How many open dice nets are there

Cube nets lesson: How many different cube nets are there? Where do you have to add the square so that it becomes a cube network?

1. Subject of the series: Cube nets

2. Structure of the series

2.1. Topic of the 1st lesson: Repetition of a cube-cube network or no cube network
2.2. Subject of the 2nd lesson: Which cube nets are there? How many different are there
it?
2.3. Subject of the 3rd lesson: How many different cube nets are there? Where do you have to add the square so that it becomes a cube network?
2.4. Subject of the 4th lesson: Where do you have to add the square so that a cube network is created?
2.5. Subject of the 5th lesson: Which are the same? Comparison of cube nets in different folded states
2.4. Subject of the 6th lesson: Additional Where is the top surface? (Specifying a base area)
2.5. Subject of the 7th lesson: Where are the arrowheads when the cube is folded?
2.6. Subject of the 8th lesson: Tipping the cube - where is the top surface now?
2.7. Subject of the 9th lesson: Tilting the cube - How does the cube get into the specified position?
2.8. Topic of the 10th lesson: cube networks / cuboid networks - similarities and differences; Tilting cuboids

3. Learning opportunities for the children in the class

The cube is a polyhedron because it is bounded by 6 congruent squares.

It has 12 edges and 8 corners. Each face is perpendicular to each of its neighboring faces and all edges meet at right angles in a corner. In addition, the cube is one of the 5 platonic solids. The surface of a cube with edge length a is A = 6 a2. Its volume is V = a3.[1]

If you want to build a cube out of cardboard or, as described later, “dress” a cube, you need a “suitable pattern”. There are several ways to do this. By differentiating cases one can show that there are exactly 11 non-congruent cube networks.

So you get 6 nets with 4 squares in a row, 4 nets with 3 squares in a row and there is 1 net with at most 2 squares in a row.[2]

In this lesson for the “Cube Nets” lesson, the students should repeat the cube nets they found from the last lesson. The teacher and the children create a poster to provide an overview. This can later be hung up in the class. The fact that the children check all cube nets themselves before the teacher sticks them on the poster creates a motivating effect.

These are all possible 11 cube nets:

Figure not included in this excerpt

These cube nets were made beforehand by the teacher from colored slips of paper in order to progress quickly during the lesson. Opposing slips of paper are of the same color. This is not yet communicated to the children, but there is an opportunity to make connections to the topic of decking.

The statement that there are eleven cube nets is at the end of the production of the poster. Some students will certainly argue that there are other dice nets. This is an incentive for everyone to deal with the dice nets at home (voluntary homework) in order to find another dice net. As with the creation of the poster, some children will probably not think about the fact that symmetrical cube networks have already been found by rotating or mirroring them.

The pupils then receive an exercise sheet on which they should complete almost finished cube nets (see appendix). In this case, one area of ​​the cube network is missing, which should be added at different points. All the different places should be found by the pupils. Here, however, the poster is removed from the wall if necessary, or the pupils are advised that they should work without the help of the poster. This ensures that the pupils think again intensively about the cube nets and the necessary conditions that they must have. As in the previous lesson, the pupils are allowed to use beer coasters and scotch tape to help them make cube nets and check them.

The compilation of the 11 cube nets and the processing of the worksheet are an excellent exercise in head geometry for training the spatial conception.[3]

In this way you will recognize many children who first have to pick up the net of dice in order to be able to check the result.

A similar exercise sheet is used as homework, on which there are again 2 tasks. This exercise sheet prepares you well for the following lesson.

The Main learning objective is the training of the spatial imagination of the students by trying to check the cube nets for their correctness.

In addition, the primary school curriculum includes the subtopics “Making and Investigating Networks; Making models of cubes and cuboids from meshes ”is provided.

But here, too, the overriding learning objective is training the spatial awareness.

[...]



[1] Cf. Schülerduden Die Mathematik I - Mannheim 5. revised. Edition - 1990 p. 497f

[2] Cf. Schülerduden Die Mathematik I - Mannheim 5. revised. Edition - 1990 p. 498

[3] See Radatz, Rickmeyer - Handbook for Geometry Lessons in Elementary Schools (Schroedel 1991) p. 56

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