committing to an answer. By the time pairs share with the class, every student has already articulated and refined their thinking at least once.
For managing student responses: write definitions on the board exactly as students say them, without tidying the language. Resist the urge to paraphrase or correct. The class will do that work through the testing activity that follows.
Think-pair-share gives every student time to formulate their own idea before being asked to share it publicly. This is particularly valuable in definition-building activities, where students who are less confident may otherwise wait to hear what others say before committing to an answer. By the time pairs share with the class, every student has already articulated and refined their thinking at least once.
For managing student responses: write definitions on the board exactly as students say them, without tidying the language. Resist the urge to paraphrase or correct. The class will do that work through the testing activity that follows.
Display Slides 5-11 of Euler’s dice Slides. Work through each object one at a time. For each object, ask students to independently classify it as a polyhedron or non-polyhedron using the class definition, then reveal the accepted classification. Where students disagree with the accepted answer, pause and ask: does our definition need patching, or did we apply it incorrectly? Use any mismatches to refine the class definition before moving on.
Work toward a class consensus: Can we agree on one definition that handles every object we just tested?
Show Slide 12 of Euler’s dice Slides which contains the accepted mathematical definition for a polyhedron: A polyhedron is a 3D object enclosed entirely by flat faces, where every face is a polygon and every vertex is a point where three or more edges meet.
Pause to name the key features of a polyhedron explicitly:
- Face: any flat surface.
- Polygon: a flat, closed shape with straight edges. A circle is not a polygon because its boundary is curved.
- Edge: where two faces meet.
- Vertex (plural: vertices): a point where edges meet.
See the embedded professional learning Mathematical and everyday language below for further guidance on introducing these mathematical terms.
Compare the accepted mathematical definition with the class definition. If they are close, celebrate it! If there is a gap, discuss:
Our definition said ____. The mathematical definition says that too, but also requires ____. Why might that extra condition matter?
Display Slide 13, containing an image of an object. Ask students: Is this a polyhedron? How do you know?, and use the animation to build a flowchart one question at time. Work through each object in turn:
- Traffic cone: Ask the class whether the traffic cone is a polyhedron. Reveal the first question (Is every face a polygon?). The cone fails here: its base is a circle, not a polygon. It is not a polyhedron.
- Open box: The flowchart now has one question visible. Ask students to apply it. The open box passes as all its faces are flat polygons. Reveal the second question (Is it fully enclosed?). The open box fails here: the missing lid means it is not fully enclosed. It is not a polyhedron.
In this sequence, the open box is treated as a mathematical surface (a collection of flat faces with no thickness). In reality, cardboard has physical thickness and the box is a composite 3D object made from rectangular prisms. The modelling decision here is to ignore that thickness and treat each face as a 2D surface. So the polyhedron is not the cardboard itself, but the idealised mathematical surface the cardboard approximates.
- Truncated pyramid: Both questions are now visible. It passes both tests and is classified as a polyhedron.
The first two examples are chosen deliberately (the cone fails at question 1, the open box passes question 1 but fails question 2) so students see both exit points used before designing their own algorithm later this task. The key ideas to surface are:
- Questions must have yes/no answers.
- Order matters.
- Passing one question does not guarantee the final classification.
Mathematical and everyday language
It is important to use correct mathematical language when describing a polyhedron.
Students may already be using everyday language like “side” and “corner”. These are useful starting points. However, sides describes the faces or edges of 2D shapes, while faces and edges are the precise terms for 3D objects. Corner and vertex are both used across 2D and 3D contexts, but vertex is the mathematical term. The goal here is to introduce the mathematical terms alongside the everyday ones so that students have both available. A helpful technique is to use both words together at first (e.g. “A cube has 8 corners, or vertices.”) before gradually shifting to the mathematical term on its own.
One more term worth avoiding: solid. Referring to 3D objects as solids can confuse students, particularly when the manipulatives they are handling are hollow. 3D object is clearer and worth using consistently throughout the sequence.
It is important to use correct mathematical language when describing a polyhedron.
Students may already be using everyday language like “side” and “corner”. These are useful starting points. However, sides describes the faces or edges of 2D shapes, while faces and edges are the precise terms for 3D objects. Corner and vertex are both used across 2D and 3D contexts, but vertex is the mathematical term. The goal here is to introduce the mathematical terms alongside the everyday ones so that students have both available. A helpful technique is to use both words together at first (e.g. “A cube has 8 corners, or vertices.”) before gradually shifting to the mathematical term on its own.
One more term worth avoiding: solid. Referring to 3D objects as solids can confuse students, particularly when the manipulatives they are handling are hollow. 3D object is clearer and worth using consistently throughout the sequence.
The definition of a polyhedron
The widely accepted definition of a polyhedron is: A polyhedron is a 3D object enclosed entirely by flat polygonal faces, where every edge borders exactly two faces.
Each part of this definition is doing specific work:
- “Flat polygonal faces” rules out any shape with a curved surface (cylinders, cones, spheres) and any shape whose faces have curved boundaries (such as a circle).
- “Enclosed entirely” rules out open surfaces. A cube with one face missing, such as an open box, is not a polyhedron. It is a surface with a boundary, not a closed object.
- “Every edge borders exactly two faces” is the rigorous version of “fully enclosed”. If an edge only borders one face, there is a gap. If it borders three or more, faces are intersecting in a way that violates the standard definition. This condition guarantees the surface closes up cleanly with no gaps or overlaps.
The student-facing flowchart uses “Is every face a polygon?” and “Is it fully enclosed?” as simplified versions of the first and third conditions. These work well for the objects in this sequence, though the “every edge borders exactly two faces” formulation is more precise and handles edge cases the simpler version might miss.
The widely accepted definition of a polyhedron is: A polyhedron is a 3D object enclosed entirely by flat polygonal faces, where every edge borders exactly two faces.
Each part of this definition is doing specific work:
- “Flat polygonal faces” rules out any shape with a curved surface (cylinders, cones, spheres) and any shape whose faces have curved boundaries (such as a circle).
- “Enclosed entirely” rules out open surfaces. A cube with one face missing, such as an open box, is not a polyhedron. It is a surface with a boundary, not a closed object.
- “Every edge borders exactly two faces” is the rigorous version of “fully enclosed”. If an edge only borders one face, there is a gap. If it borders three or more, faces are intersecting in a way that violates the standard definition. This condition guarantees the surface closes up cleanly with no gaps or overlaps.
The student-facing flowchart uses “Is every face a polygon?” and “Is it fully enclosed?” as simplified versions of the first and third conditions. These work well for the objects in this sequence, though the “every edge borders exactly two faces” formulation is more precise and handles edge cases the simpler version might miss.
Display Slide 14 of Euler’s dice Slides, which shows a set of polyhedra sorted into three groups: prisms, pyramids, and other polyhedra.
Tell students: Now it’s your turn. In your groups, design a flowchart that classifies any 3D object by first deciding whether it is a polyhedron or not, and if it is, whether it is a prism, a pyramid, or another polyhedron.
Give each group a stack of sticky notes. They write one yes/no question per sticky note and arrange them into a flowchart on a desk or on a large sheet of paper. Emphasise that sticky notes can be moved: if a test case breaks their algorithm, they can reorder or replace questions rather than starting over.
| If a group... | Try asking... |
| ...incorrectly classifies a test object | Walk me through your flowchart for that object step by step. Is the question itself the problem, or is it in the wrong place? |
| ...uses vague language in a question, for example “does it have two of the same faces?” | Your logic is right. But what does “the same” mean exactly? Could two faces be the same shape but not parallel? What if there were more than two of the same faces? |
| ...has a question early that assumes the object is already a polyhedron | What would happen if I put a cone into your flowchart at the very first question? Would it ever get caught? |
| ...has redundant questions | Your flowchart works. Now can you make it more efficient? Is there a question you could remove without breaking it? |
| ...wants to start over completely | Which one question caused the problem? Try moving or rewording just that sticky note before you start again. |
| ...finishes early | Go to polyhedra.net and find a shape you haven’t seen before. Run it through your flowchart. Does it still work? |
| ...has the same questions as another group, but in a different order | These two flowcharts have exactly the same questions but arranged differently. Does that mean one of them is wrong? Try running the same object through both. |
Once groups have a first draft of their flowchart, reveal the test objects one at a time using Slides 15-24. Before revealing each object’s classification, groups run it through their flowchart and record what their flowchart says. Then reveal the accepted answer. If a group’s algorithm gave the wrong classification, give them time to modify their flowchart before moving to the next object.
After all objects have been tested, ask: Did you have to move or rewrite any sticky notes? Which object caused the most trouble? Why?
Prisms and pyramids: formal definitions
The definitions for “prism” and “pyramid” that students arrive at through the lesson activity will likely be close to, but not identical to, the formal mathematical definitions. Here they are for reference.
A prism is a polyhedron with two congruent, parallel faces (the bases), with all remaining faces being parallelograms. When those remaining faces are rectangles, it is called a “right prism”. When they are non-rectangular parallelograms, it is called an “oblique prism”. In most school contexts, “prism” refers to a right prism unless stated otherwise.
Some examples of prisms are shown above.
A pyramid is a polyhedron with a polygonal base and triangular faces that all meet at a single point called the apex. The pyramid is named according to the shape of its base: a square pyramid has a square base, a triangular pyramid has a triangular base, and so on.
Some examples of pyramids are shown below.
Note that a triangular pyramid is the same thing as a tetrahedron. This often surprises students, and it’s a good example of how the same object can have more than one valid name depending on how you’re classifying it.
The definitions for “prism” and “pyramid” that students arrive at through the lesson activity will likely be close to, but not identical to, the formal mathematical definitions. Here they are for reference.
A prism is a polyhedron with two congruent, parallel faces (the bases), with all remaining faces being parallelograms. When those remaining faces are rectangles, it is called a “right prism”. When they are non-rectangular parallelograms, it is called an “oblique prism”. In most school contexts, “prism” refers to a right prism unless stated otherwise.
Some examples of prisms are shown above.
A pyramid is a polyhedron with a polygonal base and triangular faces that all meet at a single point called the apex. The pyramid is named according to the shape of its base: a square pyramid has a square base, a triangular pyramid has a triangular base, and so on.
Some examples of pyramids are shown below.
Note that a triangular pyramid is the same thing as a tetrahedron. This often surprises students, and it’s a good example of how the same object can have more than one valid name depending on how you’re classifying it.
Ask groups to look back at the questions in their flowcharts and think about what those questions are really saying about the shape.
Ask: If you had to turn your flowchart questions into a single sentence that describes what a prism is, what would it say?
Collect and refine definitions on the board as a class, following the same process when previously defining polyhedra, then repeat for pyramids. Use student responses to nudge toward the key features of each shape:
- Prism: two parallel bases that are the same shape and size, connected by rectangular faces.
- Pyramid: a polygon base with triangular faces meeting at a single apex.
Display Slide 25 if students need prompting toward the formal definitions.
Close with: Your flowchart was already a definition. You just wrote it in a different form.
For prisms:
- An object with the same shape at both ends.
- An object with two matching faces and rectangles connecting them.
- An object with two faces that are the same and some rectangles.
For pyramids:
- An object with a pointy top.
- An object with a flat bottom and triangles going up to a point.
- An object with one base and triangles on the sides.
- An object with a flat base and slanted faces.