YEAR 8 MATHS FOCUS
NUMBER AND ALGEBRA
MEASUREMENT AND GEOMETRY
AREA
OUTCOME
A student:
MA4-13MG:
- uses formulas to calculate the areas of quadrilaterals and circles, and converts between units of area
TEACHING POINTS | 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 πr; 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. |
| 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’. |
EXPECTATIONS OF ATTAINMENT
| Choose appropriate units of measurement for area and convert from one unit to another | choose an appropriate unit to measure the areas of different shapes and surfaces, eg floor space, fieldsCritical and creative thinking
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| 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 | develop and use the formulas to find the areas of rectangles and squares: Area of rectangle=lb where l is the length and b is the breadth (or width) of the rectangle
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| find the areas of simple composite figures that may be dissected into rectangles, squares, parallelograms and triangles |
Find areas of trapeziums, rhombuses and kites
| develop, with or without the use of digital technologies, and use the formula to find the areas of kites and rhombuses: |
develop and use the formula to find the areas of trapeziums:
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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
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Investigate the relationship between features of circles, such as the area and the radius; use formulas to solve problems involving area
| develop, with or without the use of digital technologies, and use the formula to find the areas of circles: Area of circle=πr2 where r is the length of the radius
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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 π and an approximate answer using a calculator’s approximation for π |
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