Students require ample experiences in creating and deconstructing
three-dimensional (3D) objects. Some common misunderstandings that
students have are, not being able to visualise 3D shapes, or hidden faces
when observing two-dimensional (2D) perspective drawings.
Other difficulties
arise when students are unable to describe the properties of a shape or solid,
as they merely point out the obvious features visible to them. Namely, the
faces, vertices and edges.
It is important that students describe other attributes such as shapes with
parallel faces, and begin to categorise solids based on their similar features
e.g. prisms and pyramids.
A way to help students develop their spatial
awareness and engage in mathematical thinking
and investigation when working with
3-dimensional objects, is to provide them with
opportunities to:
Use various materials to construct
3-dimensional shapes
Design their own nets rather than having
templates printed
Use key language
For example, if you wanted to focus on language development and properties
such as:
The faces of a shape you could have students use paper. When investigating
prisms paper models help students identify the bases and flat rectangular
surfaces
Identifying the vertices then toothpicks and blu tack would be ideal
Counting the edges then rolled up newspaper or straws should be used. Here
they can also discover that the edges are where two faces connect
Other activities such as deconstructing boxes will
help students understand how polygons fit
together to make a net for a 3D solid. You could
also ask students to observe the polygons used
to make the solid and construct a different net.
Victorian Curriculum
Connect three-dimensional objects with their nets and other two-dimensional representations (VCMMG198)
VCAA Sample Program: A set of sample programs covering the
Victorian Curriculum Mathematics.
VCAA Mathematics glossary: A glossary compiled from subject-specific terminology found within the content descriptions of the Victorian Curriculum Mathematics.
Achievement standards
Students use appropriate units of measurement for length, area, volume, capacity and mass, and calculate perimeter and area of rectangles and volume, and capacity of rectangular prisms. They convert between 12 and 24-hour time.
Students use a grid reference system to locate landmarks. They estimate angles, and use protractors and digital technology to construct and measure angles.
Students connect three-dimensional objects with their two-dimensional representations. They describe transformations of two-dimensional shapes and identify line and rotational symmetry.
In this activity students will be investigating cubes and their nets.
Show students the familiar template for a cube and have students
analyse its structure.
Ask them to find other ways to
create the cube net and inform
them that there are 11 different
ways. As students begin
investigating have them
observe the nets they have
found so far and look for
patterns. If they are struggling, they can discuss with their peers
which nets do not work and explain why.
For example, the nets below
each have six square faces however, one has more than three faces
meeting at a vertex and the other has six in a row which would cause
faces to overlap each other (as shown below).
Pose the question, are there only 11 types? If so find a proof or explanation. Students will find that there can only be lines of 4, 3, or 2 in a row. This activity will help students develop their spatial awareness.
Students can also use interactive sites such as cube nets, which show the nets transform into the cube to help students find all possibilities.
As an extension, students can explore pentominoes (a shape made up of five squares - think of a cube without a lid). There are 12 possibilities. Each of these shapes can be formed into a large rectangle, therefore, have students draw the shapes on grid paper and cut them out, so that they can assemble them.
In this activity, students work in groups to create
designs for houses in a new suburb. They have to follow some
guidelines which state:
The cubes must be the same size
The faces of the cubes must be touching
All houses must be a different design, i.e. rotating a house will
not count
Together students will explore all the possible designs
(there are 15). They can use blocks or draw these
images on isometric dot paper. If students struggle
with creating 2-dimensional perspective drawings,
they can use the isometric drawing tool.
As an extension, students can work out the costs for each design,
the prices for construction are listed below:
$10,000 for each square unit of land used
$4,000 to design each external wall
$6,000 for each unit to construct a roof
Students can create booklets to describe the features of each house
and the benefits of each design. Here students can get creative with
decorations and begin to think of the practicality of each design.
In this example, students could discuss the benefits of having a
two-storey house or how much they would save on land area.
You can also increase the number of cubes used and have them find
all possible designs.
In this activity, students explore the
five platonic solids and their
properties. Explain to students that
during the time of Plato 500 B.C.
these solids were once linked with
the elements. Namely, the tetrahedron
to fire, the cube (hexahedron) to
Earth, the icosahedron to water,
the octahedron to air and the
dodecahedron to the universe.
Have students use or create nets to construct all five solids.
Once they have been constructed ask students to begin describing
the properties, for instance have them count the number of faces,
vertices, edges. Then pose the question, why only five?
As students
analyse the shapes, they will identify that each face is congruent, they
are made up of regular polygons, the same number of faces meet
at every vertex, the edges are also congruent, and they have a high
level of symmetry in that no matter which angle you look at the shape
it will be identical.
In this activity from NZMaths, students will construct a variety of
polyhedrons using various materials such as straws, sticks and tape.
After they construct each design ask students to note down the
properties of each shape on a table (as shown).
If students are familiar with the prefixes they can determine the
number of faces eg tetra meaning 4, hexa meaning 6, octa meaning
8 etc. Be sure that students understand the meaning of the word
polyhedron: poly meaning many and hedron meaning faces so that
they understand that they are constructing 3-dimensional solids.
Using the materials ask students to create as many different solids as
they can and continue to fill in the table.
Ask students to analyse the table and make a generalisation.
Euler’s rule: V + F - 2 = E this indicates that if you add the number of
vertices with the number of faces and subtract 2, it will give you the
total number of edges.
To extend this, students can test the rule with
2-dimensional shapes.
In this activity, students are given a drawing off the top view of a tower.
They must draw the front and side views of the tower. They can use
blocks to build the tower or drawings on isometric dot paper. Have
students compare their models with their peers.
Once students have explored various designs show them the side
and front view.
Ask the students what is the maximum or minimum number of blocks they can use and explain their reasons.
As an extension, students can create their own towers and drawings of the different views to give to a friend to build. They can even create the nets for their towers.
In this activity, your child will create designs for houses in a new suburb using four cubes. They have to follow some guidelines which state:
The cubes must be the same size
The faces of the cubes must be touching
All houses must be a different design, i.e. rotating a house will not count
Ask your child to consider all the possible designs (there are 15). They can use blocks or draw these images on isometric dot paper. A free printable copy of isometric dot paper can be found here. If your child is struggling to create 2-dimensional perspective drawings, this online isometric drawing tool may help.
As an extension, ask your child to work out the costs for each design, the prices for construction are listed below:
$10,000 for each square unit of land used
$4,000 to design each external wall
$6,000 for each unit to construct a roof
Your child could create a booklet/brochure to describe the features of each house and the benefits of each design. They can get creative with decorations and begin to think of the practicality of each design.
You can also increase the number of cubes used and have them find all possible designs.
