Target group : 7^e – 5^e

Place : a classroom

Materials required: 100 normal dice, a few special dice, tablets or computers, Makerspace

Duration: 4-5 teaching hours

Content :

• Solids and volumes
• Basic notions of probability

Knowledge and skills

The pupil knows

• vocabulary of solids: polyhedron, vertex, edge, face
• common solids: cube, rectangular parallelepiped, prism, right prism, pyramid, cone, sphere, cylinder
• formulae for calculating the volume of a cube, rectangular parallelepiped, right prism, cylinder and pyramid
• formulae for calculating the lateral surface area and total surface area of a cube, rectangular parallelepiped, right prism and cylinder
• the units of length, area and volume.

The pupil is able to

• recognise and classify solids
• count the edges, vertices and faces of a solid
• associate a solid with its net
• calculate the volume, lateral area and total area of a solid
• use dynamic geometry software to visualise solids and make their nets.

Setup (before the lesson)

Benches and chairs are arranged in the classroom to form islands for 2 to 5 students.

First two hours of teaching

The first two hours are taught in a learning stations format. A set of dice is placed on each of the 6 tables. The 6 games are available under 3.3 Teaching materials M1. All these games are played in groups of 2 to 5 players. To optimise learning by stations, it is preferable for each group to have a similar size. The pupils can form their groups themselves, but the groups will probably be more diverse if the teacher determines their composition in advance. Students should remain in the same group until the end of the learning stations session.

It is advisable to leave at least one of the six stations unoccupied at the start of the activity. This way, the first group to finish can move straight to it. This way, the students don’t have to wait to start a new station. In principle, each group spends between 15 and 30 minutes per station. It is not compulsory for all the pupils to try out all the games. The session lasts 2 teaching hours and, depending on the pace of the pupils, a variable number of games will be explored.

The lesson begins with a short introduction (5 minutes), during which the methods of learning stations are explained. The teacher explains to the pupils that at each station they must read the rules of the game and play a game according to the rules. During the various games, the pupils use dice of different shapes and develop an intuition for the probabilistic results in relation to the throw of the dice. They also develop a strategy to win the game or maximise their winnings.

The number of each station is clearly indicated and each station is equipped with the necessary material for the corresponding game (M1). Most of the games use standard D6 dice. For the best experience, we recommend that you use around a hundred D6 dice for this sequence. The roll of the dice can also be simulated virtually at https://g.co/kgs/EFqAMnE or www.mathigon.org.

Each student receives handouts (Station 1 to Station 6 sous M1)

describing the different stations and the considerations to be made. The students complete these documents as they go along, based on their experiences at each station.

The various games can of course be replaced by other dice games. Here is a non-exhaustive list of other games:

• Würfelkönig
• King of Tokyo
• Escape: The Curse of the Temple
• Sagrada
• Tenzi
• Yahtzee
• Pikomino
• Perudo
• Can’t stop

Third and fourth hour of teaching

The next two hours are organised around project-based learning. The aim is for each pupil to create their own personalised die and possibly invent a new game.

The session begins with a brief introduction (5 minutes) in which the teacher clearly explains the project to the whole class.

The pupils then embark on the work, which is structured into 4 compulsory phases and 1 optional phase:

First phase. The pupils define the characteristics of their personalised die through structured brainstorming. They use the idea-gathering sheets (M2) to organise and clarify their concepts. During this phase, they will encounter potentially new geometric concepts. To support them, 8 mini-lessons (M3) are available on request:

1. Tinkercad tutorials
2. Vocabulary of solids
3. Right prisms
4. Area of a right prism
5. Volume of a right prism
6. Right cylinders
7. Area and volume of a cylinder
8. Dice and probabilities

The idea is not for the teacher to teach these mini-lessons in a classroom setting, but for the students to use them according to their specific needs. Each mini-lesson includes self-assessment exercises. To encourage autonomy, we recommend that the answers (3.7 Solutions) be made available in different (more or less controlled) ways: either on the teacher’s desk for consultation after completing the exercises, or directly accessible to the students (paper or digital version). The teacher may also decide to run a mini-lesson as a whole class session if the situation so requires.

Second phase. The students create their 3D die using Tinkercad software. A tutorial is available in mini-lesson 1 (to be consulted digitally, as it consists of GIF animations). The students must save their model. It will be used in two different ways in the third and fourth phases.

Third phase. The students create their die, either by 3D printing or on paper (Pepakura). The assistance of the MakerSpace manager is recommended for this stage. Printing is based on the 3D model created in phase 2.

Fourth phase. Pupils use their creation in one of the following ways:

1. Creation of a poster presenting the die with the information detailed in the sheet for phase 4 (M2).
2. Creation of a card for a “Top Trumps” game¹. Each pupil produces a card describing their die. All the cards in the class make up a complete game, the rules of which are explained in M2.
3. Production of a video in which one of the characteristics of the die is determined experimentally. Four ideas are given in M2. The students will start by researching these ideas in order to fully understand the concepts and how they can be applied concretely to the study of a die. An important point to emphasise concerns the verification of Euler’s formula: to obtain convincing results, it is preferable to test this formula on several dice of different shapes. Pupils who choose this approach will have to borrow their classmates’ dice to enrich their experimentation. In addition, students are perfectly free to propose and develop their own ideas for experiments beyond those suggested.

Note: Section 5.3 M4 of the PITT VideoMATHon module gives very precise instructions on how to make a video using a smartphone.

Fifth phase. The fifth phase is optional. Once the personalised die has been created, the work can be continued by creating a game to be played with this die. There are a number of different forms available on the Internet to help students create games. This phase could also be done in collaboration with the French or Digital Sciences class (see 3.4 Interdisciplinary ideas).

1 Top Trumps is a card game first published in 1978. Each card contains a list of numerical data, and the aim of the game is to compare these values in an attempt to beat and win an opponent’s card. https://en.wikipedia.org/wiki/Top_Trumps

During the first two hours, the groups progress at their own pace between the different stations (M1). By keeping a station free from the start, students do not need to complete their activities simultaneously. What’s more, they don’t have to explore all the stations. These two methods ensure that the learning process respects the individual pace of each group of students.

During the next two hours of teaching, the mini-lessons enable effective internal differentiation. As well as optimising classroom time management, mini-lessons (M3) are a powerful tool for differentiating teaching. Students navigate with ease between their personal work, learning stations, available resources and mini-lessons, according to their specific needs. Fully engaged in their learning journey, they develop their autonomy while the teacher adapts his or her support without imposing a uniform pace. As each pupil has different levels of attainment, mini-lessons can either be offered as a resource or made compulsory only for those who will really benefit from them.

1. Luxembourg context: This module requires the availability of tablets in class as well as access to a Makerspace within the school. Given the excellent digital infrastructure of Luxembourg schools, this module is particularly suitable for implementation in Luxembourg.
2. Differentiation: As indicated in the previous paragraph, the very structure of the module naturally incorporates elements of pedagogical differentiation.
3. Media Compass: Competences targeted by the Media Compass¹
   • Competence 2 – Communication and collaboration: 2.1 Working with others
   • Competence 3 – Creating content: 3.3. Modelling, structuring and coding
4. 4C model: communication, collaboration, creativity, critical thinking: The 4Cs are integrated into this module. During the first two hours of teaching, students are required to collaborate and communicate to play the games and answer the questions on the worksheets. The design of the personalised die is a truly creative process. What’s more, the pupils are involved in an autonomous learning process where they must decide for themselves when and which mini-lesson to use to make progress on their project. This empowerment significantly develops their critical thinking skills.
5. Link with mathematics research: The solids presented in this module represent special cases of polytopes. Polytopes, particularly in dimensions greater than 2 and 3, are a dynamic and current area of research in mathematics. See also Section 3.5 More on this topic.

1 https://edumedia.lu/wp-content/uploads/2024/12/Medienkompass_EN_web.pdf 

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