Construct an equilateral triangle with sides 16cm long using a rule and compass, and then dissect it as shown in Figure 1. The lines AE and DF are perpendicular to the line BC.
[FIGURE 1 OMITTED]
Next make the four dissected shapes from thick cardboard or masonite, cutting them out accurately. When this is finished, reposition them into the triangle shape and measure its perimeter and area. Now rearrange the four pieces to make a rectangle. Measure its perimeter and area.
Another shape task was introduced in de Mestre and Baker (1992). Ten plastic counters are placed in a triangular pattern as shown in Figure 2, and can be considered to represent a flock of birds or a school of fish.
[FIGURE 2 OMITTED]
The problem is to invert this triangular pattern by only moving three counters. The problem can be extended in many ways as a discovery lesson. More rows can be added so that the triangular numbers are obtained. Also the minimum number to move to invert the triangle can be sought for each triangular number. This requires the students to look for the maximum number not moved, as a method of finding the minimum number moved. The symmetry of the counters making up the shape not moved is of considerable help. As an illustration the minimum number to move with five rows in the triangular number is illustrated in Figure 3.
[FIGURE 3 OMITTED]
The well-known four-colour problem can be demonstrated using the following hands-on task. A 6 x 4 chart is prepared as shown in Figure 4.
[FIGURE 4 OMITTED]
There are 36 unit tiles (1 x 1) with 9 blue, 9 red, 9 yellow, 9 green. The problem is to place the tiles on the chart so that each enclosed region has only one colour, and no colour is the same on either side of any common boundary. Challenge your students to try to find a dissection of the 6 x 4 rectangle so that five colours must be needed to obey the rule in the previous sentence. They will find that this is impossible.
An excellent three-dimensional hands-on task is to take 27 unit cubes and arrange them into a 3 x 3 x 3 solid cube. The outside faces of this large cube are then painted red, say. The problem is to make a list of the number of unit cubes with 0, 1, 2 and 3 red faces respectively. This list is related to the number of vertices, faces and edges of the large cube, giving an example of Euler's theorem that
Faces + Vertices - Edges = 2
This task was described previously in de Mestre (1999). Euler's theorem can then be checked by rearranging all the 27 cubes into many different solid shapes.
The geometry of numbers is illustrated by the following hands-on task using 7 matchsticks or straight rods of equal length. The first task is to construct a table of the number of rods used to make each digit from 0 to 9 as they appear in a digital calculator or clock. Next, students can be asked to say what time for a 24-hour clock uses the maximum number of rods, and then the time for the minimum number. Finally the idea of which digits can be inverted to still give a readable digit can be utilised in various tasks.
The chessboard is a useful 8 x 8 grid of cells for hands-on tasks. One task is to place 8 pawns (or counters) on the board so that each occupies only one cell, each row has only one pawn, each column has only one pawn, and there are no pawns on the two main diagonals. A more difficult extension is to add that no pawn can be reached diagonally from any other pawn.
Here is a task which involves the shapes known as pentominoes. A pentomino is formed when 5 equal squares are joined so that every square has at least one side touching the whole side of another square. Some examples are shown in Figure 5.
[FIGURE 5 OMITTED]
The task is to use 60 square tiles to obtain all 12 different pentominoes. Pentominoes are the same if the shape can be obtained by rotating or flipping a shape already produced. An extension is to construct the 12 pentominoes from cardboard or masonite and form them into a 10 x 6 rectangle.
Finally, consider the following simple task involving unit cubes. Place 4 cubes together so that the final shape has minimum surface area. Next do 5, 6, 7 and 8 cubes respectively. Now try to discover the general result for N cubes (N is any positive integer).
de Mestre, N. & Baker, J. (1992). Pebbles, ducks and other surprises. The Australian Mathematics Teacher, 48 (3), 4-7.
de Mestre, Neville (1999). Discovery: Patterns with cubes. The Australian Mathematics Teacher, 55 (2), 8-9.
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|Author:||de Mestre, Neville|
|Publication:||Australian Mathematics Teacher|
|Date:||Sep 22, 2005|
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