Reflectional Symmetry in Cubes and Regular Tetrahedra
Teaching Note 1
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
Aims of the Learning
Activity:
1.
2.
Recognise the meaning of mirror images of 3D figures
Recognize the reflectional symmetries in cubes and regular
tetrahedra
3.
Draw the planes of reflectional symmetry in cubes and
regular tetrahedra
Recongise the relation between the number of vertices,
edges and faces in cubes and regular tetrahedral, and the
number of planes of reflectional symmetries
4.
Pre-requisite knowledge:
Recognise the meaning of reflectional symmetry in 2D figures
Background:
Year level: S2 or S3 (suggested)
Teaching aids: Worksheet 1, mirror/plastic film/“silver-coated” paper, long strip model (that can be
folded to become a regular tetrahedron) and 3 “half-cube” models
Flow of Learning Activities:
1.
The teacher, according to students’ needs, revises with them the meaning of regular tetrahedra
and how to interpret the meaning of 2D representations of 3D figures (Worksheet 0).
2.
The teacher revises with students the definition of reflectional symmetry in 2D figures and
students are requested to draw the axis of reflectional symmetry in simple 2D figures (Section
0 of Worksheet 1). After then, the teacher may use the software 2D_Sym1 to demonstrate the
effects of reflection of these figures and check students’ answers.
1
The 2D_Sym software in Exemplar 9 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.
The effect of
reflection in some 2D figures can be shown through executing the software 2D_Symmetry in the folder 2D_Sym.
The Reflectional Symmetry in Cubes and Regular Tetrahedra
3.
Teaching note 1
The teacher explains the meaning of the image of a 3D object after reflection and asks students
to draw the images of given reflections (Questions 1-3 in Section 1 of Worksheet 1).
4.
The teacher elaborates the meaning of reflectional symmetry in 3D figures and discusses with
students the similarity and difference between reflectional symmetries in 2D and 3D figures
(say, when they reflect in an axis/ a plane, the image and the object will overlap. 2D figures
are reflected in an axis but the 3D figures are reflected in a plane).
5.
Students are then asked to construct half-cubes from the nets of “half-cube model” in the
“Symmetry in 3D figures” Package. Then, they can use the sheet of “silver-coated”
paper/mirror to observe the images of these “half-cube” so as to recognise the reflectional
symmetry in cubes. Students can find that some planes are the planes of reflectional
symmetry of the cube (Questions 1-2 in Part A of Section 2 in Worksheet1).
6.
The teacher may also use the powerpoint 3D Figure_ReflnSym.ppt2 to assist in demonstrating
2
The software 3D Figure_ReflnSym.pps can be downloaded from the webpage of Mathematics Education
http://www.edb.gov.hk/index.aspx?nodeid=8200&langno=1. Nevertheless, the software Cabri3D_Plugin should
Page 2
The Reflectional Symmetry in Cubes and Regular Tetrahedra
Teaching note 1
the effects of reflectional symmetry in cubes. The teacher just clicks the right mouse button
and drags the object to view its different orientation.
7.

Drag the red button to view the images of the “half-cube” in different planes of reflection
and check whether the image and the other half of the cube overlap.

Explain why not all planes which cut the cube into 2 equal halves (in terms of volume)
are the plane of reflectional symmetry (the blue model can be used for the explanation).
The teacher then asks students to draw all the planes of reflectional symmetry in cubes
(Question 3 in Part A of Section 2 in Worksheet 1) and checks their answers one by one.
He/She may use the software Ref_cube in 3D_Sym folder 3 to show all the planes of
reflectional symmetry in one picture.
3
be installed in the computer so as to demonstrate the dynamic movement.
The 3D_Sym software can be downloaded from the webpage of Mathematics Education Section
http://cd1.edb.hkedcity.net/cd /maths/en/ref_res/MATERIAL/MSS_e/MSS_e%20content.htm. (Exemplar 16)
Page 3
The Reflectional Symmetry in Cubes and Regular Tetrahedra
8.
Teaching note 1
The teacher then summarises the meaning of reflectional symmetry in 3D figures and reiterates
that the plane cuts a 3D figure into 2 equal halves may not be the plane of reflectional
symmetry.
9.
The teacher distributes the long white strip in the “Symmetry in 3D Figures” Package and
explains to students on how to construct the regular tetrahedron (Questions 1-2 in Part A of
Section 3 in Worksheet 1).
10. Students then use a sheet of “silver-coated” paper to observe the image of the
“half-tetrahedron”. The teacher may also use the powerpoint 3D Figure_ReflnSym.pps4 to
explain the meaning of reflectional symmetry in regular tetrahedra. Through dragging the red
button to change the reflectional plane, the students can then observe their corresponding
images and discover when the object and the image overlap.
4
The software 3D Figures_ReflnSym.pps can be downloaded from the web page of the Mathematics Education
Section http://www.edb.gov.hk/index.aspx?nodeid=8200&langno=1. Nevertheless, the software Cabri3D_Plugin
should be installed in the computer so as to enable the dynamic movement.
Page 4
The Reflectional Symmetry in Cubes and Regular Tetrahedra
Teaching note 1
11. The teacher then discusses with students on whether there is reflectional symmetry in regular
tetrahedra. Students check their conjectures on the number of planes of reflectional
symmetry through working out Question 3 in Part A of Section 3 in Worksheet 1. The teacher
may also use the software Ref_tet5 in the folder 3D_Sym to show all the planes of reflectional
symmetry in the regular tetrahedron in one picture.
12. Finally, the teacher can ask students to categorise and count the number of planes of
reflectional symmetry in cubes and regular tetrahedra, and complete Part B of Sections 2 and 3
in Worksheet 1. The relation between the number of vertices, edges and faces of the 3D
figures, and the number of planes of reflectional symmetry can be further worked out.
5
The teacher can download the software 3D_Sym 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).
Page 5
The Reflectional Symmetry in Cubes and Regular Tetrahedra
Teaching note 1
The name of the 3D
figure which divide
the cube into 2 equal
halves
Number of planes of
reflectional symmetry
Type I
Cuboid
3
Number of faces  2
(i.e. 6  2)
Type II
Triangular prism
6
Number of edges  2
(i.e. 12  2)
Total number
Only
One Type
Relation with the numbers
of vertices, edges and
faces in the cube
9
The name of the 3D
figure divide the
regular tetrahedron
into 2 equal halves
Number of planes of
reflectional
symmetry
Triangular pyramid
6
Page 6
Relation with the numbers of
vertices, edges and faces in the
regular tetrahedron
Same as the number of edges