# What are some uses of cuboids

#### Maike Willms

### Build cuboids with cubes

#### Different cuboids can be built with 24 cubes. But how many are there? The children first build cuboids with 24 cubes, draw construction plans and compare representations. Finally, one assertion puzzles the children: Can you actually build as many different cuboids with 30 cubes as you can with 24 cubes?

The lesson begins with the question: “How many different cuboids can you build with 24 cubes?” After repeating the properties of a cuboid, the term “differently” is clarified. To do this, a cuboid is first tilted, then rotated. The class sees that the body does not change in either case. A tilted or rotated cuboid should therefore not be referred to as "different".

For the individual work, each child receives 24 dice and a sheet of white paper. So it can always build a cuboid and hold it with a freely chosen representation. In class 3b, everyone uses the familiar blueprint display.

This is followed by partner work. The same cuboids are sorted out and missing ones are determined. The results will be transferred to a joint poster. The first teams quickly claim to have found all possible blocks. Their reasons, however, remain superficial: “Because they are all” or “Because we can't find any more”. Some children consider specially tilted cuboids to be "different". These children are asked to (again) build the blocks and compare them.

Due to the difficulties in recognizing identical cuboids based on the construction plans, a worksheet with all six possible cuboids is distributed at the beginning of the next lesson (KV 9). The LxWxH representation is new on it. After the meaning of the letters or numbers has been discussed, the children and their partner assign the building plans from the previous day to the six blocks on the worksheet: Which building plan belongs to 2x3x4, for example?

To discuss the results, the cuboids from the worksheet are copied together and the corresponding construction plans are drawn on the board. “Can you give me the blueprint for the 1x3x8 cuboid without building it?” Asks the teacher. “There are three towers with eight dice each. You can see that in the numbers, ”says Lilly and draws the construction plan. As a control, the cuboid is built with cubes. The children take a closer look at the other examples on the board: They notice that the height corresponds to the numbers in the squares and trace both of them in yellow. “The length is how many squares are next to each other,” adds Timo. After the length and width have been marked in color (Fig.1), the construction plans for the remaining cuboids are drawn.

"If there are six different cuboids with 24 cubes, there must also be six cubes with 30 cubes." With this claim and a white A3 poster, the children are sent to work as a partner at the beginning of the next double lesson. Two groups hold that the claim is false (Fig. 2 and 3). Moritz and his partner confirm the claim with the argument that 30 is divisible by 6.

The class is gathered in front of the blackboard. The enlarged material (KV 10-12) and cubes are ready there. Six construction plans (KV 10 and Fig.4), which were also on children's posters, are placed in front of the board: “Are these all cuboids?” Although some children had previously disagreed, they all nod in agreement.

The class is asked to describe the cuboids with the construction plan “30”. “There are 30 cubes on top of each other,” says Maja. Since the construction of this cuboid is shaky, the oblique image (KV 11) is attached. The same procedure is used with the next cuboids until Mats says: "The five is the same as the six." Oke agrees: "If you unfold the five, there are six next to each other." Amy contradicts: ...

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