Mathematics

Prisms and pyramidsBookmark

Learning area: Mathematics
Year level: Year 6
Country: China, Indonesia, Japan, South Korea
General capability: Intercultural understanding, Numeracy

This learning sequence focuses on developing students' familiarity with three-dimensional shapes, in particular prisms and pyramids. The activities support the Australian Curriculum for Mathematics: Measurement and Geometry.

Students develop the geometrical language and properties associated with these shapes and construct geometrical models. They explore monuments that are pyramid-shaped and found in the Asian region.

Students can explore geometry through origami

Acknowledgements

Image: Howard Reeves

Activity 1: Developing mathematical vocabulary

In this activity you will:

• develop and refine your mathematical vocabulary
• understand that some words in common usage have specific and often quite precise meanings when used in mathematical contexts
• become familiar with names of polygons such as triangles, quadrilaterals and pentagons
• become familiar with specific polygons including isosceles and equilateral triangles, squares, rectangles and parallelograms
• use mathematical vocabulary in relation to space and shape.

1. Review the geometric terms below before beginning this activity.
2. Did you know that a polyhedron is any closed solid shape consisting of four or more polygon-shaped faces in which pairs of polygon faces join along edges and three or more edges meet at a vertex?
3. Use a search engine to find websites which display 'polyhedra' by typing 'polyhedron' or 'polyhedra' into the search engine.

1. Five polyhedra are regarded as being special and are referred to as the platonic solids. Two of them are the cube and the tetrahedron.
2. Find the names of the five platonic solids and describe what is special about them.

1. Classify and name the classroom collection of 3D shapes according to whether they are prisms, pyramids or (possibly) neither.
2. Investigate Euler's formula and see References for expansion notes for Euler's formula.;

The geometrical shape to the right is a rectangular prism and comprising:

• 6 rectangular faces [F]
• 12 edges [E]
• 8 vertices [V]
3. Does the conjecture F + V = E + 2 hold true for the prisms and pyramids in the classroom collection?
4. Practise applying the formula to different prisms and pyramids.

Geometric terms

• apex: The vertex above the base of a pyramid is called the apex of the pyramid.
• base: This is what a prism sits on and what a pyramid stands on.
• cube: A cube is a special square prism ... all six faces are identical squares. It is also called a regular hexahedron because it has six identical square faces.
• edge: Two adjacent polygonal faces meet at an edge.
• faces: Prisms and pyramids are solid geometrical shapes with faces (sides) that are polygons.
• polygons: These are many-sided shapes like triangles, quadrilaterals and pentagons.
• polygons with specific properties: Some shapes like isosceles and equilateral triangles, squares, rectangles and parallelograms have specific properties.
• polyhedra [singular: polyhedron. 'poly' = many; 'edron' = face]: A polyhedron is any closed solid shape consisting of four or more polygon-shaped faces in which pairs of polygon faces join along edges and three or more edges meet at a vertex.
• prism: A prism sits on a base and has a top parallel to the base that is the same-shaped polygon or parallelogram, and has rectangular faces (sides).
• pyramid: A pyramid stands on a base that is a polygon and has triangular faces meeting at a point or vertex (referred to as the apex).
• tetrahedron: A tetrahedron is a triangular pyramid which has four triangular faces; when the four faces are identical equilateral triangles, it is called a regular tetrahedron.
• vertex: Three or more edges meet at a point called a corner or a vertex, the plural is vertices.

Acknowledgements

Images: Howard Reeves; Triangular prism - Wikimedia Commons; Cube, triangular pyramid, and square pyramid - ClipArt ETC License

Activity 2: Nets of prisms and pyramids

In this activity you will learn about nets of solids and construct solids using nets.

Most catalogues of educational and mathematics classroom materials contain details about commercially available geometrical kits. Some are collections of ready-made geometrical solids, some are kits of interlocking geometrical shapes that may be used to construct polyhedra.

1. Choose a cereal packet and find where it has been glued. Run a blade along the glued sections and open out the packet to see the shape of its net.
2. Choose a prism or pyramid and place it on a piece of paper. Trace around the face in contact with the paper.
3. Now, along one of the edges of the solid, roll the solid onto another face and trace around that face. Repeat the rolling and tracing process until all faces have been traced.
4. Place a pyramid on a piece of paper and trace around the base in contact with the paper. Imagine cutting all the edges of the triangular faces and flattening them onto the paper.
5. Use a triangular face as a tracing template to draw the net. The same process may be used to construct the net of a prism. Don't forget to include the shape for the top of the prism as well as the base.
6. Construct prisms and pyramids by opening out the solid, making sure to keep each face connected (along an edge) to one other face.
7. Sketch the net from this physical model of the net a pyramid.
8. Check your net by folding and taping or gluing to re-form the prism or solid.

The 'net of a solid' shape is a connected group of polygons (connected along the sides of the polygons, not just at the corners) that could be folded to form the solid shape. It can also be considered as the shape of a piece of paper required to exactly cover or wrap the solid.

Alternatively, given a cardboard model of a prism or pyramid, the net is the plane shape obtained when edges are cut and the prism or pyramid is flattened.

Consider the different nets for a cube and  Notice that the net of a solid is not unique.

Unless tape is used to join edges, glue tabs need to be added to the net in order to construct the solid shape from the net. Refer to the net of a tetrahedron, including glue tabs, as shown.

Acknowledgements

Images: Howard Reeves

Activity 3: Pyramids... Where in the world?

In this activity you will:

• learn about architectural pyramids found in the countries of Asia
• use the internet to identify the location of some of these pyramid-shaped structures and find out about their history, construction, relative size and cultural significance
• give a short presentation on the pyramid of your choice.

Exploring pyramids

Three-dimensional geometrical shapes occur in nature in crystals of substances. Some of these shapes are commonly used in monuments and sculptures and in the architecture of buildings and other physical structures. Ancient man-made pyramids are found in many parts of the world and in some cases the cultural and archaeological significance of these is still being studied.

Task 1: Big pyramid-shaped structures in the world

1. List where in the world you would find big pyramid-shaped structures.
2. Check to see what your classmates have written. How many different countries or places have been suggested?
3. What are the Seven Ancient Wonders? Use the internet by typing 'Seven Ancient Wonders' into your computer's search engine.
5. Pyramids in Asia: There are some significant pyramids in Asia ... even some that are under water!
6. Make a list of pyramid-shaped structures and their country location in Asia.
7. Choose one Asian pyramid to research in some detail. Prepare some notes for a short talk on your chosen pyramid that includes:
• use of mathematical language to be precise about the shape and nature of the pyramid
• details about the size of the pyramid compared to the size of the Gaza Pyramid; for example, its base measurements and its approximate height
• the age of the pyramid
• the historical or cultural significance of the pyramid.
8. Once everyone has presented their talk, as a class compile a collaborative list of what you think are the seven ancient wonders of the Asia region.

Task 2: Tower of Hanoi puzzle

The invention of this puzzle is attributed to a French mathematician Eduoard Lucas in the 1880s.

The challenge is to move the disks from their starting position (as shown) to the right-hand pole according to the following rules:

• The disks may only be moved one at a time.
• A move consists of removing a disk from the top of a stack and moving it to another pole on top of any number of disks that may already be on the pole.
• At no stage can a disk occupy a position on top of a smaller disk.
• When played with 5 disks, the smallest number of disk moves to move the tower from the left pole to the right pole is 2m + 1 or 31 moves. Read more about the patterns involved at Math Forum: Tower of Hanoi.
• The puzzle may be played with any number of disks.

The Tower of Hanoi puzzle is also known as the Tower of Brahma. The two names relate to the origins of a religious legend, which surrounds the puzzle. The legend involves a puzzle with 64 disks to be moved. See: Tower of Hanoi (Wikipedia).

Interactive versions of the Tower of Hanoi puzzle may be found in many places on the internet. Two good websites are: Tower of Hanoi (Maze Works) and Tower of Hanoi (Maths is Fun)

Acknowledgements

Images: Candi Sukuh by dany13 (CC BY 2.0); Tower of Hanoi - Wikipedia Commons

Activity 4: Geometrical origami

In this activity you will explore Japanese origami and use origami paper to construct a range of different geometric shapes.

Japanese paper folding to produce geometrical shapes

Origami [ori = folding; kami = paper] is a traditional Japanese form of art and creativity involving the folding of paper into different shapes and structures. In its purest form it involves the folding of a single square sheet of paper without the use of scissors or glue, but the techniques can also be used with several sheets of paper to produce more complicated structures.

1. Use a search engine and type in the word 'origami'. Then, find examples of origami sculptures by scanning and searching in the images category.
2. Now it's your turn! Use origami techniques to construct some geometrical shapes and models with help from the listed internet sites below:

This learning sequence will assist students to develop the geometrical language and properties associated with shapes of prisms and pyramids and to construct geometrical models. By exploring monuments that are pyramid-shaped and found in the Asia region, students will develop an awareness of the achievements and contributions of the peoples of Asia.

Activity 1: Developing mathematical vocabulary

At the beginning of this activity find out what students already know about the vocabulary of prisms and pyramids. Geometric terms are provided for students to build students' knowledge about the terms and meanings fundamental to understanding the geometry of prisms and pyramids.

In preparation for working with prisms and pyramids it is a good idea to assemble a classroom collection. Foods such cereals and chocolate are often boxed in containers that are prism-shaped forms. Other products such as soaps, candles and sweets are sometimes boxed as pyramids. Students should be encouraged to contribute to the collection.

Expansion notes for Euler's formula

F + V = E + 2

This formula is known as Euler's formula, named after Leonhard Euler, a Swiss mathematician. It refers to the relationship between the number of faces [F], sides [S], and vertices [V] of a convex (one without dents) polyhedron.

For any convex polyhedron in the classroom collection check that the formula holds true.

Extension

Notice that for a square pyramid, F = 5, V = 5 and E = 8 and so F + V = E + 2.

Now imagine the square pyramid transforming into a pentagonal pyramid by the number of sides in the base increasing by 1. Consider what happens to F, V and E.

Clearly V will increase by 1 (as there will be a new vertex in the base). The new side in the base will result in there being an increase of 1 in the number of faces [F]. The new side in the base shape will result in an extra edge in the base and an extra edge in the triangular faces. So we get …

[F + 1] + [V + 1] = [E + 2] + 2

This is true for any chosen pyramid … If the number of sides in the polygon base is increased by 1, the number of vertices [V] will increase by 1 as will the number of faces [F] and the number of edges [E] will increase by 2.

Activity 2: Nets of prisms and pyramids

Nets of prisms and pyramids require you to become familiar with the 'language of shape' and prepare the materials for the accompanying activities..

Equipment for this activity includes geometrical models and light cardboard for model making, scissors, rulers and craft glue.

Activity 3: Pyramids ... Where in the world?

This activity develops the link between the Mathematics curriculum content and architectural pyramids found in the countries of Asia. The aim of the activity is to expand students' awareness that pyramids are also found in Asia. While research about Egypt's pyramids is extensive, there is very little known about the large pyramid-shaped structures in the Asian region, especially about their construction, history and cultural significance.

Students will use the internet to identify the location of some of these pyramid-shaped structures and investigate their relative size and cultural significance.

Associated learning

There is an opportunity to introduce students to the Tower of Hanoi (sometimes referred to as the Tower of Brahma) puzzle. The puzzle is simply constructed. It can be regarded as a stacked pyramid and its solution from simple two- or three-piece forms to a number of pieces (5, 6, 7 …) is challenging to complete and to describe. Links are provided to websites which contain interactive representations of the puzzle.

Activity 4: Geometrical origami

This activity takes students beyond the Japanese origami techniques of folding paper squares into different sculptural shapes such as animals and flowers.

Provide origami paper for students who will use origami techniques to make geometrical shapes and solids, some involving modular approaches requiring a number of sheets of paper. Links to websites are provided to display photographs of examples of such constructions and to access construction instructions.

Useful websites

It is recommended that teachers preview websites to ensure they are suitable for their students prior to use in class. Content accessed via these links is not owned or controlled by AEF and is subject to the terms of use of the associated website.

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