- Level 5
- Measurement and Geometry
- Shape
- Connect 3D objects with their nets and other 2D representations

###### Shape • Level 5

# Connect 3D objects with their nets and other 2D representations

VCMMG198

## Teaching Context

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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.

## Online Resources

## Teaching ideas

- How many ways can you make a cube net?
- Four cube houses
- Platonic solids
- Euler’s formula
- Build the tower
- Learning from home

### How many ways can you make a cube net?

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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.

### Four cube houses

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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.

### Platonic solids

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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.

### Euler’s formula

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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.

### Build the tower

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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.

### Learning from home

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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.