Rotational Symmetry in Cubes and Regular Tetrahedra
Teaching Note 2
Learning Unit:
More about 3D Figures
Learning Objectives:
Extend the idea of symmetry in 2D figures to recognise and
appreciate the reflectional and rotational symmetries in cubes
and regular tetrahedra
1.
2.
Aims of the Learning
Activity:
3.
4.
Pre-requisite knowledge:
Recognise the meaning of rotational symmetry in 3D
figures
Recognize the rotational symmetries in cubes and regular
tetrahedra
Draw the axes of rotational symmetry in cubes and regular
tetrahedra
Recongise the relation between the number of vertices,
edges and faces in cubes and regular tetrahedra , and the
number of axes of rotational symmetries
1. Recognise the meaning of rotational symmetry in 2D figures
2. Recognize the number of folds in simple 2D figures
3. Recognise the sections of cubes and regular tetrahedra
Background:
Year level: S2 or S3 (suggested)
Teaching aids: Worksheet 2, straws, the nets of punched cubes and a regular tetrahedron
Flow of Learning Activities:
1.
The teacher revises with students the definition of rotational symmetry in 2D figures and asks
them to find the numbers of folds of rotational symmetry in some common 2D figures (Section
0 of Worksheet 2). The teacher may also use the software 2D_Symmetry in the folder
2D_Sym1 to demonstrate the effect of rotation of some simple 2D figures (as shown on the
next page).
1
The software 2D_Sym can be downloaded from the web-page of Mathematics Education Section
http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/MSS_e/MSS_e%20content.htm (Exemplar 9).
Rotational Symmetry in Cubes and Regular Tetrahedra
2.
Teaching Note 2
The teacher puts a straw in the position XY as shown below. Revolving the cube for one
round around the axis XY (the straw), the teacher asks students to observe the effect of rotation.
The teacher then explains that when the 3D figure is revolved around a certain axis for one
round and it overlaps with its image more than once, there is rotational symmetry in this 3D
figure. The teacher may also use the software 3D Figure_RotateSym.pps2 to supplement the
explanation. By changing the position of red button, the position of axis of rotation can be
changed. The teacher may drag the point P for one round and discuss with students on the
topics related to the coincidence of the image and the object.
X
Y
3.
2
The teacher then discusses with students the similarity and difference between the rotational
symmetries in 2D and 3D figures (The 2D figure is revolved around a point whereas the 3D
The software 3D Figure_RotateSym.pps can be downloaded from the web-page of Mathematics Education Section
http://www.edb.gov.hk/index.aspx?nodeid=8200&langno=1. The software Cabri3D_Plugin should be instralled in
order to view the dynamic movement.
Page
2
Rotational Symmetry in Cubes and Regular Tetrahedra
Teaching Note 2
figure is revolved around an axis of rotational symmetry).
4.
The teacher asks students to construct a cube from a given punched net. They can put the
straw into different holes and then rotate the cube so as to make conjectures on the position of
the axis of rotational symmetry and the corresponding number of folds (Questions 2-4 in Part
A of Section 1 in Worksheet 2).
5.
Students are then asked to draw all the axes of rotational symmetry (Question 5 in Part A of
Section 1 in Worksheet 2). The teacher checks the answers one by one with students.
He/she may also use the software Rot_cube3 in the folder 3D_Sym to show all the axes in one
picture.
3
The software 3D_Sym can be downloaded from the web-page of the Mathematics Education Section
http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/MSS_e/MSS_e%20content.htm (Exemplar 16).
Page
3
Rotational Symmetry in Cubes and Regular Tetrahedra
6.
Teaching Note 2
The teacher summarises the meaning of rotational symmetry in 3D figures.
It is essential to
remind students that the rotational symmetry in a 3D figure exists if the figure and the image
coincide more than once when rotating around an axis in one round. It is because there are
quite a number of students who mistakenly believe that any edge/line in the 3D figures can be
an axis of rotational symmetry. For example,
D
P
C
Q
A
ABCD is a regular tetrahedron. M, N, P and Q are
the mid-points of AB, BC, DC and AC respectively.
Which of the following is the axis of rotational
symmetry of the regular tetrahedron?
N
M
B
There were around one fourth of students, participated in the Seed project, chose DM or AB as
the axis of rotational symmetry.
7.
Students are asked to fold the punched net of the regular tetrahedron to form the regular
tetrahedron (Part A of Section 2 in Worksheet 2). Students then put the straw into the holes
and rotate the tetrahedron to make conjectures on the positions of the axes of rotational
symmetry. The teacher may also use the software 3D Figure_RotateSym.pps4 to demonstrate
the effects of rotating the tetrahedron around the axis of rotational symmetry. It is important
to emphasize the requirement for the existence of rotational symmetry in a 3D figure (i.e. the
image and the figure coincide more than once when rotating around an axis in one round).
4
The software 3D Figure_RotateSym.pps can be downloaded from the web-page of Mathematics Education Section
http://www.edb.gov.hk/index.aspx?nodeid=8200&langno=1. It is required to install the software Cabri3D_ Plugin
so as to demonstrate the dynamic movement.
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4
Rotational Symmetry in Cubes and Regular Tetrahedra
8.
Teaching Note 2
The teacher then discusses with students whether there is rotational symmetry in regular
tetrahedron and asks them to make conjectures on the number of axes, types and their
corresponding number of folds. Through working out Question 1 in Part A of Section 2 in
Worksheet 2, students can check their conjectures. The teacher may use the software Rot_tetr
in the folder 3D_Sym5 to show all axes on one screen.
9.
Finally, the teacher can ask students to categorise the axes of rotational symmetry in cubes and
regular tetrahedra. The students can be asked to write down the types, the number of axes of
rotational symmetry and the number of corresponding folds in questions of Part B in Section 1
and Section 2 of Worksheet 2. The teacher can further work with students on the relation
between the numbers of vertices, edges and faces of these 3D figures, and the numbers of types
of axes of rotational symmetry.
5
The software 3D_Sym can be downloaded from the web-page of Mathematics Education Section
http://cd1.edb.hkedcity.net/cd/maths/en/ref_res/MATERIAL/MSS_e/MSS_e%20content.htm (Exemplar 16).
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5
Rotational Symmetry in Cubes and Regular Tetrahedra
Teaching Note 2
Cube
Axis of rotational symmetry
Number of folds
Point 1
Type
Vertex
Point 2
Centre
Mid-point
of face
of edge
Vertex
of rotational
Centre
Mid-point
of face
of edge

Relation
4
No. of
vertices
symmetry
3
Type I
Quantity

2
4

Type II
No. of
faces 
2

2

Type III
3
6
No. of
edges 
2

Total
13
Regular tetrahedron
Axis of rotational symmetry
Number of folds
Point 1
Type
Vertex
Type 1
Type 2
Point 2
Centre
Mid-point
of face
of edge

Vertex
of rotational
Centre
Mid-point
of face
of edge
Quantity
Relation
3
4
No. of
vertice
s/faces
2
3
No. of
symmetry


edges 
2

Total
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6
7
Rotational Symmetry in Cubes and Regular Tetrahedra
Teaching Note 2
10. The teacher may also use the approach of sections in 3D figures to elaborate further the
number of folds in different axes. Students can work out Questions 1-4 in Part A of Section 1
in Worksheet 2.
Cube
The sections cut perpendicular to this axis are
all squares. The number of folds is 4.
X
Y
Cube
The sections cut perpendicular to this axis are
equilateral triangles or special types of
hexagons. The number of folds is 3.
X
Y
Cube
The sections cut perpendicular to this axis are
rectangles or squares. The number of folds is
2.
X
Y
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7
Rotational Symmetry in Cubes and Regular Tetrahedra
Teaching Note 2
Regular tetrahedron
The sections cut perpendicular to this axis are
equilateral triangles. The number of folds is
3.
X
Y
Regular tetrahedron
The sections cut perpendicular to the axis are
rectangles or squares.
2.
X
Y
Page
8
The number of folds is
Rotational Symmetry in Cubes and Regular Tetrahedra
Teaching Note 2
11. The teacher may, according to students’ needs, revise different sections of cubes and
rectangular tetrahedra. The software Doorzien 6 can be downloaded from the web to
demonstrate different sections of 3D figures.
6
The
software
Doorzien
can
be
downloaded
http://www.fisme.uu.nl/toepassingen/00349/toepassing_wisweb.html.
from
the
web-page
As the software is written in Dutch, teachers
may use the Figuren in the pull-down menu to select solids (Kubus is cube, Vievlak is regular tetrahedron, etc.).
Select
Opties to choose the points in which the sections passing through (Select Hulppunten to get mid-points, select dvie
delen to get points of trisection, etc.)
Sections will be shown as in the above diagram if the software is used.
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