Measuring area: 2.75

• Indicator of Progress
• Teaching Strategies
• Further Resources

Supporting materials

• Related Progression Points
• Developmental Overview of Measurement Attributes (PDF - 30Kb)

 

Indicator of Progress

At this level students understand that area can be measured in formal units such as square centimetres (abbreviation: cm²). They can measure area using prepared square centimetre grids, and provide good estimates for the area of non-rectangular shapes.  They can calculate the area of a rectangle with whole number lengths using length × width.

Before this level, students are able to identify the attribute of area and can compare the area of two or more shapes using informal units, for example by covering by hand or with ‘kindergarten squares’ and then comparing the count. Through their work with measuring other attributes (e.g. length, time) they will understand the importance of a formal unit of agreed size for reporting measurements, although it is only at this level that they use formal units for area.

Later, students will generalise the formula for area of a rectangle to lengths that are not whole numbers. They will find areas of other shapes by relating to the rectangle (e.g. noting a triangle is half of a corresponding rectangle, or dividing shapes into rectangular components).  They will also measure areas using square metres, where an overlay grid, as used at this level, is not practical.

Three phases for teaching measurement – in terms of the measurement phases, this indicator is about ‘learning to measure’ area, and it also includes the first step of ‘learning to calculate’ area. Whereas learning to use the instruments for measurement is a major task for some attributes (e.g. length, time), the square grid is the only ‘instrument’ that we have for measuring area, and this is quite simple to use. In adult life, calculation is the most important way of finding areas.

 

Illustration 1: Square units measure area, but a shape does not have to be square for the area to be determined

Students need to understand that an object doesn’t have to be square or rectangular in order for area to be measured. Students learn to use grid paper of varying sized grids (cm² then mm²) to either find areas or to provide good approximations for the area of a variety of shapes.

 

Illustration 2: Fundamental aspects of units

Measuring depends on understanding certain fundamental features of units, which apply to measurement of all attributes. If students are unsure of these principles, then measuring will not make sense and they will make errors such as counting the lines on a square grid, rather than the number of squares.

The idea of a unit: (Level 2.0) – addresses concepts about features of units.

 

Illustration 3: Moving from measuring to calculating

To find the area of a rectangle, students need to move from the stage of counting squares on a superimposed grid (most likely of area 1 cm² ), to the stage of recognising that there is an array of rows and columns of squares. They will then recognise that multiplying can give the answer. Later, they will make the generalisation that the area of a rectangle can be found by multiplying length by width, even if these are not whole numbers.

 

Teaching Strategies

At this first stage of measuring area, students need to make the transition from informal units to formal units and they need to learn to measure area in these formal units. The only available instrument for measuring area is a transparent grid, which is easy to use. Students should continue to develop the habit of estimating before measuring. For this purpose, one activity aims to develop personal benchmarks for estimation of area. When measuring is well established they move to the first step of calculating area: the area of a rectangle with whole number of centimetre sides.

Wherever possible, measuring should be undertaken to describe or compare areas for a purpose, so look for opportunities that arise in normal classroom life (e.g. are our desks larger than the desks in the next classroom? A can of paint covers 16 square metres; is that enough to paint the noticeboard?)

Activity 1: The need for square centimetres checks students’ understanding of the attribute of area, their use of informal units of area and introduces the formal unit.
Activity 2: Measuring and estimating area  provides experience in measuring in formal units and builds personal benchmarks for estimating areas.
Activity 3: Calculating areas of rectangles develops understanding that multiplying length by width will provide the area of a rectangle, linking to the equal groups of squares in the rows.

 

Activity 1: The need for square centimetres

The aim of this activity is to show students the usefulness of having a formal unit for area, and how the square centimetre is a simple solution to this need. The activity begins by comparing areas using one grid, and then establishes the need for a standard grid size.

Teacher preparation:  Take a piece of 1cm graph paper which will photocopy clearly to use as a master.

Incompetech (http://incompetech.com/beta/plainGraphPaper/) – if necessary, graph paper can be downloaded from here. (Check that your photocopier keeps exact size so 1cm paper photocopies as 1cm!)

Prepare a set of ‘houses’ from Lilliput, a country where the people are very small, by colouring ‘houses’ on your grid paper s as shown on the image below. Make one copy of the page of ‘houses’ for each pair of students.

Step 1: Using the grid to put the houses in order of floor area

Give each pair of students a grid sheet, with the task of listing the houses on their grid in order of area.  Teachers can paint a scenario about these being exact house plans from a country which has tiny little people.  The shapes are presented on a grid so that students are able to easily identify the area of each shape.

Ask students to explain why particular shapes have the same area, even though they look different, and how they know some are larger than others. Focus on the attribute of area here and how it is possible to determine area using the informal unit of a square in this shape. Students should be able to state the area in terms of numbers of squares, but at this stage they are using the grid square as an informal unit, not yet as the formal unit of the square centimetre.

AREAS:  Measured in grid squares S=16; T=V=15;  P=Q=R=14; U=9.

Students now write the areas on their shapes.

Step 2: Describing the area

Conduct a group discussion brainstorming how students could tell someone over the telephone how big the houses are.  Students will probably suggest two options:

• comparing with the area of an object that both parties to the telephone call have (e.g. how many stamps or fractional parts of a stamp cover each house)
• measuring the grid so that the person being telephoned can make their own (e.g. my grid lines are 1cm apart).

Step 3: Selecting and naming the formal unit

Discuss the advantages of saying how far apart the grid lines are, and of making the grid lines 1 cm apart. Everyone around the world knows how big it is. Name the unit shape “one square centimetre”. It is not important to introduce the symbol 1 cm² now.

Step 4: Making different shapes of area 1 square centimetre

With rulers, students draw several squares each measuring 1 cm by 1 cm. Colour each one a different colour. Leave one square as is.  Cut the others into two or three pieces, and rearrange the pieces to make non-square shapes of area 1 square centimetre. Paste these into a workbook and label ‘All of these shapes have area of 1 square centimetre’.

 

Activity 2: Measuring and estimating area

In order to be able to estimate area, students need to develop personal benchmarks that are readily accessible in everyday life. This activity will develop measurement skills and has the additional intention of providing students with some benchmarks.

Small areas
Photocopy square centimetre grid paper onto transparent overhead projector sheets. Each student will use a sheet to overlay onto a shape to determine its area.

Incompetech (http://incompetech.com/beta/plainGraphPaper/) – this is one of a number of websites that offer downloads of free graph paper. Note: check that your printer and photocopier do not distort it.

Select common objects, including some which might provide useful benchmarks, such as:

• their handprint or bottom of shoe
• the base of a bottle of water or a milk carton
• a post-it note, an A4 sheet of paper, an envelope, a ruler,
• a sheet of newspaper (this will need to be measured by ‘iterations’ of the grid).

Students measure the area with the transparent grid, and select one or two that are easy for them to remember (e.g. I remember that my square post-it notes have an area of about 50 cm²).

Now students estimate the area of some small shapes (e.g. area of a book) using personal benchmarks and measure with the square grid to check. Discuss how students made their estimates and the personal benchmarks used. Discuss the difficulty of estimating area accurately and the usefulness of a rough approximation.

Large areas
The activity above can be repeated with large shapes, measuring in square metres. The square metre itself, made by joining 4 one metre rods, is probably the best ‘personal benchmark’. There is an important difference in the measuring here. Whereas the transparent grid provides many copies of the square centimetre, students will need to use their one square metre multiple times. This is called ‘iteration’.
It is also useful to have a square of material measuring 1m by 1m to show 1 m² and to display this on the classroom wall. A useful benchmark is that a double page of The Age newspaper is about 0.5 m². This can be demonstrated by showing that 2 double pages approximately cover the one square metre.

 

Activity 3: Calculating areas of rectangles

In this activity students learn to calculate the area of a rectangle by generalising patterns in results obtained by measuring the area of rectangles using grid paper. They discuss the reasons why the pattern works.

Students draw rectangles (whole number of centimetre sides only) on grid paper and find the areas by counting the number of cm². Record the findings in a table, as shown below. Encourage students to be systematic in their selection of rectangles as working systematically is a useful strategy in mathematics. A table organises results and helps students to identify the pattern.

Example of a table students use to organise their results when calculating the area of a rectangle
Drawing of a rectangle	Length (cm)	Width (cm)	Area (cm2)
	4 cm	2 cm	8 cm2

Discuss the patterns observed, especially the fact that the area is length × width. Be sure that all students understand why this is. For example, in the 2 cm by 4 cm rectangle above, we can see that there are 2 rows of squares and each row has 4 squares in it. 2 groups of 4 make 8. This links with the array model of multiplication. It is important that students do not just observe patterns, but look for the reasons behind them.

For more information: Better multiplication strategies: Level 2.75. addresses concepts about the array model.

Summarise the finding as what is probably the students’ first formula:

Area of rectangle = length × width.

Practise using multiplication to find areas of rectangles. Give students the dimensions of the rectangles, ask them to find the areas (e.g. by multiplication) and also ask them to show or say where the equal groups are.

Include some examples with mixed units such as the area of a rectangle 1 m by 4 cm. Students need not draw the whole rectangle, but they should sketch it and say that if it was divided into square centimetres, there would be 4 rows, each containing 100 squares. This visualisation is essential to prevent students ignoring the mixed metres and centimetres.

When they have the number skills in place, students move on to calculating areas of rectangles with fraction or decimal lengths.

For more information: Fraction as a number: Level 3.5 addresses concepts about calculations with fractions and decimals.

Reverse area questions
Ask students to draw a number of different rectangles with an area of 24 cm². This activity can be repeated for other values of area. Ask questions such as:

• How did you decide on the length and width for the rectangle? (Link to factors!)
• What is the perimeter of this rectangle?
• Is there another rectangle with an area of 24 cm²?
• Some students will extend this to areas with side lengths that are not whole numbers (e.g. by splitting 3 × 8 into two parts and rearranging, they can go to 1.5 × 16). This will be an important transition to multiplication of fractions and decimals.

References

Incompetech (http://incompetech.com/beta/plainGraphPaper) – online graphs paper.

Department of Education and Training, Western Australia (2005) First Steps in Mathematics – Measurement (Understand Units, Direct Measure). Rigby, Melbourne.

Department of Education and Training, Western Australia (2005) First Steps in Mathematics – Measurement (Indirect Measure, Estimate). Rigby, Melbourne.

Further Resources

The following resource contains sections that may be useful when designing learning experiences:

Digilearn *

Area counting with Coco – students find the area of rectangles on a grid. Students explore how the formula works for finding a rectangle's area. First, they estimate the area of a chosen rectangle or compound rectangular shape on a grid. Second, they work out the correct formula for finding area by placing rows and columns of squares inside the rectangles. Then, they compare the actual area of the original shape with their first estimate.
(https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4671)

* Note that Digilearn is a secure site; DEECD login required.