Australian curriculum: mathematics--a new look at a familiar task.

Swan, Paul ; Marshall, Linda

For the next few issues the centre pages of Australian Primary Mathematics Classroom will be devoted to tasks and activities that help to illustrate key ideas embedded in the new Australian Curriculum: Mathematics. In this issue we focus on linking two of the proficiency strands (Problem Solving and Reasoning) with the Measurement and Geometry content strand. Graphically this might be represented in a table:

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Specifically we believe that this small unit of work involving pentominoes would involve the following aspects of the Geometry and Measurement strand.

Year 4

* Compare and describe two-dimensional shapes that result from combining and splitting common shapes, with and without the use of digital technologies. (ACMMG088)

Further information and digital resources relating to pentominoes may be found at:

* http://illuminations.nctm.org/ LessonDetail.aspx?ID=L853

* http://nlvm.usu.edu/en/nav/frames_asid_114_g_2_t_2.html

Year 5

* Connect three-dimensional objects with their nets and other two-dimensional representations. (ACMMG111)

* Apply the enlargement transformation to familiar two-dimensional shapes and explore the properties of the resulting image compared with the original. (ACMMG115)

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Pentominoes

Materials

* Square tiles

* 10 mm grid paper

* Scissors

A pentomino is a geometric figure consisting of 5 unit squares, each square having at least one side in common with another square.

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Pentominoes are a particular group of polyominoes, a mathematical generalisation of the idea of a domino.

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There are two different pentominoes shown above. Use square tiles to create more pentominoes.

Record using grid paper.

How many different pentominoes are there?

Watch out for duplications, that is, reflections and rotations.

How will you know when you have found them all?

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Scale

Materials

* 10 mm grid paper

* Scissors

The lesson

1. Assign students a different pentomino shapes. (Note there are only 12, see Figure 1, p. 16.)

2. Ask the children to draw it again but this time in double scale.

3. Ask, "What do you notice about the perimeter of the new shape?" (It doubled: 12 units became 24 units of length) "What about the area, has that doubled too?"

4. Children calculate areas, and of course discover that all the enlarged pentominoes have an area of 20 square units, four times the size of the original. This can be quite a revelation, i.e., the perimeter doubles while the area quadruples.

5. On grid paper, draw a double scale model of the U pentomino. Fill it with four of your pentominoes. Can you fill your double scale model using some of the other pentominoes? How many are required? Repeat this exercise with the other pentomino shapes.

6. The triplication problem: select any pentomino and then using only the remaining shapes form a triple scale model of the first piece. An outline on grid paper may help. How many shapes did you have to use?

7. If you made a quadruplicate pentomino, how many pentomino shapes would you need to cover the quadruplicate? Note: You will require more than one set of pentominoes.

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Nets

Materials

Pupils will need the double scale drawings from the last lesson, and scissors. The Teacher will need one cut out pentomino for demonstration purposes.

The lesson

1. Fold your pentomino to demonstrate how it can be made into an open cube.

2. Pose the problem: "If each pentomino were cut out, which ones would make an open box--a cube without a lid?" Give small groups of children two or three minutes to discuss this and write down their predictions. (This visualisation stage is vital.)

3. Next, ask pupils to cut out their pentominoes and check their predictions. (Correct answers are L, X, F, T, Y, Z, N, W. See page 16 for all 20 pentominoes.)

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Samples can be displayed in books by taping down the base square.