Area Units of Work

Area is a two-dimensional concept related to the geometric concept of an enclosed region. It is defined in the maths curriculum as the size of a surface expressed as a number of square units. Investigations of the size of an area should begin with comparisons between different surfaces and progress to the use of non-standard, and then standard, units. The use of formulae to calculate the areas of common polygons is the final stage of the learning sequence.

Stage One: Identifying the attribute

Early area experiences must develop an awareness of what area is, and of the range of words that can be used to discuss it. Awareness of area as the "amount of surface" is developed by "covering" activities such as wrapping parcels, colouring in, and covering tables with paper. The use of words such as greater, larger and smaller, focus attention on the attribute of area. The awareness of the attribute of area is extended as comparisons of areas are made at the next stage.

Stage Two: Comparing and ordering

It is important that students experience activities in which they compare and order attributes as these extend their understanding of the attribute and introduce them to informal measuring processes. Begin by comparing identical shapes of different size so that one shape fits inside the boundaries of others. Most classrooms have attribute shapes that are ideal for this purpose.

The next step is to compare different-shaped objects, where it is possible to lay one object on top of the other. For example, a circular card can be placed on a sheet of paper.

Shapes which cannot be put together can be compared indirectly by cutting paper to cover one surface and then comparing the (size of the) paper with the surface of the second shape.

When students can compare two areas they should be given the opportunity to order three or more areas. The process of ordering three or more areas is not a simple extension as it involves thinking that if A is larger than B and B is larger than C then A is larger than C.

Now A > B, B > C implied A > C is a transitive relation. These are extremely important in life and even more so in maths. In life we actually put too much store by them and wonder why, when the All Black beat the Springboks and the Springboks beat the Wallabies, that the All Blacks can’t beat the Wallabies.

Stage 3: Non-Standard units

Measuring the area of objects using non-standard or informal units is the third stage in the learning sequence. Beginning with non-standard but familiar units, allows the students to focus on the process of repeatedly using a unit as a measuring device. Parts of the body provide interesting units for introductory use, for example, handprints and fingerprints.

Students should therefore be given lots of opportunities to cover the surface of a wide range of objects using, for example, their hands and sheets of paper and then counting how many are used.

Covering surfaces with a single unit will also lead to discussion about shapes that tessellate, and are therefore useful for covering surfaces, and those that don’t. For example, rectangles and squares tessellate the plane, whereas circles don’t. Tessellating with non-standard units establishes the need to cover surfaces without leaving gaps and without overlapping. It also demonstrates the advantages of using arrays that can be readily counted by using multiplication, for example, 3 rows of 6 tiles gives an area of 18 tiles.

From the earliest of these experiences, students should be encouraged to estimate. Initially these estimates may be no more than guesses, but estimating involves the students in developing a sense of the size of the unit. As everyday life involves estimating at least as frequently as finding exact measures, the skill of estimating is important.

At this stage students can also be introduced to the appropriateness of units of measure. For example, a hand is more appropriate that a finger tip for measuring the area of a desktop.

Although non-standard units reinforce most of the basic measuring principles students need to realise that they are limited as a means of communication. This can be highlighted through activities that involve the students measuring the surface of an object using non-standard units, for example, hands, and discussing the different results.

Stage 4: Standard units

When students can measure areas effectively using non-standard units, they are ready to move to the use of standard units. The motivation for moving to this stage, often follows from experiences where the students have used different non-standard units for the same area and have realised that consistency in the units used would allow for the easier and more accurate communication of area measures.

Students’ measurement experiences must enable them to:

1. develop an understanding of the size of a square metre and a square centimetre;
2. estimate and measure using square metres and square centimetres.

The usual sequence used in primary school is to introduce the square centimetre and then the square metre.

The square centimetre is introduced first, because it is small enough to measure common objects. The size of the square centimetre can be established by constructing it, for example by cutting 1-centimetre pieces of paper. Most primary classrooms also have a supply of 1-cm cubes that can be used to measure the area of objects. An appreciation of the size of the unit can be built up through lots of experience in measuring everyday objects. The students should be encouraged to develop their own reference for a centimetre, for example, a fingernail or a small button.

As the students become familiar with the size of the square centimetre they should be given many opportunities to estimate before using precise measurement. They can also be given the task of using centimetre-squared paper to create different shapes of the same area.

A square metre can be established using a similar sequence of experiences in constructing the unit and then using it to measure appropriate objects. An important learning point is that one square metre of area can take many shapes whereas a one-metre square must be a square with an area of one square metre.

The square kilometre can be illustrated on a local map so that the students can establish a real reference for a unit of this size. Alternatively the number of rugby fields needed to form a square kilometre could help establish a reference.

Stage 5: Applying and Interpreting

When the students are able to measure efficiently and effectively using standard units, their leaning experiences can be directed to situations that encourage them to "discover" measurement formula. In area work, the students may realise as they count squares to find the area of a rectangle, that it would be quicker to find the number of squares in one row and multiply this by the number of rows. In the same way, the students might find a formula for calculating the area of a right-angled triangle by seeing it as half of a rectangle.

Dissecting and reassembling parallelograms, trapezia and triangles, allow students to "discover" and understand area formulae.

Summary of Area Formulas