NSW Syllabuses

# Mathematics K–10 - Stage 4 - Measurement and Geometry Area

## Outcomes

#### A student:

• MA4-1WM

communicates and connects mathematical ideas using appropriate terminology, diagrams and symbols

• MA4-2WM

applies appropriate mathematical techniques to solve problems

• MA4-13MG

uses formulas to calculate the areas of quadrilaterals and circles, and converts between units of area

Related Life Skills outcome: MALS-29MG

## Content

• Students:
• Choose appropriate units of measurement for area and convert from one unit to another (ACMMG195)
• choose an appropriate unit to measure the areas of different shapes and surfaces, eg floor space, fields
• use the areas of familiar surfaces to assist with the estimation of larger areas, eg the areas of courts and fields for sport (Problem Solving)
• convert between metric units of area using 1 cm2 = 100 mm2, 1 m2 = 1 000 000 mm2, 1 ha = 10 000 m2, 1 km2 = 1 000 000 m2 = 100 ha
• Establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving (ACMMG159)
• develop and use the formulas to find the areas of rectangles and squares:
$$\mbox{Area of rectangle} = lb\,$$ where l is the length and b is the breadth (or width) of the rectangle
$$\mbox{Area of square} = s^2\,$$ where s is the side length of the square
• explain the relationship that multiplying, dividing, squaring and factoring have with the areas of rectangles and squares with integer side lengths (Communicating)
• explain the relationship between the formulas for the areas of rectangles and squares (Communicating)
• compare areas of rectangles with the same perimeter (Problem Solving)
• develop, with or without the use of digital technologies, and use the formulas to find the areas of parallelograms and triangles, including triangles for which the perpendicular height needs to be shown outside the shape:
$$\mbox{Area of parallelogram} = bh\,$$ where b is the length of the base and h is the perpendicular height
$$\mbox{Area of triangle} = \frac{1}{2}\!bh\,$$ where b is the length of the base and h is the perpendicular height
• identify the perpendicular heights of parallelograms and triangles in different orientations (Reasoning)
• find the areas of simple composite figures that may be dissected into rectangles, squares, parallelograms and triangles
• develop, with or without the use of digital technologies, and use the formula to find the areas of kites and rhombuses:
$$\mbox{Area of rhombus/kite} = \frac{1}{2}\!xy\,$$ where x and y are the lengths of the diagonals
• develop and use the formula to find the areas of trapeziums:
$$\mbox{Area of trapezium} = \frac{1}{2}\!h \left( a+b \right)\,$$ where h is the perpendicular height and a and b are the lengths of the parallel sides
• identify the perpendicular heights of various trapeziums in different orientations (Reasoning)
• select and use the appropriate formula to find the area of any of the special quadrilaterals
• solve a variety of practical problems relating to the areas of triangles and quadrilaterals
• convert between metric units of length and area as appropriate when solving area problems (Problem Solving)
• Investigate the relationship between features of circles, such as the area and the radius; use formulas to solve problems involving area (ACMMG197)
• develop, with or without the use of digital technologies, and use the formula to find the areas of circles:
$$\mbox{Area of circle} = \pi r^2\,$$ where r is the length of the radius
• find the radii of circles, given their circumference or area (Problem Solving)
• find the areas of quadrants, semicircles and sectors
• solve a variety of practical problems involving circles and parts of circles, giving an exact answer in terms of $$\pi$$ and an approximate answer using a calculator's approximation for $$\pi$$

### Background Information

The area formulas for the triangle, the special quadrilaterals and the circle should be developed by practical means and/or by the use of dynamic geometry software, such as prepared applets.

The area formulas for the triangle and the parallelogram should be related to the area of a rectangle. Applets may be particularly useful in demonstrating that the respective formulas hold for all triangles and parallelograms, including those for which the perpendicular height needs to be drawn outside the shape.

The area formula for the rhombus or kite depends upon the fact that the diagonals are perpendicular, and so is linked with the geometry of special quadrilaterals. The formula applies to any quadrilateral in which the diagonals are perpendicular. Students should also be aware that because the rhombus is a special type of parallelogram, the area can be found using the formula $$A = bh$$.

The area formula for the trapezium can be developed using various dissections and techniques. Students need to be able to apply the area formula for the trapezium appropriately to trapeziums in any form or orientation.

The area formula for the circle may be established by using one or both of the following dissections:

• cut a circle into a large number of sectors and arrange the sectors alternately point-up and point-down to form a rectangle with height $$r$$ and base length $$\pi r$$
• inscribe a number of congruent triangles in a circle, all with corresponding vertex at the centre, and show that the area of the inscribed polygon is half the length of the perimeter times the perpendicular height of the triangles.

Students should be made aware that the perpendicular height of a triangle is the shortest distance from the base to the opposite angle. They may also need to be explicitly taught that the shortest distance between the parallel sides of a quadrilateral is the perpendicular distance between these sides.

Finding the areas of rectangles and squares with integer side lengths is an important link between geometry and multiplying, dividing, factoring and squaring. Expressing a number as the product of two of its factors is equivalent to forming a rectangle with those factors as the side lengths, and (where possible) expressing a number as the square of one of its factors is equivalent to forming a square with that factor as the side length.

Graphing the relationship between the length of a rectangle with a constant perimeter and possible areas of the rectangle links to non-linear graphs.

#### Purpose/Relevance of Substrand

The ability to determine the areas of two-dimensional shapes and solve related problems is of fundamental importance in many everyday situations, such as carpeting a floor, painting a room, planting a garden, establishing and maintaining a lawn, installing concrete and paving, and measuring land for farming or building construction. Knowledge and understanding with regard to determining the areas of simple two-dimensional shapes can be readily applied to determining the surface areas of simple (and composite) three-dimensional objects such as cubes, other rectangular prisms, triangular prisms, cylinders, pyramids, cones and spheres.

### Language

Teachers should reinforce with students the use of the term 'perpendicular height', rather than simply 'height', when referring to this attribute of a triangle. Students should also benefit from drawing and labelling a triangle when given a description of its features in words.

Students may improve their understanding and retention of the area formulas by expressing them in different ways, eg 'The area of a trapezium is half the perpendicular height multiplied by the sum of the lengths of the parallel sides', 'The area of a trapezium is half the product of the perpendicular height and the sum of the lengths of the parallel sides'.

The use of the term 'respectively' in measurement word problems should be modelled and the importance of the order of the words explained, eg in the sentence 'The perpendicular height and base of a triangle are 5 metres and 8 metres, respectively', the first attribute (perpendicular height) mentioned refers to the first measurement (5 metres), and so on.

The abbreviation m2 is read as 'square metre(s)' and not 'metre(s) squared' or 'metre(s) square'. Similarly, the abbreviation cm2 is read as 'square centimetre(s)' and not 'centimetre(s) squared' or 'centimetre(s) square'.

When units are not provided in an area question, students should record the area in 'square units'.

### National Numeracy Learning Progression links to this Mathematics outcome

When working towards the outcome MA4‑13MG the sub-elements (and levels) of Number patterns and algebraic thinking (NPA9), Understanding units of measurement (UuM7-UuM9) and Understanding geometric properties (UGP6) describe observable behaviours that can aid teachers in making evidence-based decisions about student development and future learning.

The progression sub-elements and indicators can be viewed by accessing the National Numeracy Learning Progression.