Lesson9 - ENJOY MATHEMATICS in 3D

Outline: "based on the characteristics of a regular hexagon and regular pentagon, explain that the cut surface of a cube can not be a regular pentagon"

... Learning activities

... Teacher's instructions/guide

... Evaluation (expected student responses)

Regarding the necessary software data for using "Cabri 3D File" and "Flash Movie" on the pages, please review [Installing necessary software] .

Learning content

Present a regular hexagon and regular pentagon, discuss each figure's characteristics, and request the students to think about each figure's characteristics and the points that differ from each other.

As focus points for observing the regular hexagon and regular pentagon, take up the "existence and positional relationship of sides facing each other" from the worksheet used the previous time.

Using the regular hexagon and regular pentagon shown in the worksheet, think about each figure's characteristics and different points.

Describe what he or she noticed.

Point out a student thinking about the positional relationship by extending each side, and incorporate him or her in to the inquiries as a whole class.

Using Plasmavision, present a regular hexagon and regular pentagon created by GSP (geometer's sketchpad), and request the whole class to observe the appearance of the straight lines extended from each side.

Confirm that a regular hexagon has three sets of sides facing and parallel to each other, while the regular pentagon has no sides facing each other.

With the point of view confirmed by the whole class, instruct observation of the cut surfaces using Cabri 3D.

While operating Cabri 3D, confirm that the hexagons shown in the cut surfaces always have a set of sides facing and parallel to each other.

Confirm that pentagons also appear and have two sets of sides facing and parallel to each other. Based on these points, summarize that they cannot become regular pentagons .

A hexagon-shaped cut surface
A cube's hexagon-shaped cut surface
A pentagon-shaped cut surface
A cube's pentagon-shaped cut surface

By relating to the positional relation ship s of the surfaces making up the cube, explain that the sides facing each other will always b e parallel.

Since a regular pentagon has no (parallel) sides facing each other, explain that regular pentagon s can never be created.

Present again the sketch shown in the beginning by the teacher, and ask: "what is strange about this?"

Describe in the worksheet what he or she has learned, what convinced him or her of that , what appeared strange, and any other matter.

Present again the "impossible figure" (the figure to the left) shown in the beginning, and request review of its strange point s .

[Note]
This class confirms that the proposition, "If it is a regular pentagon, there is no set of parallel sides facing each other," is true , and explains why a regular pentagon cannot be created using the contra sting proposition, "If there is a set of parallel sides facing each other, it is not a regular pentagon. This means, "The correct proposition's contraposition is true ," and offers the opportunity for demonstration.

Expected responses from the students

A	Regarding the case of a regular hexagon, the sides face each other; however, regarding the case of a regular pentagon, each side and vertex face each other.
B	The positional relationship of the sides can be understood when each side of the regular hexagon and regular pentagon drawn in the worksheet is extended.
C	Regarding the case of the regular hexagon, the sides facing each other are always parallel; however, regarding the case of the regular pentagon, there are no sides parallel to each other.
D	Even when the size of the figure is changed in various manners by using the drawing tool (GSP), the same thing can always be said of the positional relationship of the sides.
E	Watch the hexagon-shaped cut surface of a cube [Cabri 3D File ] Regarding the cut surfaces of the cube, when they are observed from the viewpoint of the characteristics of a regular hexagon and regular pentagon, it appears possible to understand why a regular pentagon cannot be created.
F	Even when the plane surface is moved, the hexagon's sides facing each other are always parallel.
G	Watch the pentagon-shaped cut surface of a cube [Cabri 3D File] Since the sides facing each other are parallel in the case of the regular hexagon, the figure appears on the cut surface. Regarding hexagons other than the regular hexagon, it is a common point that the sides facing each other will be parallel.
H	Of the five sides, there will always be two sides facing each other.
I	In the case of a regular pentagon, each side and vertex face each other but the sides do not face each other, so the figure can not form .
J	The whole class observes the file [Cabri 3D File] When a cube is rotated, it can be understood that the sides facing each other are on the same plane. When two parallel planes facing each other are cut by another plane, it can be understood that two parallel straight lines facing each other will appear.
K	In the case of a regular hexagon, there are three sets of parallel sides facing each other, so the figure will appear on the cut surface. Thus, it becomes possible to explain why the regular hexagon can be created while a regular pentagon cannot.
L	Since the sketch drawn by the teacher has some sides that are not on the same plane, this figure does not exist.