Teaching area through graphs

A vast majority of mathematics teachers follow the deductive method in teaching the concept of area of squares, rectangles, or triangles. The area of a shape is explained simply as the product of two sides (square), or length and breadth (rectangle), or half the base and height (triangle), and so on. This method of teaching hinders the development of their ability to use the concept for practical purposes. It may be interesting to note that most students who are able to calculate the area correctly will not be able to explain the meaning of one square centimetre or to shade a definite area on a graph sheet. This indicates that while teaching the concept, the level of knowledge and understanding achieved is generally not considered.
The concept of area
The term ‘area’ may be explained as ‘a space bounded by a curved line or line segments’. Any area bounded by line segments can be best explained by using a graph sheet. For example:

Fig. (i) shows a line segment AB of 1 cm. length.
Fig. (ii) shows the shaded part ABCD, an area of 1 sq. cm. bounded by four line segments AB, BC, CD and DA.
The shaded part ABC in Fig. (iii) too shows an area of 1 sq. cm. bounded by three line segments, AB, BC, and CA.  Note that the latter figure is using two halves from two adjacent squares.
Figs. (ii) and (iii) show different shapes with the same area.

Similarly, we can show larger line segments and areas as in Figs. (iv) and (v), where the shaded parts show an area of 2 sq. cm.

Area of a square

As we can see from Figs. (vi) and (vii), the graph sheet is needed to enable the students to visually relate the side of the square to its area.                      

Consequently, the formula ‘area = side × side’ is arrived at through the inductive method.

Area of a rectangle

The same inductive method may be followed in the case of a rectangle also.

Figs. (viii), (ix) and (x) show the possibility of drawing rectangles having the same area with various dimensions. The students can see this for themselves and understand that this is not possible in the case of a square.

Counting the number of unit squares in each figure will give the area of the rectangles in Fig. (viii), Fig. (ix) and Fig (x). Here, the area of the each rectangle is visualised in relation to its length and breadth. Thus the formula ‘area = length × breadth’ is arrived at by inductive method using a graph sheet.

Triangles and parallelograms

This process of learning the concept of unit area and the area of a square and a rectangle at the primary level may be further extended into middle school to understand the measurement of area of other geometrical figures, such as the area of a triangle or a parallelogram.

Consider Fig. (xi). The area of the square in the shaded part is 1 sq. cm and the sum of the area of the two shaded triangles (two half squares) is 1 sq. cm. Therefore the area of the triangle ABD = 2 sq. cm.

Area of the square ABCD = 4 sq. cm.
Area of the right-angled isoceles triangle
= ½ the area of the square
= ½ a × a (a = side)
= ½ a2 sq. units.

Similarly counting each unit, the area of the rectangle ABCD is found to be 8 sq. cm. In triangle ABE, the area of the 4 half squares and the 2 squares are 4 sq. cm.

Area of the triangle ABE = ½ of the area of the rectangle ABCD
= ½ of AB × BC
= ½ AB × FE (BC = FE)
= ½ base × height

Through this it can be made clear that the area of the isoceles right-angled triangle is ½ a2 (half of the product of equal sides) and the area of the triangle is (½ base × height), i.e., half of the product of base and height.

Look at the parallelogram ABCD in Fig. (xiii). It has six unit square centimetres. There are also four half squares with a total area of 2 sq. cm.

Therefore the area of the parallelogram ABCD = 8 sq. cm.

One can also calculate the area of the parallelogram by dividing into squares, rectangles and triangles, or by extending it as shown in the above figure.
Area of the parallelogram = area of the rectangle (because we have only moved the outer triangle into the figure at the opposite end).

Area of the rectangle EFCD = length × breadth
= EF × FC
= AB × DE (since EF = AB and CF = DE)
= base × height
= 8 sq. cm.

To calculate the area of irregular figures using a graph sheet it must be pointed out that 1 cm = 10 mm., whereas 1 sq. cm. = 100 sq. mm. First all the square centimetres should be counted, to which are added the remaining square millimetres.
Linear measurement is made in terms of centimetre or metre or kilometre. Area is measured in terms of square centimetre or square metre or square kilometre.

This article first appeared in Teacher Plus, Issue No.76, January-February 2002 and has been adapted here with changes.

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