ENJOY MATHEMATICS in 3D

Lesson 1

Create a cube and observe the positional relationships among the surfaces, sides, and vertexes.

Set the learning activity of creating the cube for the purpose of using it as a die in a game.
By using polyhedrons, set up the time of observing the positional relations among such matters as the development types and surfaces while students open them up and close them again.

Clarify a difference between the developments capable of creating a cube (correct developments) and those incapable of doing so (incorrect developments).

Clarify the common characteristics of the developments capable of creating a die, and set the learning activity of explaining why some developments are incapable of creating a die.
* The situation provides an opportunity to see a converse proposition B of a true proposition A is not necessarily true. It is considered as the learning of fostering a basis for demonstration.

While a cone is drawn with Cabri 3D, understand the basic operations of Cabri 3D through observation of three-dimensional figures.

Set a learning of drawing the cone by using Cabri 3D.
Deal carefully with the method of taking a point in a given space.
* By dealing with such methods as changing a radius of a circle, changing a point's position, and observing from various directions, this hour aims at ensuring that subsequent learning activities are smoothly carried out.

Expand perspective regarding the method of constructing three-dimensional figures. (Part 1)
* Columns and oblique prisms are seen as the three-dimensional figures constructed by movements of planes.

Present a thick book and a stack of paper, and expand images.
Prepare simulation of constructing a quadrangular prism from a square.
Arrange the learning environments to ensure that the observation can be carried out while Cabri 3D is used to create the columns and oblique prisms by moving planes parallel.

Expand perspective regarding the method of constructing three-dimensional figures. (Part 2)
* Such figures as spheres, cylinders, and cones are seen as the three-dimensional figures constructed by rotating a plane.

Present the model of a rotating body, and confirm that it is a three-dimensional figure that can always be seen as a circle from a certain direction.
Show simulation of a cylinder as a rotation body example.
Arrange the learning environments to ensure that three dimensional figures are created by rotating various figures around the rotation axes by using Cabri 3D.

Expand perspective regarding the method of constructing three-dimensional figures. (Part 3)
* Columns and oblique prisms are seen as the three-dimensional figures constructed by moving line segments.

Arrange the learning environments to ensure that the columns and conic solids are seen as a set of line segments (generating lines) of the same length.
Arrange the learning environments to ensure that a column can be created by rotating, for one time, a line segment vertical to the base, along the circumference of the base, and that a conic solid can be created by fixing one tip of a line segment while rotating another tip along the circumference of the base.
Prepare a model allowing students to see that something is inside of the three-dimensional figures in order to facilitate their understanding: "The insides are not stuffed." This is one of the differences from three-dimensional figures created by rotation or parallel translation of (a) plane(s).

Set a learning activity of creating stamps by cutting the cubes.
Arrange the learning environments to ensure that the observation can be carried out while the cubes are freely cut with Cabri 3D.

Take up polygons that can and cannot be created on the cut surfaces, and think about why.

Set up a situation to take up the challenge of creating a stamp in the form of a regular pentagon.
Let the students focus on the outlines created by crossing two planes and on the positional relationships among the surfaces making up the cubes.

Based on the characteristics of a regular pentagon and a regular hexagon, set the learning activity of thinking about why a regular pentagon does not appear on a cut surface of a cube.
Prepare the contents facilitating the observation of the cases where a cut surface becomes a regular hexagon and where a cut surface becomes a pentagon.
* The situation provides an opportunity to actually see that a contrasting proposition B of a true proposition A is also true (true and false coincide). It is considered as the learning of fostering a basis for demonstration.

Cut three-dimensional figures other than a cube, and observe figures appearing the cut surfaces.

Set a learning activity of creating stamps by cutting the three-dimensional figures other than a cube.
Set learning environments to ensure that the three-dimensional figures such as a sphere, cylinder, and cone can be observed as they are cut with Cabri 3D.

Observe the figures that appear when a cone is cut by various planes, and summarize the characteristics.

Prepare two congruent three-dimensional figures whose vertexes are connected and which are symmetrically placed with respect to the point. Set learning activities of cutting the figures with a plane parallel to the generating line and with a plane parallel to the rotation axis, and of summarizing the characteristics.
Use Flash Movie, and prepare the contents facilitating understanding of existence of a hyperbola.
* A parabola also appears. But do not deal with the further details. However, when the figures are cut by a plane parallel to the rotation axis and with a place parallel to the generating line, expect that there are students who notice their differences.

Introduce products that effectively use the methods of cutting a cylinder and a cone.
Introduce situations where designs are contrived in terms of functionality.
Introduce situations where various three-dimensional figures can be observed (for example, a photo of the head office of Fuji Television Network, Inc.), and set a learning of drawing it with Cabri 3D.
* The hour is to freely use Cabri 3D.

Take up a cylinder-shaped container and a cone-shaped container which have the same base area and height, and set a learning situation where the volumes are compared.
Through actually creating each three-dimensional figure, emphasize the relationship of the actual measurements and calculation methods.
Through the comparison of the volumes of the cylinder and a cone, summarize inductively how to obtain volumes of prisms and of conic solids.

Regarding the cylinder and the cone created during the previous hour, set a learning situation to think which container can be created with smaller paper.
Confirm that the areas of the developments coincide with the surface areas of the figures.
Summarize how to obtain the area of the sector.
Confirm the meanings of “surface area,” “base area,” and “lateral area.”
Through the comparison of the volumes of the cylinder and the cone, summarize inductively that the surface areas of the three-dimensional figures coincide with the areas of the developments.
* When the areas of the developments are obtained, the areas of the known figures need to be obtained and totaled. Thus, it is considered that this point can be summarized as the matter common to all three-dimensional figures.